Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A$^*$

Here we briefly review the calculation of the shadow radius in spherically symmetric space-times. If such a space-time possesses a photon sphere, the gravitationally lensed image thereof as viewed by an observed located at infinity will constitute the BH shadow ²⁷ . Let us consider a generic static, spherically symmetric, asymptotically flat space-time, i.e. one admitting a global, non-vanishing, time-like, hypersurface-orthogonal Killing vector field, with no off-diagonal components in the matrix representation of the metric tensor. Without loss of generality, the space-time line element in Boyer–Lindquist coordinates can be expressed as:

$$\begin{align} \mathrm{d}s^2 = -A(r)\mathrm{d}t^2 + B(r)\mathrm{d}r^{\,2}+C(r)\mathrm{d}\Omega^2\,, \end{align} \tag{ 9 }$$

where $d\Omega$ is the differential unit of solid angle ²⁸ . In the following, we shall henceforth refer to the function $A(r) = -g_{tt}(r)$ as the 'metric function', and occasionally to $C(r) = g_{\theta\theta} = g_{\phi\phi}/\sin^2\theta$ as the 'angular metric function' (though we note that the latter terminology is not a widespread one). If the space-time is asymptotically flat, $A(r) \to 1$ as $r \to \infty$ in equation (9). Note also that in most cases of interest $C(r) = r^{\,2}$, although here we choose to keep the discussion as general as possible.

Let us then define the function h(r) (see e.g. [43]):

$$\begin{align} h(r) \equiv \sqrt{\frac{C(r)}{A(r)}}\,. \end{align} \tag{ 10 }$$

If the metric in equation (9) possesses a photon sphere, the radial coordinate thereof, $r_\mathrm{ph}$, is given by the solution to the following implicit equation:

$$\begin{align} \frac{\mathrm{d}}{\mathrm{d}r} \left [ h^2(r_\mathrm{ph}) \right ] =0\,, \end{align} \tag{ 11 }$$

which can be arranged to the following form:

$$\begin{align} C^{^{\prime}}(r_\mathrm{ph})A(r_\mathrm{ph})-C(r_\mathrm{ph})A^{^{\prime}}(r_\mathrm{ph})=0\,. \end{align} \tag{ 12 }$$

Here, the prime denotes a derivative with respect to r. If, as in the majority of cases we will consider, the angular metric function is $C(r) = r^{\,2}$, then equation (12) reduces to the following well-known expression:

$$\begin{align} A(r_\mathrm{ph})-\frac{1}{2}r_\mathrm{ph}A^{^{\prime}}(r_\mathrm{ph})=0\,. \end{align} \tag{ 13 }$$

The shadow radius $r_\mathrm{sh}$ corresponds to the gravitationally lensed image of the surface defined by $r_\mathrm{ph}$, and is therefore given by (see e.g. [41–43, 135, 211]):

$$\begin{align} r_\mathrm{sh} = \sqrt{\frac{C(r)}{A(r)}}\Bigg\vert_{r_\mathrm{ph}}\,. \end{align} \tag{ 14 }$$

For the 'standard' angular metric function $C(r) = r^{\,2}$ equation (14) reduces to the following well-known expression:

$$\begin{align} r_\mathrm{sh} = \frac{r}{\sqrt{A(r)}}\Bigg\vert_{r_\mathrm{ph}}\,. \end{align} \tag{ 15 }$$

In the case of the Schwarzschild BH, whose metric function is $A(r) = 1-2M/r$ and whose angular metric function is $C(r) = r^{\,2}$ [1], combining equations (13) and (15) leads to the well-known result $r_\mathrm{sh} = 3\sqrt{3}M$. For obvious symmetry reasons, the BH shadow for a spherically symmetric space-time is a circle of radius $r_\mathrm{sh}$ on the plane of a distant observer, irrespective of the inclination angle.

Note that the computation we have outlined is strictly valid for asymptotically flat space-times only. For other metrics, including but not limited to those matching on to the cosmological accelerated expansion at large distances and hence asymptotically de Sitter, the size of the BH shadow can explicitly depend on the radial coordinate of the distant observer, and on whether the observer is static or comoving (see e.g. [43, 212] for extensive discussions on this point). An example of such a situation occurs with the Kottler metric [213]. Given Sgr A$^*$'s proximity to us, for practical purposes we focus on a static observer (i.e. one which is not in the Hubble flow with respect to the BH) located at a distance $r_\textrm O$. For simplicity we also focus on the case where the angular metric function is $C(r) = r^{\,2}$. Then, the angular size of the BH shadow $\alpha_\mathrm{sh}$ is given by (see e.g. [43]):

$$\begin{align} \sin^2\alpha_\mathrm{sh} = \frac{r^{\,2}_\mathrm{ph}}{A(r_\mathrm{ph})}\frac{A(r_\textrm O)}{r_\textrm O^{\,2}}\,. \end{align} \tag{ 16 }$$

In the physically relevant small-angle approximation, it is easy to see that the shadow size is given by (see [43] for more detailed discussions):

$$\begin{align} r_\mathrm{sh} = r_\mathrm{ph}\sqrt{\frac{A(r_\textrm O)}{A(r_\mathrm{ph})}}\,. \end{align} \tag{ 17 }$$

The explicit dependence of the shadow size on the observer's position is clear in equation (17). It is trivial to see that for a distant observer viewing a BH described by an asymptotically flat metric, equation (17) reduces to equation (15), since $A(r_\textrm O) \approx 1$ as long as the observer is sufficiently distant from the BH. For instance, for the Schwarzschild metric this condition is satisfied as long as $r_\textrm O \gg M$, i.e. the distance between the BH and the observer is much larger than the BH gravitational radius. This condition is satisfied for the case of Sgr A$^*$, whose distance from us of ${\approx} 8\,\mathrm{kpc}$ should be compared to its gravitational radius $r_\textrm g \approx {\cal O}(10^{-7})\,\mathrm{pc}$.

As a further caveat, it is also worth noting that the computation outlined in equations (10)–(15) does not (necessarily) hold in theories with electromagnetic Lagrangian other than the Maxwell one (for instance non-linear electrodynamics (NLED)), since in this context photons do not necessarily move along null geodesics of the metric tensor, but along null geodesics of an effective geometry [214, 215], a fact which has only been recently appreciated in the literature. We will consider certain classes of NLED theories in this work, carefully accounting for the effective geometry.

As a final caveat, it is worth pointing out that the 'mathematical' result for the shadow radius obtained through the procedure discussed previously is 'astrophysically' relevant only if the resulting shadow size is located outside the event horizon for a BH, or the throat for a wormhole (WH). In the following, we have explicitly checked that this is indeed the case for all space-times discussed, or added caveats as appropriate.

In what follows, we will compute the shadow radius of various space-times of interest, and then compare them to the image of Sgr A$^*$ by imposing the bounds of equations (7) and (8) ²⁹ . These, as discussed earlier, already take into account various observational, theoretical, and modeling uncertainties, including the potential multiplicative offset between the radius of the bright ring of emission and the underlying shadow radius. We will show that the EHT image of Sgr A$^*$ sets particularly stringent constraints on theories and frameworks which predict a shadow radius larger than the Schwarzschild radius $3\sqrt{3}M$. Although the effect of new physics in many of the cases we will consider is that of decreasing the shadow radius with respect to the Schwarzschild case, a few will actually turn out to increase it, and these will be the models whose parameters are most tightly constrained. The key difference between these two classes of scenarios can easily be understood in terms of the effective potential experienced by photons which, assuming spherical symmetry and considering motion in the equatorial plane ($\theta = \pi/2$, which simplifies the calculations and is consistent with the assumed symmetries), is proportional to $V_\mathrm{eff}(r) \propto A(r)/r^{\,2}$. The shadow radius is then directly related to the maximum of the effective potential, and this can easily be shown to move towards the left/right within the scenarios leading to a decrease/increase of the shadow radius with respect to the Schwarzschild case.

Before moving on to actually performing our analysis, we will give a very brief overview of the space-times we consider, along with motivation for moving beyond the Schwarzschild space-time. We will consider a wide range of gravity theories, fundamental physics scenarios, and space-times beyond those of the Schwarzschild BH, for which $r_\mathrm{sh} = 3\sqrt{3}M$. The NHT [64–68] states that the only possible stationary, axisymmetric, and asymptotically flat BH solutions of the 4D electrovacuum Einstein–Maxwell equations are described by the Kerr–Newman family of metrics [217, 218]. Recall that the latter describes electrically charged, rotating BHs, and reduces to the Kerr metric in the absence of electric charge, to the Reissner–Nordström (RN) metric in the absence of rotation, and finally to the Schwarzschild metric in the absence of both charge and rotation. As John Archibald Wheeler phrased it, 'black holes have no hair', with the term 'hair' referring generically to parameters other than the BH mass M, spin J, and electric charge Q, which are required for a complete description of the BH solution.

At present, there is no sign of tension between the NHT and observations of astrophysical BHs. Why, then, is going beyond the Kerr–Newman family of metrics a well-motivated endeavor? The first reason is tied to the fact that a wide variety of theoretical and observational issues hint towards the possibility that our understanding of gravity as provided by general relativity (GR) is likely to be incomplete. These hints range from the quest for unifying gravity and QM, to the conflict between unitary evolution in QM and Hawking BH radiation as encapsulated in the BH information paradox [219], and finally to cosmological observations requiring the existence of dark matter (DM) and dark energy, likely a phase of cosmic inflation, and possibly a phase of early dark energy around the time of recombination. In the presence of new physics which eventually will address these issues, it is perfectly reasonable to assume that the NHT may only be an approximation, valid to the current level of precision. In fact, several theoretical approaches towards addressing the above issues posit the existence of new fields (particularly scalar fields) or modifications to GR (often introducing effective scalar degrees of freedom), see e.g. [220–253] for an inevitably incomplete selection of examples. All of these scenarios can very naturally lead to violations of the NHT ³⁰ . Moving beyond astrophysical systems, the study of controlled violations of the NHT is well-motivated by developments in our understanding of the gauge/gravity duality [257], with important applications to various condensed matter systems [258], including holographic superconductors [259] and quantum liquids [260].

The second reason is possibly even more fundamental, and is related to the well-known Penrose–Hawking singularity theorems, i.e. the fact that in GR continuous gravitational collapse leads to the inevitable, but at the same time arguably undesirable, appearance of singularities [2, 261]. For instance, the Kerr–Newman family of metrics possesses a well-known physical (non-coordinate) singularity at r = 0. The cosmic censorship conjecture [262, 263] notwithstanding, the mere existence of singularities has prompted a long-standing search for 'regular' BH solutions, which regularize the central singularity (see e.g. [264–318] for examples). Therefore, testing the metrics of regular BHs, or BH mimickers which address at least in part the existence of singularities, is a very well-motivated direction, particularly at present time with the availability of horizon-scale BH images. Note that many of these regular BH solutions can be obtained as solutions to the Einstein field equations coupled to a suitable NLED source. Finally, as the nature of DM, dark energy, and the origin of structure (whether through a phase of inflation or an alternative mechanism) is not fully understood at present, one should be open-minded to the possibility that one or more of these phenomena may be connected to potential modifications of GR.

With these considerations in mind, in this work we will consider a wide range of space-times beyond the Schwarzschild BH, testing them against the EHT horizon-scale image of Sgr A$^*$. The space-times we consider broadly speaking fall within these (by no means mutually exclusive) categories:

• 1.  
  Regular BHs, whether arising from specific theories or constructed in a phenomenological setting;
• 2.  
  BHs in modified theories of gravity, modified electrodynamics, and string-inspired settings;
• 3.  
  BHs in theories with additional matter fields (usually scalar fields), typically bringing about violations of the NHT;
• 4.  
  BH mimickers such as WHs (or effective WH geometries) and naked singularities (NSs);
• 5.  
  Modifications induced by novel fundamental physics frameworks (e.g. generalized or extended uncertainty principles (EUPs), non-commutative geometries (NCGs), or entropy laws beyond the Bekenstein–Hawking entropy).

Of course, several space-times we will consider fall within more than one of the above categories at the same time (especially 1. and 2., or 1. and 3.). Most of these space-times are described by one or more extra parameters, which in most cases we shall generically refer to as 'hair', 'hair parameters', or 'charges', in the latter case following the language adopted by [20, 144].

Before moving on to the results, one final clarification is in order. In all the cases we will consider, the hair parameters discussed above essentially fall into one of two categories. In the first case, they are 'universal', i.e. the same for each BH in the Universe regardless of its other hairs (mass, spin, electric charge). In the second case, they are 'specific', i.e. they can vary from BH to BH. With a slight abuse of language, we shall refer to these as being universal hairs and specific hairs respectively: note that the two should not be confused with the concepts of primary hair and secondary hair, although they are to some extent related.

An example of universal hair could be one which is exclusively determined by a parameter (e.g. a coupling) of the underlying Lagrangian of the theory (which is therefore uniquely determined), or by another fundamental parameter introduced at a phenomenological level. A very simple example of specific hair is the electric charge of a BH appearing in the well-known RN metric: this is a conserved charge which essentially emerges as an integration constant, and which can therefore take different values from BH to BH. As far as we can tell, the distinction between these two types of hairs has not been sufficiently emphasized in the literature on tests of fundamental physics from BH shadows.

We note that the comparison between constraints on specific hairs obtained from BH shadows and other astrophysical probes (for instance gravitational waves, stellar motions, and cosmology) is meaningful only if the constraints are referred to the same source. On the other hand, in the case of universal hairs such a comparison can be made even without reference to any specific source. In what follows, for each BH solution we will study, we will clearly indicate whether the additional hair parameter which controls deviations from the Schwarzschild metric is a specific hair or an universal hair.

The first geometry we consider beyond the Schwarzschild BH, and arguably the simplest extension thereof, is the RN metric describing an electrically charged, non-rotating BH. Explicitly introducing the BH mass (which we will later set to M = 1 for convenience), and recalling that we are setting $4\pi\epsilon_0 = \hbar = c = G_N = 1$, the metric function of the RN BH with electric charge Q is given by [319–322]: ³¹

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{Q^2}{r^{\,2}}\,, \end{align} \tag{ 18 }$$

where Q describes a specific hair. Henceforth, if not explicitly stated, the angular metric function will be implicitly assumed to be given by $C(r) = r^{\,2}$. Setting M = 1, ³² it is easy to show that this space-time possesses an event horizon only for $Q \leqslant 1$. However, even for larger values of the electric charge the space-time, which is now a NS, can still possess a photon sphere (and thereby cast a shadow-like feature), provided $1\lt Q \leqslant \sqrt{9/8} \approx 1.06$: this region of parameter space describes what is commonly referred to as the 'RN naked singularity'.

Using equations (13) and (15), we straightforwardly obtain the following expression for the radius of the shadow cast by the RN space-time, which is valid within both the BH and NS regimes:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{2} \left ( 3+\sqrt{9-8Q^2} \right ) }{\sqrt{4+\frac{\sqrt{9-8Q^2}-3}{Q^2}}}\,. \end{align} \tag{ 19 }$$

It is at first glance not obvious that equation (19) does indeed reduce to the Schwarzschild result of $r_\mathrm{sh} = 3\sqrt{3}$ in the Q → 0 limit, due to the apparent divergence in the denominator. However, the proper way of taking this limit is to Taylor-expand the denominator: when doing so, one indeed finds that the denominator of equation (19) tends to $\sqrt{8/3}$, whereas the numerator straightforwardly tends to $6\sqrt{2}$, leading overall to the correct limit of $3\sqrt{3}$.

The evolution of the shadow radius as a function of the electric charge is shown in figure 3, for both the BH (solid) and NS (dashed) regimes of the RN space-time, alongside the observational constraints imposed by the EHT image of Sgr A$^*$ (equations (7) and (8)). We see that as the electric charge increases, the shadow radius decreases, which can be understood by studying how the electric charge affects the effective potential felt by test particles. This is a feature which is common to a number of extensions of the Schwarzschild metric, as we shall see throughout the paper: most extensions will lead to a shadow radius which decreases with increasing charge/hair parameter (although notable exceptions exist, which we shall also discuss in this paper).

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From figure 3, we observe that the EHT observations set the 1σ upper limit $Q \lesssim 0.8M$, and the 2σ upper limit $Q \lesssim 0.95M$. Therefore, within more than 2 standard deviations, the EHT observations rule out the possibility of Sgr A$^*$ being an extremal RN BH (Q = M). Of course, the NS regime ($M\lt Q \leqslant \sqrt{9/8}M$) is completely ruled out: therefore, the EHT observations rule out the possibility of Sgr A$^*$ being one of the simplest possible NSs (although other NS space-times are allowed as we shall see later). Finally, re-inserting SI units, we see that the previous 2σ upper limit translates to $Q \lesssim 0.95\sqrt{4\pi\epsilon_0G}M \sim 6.5 \times 10^{26}\,\mathrm{C}$. In practice, this limit is significantly weaker than the limit $Q \lesssim 3 \times 10^8\,\mathrm{C}$, or equivalently $Q \lesssim {\cal O}(10^{-19})$ in units of mass of Sgr A$^*$, obtained from astrophysical considerations in [323, 324] (see also [325–329]).

We now consider regular BH solutions, which are free of the r = 0 core singularity present within the Kerr–Newman family of metrics. Several approaches towards constructing regular BH metrics replace the Kerr–Newman core with a different type of core, such as a de Sitter or Minkowski core, or phenomenologically smearing the core over a larger surface. The Bardeen space-time is one of the first regular BH solutions to the Einstein equations ever proposed [330], and describes the space-time of a singularity-free BH with a de Sitter core [331]. The Bardeen space-time carries a magnetic charge $q_\textrm m$, and can be thought of as a magnetic monopole satisfying the weak energy condition [332]. It also emerges in the context of GR coupled to a very specific NLED Lagrangian [333], and can further be interpreted as a quantum-corrected Schwarzschild BH [334].

The metric function for the Bardeen magnetically charged BH is given by [330]:

$$\begin{align} A(r) = 1-\frac{2Mr^{\,2}}{(r^{\,2}+q_\textrm m^2)^{3/2}}\,, \end{align} \tag{ 20 }$$

with $q_\textrm m$ characterizing a specific hair and satisfying the condition $q_\textrm m \leqslant \sqrt{16/27} \approx 0.77$. We compute the corresponding shadow radius numerically, and show its evolution as a function of the magnetic charge $q_\textrm m$ in figure 4. As with the RN BH, we see that increasing $q_\textrm m$ decreases the shadow radius. Within 2σ, the EHT observations are consistent with Sgr A$^*$ being a Bardeen BH for any (allowed) value of the magnetic charge, including the extremal value $q_\textrm m = \sqrt{16/27}M$.

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A comment, relevant for several other space-times, is in order here. Since the Bardeen regular BH can emerge within a specific NLED theory, there is potentially some ambiguity as to how one computes the corresponding shadow: one can compute it (a) using the 'standard' formulae (equations (12)–(14)), thus requiring only knowledge of the metric function equation (20), or (b) accounting for corrections to the effective geometry arising from the underlying NLED theory. Approach (a) can be considered model-agnostic, in the sense that the metric is taken as a phenomenological input without further questioning its origin, whereas approach (b) assumes a specific microscopical origin for the metric. For the Bardeen BH, we have chosen to adopt approach (a) for three reasons:

• Historically, Bardeen first proposed this specific metric on a phenomenological basis as a toy model, and only at a later stage was the associated NLED theory obtained;
• The corresponding underlying NLED theory is arguably quite baroque, and was 'reconstructed' to give the desired solution;
• NLED is not the only possible microscopical origin for the Bardeen BH.

 Regarding the second point in particular, it is worth noting that it is almost always possible to reconstruct a possible NLED source for a given (regular or non) BH metric. The corresponding Lagrangian is not always guaranteed to be particularly motivated, and in fact more often than not will be quite baroque. While this is somewhat a matter of taste, in our view a more aesthetically appealing approach is a 'top-down' one starting from a specific well-motivated NLED Lagrangian, and finding the corresponding solution. For this reason, if a specific metric can be supported by a NLED source, but the latter is quite baroque or was reconstructed a posteriori, we choose to compute the shadow following the phenomenological/model-agnostic approach (a). This is the case for the Bardeen BH, which is the reason why only one curve, obtained from the standard formulae equations (12)–(14), is present on figure 4, and similar considerations hold for other metrics we study later.

Another well-known regular BH solution has been proposed by Hayward [335]. This space-time replaces the r = 0 GR singularity by a de Sitter core with effective cosmological constant $\Lambda = 3\ell^2$. In doing so, one is assuming that an effective cosmological constant plays an important role at short distances, with Hubble length $\ell$. Such a behavior, while introduced at a phenomenological level for the Hayward BH, has been justified within the context of the equation of state of matter at high density [331, 336], or an upper limit on density or curvature [337–339], the latter expected to emerge within a quantum theory of gravity [340].

The metric function for the Hayward regular BH is given by [335]:

$$\begin{align} A(r) = 1-\frac{2Mr^{\,2}}{r^3+2\ell^2M} \,, \end{align} \tag{ 21 }$$

with the Hubble length associated to the effective cosmological constant satisfying $\ell \leqslant \sqrt{16/27}M$ and, as originally envisaged by Hayward, characterizing an universal hair (although, since the space-time is introduced in a phenomenological way, nothing prevents the hair from being specific). We compute the corresponding shadow radius numerically, showing its evolution as a function of the Hubble length $\ell$ in figure 5. Again, we see that increasing $\ell$ decreases the shadow radius. Within the present constraints, we see that the EHT observations are consistent with Sgr A$^*$ being a Hayward BH for any (allowed) value of the Hubble length associated to the effective cosmological constant, including the extremal value $\ell = \sqrt{16/27}M$.

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A well-known generalization of the Hayward BH, with the same quantum gravity-inspired theoretical motivation and featuring an additional charge parameter, has been proposed by Frolov [341]. The metric function for the Frolov BH is given by:

$$\begin{align} A(r) = 1-\frac{(2Mr-q^2)r^{\,2}}{r^{\,4}+(2Mr+q^2)\ell^2} \,, \end{align} \tag{ 22 }$$

where the additional charge parameter q characterizes a specific hair and satisfies $0\lt q \leqslant 1$, and the length below which quantum gravity effects become important satisfies $\ell \leqslant \sqrt{16/27}$ as with the Hayward BH. In principle, q admits an interpretation in terms of electric charge measured by an observer situated at infinity, where the metric is asymptotically flat.

Here, we focus on the charge q, while fixing $\ell = 0.3$. This is similar to the choice reported in [20], which instead fixed $\ell = 0.4$, and can be motivated by the fact that the shadow radius changes slowly with $\ell$, as is clear from figure 5. Therefore, the limits on q we will report can be interpreted as constraints along a particular slice of the full parameter space (see also the earlier work of [103] by some of us). We compute the corresponding shadow radius numerically, showing its evolution as a function of the charge q in figure 6. We see that, for $\ell = 0.3$, the EHT observations set the upper limits $q \lesssim 0.8M$ (1σ) and $q \lesssim 0.9M$ (2σ), qualitatively similar to the constraints on the electric charge of the RN metric Q we obtained in section 3.1. Recall once more that these constraints are valid for $\ell = 0.3$. From the results reported in section 3.3 and figure 5 for the Hayward BH, we can expect that fixing $\ell$ to a smaller/larger value would result in weaker/tighter constraints on q, as increasing $\ell$ leads to a slight decrease in the shadow size: in other words, we can expect a negative correlation/parameter degeneracy between $\ell$ and q.

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NLED theories are well-motivated extensions of Maxwell's Lagrangian in the high-intensity regime [342], and appear in the low-energy limit of various well-known theories, including Born–Infeld electrodynamics [343], as well as several string or supersymmetric constructions [344–348]. Another important class of regular BHs arises in the context of GR coupled to Bronnikov NLED, where the Lagrangian of the latter is given by [349]:

$$\begin{align} \mathcal{L}(U) = F_{\mu\nu}F^{\mu\nu}\cosh^{-2}\left[ a \left( F_{\alpha\beta}F^{\alpha\beta}/2\right)^{1/4}\right]\,. \end{align} \tag{ 23 }$$

We recall that the relativistic invariants U and W are constructed from the electromagnetic field-strength tensor $F_{\alpha\beta}$ and its dual $F^{\star}_{\alpha\beta}$ as follows:

$$\begin{align} U & =F^{\alpha\beta}F_{\alpha\beta}\,, \quad W=F^{\alpha\beta}F_{\alpha\beta}^{\star}\,, \nonumber \\ F^{\star}_{\alpha\beta}& =\frac{1}{2}\epsilon_{\alpha\beta\mu\nu}F^{\mu\nu}\,, \quad F_{\mu \nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,, \end{align} \tag{ 24 }$$

where $\star$, $\epsilon_{\alpha\beta\mu\nu}$, and A_µ denote respectively the Hodge dual operator, the Levi-Civita symbol, and the electromagnetic gauge field.

Einstein–Bronnikov gravity admits regular BHs only if these carry magnetic and not electric charge, i.e. choosing a purely magnetic configuration for the gauge field:

$$\begin{align} A_{\mu} = q_\textrm m\cos\theta\delta_{\mu}^{\phi}\,, \end{align} \tag{ 25 }$$

where $q_\textrm m$ is the magnetic charge. In this case, the central singularity is removed if the parameter $a = q_\textrm m^{3/2}/M$ is different from zero. The metric function can be written explicitly as [349]:

$$\begin{align} A(r) = 1-\frac{2M}{r} \left ( 1-\tanh \frac{q_\textrm m^2}{2Mr} \right ) \,, \end{align} \tag{ 26 }$$

where $q_\textrm m$ characterizes a specific hair. As anticipated earlier, however, in NLED theories photons do not move along null geodesics of the metric tensor, but of an effective geometry which depends on the details of the NLED theory. Therefore, knowledge of the metric function alone is in principle not sufficient to compute the shadow cast by BHs within NLED theories.

To proceed, we follow Novello's method [214, 215] to compute the effective geometry. Since these can now be considered standard references in the field, here we very succinctly review the basic steps of Novello's method, and simply refer the reader to [214, 215] for further details. Novello's method is itself based on Hadamard's method to obtain the equations for the propagation of the electromagnetic field, where the discontinuity surface Σ in the electromagnetic field is identified with the wavefronts. In short, one starts from the Bianchi identities involving the field-strength tensor $F_{\mu\nu}$ and its dual. It is assumed that the field-strength is continuous across Σ, while its first derivative is discontinuous, with the discontinuity parametrized by $f_{\alpha\beta}k_{\mu}$, where k_µ is the propagation vector. These conditions are then applied to the Bianchi identities and the resulting equation(s), taking the form of a cyclic identity (involving the tensor f and the propagation vector) with indices, say, $_{\alpha\beta\mu}$, are contracted with $F^{\alpha\beta}k^{\mu}$ to obtain a scalar equation. These equations are evaluated on-shell and, after simple manipulations, can be expressed in the form $g_{\mu\nu}^\mathrm{eff}k^{\mu}k^{\nu} = 0$, showing that photons move along null geodesics of $g_{\mu\nu}^\mathrm{eff}$, itself depending on the NLED Lagrangian.

For the intermediate steps leading to the computation of the effective geometry, we refer the reader to equations (2.10)–(2.21) of [104] by some of us (see also [350]), where these steps are explicitly spelled out. We compute the shadow radius numerically, following the procedure outlined in [104]. In figure 7, we show the evolution of the shadow size as a function of the only free parameter, $\vert q_\textrm m \vert$. We see once more that increasing the magnetic charge $q_\textrm m$ acts to decrease the shadow radius. We also see that the EHT observations restrict the magnetic charge to $\vert q_\textrm m \vert \lesssim 0.8M$ (1σ) and $\vert q_\textrm m \vert \lesssim M$ (2σ), while noting that within Bronnikov NLED the magnetic charge is in principle allowed to take values $q_\textrm m\gt M$.

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The GCSV BH was introduced in [351–353], including work by one of us, as toy model of a regular BH in the sense of Bardeen, where the core is asymptotically Minkowski rather than de Sitter ³³ : stated differently, the associated energy density and pressure asymptote to zero rather than to a final value determined by the effective cosmological constant of the de Sitter case (see also [131, 354–362]). While the GCSV BH is introduced in a purely phenomenological way, Simpson and Visser (SV) argued in [353] that this space-time is mathematically interesting due to its tractability (the curvature tensors and invariants take forms which are much simpler than those of the Bardeen, Hayward, and Frolov BHs), and physically interesting because of its non-standard asymptotically Minkowski core. The metric function describing the GCSV space-time is given by [351–353]:

$$\begin{align} A(r) = 1-\frac{2Me^{-g^2/2Mr}}{r}\,, \end{align} \tag{ 27 }$$

where depending on the underlying origin for this space-time, g can characterize either an universal or specific hair. As shown by some of us in [363, 364], the space-time described by equation (27) can be obtained in the context of GR coupled to a suitable NLED source, with g associated to the NLED charge.

We compute the shadow size numerically, and show its evolution against $\vert g \vert$ in figure 8. As for all solutions considered so far, we see that increasing the hair parameter decreases the shadow size, as also observed in the earlier work of [131] by some of us. The EHT observations set the limits $\vert g \vert \lesssim 0.8M$ (1σ) and $\vert g \vert \lesssim M$ (2σ). We are not aware of any a priori theoretical restriction on the value of g, given that the metric has been introduced as a toy model.

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As a further example of regular BH, we consider the KS BH, which arises within a string-inspired model where spherically symmetric quantum fluctuations of the metric and the matter fields are governed by the 2D dilaton-gravity action [365]. The KS BH then provides a well-motivated example of quantum deformation of a Schwarzschild BH. The metric function for this space-time is given by:

$$\begin{align} A(r) = -\frac{2M}{r}+\frac{\sqrt{r^{\,2}-\ell^2}}{r} \,, \end{align} \tag{ 28 }$$

where the hair parameter $\ell\gt0$ sets the scale over which quantum deformations of the Schwarzschild BH shift the central singularity to a finite radius, and characterizes an universal hair. Physically speaking, what effectively occurs is that the singularity is smeared out over a two-dimensional sphere of area $4\pi\ell^2$. While $\ell$ is expected to be of the order of the Planck length $\ell_\mathrm{Pl}$, in a phenomenological approach it can in principle take any positive value, as done for instance in [366].

We find that a closed-form expression for the shadow radius can be provided, and is given by:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt[4]{\left ( 9+3\ell^2+3\sqrt{9+2\ell^2} \right ) ^3}}{{\sqrt{-4\sqrt{2}+2\sqrt{9+\ell^2+3\sqrt{9+2\ell^2}}}}}\,, \end{align} \tag{ 29 }$$

which correctly reduces to $3\sqrt{3}$ in the limit $\ell \to 0$. The evolution of the shadow size $r_\mathrm{sh}$ is shown against $\vert \ell \vert/M$ in figure 9. The KZ BH provides the first example, among those considered, of a space-time where the size of the BH shadow increases with the hair parameter. As per our previous discussion, it is easy to show that the location of the maximum of the effective potential for the KS space-time moves to the right with respect to the Schwarzschild case for any value of $\ell\gt0$. The EHT observations set the limits $\vert \ell \vert \lesssim 0.2M$ (1σ) and $\vert \ell \vert \lesssim M$ (2σ). The physical meaning of this limit is that the singularity cannot be spread out across a scale larger than the gravitational radius, and in any case has to be confined within the event horizon. This example also highlights how the EHT near-horizon image of Sgr A$^*$ places particularly tight limits on fundamental physics scenarios which lead to a larger shadow compared to the Schwarzschild BH.

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Another interesting BH solution, which we shall refer to as the GK BH, was proposed by some of us in [367], and is described by the following metric function:

$$\begin{align} A(r) = 1-\frac{2M}{\sqrt{r^{\,2}+k^2}} \,, \end{align} \tag{ 30 }$$

where depending on the underlying origin for this space-time, k can characterize either an universal or specific hair. Although the above metric function is regular throughout the whole space-time, there is nonetheless a scalar polynomial singularity at r = 0, which can be seen by computing the Kretschmann and Ricci scalars [367]. In addition, the space-time exhibits an interesting reflection symmetry $r \to -r$. This can be viewed as a minimal deformation of the Schwarzschild BH, and is similar to the SV space-time we will consider later in section 3.9, albeit with a crucial difference in the angular metric function, which here is the standard $C(r) = r^{\,2}$.

We find that a closed-form expression for the shadow radius exists and is given by:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{{\cal X}-k^2 \left ( 1+\frac{2}{{\cal X}}\right ) +\frac{9}{{\cal X}}+3}}{\sqrt{1-\frac{2}{\sqrt{{\cal X}+\frac{9}{{\cal X}}-2\frac{k^2}{{\cal X}}+3}}}}\,, \end{align} \tag{ 31 }$$

where the quantity ${\cal X}$ is defined as:

$$\begin{align} {\cal X} \equiv \sqrt[3]{\frac{k^4-18k^2+\sqrt{k^6 \left ( k^2-4 \right ) } +54}{2}}\,. \end{align} \tag{ 32 }$$

It is easy to show that, with ${\cal X}$ defined as in equation (32), $r_\mathrm{sh}$ correctly approaches $3\sqrt{3}$ in the limit k → 0, where ${\cal X} \to 3$. We show the evolution of the shadow size against $\vert k \vert/M$ in figure 10. We see that increasing $\vert k \vert$ decreases the shadow size, as with most BH solutions studied so far. We find that the EHT observations set the upper limits on the NLED charge $\vert k \vert \lesssim 1.4M$ (1σ) and $\vert k \vert \lesssim 1.6M$ (2σ). To the best of our knowledge, these are the first ever constraints on this space-time ³⁴ .

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We now start moving away from regular BH metrics, considering a metric encompassing both a regular BH as well as a traversable WH regime. A WH is a (typically tunnel-like) space-time structure with a non-trivial topology, which can connect either two distant regions of the same Universe, or two different Universes (see e.g. [368] for a review of astrophysical WHs). In most cases, the field content needed to sustain the WH geometry requires a violation of the null energy condition [369, 370], either in the form of exotic matter or a modification to Einstein's gravity ³⁵ . The simplest WH configurations feature two 'mouths' connected by a 'throat', but more complex structures are possible.

In [417], SV proposed a one-parameter extension of the Schwarzschild space-time which interpolates between the latter and a traversable WH, while passing through a regular BH geometry. The SV space-time is described by the following metric functions:

$$\begin{align} A(r) = 1-\frac{2M}{\sqrt{r^{\,2}+a^2}}\,,\quad C(r)=r^{\,2}+a^2 \,, \end{align} \tag{ 33 }$$

where the new parameter is required to satisfy a > 0, and characterizes a specific hair. For a = 0, equation (33) reduces to the Schwarzschild space-time. On the other hand, for $0\lt a\lt2M$, the metric describes the space-time of a regular BH with a one-way space-like throat. For $a = 2M$, the space-time corresponds to that of a one-way WH with an extremal null throat. Finally, for $a\gt2M$, equation (33) describes a traversable WH with a two-way time-like throat. We also note that when M = 0, equation (33) reduces to the metric of the Ellis–Bronnikov WH [418, 419]. The SV space-time and extensions thereof have received significant attention in the literature, attracting several follow-up studies, see [420–449] for examples of representative works.

As one sees from equation (33), the SV space-time is the first example of space-time we consider where $C(r) \ne {r^{{\kern 1pt} 2}}$. As a consequence, the coordinate r is in general not the areal radius, unless a = 0. This is also the key difference between the SV space-time and the GK BH we studied previously, which are otherwise very similar, and underlies several peculiar features of the SV space-time we shall now discuss. The properties of the SV space-time are easy to compute analytically. To compute the location of the photon sphere we use equation (12), which upon inserting A(r) and C(r) as given by equation (33), and working in units where M = 1, reduces to:

$$\begin{align} 2r\left( {1 - \frac{2}{{\sqrt {{r^{{\kern 1pt} 2}} + {a^2}} }}} \right) - ({r^{{\kern 1pt} 2}} + {a^2})\frac{{2r}}{{\sqrt {{{({r^{{\kern 1pt} 2}} + {a^2})}^3}} }} = 2r\left( {1 - \frac{3}{{\sqrt {{r^{{\kern 1pt} 2}} + {a^2}} }}} \right) = 0{\mkern 1mu} , \end{align} \tag{ 34 }$$

from which we trivially find the photon sphere location as being $r_\mathrm{ph} = \sqrt{9-a^2}$, as also obtained in [417]. Note that these results are only valid for a < 3: for $a \geqslant 3$, the space-time does not possess a photon sphere (although, we stress, this does not necessarily mean that it cannot cast a shadow).

To obtain the shadow size, we use the more complete expression given by equation (14), finding that $r_\mathrm{sh} = 3\sqrt{3}$, with no dependence on the parameter a. Therefore, the shadow cast by the SV space-time is in perfect agreement with the EHT observations within all four the singular BH, regular BH, one-way WH, and traversable WH regimes, as shown in figure 11. While phenomenological, this is therefore the first example (among those we have considered) of a BH mimicker space-time which could be a viable candidate in light of Sgr A$^*$'s shadow ³⁶ . On the other hand, complementary x-ray reflection spectroscopy constraints [451] obtained by some of us set the very tight limit $a \lesssim 0.5M$ at $90\%$ C.L. [308], which is particularly tight since it is obtained from observations of the stellar-mass x-ray binary candidate BH EXO 1846-031, whose mass is 6 orders of magnitude lower than Sgr A$^*$.

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The somewhat surprising result that the shadow cast by the SV space-time is identical in size to that of the Schwarzschild BH deserves further comments. The fact that the SV and Schwarzschild space-times are shadow-degenerate had already been noted in [452], and the easiest way of seeing this is to perform a coordinate transformation and express the metric described by equation (33) in terms of the areal radius $R = \sqrt{r^{\,2}+a^2}$. In these new coordinates, g_tt , $g_{\theta\theta}$, and $g_{\phi\phi}$ are exactly of the Schwarzschild form, whereas the only deviations from the Schwarzschild form appear in g_RR , which however does not enter in the computation of the shadow radius as is clear from equations (10)–(15). More generally, [452] shows that the condition for a space-time to be shadow-degenerate with the Schwarzschild space-time is that the dominant photon sphere of both has the same impact parameter, after correcting for the potentially different redshift of comparable observers in the two space-times. Besides the SV space-time, [452] provides another example of a space-time which is shadow-degenerate with the Schwarzschild space-time, termed 'Class II'.

Finally, as anticipated earlier, we note that the absence of a photon sphere for a > 3 does not necessarily imply that the SV space-time cannot cast a shadow in this regime. In fact, as noted in [453], for a > 3 the WH throat can act as photon sphere. For the specific case of a = 3, two photon spheres and an anti-photon sphere at the throat degenerate into an effective marginally unstable photon sphere. This allows the SV space-time to cast a shadow. Given the significantly more complicated behavior of the system in the $a \geqslant 3$ regime, here we choose not to cover it further, and refer the reader to [453] for more detailed discussions on the subject.

We now consider what is arguably the simplest example of non-rotating, asymptotically flat, traversable WH, described by the well-known MT metric [454]:

$$\begin{align} \mathrm{d}s^2 = -e^{\Phi(r)}\mathrm{d}t^2+\frac{\mathrm{d}r^{\,2}}{1-\Psi(r)}+r^{\,2}\mathrm{d}\Omega^2\,, \end{align} \tag{ 35 }$$

where $\Phi(r)$ is referred to as the redshift function, and $\Psi(r)$ as the shape function (although the reader should note that different definitions and conventions for the redshift and shape functions exist in the WH literature). One of the simplest choices for these two in principle free functions consistent with a WH geometry is to set $\Phi = -r_0/r$ and $\Psi(r) = (r_0/r)^{\gamma}$ [454–458], where r₀ is the radial coordinate of the WH throat, and γ is a constant which we shall henceforth fix to γ = 1. For this choice of redshift and shape functions, it is easy to show that the Arnowitt–Deser–Misner (ADM) mass [459, 460] is $M = r_0/2$, so the metric function can be rewritten as:

$$\begin{align} A(r) = e^{-\frac{2M}{r}}\,. \end{align} \tag{ 36 }$$

Importantly, the size of the WH shadow in units of the ADM mass is then uniquely determined and does not depend on additional hair parameters, given that the only free parameter in the theory (γ) enters exclusively in g_rr , which does not affect the computation of the shadow ³⁷ .

For the choice of redshift function we have adopted, the equations governing the location of the photon sphere and the corresponding shadow size simplify considerably. The equation for $r_\mathrm{ph}$ takes the form $r_\mathrm{ph}(1-r_\mathrm{ph}) = 0$, for which the physical solution is $r_\mathrm{ph} = 1$. Then, the shadow size is given by $r_\mathrm{sh} = 1/\sqrt{e^{-2}} = e$. In other words, the size of the MT WH shadow in units of ADM mass is exactly equal to Euler's number, $r_\mathrm{sh}/M = e \approx 2.718$, i.e. almost a factor of 2 smaller than the size of the shadow of a Schwarzschild BH with the same ADM mass. By direct comparison, and as is visually clear in figure 12, the EHT measurements rule out at very high significance (${\gt}5\sigma$) the possibility of Sgr A$^*$ being a MT traversable WH, for the particular well-motivated choice of redshift function $\Phi = -r_0/r$. Finally, we note that this result has been also been discussed by the EHT collaboration in [20]. ³⁸

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Another interesting WH solution was proposed by DS [461] (see also [462–466] for further relevant work), with the goal of studying the extent to which globally static WHs can mimic observational features of BHs, including accretion of matter, no-hair properties, ringing of quasi-normal modes, and dissipative properties associated to the event horizon. The DS WH is described by the following metric [461]:

$$\begin{align} \mathrm{d}s^2 = - \left ( 1 - \frac{2M}{r} + \lambda^2 \right ) \mathrm{d}t^2+\frac{\mathrm{d}r^{\,2}}{1-\frac{2M}{r}}+r^{\,2}\mathrm{d}\Omega^2 \,, \end{align} \tag{ 37 }$$

and differs from the Schwarzschild BH because of the presence of the dimensionless parameter λ, which characterizes an universal hair. When λ ≠ 0, equation (37), the would-be horizon at $r = 2M$ actually corresponds to a throat joining two isometric regions. Note that the would-be Killing horizon where $g_{tt} = 0$ would actually be located at $r = 2M/(1+\lambda^2)\lt2M$, so behind the throat. As discussed in [461], this space-time provides an example of Lorentzian WH ³⁹ , where the throat describes a brane-like structure (a high-tension distribution), with radial and tangential tensions proportional to $1/\lambda^2$ but with zero energy density, localized in a thin shell around the throat. We also note that a similar space-time was studied in 3D in [471], with the goal of restoring Poincaré recurrence in BHs: in this specific case, $\lambda \sim e^{-4\pi GM^{\,2}}$ is exponentially small, so as to mimic the expected dependence of the Poincaré recurrence time on the BH entropy. In [461], it was argued that choosing exponentially small values of λ allows the resulting WH space-time to mimic both classical and quantum properties of Schwarzschild BHs.

In the following, we treat λ as a free parameter and simply ask what values are compatible with the observed shadow of Sgr A$^*$, assuming that the latter is indeed described by a DS WH. Note that as soon as λ ≠ 0 the space-time is no longer asymptotically flat, so in order to compute its shadow we adopt the more general computation given by equations (16) and (17). Following this procedure, we can easily obtain a closed-form expression for the shadow radius as follows:

$$\begin{align} r_\mathrm{sh} \simeq \frac{3\sqrt{3}}{1+\lambda^2}\,. \end{align} \tag{ 38 }$$

where the $\simeq$ sign is included as we have made the approximation $r_\textrm O \gg 1$ (in units M = 1), valid for Sgr A$^*$. We plot the evolution of the shadow size against λ in figure 13, where we see that increasing λ decreases the size of the WH shadow. In particular, we find that the EHT observations set the upper limits $\lambda \lesssim 0.4$ (1σ) and $\lambda \lesssim 0.5$ (2σ). To the best of our knowledge, these are the first robust observational constraints on λ. We note of course that our limits are consistent with the exponentially small values of λ suggested in [461, 471], for which the impact on the shadow size is negligible as we could have expected.

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Having discussed a few interesting WH solutions, we now turn our attention to another fascinating class of BH mimickers: NSs (see [472] for a review). The cosmic censorship conjecture, taken at face value, appears to forbid the existence of NSs [2]. In addition, the end result of gravitational collapse is commonly believe to be a BH, with the central singularity (if any) surrounded by an event horizon, a result which seemingly enjoys support from Birkhoff's theorem as well [473]. The first gravitational collapse solution of a spherically symmetric and homogeneous dust cloud (the so-called OSD collapse) was derived in 1939 by Oppenheimer, Snyder, and Datt [474, 475], and does indeed recover the Schwarzschild BH as final state. Within the OSD collapse scenario, trapped surfaces will form around the center before the formation of the central space-like singularity, which is therefore causally disconnected from all other points of space-time.

However, the OSD collapse model (as well as subsequent refinements thereof) is highly idealized in its assuming homogeneous density and zero pressure within the massive collapsing star: this is not expected to be the case for realistic astrophysical stars. Under physically more realistic situations, the introduction of inhomogeneities in the matter density profile and a non-zero pressure within GR leads to the possibility of dynamical collapse resulting in a strong but non-space-like curvature singularity, not hidden behind an event horizon [476–485]. Therefore, given suitable initial conditions, continuous gravitational collapse of an inhomogeneous matter cloud can lead to the formation of a class of exotic horizonless compact objects known as NSs, which have received significant attention, particularly in the recent literature (see e.g. [486–505]).

One of the simplest known NS solutions is the JNW NS. This is a solution of the Einstein–Klein–Gordon equations for a massless scalar field [506] (see also [507–510]). Despite being a NS, it possesses a photon sphere and therefore casts a shadow. The JNW space-time is characterized by a scalar charge which one can continuously adjust to recover the Schwarzschild solution. Given its importance, this space-time has been the subject of a large number of studies, particularly regarding the possibility of observationally distinguishing it and other NSs from the Schwarzschild BH, see e.g. [102, 485, 511–519].

The JNW space-time can be characterized by the following metric functions [506]:

$$\begin{align} A(r) = \left [ 1-\frac{2M}{r(1-\overline{\nu})} \right ] ^{1-\overline{\nu}}\,,\quad C(r)=r^{\,2} \left [ 1-\frac{2M}{r(1-\overline{\nu})} \right ] ^{\overline{\nu}}\,. \end{align} \tag{ 39 }$$

Here, $\overline{\nu} = 1-\sqrt{1+q/M}$, q is the scalar charge of the NS, and the scalar field radial profile is given by the following:

$$\begin{align} \phi(r) = \frac{q}{2\sqrt{M^{\,2}+q^2}} \left ( 1-\frac{2\sqrt{M^{\,2}+q^2}}{r} \right )\,. \end{align} \tag{ 40 }$$

Overall, the parameter $\overline{\nu}$ characterizes a specific hair. Note that, since $C(r) \neq r^{\,2}$, the coordinate r appearing in equation (39) is not the areal radius, which can instead be obtained through an appropriate transcendental coordinate transformation. We also note that the JNW space-time can only cast a shadow for $0 \leqslant \overline{\nu} \leqslant 0.5$.

We compute the shadow radius numerically, and plot its evolution against the hair parameter $\overline{\nu}$ in figure 14. There is a wide range of parameter space where Sgr A$^*$'s shadow is consistent with the corresponding compact object being a JNW NS. In fact, the EHT observations set rather weak limits on $\overline{\nu}$, with $\overline{\nu} \lesssim 0.4M$ (1σ) and $\overline{\nu} \lesssim 0.45M$ (2σ), barely excluding the extremal JNW NS, $\overline{\nu} = 0.5$. These limits can of course be translated into rather weak limits on the strength of the scalar field from equation (40). More importantly, among all the examples of BH mimickers we have considered so far, the JNW NS is one of the few which is in excellent agreement with the EHT observations given the metric tests we have performed in this work.

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Another well-known NS solution is the JMN NS, developed by one of us in [520]. To distinguish it from another NS solution developed by the same authors in [521] we shall refer to the solution considered here as JMN singularity of the first type, or JMN-1 for short, as common practice in the literature. The JMN-1 NS can be formed as the end state of gravitational collapse given zero radial pressure but non-zero tangential pressure, i.e. in the presence of an anisotropic fluid. As shown in [520], the non-zero tangential pressure can prevent trapped surfaces from forming around the high-density core region of the collapsing matter cloud, resulting in the formation of a central NS over the course of a large co-moving time (see also [522, 523]).

The JMN-1 space-time contains a compact region filled with an anisotropic fluid for $r\lt R_b$, which is then usually matched to an exterior space-time. The interior JMN-1 space-time is described by the following metric:

$$\begin{align} \mathrm{d}s^2 = (\chi-1) \left ( \frac{r}{R_b} \right ) ^\frac{\chi}{(1- \chi)}\mathrm{d}t^2 + \frac{\mathrm{d}r^{\,2}}{(1 - \chi)} + r^{\,2}\mathrm{d}\Omega^2\,, \end{align} \tag{ 41 }$$

whose associated ADM mass is $M = \chi R_b/2$, with χ characterizing specific hair, and whose energy density ρ, radial pressure $p_ r$, and tangential pressure p_θ are given by:

$$\begin{align} \rho = \frac{\chi}{r^{\,2}}\,, \quad p_r=0\,, \quad p_{\theta}=\frac{\chi}{4(1-\chi)}\rho\,, \end{align} \tag{ 42 }$$

satisfying all the energy conditions. It can be shown that the interior JMN-1 space-time can be matched smoothly at $r = R_b$ to an exterior Schwarzschild space-time with the same mass M, since the induced metrics and extrinsic curvatures of the two space-times can be matched on the matching hypersurface (see [524]).

The shadow cast by the JMN-1 space-time was studied in a number of works by some of us, including [78, 485, 525]. It can be shown that the JMN-1 space-time does not support a photon sphere and cannot cast a shadow for $0\lt\chi\lt2/3$ (which can be re-written in the somewhat more illuminating form $R_b\gt3M$). On the other hand, for $2/3 \leqslant \chi\lt1$, the JMN-1 space-time casts a shadow of size $r_\mathrm{sh} = 3\sqrt{3}\chi R_b/2 = 3\sqrt{3}M$, i.e. identical to that of a Schwarzschild BH with the same mass M. This is shown in figure 15. The physical reason for this is that 'shadow photons' are actually traveling in the exterior Schwarzschild metric, since the interior metric does not support a photon sphere, and therefore the shadow size is identical to that of a Schwarzschild BH, thus trivially ensuring consistency with the findings of [452]. The photon sphere associated to the exterior Schwarzschild metric ($r_\mathrm{ph} = 3M$) is astrophysically relevant only if it is located outside of the matching radius (R_b ): when setting $M = \chi R_b/2$, the condition $3M \geqslant R_b$ becomes $\chi \geqslant 2/3$, recovering the previously discussed condition for the JMN-1 space-time to cast a shadow. Summarizing, based on the metric test considered in this work, the JMN-1 NS is completely indistinguishable from a Schwarzschild BH, and is in perfect agreement with the EHT observations, which therefore allow for the possibility of Sgr A$^*$ being a NS.

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In closing, we note that one of us studied the effect of spherical accretion upon a JMN-1 NS, finding that the corresponding images remain indistinguishable from those of a Schwarzschild BH, even when considering the accretion flow spectra at ${\approx} 200\,\mathrm{GHz}$ [485]. However, a subsequent study by some of us in [525] showed that the two can in principle be distinguished by the intensity distribution of light inside the shadow region: although extremely challenging, this opens up the intriguing possibility of discriminating a JMN-1 NS from a Schwarzschild BH with the same mass, which is otherwise not possible with the test considered in this work. Finally, we also note that in the $0\lt\chi\lt2/3$ regime, for which no photon sphere exists, the central singularity can cast a peculiar full-moon-type image [485] which, although inconsistent with the EHT image of Sgr A$^*$, is certainly an interesting signature to look for in upcoming SMBH images.

In all the space-times we have discussed so far, the existence of a shadow (if any) was tied to the existence of a photon sphere, which one could argue to be responsible for casting the shadow itself. However, as shown in a number of recent works, the existence of a photon sphere is a sufficient but not necessary condition for a space-time to cast a shadow. One particularly interesting example in this context is that where the existence of a thin-shell of matter is responsible for casting a shadow. Here, we consider a NS space-time surrounded by a thin shell of matter, studied by some of us in [526]. Specifically, we glue two JMN-1 NS space-times (see section 3.13), an interior one (JMN-1$_\textrm i$) characterized by the parameter $\chi_\textrm i$ and an exterior one (JMN-1$_\textrm e$) characterized by the parameter $\chi_\textrm e$, at $r = R_{b1}$. The JMN-1$_\textrm e$ space-time is then further glued to an external Schwarzschild space-time with mass $M = \chi_\textrm eR_{b2}/2$ at $r = R_{b2}$. Explicitly, the metric of the JMN-1$_\textrm i$ space-time is given by:

$$\begin{align} \mathrm{d}s^2= (\chi_\textrm e-1) \left ( \frac{R_{b1}}{R_{b2}} \right ) ^\frac{\chi_\textrm e}{(1- \chi_\textrm e)} \left ( \frac{r}{R_{b1}} \right ) ^\frac{\chi_\textrm i}{(1- \chi_\textrm i)}\mathrm{d}t^2 + \frac{\mathrm{d}r^{\,2}}{(1 - \chi_\textrm i)} + r^{\,2}\mathrm{d}\Omega^2\,, \end{align} \tag{ 43 }$$

whereas the metric of the JMN-1$_\textrm e$ space-time is given by:

$$\begin{align} \mathrm{d}s^2 = (\chi_\textrm e-1) \left ( \frac{r}{R_{b2}} \right ) ^\frac{\chi_\textrm e}{(1- \chi_\textrm e)}\mathrm{d}t^2 + \frac{\mathrm{d}r^{\,2}}{(1 - \chi_\textrm e)} + r^{\,2}\mathrm{d}\Omega^2\,. \end{align} \tag{ 44 }$$

As shown by some of us in [526] using Israel's junction conditions [527], the gluing procedure is possible even if $\chi_\textrm i \neq \chi_\textrm e$, provided a thin shell of matter exists at the time-like hypersurface $r = R_{b1}$. Note that $\chi_\textrm i$ and $\chi_\textrm e$ characterize specific hair.

If $\chi_\textrm e\lt2/3$, based on our earlier discussions in section 3.13 it is clear that the space-time does not support the existence of a photon sphere in the usual sense. Nonetheless, if $\chi_\textrm i\gt2/3$, the space-time can still cast a shadow, thanks to the presence of a cusp-like feature in the effective potential which leads to the existence of a critical impact parameter (much like in the case where a photon sphere is present), as shown in [526]. One can then show that the shadow size is given by:

$$\begin{align} r_\mathrm{sh} = \frac{R_{b1}}{\sqrt{1-\chi_\textrm e} \left ( \frac{R_{b1}}{R_{b2}} \right ) ^{\frac{\chi_\textrm e}{2(1-\chi_\textrm e)}}}\,, \end{align} \tag{ 45 }$$

which depends on the location of the thin shell of matter ($R_{b1}$), the radius where the JMN-1_e space-time is matched to the external Schwarzschild space-time ($R_{b2}$), and the value of χ corresponding to the JMN-1$_\textrm e$ space-time ($\chi_\textrm e$), recalling that $M = \chi_\textrm eR_{b2}/2$, and subject to the conditions $R_{b2}\gt R_{b1}$ and $\chi_\textrm e\lt2/3$ (or equivalently $R_{b2}\gt3M$), with the latter required for the space-time to not possess a photon sphere, and therefore for the shadow to be cast exclusively by the thin shell of matter.

For illustrative purposes, we consider an example where we fix $\chi_\textrm e = 0.6$ and $R_{b2} \approx 0.33$ (implying that $\chi_\textrm eR_{b2} \approx 2$ for consistency, as we are working in units of M), so that the shadow size only depends on the physically most interesting parameter $R_{b1}$, i.e. the location of the thin shell of matter. We show the evolution of the shadow size against $R_{b1}$ in figure 16. We immediately note that the space-time can cast a shadow only for non-zero values of $R_{b1} \gtrsim 1.5M$. This is consistent with our expectations: in fact, the limit $R_{b1} \to 0$ corresponds to the case where the JMN-1$_\textrm i$ space-time, and correspondingly the thin-shell of matter, is not present. This case reduces to that of a single JMN-1 space-time already studied in section 3.13, for which we know that no shadow is cast for $\chi\lt2/3$ (see figure 15), which is clearly the case here since $\chi_\textrm e = 0.6$. This result confirms that it is genuinely the thin shell of matter which is responsible for the space-time casting a shadow, even in the absence of a photon sphere. Of course, it is clear from equation (45) that the shadow size depends on $\chi_\textrm e$ and $R_{b2}$ separately, so that different choices of $\chi_\textrm e$ and $R_{b2}$ such that $\chi_\textrm eR_{b2} = 2$ would lead to results which differ from those we have shown here. Nonetheless, the qualitative features we have shown and discussed would remain unchanged: this is sufficient for the purposes of the present discussion with the aim of showing that a thin shell of matter can cast a shadow.

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In the absence of a photon sphere, a thin shell of matter is not the only possibility for casting a shadow. A relevant example in this sense was studied by some of us in [74], in the context of a novel NS space-time described by the following metric:

$$\begin{align} \mathrm{d}s^2 = -\frac{\mathrm{d}t^2}{\left(1+\frac{M}{r}\right)^2}+\left(1+\frac{M}{r}\right)^2\mathrm{d}r^{\,2} + +r^{\,2}\mathrm{d}\Omega^2\,. \end{align} \tag{ 46 }$$

The above metric describes a space-time of ADM mass M which approaches the Schwarzschild space-time as $r \to \infty$, possesses a strong curvature singularity at r = 0 (as can be verified by computing the Kretschmann and Ricci scalars), and is manifestly horizon-less. Note that this space-time is not characterized by additional hairs. Since the singularity is null-like, we shall refer to the space-time described by equation (46) as 'null NS' (NNS). We note that this particular NNS space-time was introduced on purely phenomenological grounds, as a toy model to explore the possibility of a shadow being cast in the absence of a photon sphere and a thin shell of matter. We leave open the question of whether gravitational collapse from suitable initial conditions can lead to the NNS space-time as end product. We note, however, that the formation of null- and even time-like singularities from the gravitational collapse of matter clouds is known to be possible and has been studied extensively in the literature (see e.g. [478, 522, 528–535] for relevant works).

From equation (13), it is clear that the NNS does not possess a photon sphere—formally, the solution to equation (13) would be $r_\mathrm{ph} = 0$. In other words, the minimum turning point radius for a photon fired from infinity is r = 0, where the effective potential for null geodesics has a finite value. Nonetheless, the impact parameter corresponding to the minimum turning point radius is non-zero, and this allows the NNS to cast a shadow of size $r_\mathrm{sh} = M$. In other words, despite the absence of an event horizon, null geodesics reaching the NNS from infinity with an impact parameter b < M will be trapped closer and closer to the central singularity, and will not be able to return to the observer: this leads to the appearance of a central dark region in the observer's sky, which we associate to the NNS shadow. The situation is perhaps visually easier to grasp from figure 1(f) of [74], which shows examples of light trajectories around the NNS, making the physical origin of the shadow clear.

The shadow cast by the NNS is of course too small to be consistent with the EHT observations of Sgr A$^*$, as is clear from figure 17. Therefore, much as the MT traversable WH, the EHT observations rule out the possibility of Sgr A$^*$ being a NNS at very high (${\gt}5\sigma$) significance. Of course, as we have discussed earlier in sections 3.12–3.14, other NS space-times remain potentially viable descriptions of Sgr A$^*$, leaving open the intriguing possibility that the latter might not be a BH but an exotic horizonless object.

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So far we have discussed a few well-motivated regular BH solutions, as well as BH mimickers such as WHs and NSs. To continue our journey into horizon-scale tests of alternatives to the Schwarzschild metric, we now move on to BH solutions which violate the NHT in a controlled way, in the presence of additional fields or exotic interactions. As is well known, one of the most straightforward (and well-motivated) ways of violating the NHT is through the presence of additional matter fields, such as a scalar field. The following subsections will therefore be devoted to the study of a number of 'hairy' BH solutions arising broadly speaking in scenarios of 'new physics', which include but are not limited to the presence of additional fields, modified theories of gravity (these two scenarios are not mutually exclusive), or new high-energy physics models.

We begin by studying the simplest possible scenario where the NHT can be violated in a controlled way, considering a minimally coupled real scalar field φ featuring a potential. The dynamics of the field are thus governed by the following Lagrangian:

$$\begin{align} \mathcal{L} \supset \frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-V(\phi)\,, \end{align} \tag{ 47 }$$

where here and in what follows the symbol $\supset$ indicates that the Lagrangian contains at least the terms on the right, plus potentially other terms which are irrelevant for the determination of the corresponding BH metric. Following [128, 536, 537], including work by some of us, we take the following form for the potential:

$$\begin{align} V(\phi) = \frac{1}{2\nu^4}e^{-2\sqrt{2}\phi}g(\phi)\,, \end{align} \tag{ 48 }$$

where $g(\phi)$ is given explicitly in [128]. The specific choice of potential is not particularly motivated from a high-energy theory perspective. However, in the context of BH physics, it is particularly interesting because it provides a toy example of a controlled violation of the NHT, which proceeds by explicitly introducing a new energy scale in the scalar sector. More specifically, the scalar field profile is given by [128, 536, 537]:

$$\begin{align} \phi(r) = \frac{1}{\sqrt{2}} \ln \left ( 1+\frac{\nu}{r} \right ) \end{align} \tag{ 49 }$$

where the scale ν > 0 governs the scalar field fall-off, and effectively acts as a hair parameter. The metric functions read (see e.g. [128, 536, 537]):

$$\begin{align} A(r) &= -\frac{2r\nu+6M(2r+\nu)}{\nu^2}-\frac{2r(r+\nu)(6M+\nu)}{\nu^3}\ln\frac{r}{r+\nu}\,, \nonumber \\ C(r) &= r(r+\nu)\,. \end{align} \tag{ 50 }$$

Although it is not straightforward to see from equation (50), the metric function A(r) does indeed reduce to the Schwarzschild limit $1-2M/r$ as the hair parameter vanishes. This is best seen by Taylor-expanding the metric function in ν, yielding:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{3M-r}{3r^{\,2}}\nu+{\cal O}(\nu^2)\,, \end{align} \tag{ 51 }$$

from which one easily sees that the limit ν → 0 indeed recovers the Schwarzschild metric, and ν characterizes an universal hair. We compute the shadow radius of BHs within this theory numerically, as done earlier in [128] by one of us. We show the evolution of the shadow size against the hair parameter ν in figure 18, finding that the shadow size increases quickly with increasing hair parameter. As in other similar examples we have already seen, the EHT observations set particularly tight constraints on this scenario, with limits on the hair parameter of order $\nu \lesssim 0.01M$ (1σ) and $\nu \lesssim 0.4M$ (2σ). This effectively constrains the energy scale at which violations of the NHT become important, within this particular theory.

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We now study another scenario featuring a scalar field, which leads to a controlled violation of the NHT. We consider a real scalar field φ, conformally coupled to gravity, through the following Lagrangian:

$$\begin{align} \mathcal{L} \supset \partial^{\mu}\phi\partial_{\mu}\phi+\frac{1}{6}\phi^2R\,, \end{align} \tag{ 52 }$$

where R is the Ricci scalar, and the $1/6$ factor ensures that the coupling is conformal (in 4 dimensions). BH solutions within this theory have been obtained by Astorino [538] (see also [539–546] for BH and black brane solutions within related models), and contain a hair parameter S related to the scalar field profile, and directly connected to the value of the scalar field at the horizon. The hair parameter, or scalar charge, emerges as an integration constant and characterizes primary hair, i.e. it cannot be expressed as a function of the other hair parameters. The metric function reads:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{S}{r^{\,2}}\,, \end{align} \tag{ 53 }$$

where the scalar charge S characterizes a specific hair and can take both positive and negative values, with $S\lt M^{\,2}$ required in the former regime for the solution to describe a BH. The space-time described by equation (53) is closely related to the well-known 'BBMB black hole' [547, 548], and appears very similar to the RN metric [equation (18)]. However, there is a crucial difference in that the scalar hair parameter can take negative values, which are instead effectively precluded within the RN metric as $Q^2\gt0$. The S < 0 regime is in fact better known as mutated RN BH, and effectively describes the geometry of a traversable WH [549] (see also [128]). Formally, this corresponds to an analytic continuation of the RN metric for imaginary values of the electric charge (thus negative values of Q²), although we discourage the reader from taking this interpretation too far. We also note that the RN-like form of the metric is no coincidence, and is directly related to the conformal symmetry enjoyed by the Lagrangian in equation (52), see in addition [550] for earlier relevant results. Finally, it is worth noticing that a similar metric arises in the context of a conformal massive gravity model where Lorentz invariance is spontaneously broken (see [301, 551, 552]), although only for a very specific choice of the function ${\cal F}$ appearing in equation (10) of [551].

Since the metric is RN-like, the shadow size takes the same functional form as equation (19):

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{2} \left ( 3+\sqrt{9-8S/M^{\,2}} \right ) }{\sqrt{4+\frac{\sqrt{9-8S/M^{\,2}}-3}{S/M^{\,2}}}}\,, \end{align} \tag{ 54 }$$

valid within both the BH (S > 0) and WH (S < 0) regimes. We show the evolution of the shadow size against the scalar charge S in figure 19. We see that within the regime S > 0 the shadow size decreases with increasing scalar charge, analogously to the case with the electric charge in the RN metric. In this regime, the EHT observations set the limits $S \lesssim 0.65M^{\,2}$ (1σ) and $S \lesssim 0.9M^{\,2}$ (2σ). In the mutated RN BH/WH regime S < 0, the shadow size instead increases when increasing the absolute value of the scalar charge, quickly bringing the shadow size outside the range allowed by the EHT observations, which set the lower limits $S \gtrsim -0.04M^{\,2}$ (1σ) and $S \gtrsim -0.4M^{\,2}$ (2σ).

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Generalizations and extensions of Einstein's GR have a long and rich history, but undoubtedly one of the most studied theories in this sense is f(R) gravity, where the Ricci scalar R in the Einstein–Hilbert action is replaced by an arbitrary function f(R). Several f(R) models have been constructed in the literature with the aim of describing the late-time accelerated expansion of the Universe (and/or an early accelerated phase of inflation), which requires satisfying viability conditions including the positivity of effective gravitational couplings, stability of cosmological perturbations, asymptotic ΛCDM behavior in the high curvature regime, stability of the late-time de Sitter point, as well as observational constraints from the equivalence principle, Solar System, and cosmological tests. For an inevitably incomplete list of important works on a wide range of aspects of f(R) gravity, with particular reference to BH solutions therein, we refer the reader to [553–608].

Clifton and Barrow considered a particularly interesting f(R) model in [609], whose action is given by:

$$\begin{align} S = \int \mathrm{d}^4x\sqrt{-g}\,M_\mathrm{Pl}^{2-2\delta}R^{1+\delta}\,, \end{align} \tag{ 55 }$$

where δ is a dimensionless parameter, $M_\mathrm{Pl}$ is the Planck mass, and GR is recovered in the limit δ → 0. This theory was argued to possess several appealing properties, including its being a one-parameter extension of GR not introducing any additional scale besides the already present Planck mass, its providing a testing ground for new developments in particle production, holography and gravitational thermodynamics, as well as its being one of the few f(R) models possessing simple exact solutions for Friedmann-Lemaître-Robertson-Walker (FLRW)-like cosmological models and exact static spherically symmetric solutions generalizing the Schwarzschild metric. Clifton and Barrow [609] does indeed identify such a solution, whose metric function is given by:

$$\begin{align} A(r)=r^{2\delta\frac{1+2\delta}{1-\delta}}-\frac{2M}{r^{\frac{1-4\delta}{1-\delta}}} \,, \end{align} \tag{ 56 }$$

where δ characterizes an universal hair. It is easy to see that the above metric is asymptotically flat only if $-1/2\lt\delta\lt0$ or δ > 1. We exclude the regime δ > 1, as we find a posteriori that within this range the shadow size is by far too small and completely excluded by observations, and therefore focus on the region $-1/2\lt\delta\lt0$.

We compute the shadow size numerically, and show its evolution against δ in figure 20. ⁴⁰ We see that the shadow size decreases with increasing $\vert \delta \vert$ (i.e. with δ becoming more negative), and that the EHT observations set the limits $\delta \gtrsim -0.15$ (1σ) and $\delta \gtrsim -0.20$ (2σ). In the δ < 0 regime, the existence of a stable matter-dominated period of evolution, which is itself a prerequisite for a viable cosmological model, requires $\delta\lt-1/4$ [609]. This makes our limits particularly interesting, as they exclude this region, and therefore would appear to exclude this class of models, highlighting a very appealing complementarity between BH shadows and cosmological constraints.

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We now move on to a well-known and much more general class of scalar-tensor theories: Horndeski gravity. On very general grounds, scalar-tensor theories are an appealing playground for theorists as they encapsulate the key characteristics of several theories of modified gravity, which typically possess a scalar-tensor limit. Back in 1974, Horndeski wrote down the most general scalar-tensor theory with equations of motion at most of second order [610], with the latter requirement sufficient to ensure that the theory is free of the Ostrogradsky ghost [611]. ⁴¹ After living almost ignored for over 30 years, the theory, which encompasses many well-known theories of gravity (including f(R) gravity) as specific cases, was 'rediscovered' in [622, 623], making it a beautiful example of 'sleeping beauty' theory which is now attracting a tremendous amount of attention (see e.g. [624–638]) after decades of dormancy.

Here, we shall focus on the shift-symmetric subclass of Horndeski gravity, where the scalar field φ enjoys a shift symmetry $\phi \to \phi+\textrm{const}$. This sector is of particular interest since a well-known no-go theorem by Hui and Nicolis dictates that static, asymptotically flat BH solutions thereof are isometric to the Schwarzschild BH, with a trivial (constant) profile for the scalar field [639]. Of course, this no-go theorem comes with several assumptions, which we shall here break (see e.g. [272, 640–644]) in order to play with hairy BH solutions. The shift-symmetric sector of Horndeski gravity we shall consider is described by the following action:

$$\begin{align} S = \int \mathrm{d}^4x\sqrt{-g}\, \left ( {\cal L}_2+{\cal L}_3+{\cal L}_4+{\cal L}_5 \right ) \,, \end{align} \tag{ 57 }$$

where the four Lagrangian terms ${\cal L}_2$, ${\cal L}_3$, ${\cal L}_4$, and ${\cal L}_5$ are given by the following [610]:

$$\begin{align} {\cal L}_2 &= G_2(X)\,, \end{align} \tag{ 58 }$$

$$\begin{align} {\cal L}_3 &= -G_3(X) \Box \phi\,, \end{align} \tag{ 59 }$$

$$\begin{align} {\cal L}_4 &= G_4(X) R + G_{4,X} \left [ (\Box \phi)^2 -(\nabla_\mu\nabla_\nu\phi)^2 \right ] \,, \end{align} \tag{ 60 }$$

$$\begin{align} {\cal L}_5 &= G_5(X) G_{\mu\nu}\nabla^\mu \nabla^\nu \phi - \frac{1}{6} G_{5,X} \big [ (\Box \phi)^3 - 3\Box \phi(\nabla_\mu\nabla_\nu\phi)^2 \nonumber \\ & \quad + 2(\nabla_\mu\nabla_\nu\phi)^3 \big ]\,, \end{align} \tag{ 61 }$$

which involve four arbitrary functions (G₂, G₃, G₄, and G₅) of the kinetic term $X\equiv-\partial^\mu\phi\partial_\mu\phi/2$, $(\nabla_\mu\nabla_\nu\phi)^2 \equiv \nabla_\mu\nabla_\nu\phi \nabla^\nu\nabla^\mu\phi$, and $(\nabla_\mu\nabla_\nu\phi)^3 \equiv \nabla_\mu\nabla_\nu\phi \nabla^\nu\nabla^\rho\phi \nabla_\rho\nabla^\mu\phi$, and where $G_{4,X}$ and $G_{5,X}$ are respectively the derivatives of the functions G₄ and G₅ with respect to X.

As discussed in [644], the no-go theorem of [639] essentially rests upon three assumptions:

• 1.  
  The derivatives with respect to X of the G_i s only contain positive or zero powers of X in the limits X → 0, $r \to \infty$;
• 2.  
  The kinetic term X is canonical;
• 3.  
  The norm of the Noether current associated to the shift symmetry is finite down to the horizon.

 Clearly, then, in order to find hairy BH solutions within Horndeski gravity, one must at the very least break one of the above assumptions—the solutions which we shall consider are obtained by breaking the first one. As shown in [644], doing so is possible at least for the following particular choices of the G_i functions:

$$\begin{align} G_2 &\supseteq \sqrt{-X}\,,\quad G_3 \supseteq \ln \vert X \vert \,, \nonumber \\ \quad G_4 &\supseteq \sqrt{-X}\,,\quad G_5 \supseteq \ln \vert X \vert \,. \end{align} \tag{ 62 }$$

A choice of the G_i functions satisfying equation (62) has the potential to admit non-Schwarzschild-isometric, asymptotically flat BH solutions, with a non-trivial scalar field profile, and with a regular Noether current. In the following, we shall consider two specific solutions obtained in [644, 645] for a particular choice of the G_i functions satisfying equations (62), which we shall overall refer to as 'sqrt quartic Horndeski gravity ⁴² ', while simply referring to the solutions of [644] and [645] as 'Case 1' and 'Case 2' respectively.

3.19.1. Case 1.

The first BH solution in the shift-symmetric sector of Horndeski gravity we consider was studied in [644]. In the notation of equations (58)–(61), and within the system of units we have adopted, the particular choice of G_i functions is given by:

$$\begin{align} G_2=\eta X\,,\quad G_4=1/16\pi+\beta\sqrt{-X}\,,\quad G_3=G_5=0\,. \end{align} \tag{ 63 }$$

Clearly, G₂ corresponds to a canonical kinetic term, whereas it is the particular X-dependence of G₄ (the $1/16\pi$ term simply leads to an Einstein–Hilbert term in the action, since it is multiplied by R) which allows for hairy BH solutions, by satisfying the specific set of conditions laid out in equation (62).

Hairy BH solutions within this theory are described by the following metric function (again within the system of units we have adopted) [644]:

$$\begin{align} A(r) = 1-\frac{2M}{r}-\frac{8\pi p_\mathrm{eff}}{r^{\,2}}\,, \end{align} \tag{ 64 }$$

which is clearly asymptotically flat, and where we have defined the hair parameter $p_\mathrm{eff} \equiv \beta^2/\eta$ which characterizes an universal hair. Note that, depending on the sign of η (which, in principle, can be either positive or negative), $p_\mathrm{eff}$ can carry either sign: in particular, positive η corresponds to positive $p_\mathrm{eff}$, and vice-versa ⁴³ .

Note that this BH solution is of the RN-like form. In [644], this was argued to be possibly related to the remnant of global conformal invariance enjoyed by the action defined by equation (63), as the extremal RN space-time itself is known to possess a discrete conformal isometry related to spatial inversion ⁴⁴ . Within the negative $p_\mathrm{eff}$ regime, $p_\mathrm{eff}\gt-M^{\,2}/8\pi$ is required (else the solution describes a NS), whereas no such restrictions are present in the positive $p_\mathrm{eff}$ regime. Finally, we note that the positive $p_\mathrm{eff}$ regime is expected to described a mutated RN WH, as the S < 0 regime of the BH/WH with conformal scalar hair considered in section 3.17 (although we note that such an interpretation was not considered in [644]).

By virtue of the RN-like form of the metric, we can immediately borrow upon the previous results for RN-like space-times (sections 3.1 and 3.17) to obtain the shadow radius as a function of $p_\mathrm{eff}$ (see e.g. equation (54)):

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{2} \left ( 3+\sqrt{9+64\pi p_\mathrm{eff}} \right ) }{\sqrt{4+\frac{3-\sqrt{9+64\pi p_\mathrm{eff}}}{8\pi p_\mathrm{eff}}}}\,, \end{align} \tag{ 65 }$$

which, we recall, is valid within both the $p_\mathrm{eff}\lt0$ and $p_\mathrm{eff}\gt0$ regimes. We show the evolution of the shadow size against the scalar charge $p_\mathrm{eff} = \beta^2/\eta$ in figure 21, which of course is qualitatively identical to figure 19, albeit with the x axis inverted and rescaled. We see that the EHT observations set very stringent constraints on $p_\mathrm{eff}$. In the $p_\mathrm{eff}\gt0$ regime, we find the limits $p_\mathrm{eff} \lesssim 0.002M^{\,2}$ (1σ) and $p_\mathrm{eff} \lesssim 0.01M^{\,2}$ (2σ). On the other hand, in the $p_\mathrm{eff}\lt0$ regime, we find the limits $p_\mathrm{eff} \gtrsim -0.005M^{\,2}$ (1σ) and $p_\mathrm{eff} \gtrsim -0.01M^{\,2}$ (2σ). These constraints limit the combination $\beta^2/\eta$, with β governing the strength of the $\sqrt{-X}$ term in G₄ and η governing the strength of the kinetic term in G₂. A detailed study of the theoretical and model-building implications of these constraints as well as the complementarity with cosmological, astrophysical, and GW constraints (which is the subject of a rich and rapidly growing body of literature, see e.g. [651–680] for examples of these constraints and observational signatures of Horndeski gravity) would be very interesting, but goes significantly the scope of the paper, and is therefore left to future work.

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3.19.2. Case 2.

We now consider a different BH solution in the shift-symmetric sector of Horndeski gravity, studied in [645], and whose shadow was first computed in [154] by some of us. Using the same notation of equations (58)–(61), the authors begin from the following particular choice of G_i functions [645]:

$$\begin{align} G_2&=\alpha_{21}X+\alpha_{22}(-X)^{\omega_2}\,,\quad G_3=-\alpha_{31}(-X)^{\omega_3}\,,\nonumber \\ G_4&=1/16\pi+\alpha_{42}(-X)^{\omega_4}\,,\quad G_5=0\,, \end{align} \tag{ 66 }$$

which clearly generalizes equation (63), with the latter recovered for $\alpha_{22} = \alpha_{31} = 0$ and $\omega_4 = 1/2$. Note also that the theory described by equation (66) is known to be weakly hyperbolic for $\alpha_{21} \neq 0$ [681], but its stability has not been examined for $\alpha_{21} = 0$. The authors of [645] at a later stage focus on a particular choice of the α parameters, which still preserves the regularity of the Noether current associated to the shift symmetry, and the finiteness of the energy. More specifically, the following choice of G_i functions is studied [645]:

$$\begin{align} G_2&= \alpha_{22}(-X)^{\frac{3}{2}}\,,\quad G_4=1/16\pi+\alpha_{42}\sqrt{-X}\,,\nonumber \\ G_3&= G_5=0\,. \end{align} \tag{ 67 }$$

The above is clearly not a generalization of equation (63): while the G₄ functions are identical, there is a crucial difference in the G₂ functions. Therefore, we should not expect the BH solution derived within this sector to (necessarily) reduce to the one studied in section 3.19.1.

The corresponding BH solution is indeed qualitatively very different from the one considered in section 3.19.1, and is described by the following metric function:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{2Mh}{r} \ln \left ( \frac{r}{2M} \right ) \,, \end{align} \tag{ 68 }$$

where the hair parameter h characterizes an universal hair and is related to the fundamental Lagrangian parameters α₂₂ and α₄₂ appearing in equation (67) through the following:

$$\begin{align} h \equiv \frac{2\sqrt{2}}{3\sqrt{3}}\alpha_{42}\sqrt{-\frac{\alpha_{42}}{\alpha_{22}}}\,. \end{align} \tag{ 69 }$$

We see that the above space-time is asymptotically flat, while computing the Kretschmann and Ricci scalars signals the presence of a scalar polynomial singularity at r = 0. In the h > 0 regime, which for consistency requires $\alpha_{22}\lt0$ and $\alpha_{42}\gt0$, the space-time only contains one event horizon, analogously to the Schwarzschild metric. In the h < 0 regime (and more precisely for $h \in [-2;0]$), which for consistency requires $\alpha_{22}\gt0$ and $\alpha_{42}\lt0$, the space-time contains two horizons: an outer event horizon, and an inner Cauchy horizon.

Unlike the earlier [154] by some of us, where the analysis was restricted to the h < 0 regime, here we do not set any prior restriction on the sign of the hair parameter h. While slightly cumbersome, a closed-form expression for the shadow radius exists and is given by:

$$\begin{align} r_\mathrm{sh} = \frac{3h{\cal X}}{\sqrt{1+\frac{2 \left ( -1+h\ln{\cal X} \right )}{3h{\cal X}}}}\,, \end{align} \tag{ 70 }$$

where the quantity ${\cal X}$ is defined as:

$$\begin{align} {\cal X} \equiv W \left [ \frac{2e^{\frac{1}{3}+\frac{1}{h}}}{3h} \right ]\,, \end{align} \tag{ 71 }$$

with W(z) denoting the principal branch of the Lambert W function, also known as omega function or product logarithm function. In figure 22 we show the evolution of the shadow size against h. We see that the shadow radius quickly increases with increasing $\vert h \vert$ in the h < 0 regime, while decreasing more slowly with increasing h in the h > 0 regime. As a result, negative values of the hair parameter are very tightly constrained: we find that consistency with the EHT observations requires $-0.01 \lesssim h \lesssim 0.40$ (1σ) and $-0.15 \lesssim h \lesssim 0.60$ (2σ). As with the BH in Horndeski gravity studied in section 3.19.1, these constraints limit a particular combination of two fundamental Lagrangian parameters (in this case $\alpha_{42}\sqrt{-\alpha_{42}/\alpha_{22}}$), and a detailed study of the theoretical and model-building implications of these constraints, as well as the complementarity with cosmological and astrophysical constraints, while interesting, is left to future work.

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A particularly interesting modified theory of gravity was put forward by Moffat [682], and typically goes by the name of scalar-tensor-vector gravity, but is also often referred to as MOG (short for MOdified Gravity), as we shall do here. MOG is a covariant scalar-tensor-vector theory of gravity which introduces an additional massive scalar field and an additional massive vector field, while allowing the coupling strength of the latter, as well as the gravitational constant, to vary in space and time. The theory effectively adds a repulsive Yukawa force to Newton's acceleration law. For the full action of MOG, we refer the reader to [682]. We also note that MOG has a long history of successfully explaining several astrophysical and cosmological observations: for an inevitably incomplete list of important works on MOG, see for example [683–707].

A particular BH solution within MOG was derived in [708], assuming that the gravitational coupling G is constant (although not necessarily taking its Newtonian value), i.e. $\partial_{\mu}G = 0$ (see also [709–711]). In this configuration, the metric function of the resulting BH solution is given by [708]:

$$\begin{align} A(r)=1-\frac{2M(1+\alpha)}{r}+\frac{M^{\,2}\alpha(1+\alpha)}{r^{\,2}} \,, \end{align} \tag{ 72 }$$

where α is a parameter controlling the strength of the effective gravitational coupling $G = G_N(1+\alpha)$, and characterizes an universal hair. We immediately notice that the space-time is of the RN form, with effective charge proportional to the BH mass. Clearly, one of the effects of MOG is modify the BH ADM mass as $M \to M(1+\alpha)$, as already noticed in many works [690, 697, 708]. In the following, however, we shall opt for the same strategy adopted by many earlier works, and report the shadow size in units of the 'bare'/source/Newtonian mass M rather than the ADM mass. The rationale is that this is the ADM mass quantifies the energy measured by an observer at infinity, but dynamical measurements of the BH mass sufficiently close to the horizon will actually probe a quantity which is much closer to the Newtonian mass. This is indeed the case for the S-stars which underlie our mass-to-distance ratio prior, as explicitly discussed in [697]. In addition, one can envisage constraining α through independent tests of the strength of Newton's constant, allowing to disentangle the Newtonian mass from the ADM mass.

Having made these considerations (see section 3.38.1 for further considerations of this sort in the context of another model), we compute the shadow size in units of M as done in various earlier works (we refer especially to [697]), and show its evolution against the MOG parameter α in figure 23. Moreover, we note that a closed-form expression for the shadow radius can be found, and is given by the following:

$$\begin{align} r_\mathrm{sh} = \sqrt{2\alpha}\frac{3+3\alpha+\sqrt{(1+\alpha)(9+\alpha)}}{\sqrt{-3+\alpha+\sqrt{(1+\alpha)(9+\alpha)}}}\,, \end{align} \tag{ 73 }$$

which, although not obvious at first glance, does indeed reduce to $3\sqrt{3}$ as α → 0, as can be seen by Taylor expanding equation (73) in α. We see that the effect of increasing α is that of very quickly increasing the shadow size. Despite the RN-like form of the space-time, this is a result of our choice of reporting the shadow size in units of M rather than $M(1+\alpha)$, for the reasons discussed above.

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We note that the EHT observations set extremely tight upper limits on α, of the order of $\alpha \lesssim 0.01$ (1σ) and $\alpha \lesssim 0.1$ (2σ). These limits are much tighter than the limit $\alpha \lesssim 0.662$ obtained from the motion of the S2-star in [705], and safely exclude the range of values of α required for MOG to explain the rotation curves of dwarf galaxies, $2.5 \lesssim \alpha \lesssim 11.2$ [683]. On the other hand, they are competitive with the upper limits of $\alpha \lesssim 0.1$ arising from neutron stars [693], and $\alpha \lesssim 0.03$ coming from globular clusters [686]. Overall, the limits we have provided on the MOG parameter of order $\alpha \lesssim 0.01$ are therefore among the tightest in the literature.

We now consider one of the most well-motivated high-energy physics models: the Randall–Sundrum II (RSII) brane-world model, which has been tested over a wide range of scales and using an enormous variety of probes (see e.g. [712–726]). This is an AdS₅ brane-world model where the extra dimension has an infinite size and negative bulk cosmological constant [727]. When projecting the full 5-dimensional BH solution of the theory onto our 4D space-time, BHs effectively inherit a tidal charge q which carries projected information about 5D bulk (and in particular the AdS₅ curvature radius $\ell$), while characterizing the effect of the bulk on the brane. More specifically, the tidal charge controls the strength of projected Weyl tensor ${\cal E}_{\mu\nu}$ which transmits tidal charge stresses from the bulk to the brane, and takes the form [728]:

$$\begin{align} {\cal E}_{\mu\nu} = - \left ( \frac{q}{\widetilde{M}_\mathrm{Pl}^2} \right ) \frac{1}{r^{\,4}} \left [ u_{\mu}u_{\nu}-2r_{\mu}r_{\nu}+h_{\mu\nu} \right ]\,, \end{align} \tag{ 74 }$$

where $\widetilde{M}_\mathrm{Pl}$ is the 5-dimensional Planck mass, u_µ is the velocity 4-vector, r_µ is an unit radial vector, and $h_{\mu\nu} = g_{\mu\nu}+u_{\mu}u_{\nu}$. Physically speaking, a negative tidal charge amplifies gravitational effects of the bulk on the brane.

BH solutions within the RSII model have been studied in a large number of works (see e.g. [589, 729–738], as well as [739, 740]). The metric function for spherically symmetric solutions takes the following form:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{4M^{\,2}q}{r^{\,2}} \,, \end{align} \tag{ 75 }$$

where q characterizes a specific hair. We note that the metric function is again RN-like as with a few other metrics discussed earlier. Therefore, we can immediately borrow from the previous results, and write the shadow radius as a function of the tidal charge in a closed form:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{2} \left ( 3+\sqrt{9-32q} \right ) }{\sqrt{4+\frac{\sqrt{9-32q}-3}{4q}}}\,. \end{align} \tag{ 76 }$$

In figure 24 we plot the size of the shadow radius as a function of the tidal charge: the plot is of course analogous to figures 19 and 21. It is clear that the EHT observations place very tight limits on the tidal charge, $q \lesssim 0.15$ (1σ) and $q \lesssim 0.2$ (2σ). The same observations also place very tight limits on the negative tidal charge regime, $q \gtrsim -0.01$ (1σ) and $q \gtrsim -0.1$ (2σ). This is in contrast with the shadow of M87$^*$, which instead exhibited a very slight preference for negative tidal charge, as noted earlier in [101, 124]. This is due to the fact that M87$^*$'s shadow size is slightly larger than what would be expected given its mass-to-distance ratio, or conversely one could argue that its mass-to-distance ratio is biased low. At any rate, this fact again highlights how Sgr A$^*$'s shadow places particularly tight constraints on scenarios which tend to enlarge the shadow radius compared to the Schwarzschild BH. We note that the negative tidal charge regime of the RSII BH would in principle appear to admit an interpretation in terms of mutated RN BH/WH (just as the space-time discussed in section 3.17), though to the best of our knowledge we are not aware of this possibility having being discussed in the literature on brane-world BHs.

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Earlier in section 3.5 we discussed regular magnetically charged BHs arising from Einstein–Bronnikov NLED. Another well-motivated NLED theory is Einstein–Euler–Heisenberg NLED [741], which appears in the low-energy limit of Born–Infeld electrodynamics [343], and is described by the following Lagrangian:

$$\begin{align} \mathcal{L}(U,W)=-\frac{1}{4}U+\frac{\mu}{4}\left(U^2+\frac{7}{4}W^2\right)\,. \end{align} \tag{ 77 }$$

Choosing the same gauge field configuration as in equation (25), and carefully accounting for the NLED-induced effective geometry, BH shadows within the Einstein–Euler–Heisenberg model were studied by some of us in [104], whose procedure we follow. For completeness, we report the metric function for magnetically charged BH solutions within this theory, given by:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{q_\textrm m^2}{r^{\,2}}-\frac{2\mu}{5}\frac{q_\textrm m^4}{r^6}\,, \end{align} \tag{ 78 }$$

where $q_\textrm m$ is the BH magnetic charge and characterizes a specific hair, and µ is the NLED coupling appearing in equation (77). We note that as long as µ ≠ 0, values of the magnetic charge $q_\textrm m\gt1$ are allowed.

We compute the shadow radius numerically, following [104] by some of us and once more carefully taking into account the effective geometry induced by the NLED coupling by adopting Novello's method [214, 215] (again, as in section 3.5), for the intermediate steps of this calculation we refer the reader to [104]. Note that we now have 2 free parameters: $q_\textrm m$ and µ. However, for simplicity, we fix the Lagrangian coupling to µ = 0.3. The reason is two-fold: (a) as noted in [104], most of the effect on the shadow size come from the magnetic charge $q_\textrm m$ rather than the NLED coupling µ, and (b) $\mu \sim 0.3$ is approximately the largest allowed coupling before the perturbative approach of the theory around the Maxwell Lagrangian for µ → 0 ceases to be meaningful.

In figure 25 we plot the evolution of the shadow radius against $q_\textrm m$, at fixed µ = 0.3. We immediately note that the behavior of the shadow radius, and therefore the constraints from the EHT observations, depend on whether we are in the $q_\textrm m\lt1$ or $q_\textrm m\gt1$ regime. For $q_\textrm m\lt1$, we find the upper limits $q_\textrm m \lesssim 0.7M$ (1σ) and $q_\textrm m \lesssim 0.8M$ (2σ). On the other hand, for $q_\textrm m\gt1$, we observe a sharp discontinuity in the shadow radius, with the shadow size apparently diverging. This feature is due to the appearance of a singularity in the NLED-induced effective geometry: for more details see [104].

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Leaving aside the issue of whether the $q_\textrm m\gt1$ regime is physical in first place, given this singularity, in this regime we find that the shadow size varies drastically as $q_\textrm m$ changes. For µ = 0.3, consistency with the EHT observations requires $q_\textrm m \sim 1.25M$. This number depends to some extent on the value to which µ is fixed (see [104] for further discussions), although for values of $\mu \lesssim 0.3$, following the results of [104], we can expect consistency with the EHT observations for values of the magnetic charge $1.1M \lesssim q_\textrm m \lesssim 1.5M$. We stress once again, however, that it is unclear whether we can consider this range of parameter space physical.

The Sen BH is a non-regular space-time appearing in the low-energy limit of heterotic string theory [742], whose field content includes the electromagnetic field A^µ with field-strength tensor $F^{\mu\nu}$, the second-rank antisymmetric tensor gauge field $B_{\mu\nu}$ (also known as Kalb–Ramond field [743]), and the dilaton field Φ, ⁴⁵ as described by the following action in the string frame [744] (see also [745–748] for other examples of BH and black p-brane solutions in the low-energy limit of string theory):

$$\begin{align} S = \int \mathrm{d}^4x\sqrt{-g}\,e^{-\phi}\,{\cal L} \,, \end{align} \tag{ 79 }$$

where the Lagrangian ${\cal L}$ is given by:

$$\begin{align} {\cal L} = R-\frac{1}{8}F^{\mu\nu}F_{\mu\nu}+\partial^{\mu}\Phi\partial_{\mu}\Phi-\frac{1}{12}H_{\kappa\lambda\mu}H^{\kappa\lambda\mu}\,, \end{align} \tag{ 80 }$$

with $H_{\kappa\lambda\mu}$ the third-rank tensor field (Kalb–Ramond 3-form) defined in terms of the tensor gauge field and the field-strength tensor of the electromagnetic field as:

$$\begin{align} H_{\kappa\mu\nu} \equiv \partial_{\kappa}B_{\mu\nu}+\partial_{\nu}B_{\kappa\mu}+\partial_{\mu}B_{\nu\kappa} -\frac{1}{4} \left ( A_{\kappa}F_{\mu\nu}+A_{\nu}F_{\kappa\mu}+A_{\mu}F_{\nu\kappa} \right )\,. \end{align} \tag{ 81 }$$

The metric function derived in this configuration was found by Sen [744] and reads:

$$\begin{align} A(r) = 1-\frac{2M}{r+\frac{q_\textrm m^2}{M}} \,, \end{align} \tag{ 82 }$$

where $q_\textrm m$ is an effective charge (including both the electric charge and an effective charge associated with the dilaton field) measured by a static observer at infinity, which characterizes specific hair, and is bound by the requirement $q_\textrm m \leqslant \sqrt{2}$. Various studies have been devoted to the Sen BH and its rotating generalization (the Kerr–Sen BH) in the literature, see for instance [749–758].

A closed-form expression exists for the shadow radius, and is given as follows:

$$\begin{align} r_\mathrm{sh} = \frac{3-2q_\textrm m^2+\sqrt{9-4q_\textrm m^2}}{2\sqrt{\frac{\sqrt{9-4q_\textrm m^2}+q_\textrm m^2-3}{q_\textrm m^2}}}\,. \end{align} \tag{ 83 }$$

Although not obvious, equation (83) reduces to $3\sqrt{3}$ as $q_\textrm m \to 0$, as can be seen by Taylor expanding it in $q_\textrm m$. We plot the evolution of the shadow radius against $q_\textrm m$ in figure 26, from which we see that the shadow radius quickly decreases with increasing charge. We also see that the EHT observations set the upper limits $q_\textrm m \lesssim 0.6M$ (1σ) and $q_\textrm m \lesssim 0.75M$ (2σ), ruling out the possibility of Sgr A$^*$ being an extremal Sen BH with $q_\textrm m = \sqrt{2}M$. We further note that complementary x-ray reflection spectroscopy constraints obtained by one of us set a much tighter limit, $q_\textrm m \lesssim 0.6M$ at $90\%$ C.L. [759], where as in section 3.9 M denotes the mass of EXO 1846-031.

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Remaining in string-inspired scenarios, another interesting BH solution has been derived in a scenario where the Kalb–Ramond field $B_{\mu\nu}$ plays a key role in driving spontaneous breaking of Lorentz symmetry [743]. Specifically, the scenario considered by [760] envisages a field content including the Kalb–Ramond field $B_{\mu\nu}$ and the Kalb–Ramond 3-form $H_{\kappa\mu\nu}$. ⁴⁶ Inspired by the Standard Model Extension (an effective field theory containing all possible Lorentz breaking operators in addition to the Standard Model and GR Lagrangian terms [761–763]) and by heterotic string theory [760], considered a scenario where the Kalb–Ramond field acquires a non-vanishing vacuum expectation value (VEV) $b_{\mu\nu}$ (with $b^2 = b_{\mu\nu}b^{\mu\nu}$), as well as a non-minimal coupling to the Ricci tensor with strength ξ₂ (for the full action of the theory, see [760]). The non-zero VEV spontaneously breaks Lorentz invariance, leading to a host of interesting observational signatures [764–766].

BHs within this configuration have been studied by [760], and found to be described by the following metric function:

$$\begin{align} A(r)=1-\frac{2M}{r}+\frac{\Gamma}{r^{\frac{2}{k}}} \,, \end{align} \tag{ 84 }$$

where Γ is an integration constant, and k is related to the Kalb–Ramond VEV and non-minimal coupling strength through $k = \vert b \vert^2\xi_2$ and therefore characterizes an universal hair. For illustrative purposes, here we choose to focus our attention on the parameter k which is of more direct interest (since it enjoys a direct physical interpretation) and is expected to be small. To this end, we fix $\Gamma = 0.5$: although there is no deep motivation for this value, this allows us to also compare our results to the earlier work of [108] by some of us.

We compute the size of the Kalb–Ramond BH numerically, and show its evolution against k in figure 27. As already found earlier in [108], we see that increasing k decreases the size of the BH shadow (see also [767]. For this particular choice of Γ, we find that the EHT observations set the upper limits $k \lesssim 1.2$ (1σ) and $k \lesssim 1.5$ (2σ). If independent limits on the Kalb–Ramond VEV are available, these limits can be translated to limits on the Kalb–Ramond non-minimal coupling strength (and vice-versa). However, we stress that these results only correspond to a slice of the full Γ–k parameter space, and that smaller (larger) values of Γ would lead to weaker (tighter) constraints on k (for more details see [108]).

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In closing, it is worth noting that the space-time described by equation (84) is closely related to the 4D string-corrected BH solution found by one of us in [768], which generalizes the Callan–Myers–Perry BH [769] through a dimensional regularization-like procedure akin to the one considered in 4D Einstein–Gauss–Bonnet gravity, which we will discuss later in section 3.27. The solution derived in [768] corresponds to equation (84) with $k = 1/2$ (thus leading to a $1/r^{\,4}$ correction to the Schwarzschild BH), and requires $\Gamma \leqslant 27/16M^4 \sim 1.69M^4$. However, it is easy to show (as we have explicitly checked) that within this particular space-time, and given the above upper limit on Γ, the EHT observations are not able to set any meaningful limit on Γ: thus, we will not discuss this space-time further ⁴⁷ .

Still remaining in the string-inspired realm, one of the models in this context which enjoys particularly strong theory motivation is EMDG, which arises as an effective theory of the heterotic string at low energies once the additional six dimensions of the ten-dimensional manifold are compactified onto a torus [744], see also section 3.23. In EMDG, a dynamical dilaton field Φ evolving under a potential $V(\Phi)$ is included in the GR Lagrangian as:

$$\begin{align} {\cal L} \supset \frac{1}{16\pi} \left ( {\cal R}+\frac{1}{2}(\nabla^\mu\Phi)^2+V(\Phi) \right ) +\frac{\alpha^{^{\prime}}}{8}\Phi F^{\mu\nu}F_{\mu\nu}\,, \end{align} \tag{ 85 }$$

where $\alpha^{^{\prime}}$ regulates the coupling of the dilaton to photons. Given a BH of dilaton charge Q, the corresponding metric function of the so-called Einstein–Maxwell-dilaton-1 BH is given by [745, 746, 770]:

$$\begin{align} A(r) = 1-\frac{2M}{r}\left(\sqrt{1 + \frac{q^4}{4M^{\,2}r^{\,2}}}-\frac{q^2}{2Mr}\right) \,, \end{align} \tag{ 86 }$$

with the requirement that the parameter $q = Q\sqrt{\alpha^{^{\prime}}/8}$ associated to the dilaton charge, which characterizes a specific hair, satisfies $q \leqslant \sqrt{2}$.

A closed-form expression exists for the shadow radius, and is given as follows:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{6 \left ( \tilde{q}+6 \right ) -q^2 \left ( q^2+\sqrt{q^4-20 q^2+36}+16 \right ) }}{2 \sqrt{2-\frac{16}{2 q^2+\sqrt{2} \sqrt{q^4- \left ( \tilde{q}+16 \right ) q^2+6 \left ( \tilde{q}+6 \right ) }}}}\,, \end{align} \tag{ 87 }$$

where the quantity $\tilde{q}$ is defined as:

$$\begin{align} \tilde{q} \equiv \sqrt{q^4-20 q^2+36}\,. \end{align} \tag{ 88 }$$

Although not obvious, equation (87) reduces to $3\sqrt{3}$ as q → 0, as can be seen by Taylor expanding it in q. We show the evolution of the shadow radius as a function of the charge parameter q in figure 28. We see that the EHT observations set the upper limits $q \lesssim 0.8M$ (1σ) and $q \lesssim M$ (2σ), ruling out the extremal Einstein–Maxwell-dilaton-1 BH ($q = \sqrt{2}M$).

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Einstein–æther gravity is a theory of gravity that dynamically violates Lorentz symmetry by means of a unit norm time-like vector field (the 'æther'), which defines a preferred time-like direction at each point of space-time. The Einstein–æther gravity Lagrangian is governed by 4 free parameters c_i with $i = 1,2,3,4$, and is given by the following [771–773] (see also [774–783] for other important works):

$$\begin{align} {\cal L} \supset -h^{\alpha\beta}_{\mu\nu} \left ( \nabla_{\alpha}u^{\mu} \right ) \left ( \nabla_{\beta}u^{\nu} \right ) +\lambda \left ( u^2+1 \right )\,, \end{align} \tag{ 89 }$$

where u^µ is the æther field, λ is a Lagrange multiplier enforcing the condition $u^{\mu}u_{\mu} = -1$, and the tensor $h^{\alpha\beta}_{\mu\nu}$ is given by:

$$\begin{align} h^{\alpha\beta}_{\mu\nu} = c_1g^{\alpha\beta}g_{\mu\nu}+c_2\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+c_3\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}-c_4u^{\alpha}u^{\beta}g_{\mu\nu}\,. \end{align} \tag{ 90 }$$

In what follows, we will adopt the widely used shorthand notation $c_{ij} = c_i+c_j$, $c_{ijk} = c_i+$ $c_j+c_k$.

Two exact static spherically symmetric BH solutions within Einstein–æther gravity are known [784, 785]. The first solution, commonly referred to as Einstein–æther type 1, holds for $c_{14} = 0$ and $c_{123} \neq 0$. The metric function for this space-time is given by:

$$\begin{align} A(r) = 1-\frac{2M}{r}-\frac{27c_{13}M^4}{16(1-c_{13})r^{\,4}} \,, \end{align} \tag{ 91 }$$

with the requirement $0\lt c_{13}\lt1$, where c₁₃ obviously characterizes an universal hair. We compute the shadow radius numerically, plotting it as a function of the Lagrangian parameter c₁₃ in figure 29. We see yet again an example of shadow radius increasing with increasing fundamental parameter, which leads to this scenario being tightly constrained by the ETH observations, which set the upper limits $c_{13} \lesssim 0.1$ (1σ) and $c_{13} \lesssim 0.75$ (2σ)—note that the c_i s are dimensionless. While this is an independent constraint on c₁₃ obtained in the strong-field regime, we note that it is still significantly weaker than constraints arising from GWs. For instance, the multi-messenger GW event GW170817 sets the very stringent limit $\vert c_{13} \vert \lesssim 10^{-15}$ [786].

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The second exact BH solution in this theory holds for $c_{14} \neq 0$ and $c_{123} = 0$, and is commonly referred to as Einstein–æther type 2. The metric function for this space-time is given by:

$$\begin{align} A(r) = 1-\frac{(2-c_{14})M}{r}-\frac{(2-c_{13}-c_{14})(2-c_{14})^2M^{\,2}}{8(1-c_{13})r^{\,2}} \,, \end{align} \tag{ 92 }$$

with the requirement $0 \leqslant c_{14} \leqslant 2c_{13}\lt2$, where c₁₃ and c₁₄ obviously characterize universal hairs. In this case, the space-time is described by 2 free parameters. For illustrative purposes, we fix $c_{13} = 0.99$, close to the maximum value allowed, which as argued earlier is completely excluded in the case of the Einstein–æther type 1 BH. The purpose is simply to illustrate how a non-zero value of c₁₄ can in principle help bring a previously excluded range of c₁₃ values into agreement with the EHT observations. We compute the shadow radius numerically (see also [787]), plotting it as a function of the Lagrangian parameter c₁₃ in figure 30. In this case we see that in principle a value of $1.3 \lesssim c_{14} \lesssim 1.4$ is allowed by the EHT observations, when fixing $c_{13} = 0.99$. However, we note that these values of c₁₄ are excluded by GW observations, with the GW170817 event setting the very stringent limit $c_{14} \lesssim 2.5 \times 10^{-5}$ [786].

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In closing, we discuss important relations between Einstein–æther gravity and other well-known theories of gravity, to which our results could therefore carry over. We first note that Einstein-æther gravity is closely related to Hořava–Lifshitz (HL) gravity [788, 789], a well-known and well-studied candidate theory of quantum gravity which achieves power-counting renormalizability by adding higher order spatial derivatives but not time derivatives, thus taming the UV behavior of the graviton propagator, at the cost of explicitly breaking diffeomorphism invariance ⁴⁸ . Einstein–æther gravity is known to appear in the IR limit of HL gravity (when operators higher than second order are neglected), provided one requires the æther field to be hypersurface orthogonal at the level of the action [800]. In fact, the Einstein–æther BH solutions we have studied are also solutions of the IR limit of HL gravity, as discussed in [785] (see also [801] for other BH solutions in HL gravity). Another well-known modified theory of gravity appearing in the IR limit of (projectable) HL gravity is mimetic gravity [802–804]: see [805] for a full proof of this equivalence, and [806, 807] for discussions on relations to Einstein–æther and HL gravity. Mimetic gravity is related to GR by a singular (non-invertible) disformal transformation, and has gained significant attention recently (see e.g. [808–833], and [834] for a review). Spherically symmetric solutions in mimetic gravity have been studied in [835–846], though the issue remains open as to whether mimetic gravity can actually support non-pathological BH solutions. A full investigation of the relation between Einstein–æther and mimetic gravity as far as BH solutions are concerned is therefore warranted, and is left to future work, where we also plan to study the shadows cast by spherically symmetric space-times (BHs, WHs, or NSs) in mimetic gravity.

The Gauss–Bonnet invariant ${\cal G}$ is a very important quantity, which in more than four dimensions plays a role in determining the allowed Lagrangian terms (Lovelock invariants) of diffeomorphism-invariant, metric theories of gravity with equations of motion of second order [847], and is given by:

$$\begin{align} {\cal G} = R^{\mu\nu}_{\rho\sigma}R^{\rho\sigma}_{\mu\nu}-4R^{\mu}_{\nu}R^{\nu}_{\mu}+R^2\,. \end{align} \tag{ 93 }$$

Crucially, in four dimensions ${\cal G}$ amounts to a total derivative, and therefore cannot contribute to the gravitational dynamics. However, in [848] Glavan and Lin proposed what effectively amounts to a dimensional regularization procedure applied to the following action:

$$\begin{align} S = \int \mathrm{d}^Dx\sqrt{-g}\,\alpha{\cal G}\,, \end{align} \tag{ 94 }$$

by rescaling the coupling constant as $\alpha \to \alpha/(D-4)$ (see also the generalization of [849]). This procedure is shown to lead to non-trivial contributions even in D = 4, and we refer to the resulting theory as 4D Einstein–Gauss–Bonnet gravity (4DEGB). See e.g. [850] for a recent review of the theory, [851–860] for important discussions on possible pathologies associated to the rescaling procedure, and [861–880] for other important works on the theory.

With all the caveats discussed in [851, 853, 855–859] in mind, static spherically symmetric solutions within 4DEGB gravity have been studied, and the metric function is given by the following (see e.g. [112, 115–117, 848, 881–890]):

$$\begin{align} A(r) = 1+\frac{r^{\,2}}{32\pi\alpha} \left ( 1\pm\sqrt{1+\frac{128\pi\alpha M}{r^3}} \right ) \,, \end{align} \tag{ 95 }$$

which, although α characterizes universal hair, requires $\alpha \leqslant M^{\,2}/16\pi \sim 0.02M^{\,2}$, i.e. $\alpha \leqslant {\cal O}(10^{18})\,\mathrm{m^2}$ for Sgr A$^*$. We need to take the negative branch of equation (95), as it is the only one which recovers the Schwarzschild metric function $A(r) = 1-2M/r$ in the limit $r \to \infty$.

We compute the shadow radius numerically, and show its evolution as a function of α in figure 31. ⁴⁹ We see that the shadow radius decreases with increasing α. However, for all values of $\alpha \leqslant 1/16\pi$, the shadow radius is always consistent with the EHT observations within better than 1σ. Therefore, the size of Sgr A$^*$'s shadow does not place meaningful constraints on the 4DEGB coupling α, which is instead constrained by alternative astrophysical probes. For instance, the tightest constraint on α comes from binary BH systems, which limit $\alpha \lesssim 10^8\,\mathrm{m}^2$ [891]. For comparison, it is useful to recast the previous limit in units of Sgr A$^*$'s mass: noting that Sgr A$^*$'s gravitational radius is of order $r_\textrm g \approx {\cal O}(10^7)\,\mathrm{km}$, the previous constraint translates to the extremely stringent limit $\alpha \lesssim 10^{-28}M^{\,2}$, much tighter than any constraint realistically achievable from BH shadows.

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The issue of the non-renormalizability of GR, currently one of the biggest obstacles towards constructing a full quantum theory of gravity, has prompted a number of interesting ideas, many of which conjecture the existence of novel symmetries. AS gravity broadly falls within this category, in that it provides a quantum realization of scale symmetry. While, classically, the absence of dimensionful couplings is enough to ensure the realization of scale symmetry, this is no longer true at the quantum level, due to quantum fluctuations in the vacuum. In AS gravity, quantum scale symmetry is realized thanks to the presence of an interacting fixed point of the renormalization group (RG) [892, 893]: beyond a certain energy scale, a transition to a quantum scale-invariant regime occurs, and the strength of the gravitational constant flows to a constant but non-zero value (the case where this constant is zero would essentially correspond to a realization of asymptotic freedom). Therefore, within AS gravity quantum scale symmetry is achieved at fixed points of the RG, enabling an UV extension of GR (interpreted as an effective field theory) and restoring the predictivity of the latter in the UV. Currently, AS gravity is considered a very promising approach towards a theory of quantum gravity (see [894–897] for reviews), and has been the subject of study of an enormous body of literature, see e.g. [898–929] for important examples.

RG-improved BH space-times within the framework of AS gravity were first studied in detail by Bonanno and Reuter in [930], considering the following expression for the running of the gravitational constant as a function of momentum k:

$$\begin{align} G(k) = \frac{G(k_0)}{1+\omega G(k_0)(k-k_0)^2}\,, \end{align} \tag{ 96 }$$

where k₀ is a reference scale later set to $k_0 \to 0$, so that $G(k_0)$ effectively corresponds to Newton's constant G_N as measured by low-energy observations. Close to the RG fixed point, i.e. for $k^2 \gg G_N^{-1}$, the gravitational constant runs as $G(k) = 1/\omega k^2$, where ω is the inverse dimensionless fixed point value: thus, the gravitational constant 'forgets' its IR value when entering the quantum scale-invariant regime. To consider the impact of equation (96) on BH space-times, one needs a prescription for converting k to a position-dependent IR cutoff. Various well-motivated prescriptions were studied in [930]. Here, we simply quote the final expression, leaving the reader to [930] for further details:

$$\begin{align} G(r) = \frac{G_Nr^3}{r^3+\widetilde{\omega}G_N \left ( r+\gamma G_NM \right )}\,, \end{align} \tag{ 97 }$$

where M is the BH mass, and $\widetilde{\omega}$ is an in principle free parameter, related to the inverse dimensionless fixed point value ω, and expected to be of order unity from first principles considerations ⁵⁰ . To obtain the RG-improved AS-driven corrections to the Schwarzschild space-time, one therefore simply replaces G in the latter with equation (97), after identifying r with the usual radial coordinate (see also [83, 932–947] for other relevant works on BHs in AS gravity). As argued explicitly in [930], γ plays a very minor role in the phenomenology of the model, especially at the scales relevant for BH observations. Therefore, in the following, we shall set γ = 0, which effectively corresponds to considering the IR limit ($r \gg \sqrt{\gamma}$) of equation (97). Explicitly, the metric function we consider is therefore given by (where we have obviously set $G_N = 1$ for convenience):

$$\begin{align} A(r) = 1-\frac{2Mr}{r^{\,2}+\widetilde{\omega}} \,, \end{align} \tag{ 98 }$$

where $\widetilde{\omega}$ characterizes an universal hair. One can easily show that the Kretschmann and Ricci scalars associated to the above metric are finite throughout the entire space-time, which therefore describes a globally regular BH. Because the curvature scalars are finite everywhere, including r = 0, the metric describes a globally regular BH space-time. The impact of this and related space-times on BH observables such as shadows and gravitational deflection of light have been recently studied in [83, 948].

Here, we shall focus on the region $\widetilde{\omega}\lt M^{\,2}$, for which the BH possesses two horizons. We compute the shadow size numerically, and show its evolution against $\widetilde{\omega}$ in figure 32. We see that the EHT observations set very weak limits on $\widetilde{\omega}$, with the upper limit $\widetilde{\omega} \lesssim 0.9M^{\,2}$ at 1σ. Converting this to the dimensionless $\widetilde{\omega}$, we find that this corresponds to a very weak limit, and we are left to conclude that based on this metric test the EHT observations are unable to set strong constraints on AS gravity (though the situation may be different with other types of tests, see e.g. [942, 943, 945–947]). On the other hand, complementary x-ray reflection spectroscopy constraints obtained by one of us set a much tighter limit, $\widetilde{\omega} \lesssim 0.05$ at 1σ [949] from observations of GRS 1915+105, an x-ray binary system with mass of order ${\cal O}(10)M_{\odot}$.

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Rastall's theory of gravity is perhaps one of the best known and studied non-conservative theories of gravity, i.e. theories where the stress-energy tensor is not covariantly conserved, $\nabla_{\mu}T^{\mu\nu} \neq 0$. The theory was introduced by Rastall in [950] with the motivation that in the strong curvature regime the covariant derivative of the stress-energy tensor may not vanish but could depend on the space-time curvature. Based on phenomenological arguments, Rastall proposed that $\nabla_{\mu}T^{\mu\nu} \propto \lambda g^{\mu\nu}\nabla_{\mu}R$, with Einstein's equations being modified to:

$$\begin{align} R_{\mu\nu} + \left ( \kappa-\frac{1}{2} \right ) g_{\mu\nu}R = T_{\mu\nu}\,, \end{align} \tag{ 99 }$$

where we refer to κ as the Rastall coupling parameter, and Einstein's equations are recovered in the limit κ → 0. Rastall gravity has been at the center of a surge of interest in recent years, with applications to a number of interesting scenarios from thermodynamics to cosmology (see e.g. [170, 951–969]), although we note that concerns have been raised about its possibly being equivalent to GR in the gravity sector, modulo an artificially isolated part of the physical conserved stress-energy tensor in the matter sector [970].

By construction, while all vacuum solutions of GR are also solutions of Rastall gravity, the non-vacuum solutions depend on the Rastall coupling parameter and can be significantly different from their GR counterparts. BH solutions in Rastall gravity have been studied in several works [971–973]. Here, we shall consider the solution obtained by Heydarzade and Darabi [972], describing BHs in Rastall gravity surrounded by a perfect fluid with equation of state ω. In this case, the metric function is given by:

$$\begin{align} A(r)=1-\frac{2M}{r}-N_\textrm sr^{\frac{6\kappa(1+\omega)-1-3\omega}{1-3\kappa(1+\omega)}} \,, \end{align} \tag{ 100 }$$

where $N_\textrm s$ is the field structure parameter, which is related to the energy density of the surrounding fluid and therefore characterizes a specific hair, as does ω, whereas κ characterizes an universal hair. The reason why it is interesting to consider a surrounding fluid is that, as is clear from equation (100), depending on the value of κ the surrounding field acts as if it had a different effective equation of state. In the following, we aim to focus on the Rastall coupling κ, while also exploring the role of the equation of state of the surrounding fluid. In our baseline cases, we shall therefore fix $N_\textrm s = 0.005$ (so that it has little impact on the resulting shadow size), considering the cases where the BH is surrounded by dust (ω = 0), which we later extend to a stiff fluid (ω = 1, which can be generated during a period of kination, where the energy is dominated by the kinetic energy of a scalar field), radiation ($\omega = 1/3$), and a fluid with equation of state $\omega = -1/3$ (which can be associated to cosmic strings, and is the effective equation of state corresponding to spatial curvature).

We compute the resulting shadow size numerically, and show its evolution as a function of the Rastall coupling κ in figure 33. For the cases where the equation of state of the surrounding fluid is $\omega \geqslant 0$, we set the upper limit $\kappa \leqslant 1/6$ [212], ensuring that the energy conditions are not violated (by construction this cannot be done for the $\omega = -1/3$ fluid) and the space-time is asymptotically flat. In these cases, and in particular for our baseline case with $N_\textrm s = 0.005$ and ω = 0, we see that the shadow size grows extremely slowly with increasing κ, in a way which is hardly visible by eye. We find that the largest correction to the shadow size relative to the Schwarzschild counterpart within the whole $\kappa \leqslant 1/6$ regime is at most ${\cal O}(10^{-2})$, and as a result the EHT observations are unable to set meaningful constraints on the Rastall coupling parameter κ. On the other hand, for $\omega = -1/3$ the shadow size starts increasing quickly once $\kappa \gtrsim 0.25$, and we find the upper limits $\kappa \lesssim 0.01$ (1σ) and $\kappa \lesssim 0.25$ (2σ). To further show the role of $N_\textrm s$ instead, we return to our baseline case with ω = 0, but setting $N_\textrm s = 0.1$. In this case we see that the shadow size increases more markedly, and is excluded by the EHT observations within 1σ, while at 2σ we find the upper limit $\kappa \lesssim 0.07$.

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LQG is arguably one of the leading approaches towards constructing a quantum theory of gravity. One of the main hurdles towards constructing such a theory is the difficulty in constructing a quantum Riemannian geometry, going beyond the classical (fixed) reference space-time. The first steps towards this task were completed by reformulating gravity without reference to any background field (not even a metric), and subsequently introducing non-perturbative techniques from gauge theories, once more without reference to any background (see e.g. [974–978]). A theory with no background fields only has access to an underlying manifold and is therefore inevitably diffeomorphism-covariant: this, together with the non-perturbative methods mentioned previously, leads to a fundamentally discrete geometry, whereas the familiar continuum space-time of GR results as an emergent phenomenon upon a coarse grained description of the fundamental quantum Riemannian discrete structures. Summarizing the main features of LQG in a few lines is close to impossible: loosely speaking, we can say that LQG leads to a granular space-time structure, with a minimum length of order the Planck length, and 'woven' out of Planck-scale loops whose network and evolution thereof are referred to as spin network and spin foam respectively. A significant amount of work has been devoted to LQG in recent years, and we refer the reader to [979–1008] for examples thereof (including cosmological applications), as well as [1009] for a comprehensive review of the status of LQG.

BHs in LQG have been explored in several works [985, 1010–1013]. Here we shall consider the LQG BH constructed in [1014], whose shadow was studied by one of us in [111], and described by the following metric functions:

$$\begin{align} A(r)=\frac{(r-r_+)(r-r_-)(r+r_*)^2}{r^{\,4}+a_0^2}\,,\quad C(r)=r^{\,2}+\frac{a_0^2}{r^{\,2}}\,, \end{align} \tag{ 101 }$$

where $r_+ = 2 M/(1+P)^2$ and $r_{-} = 2MP^2/(1+P)^2$ are the BH's two horizons, while $r_* = \sqrt{r_+r_-} = 2 MP/(1+P)^2$. On the other hand $P = (\sqrt{1+\epsilon^2}-1)/(\sqrt{1+\epsilon^2}+1)$ is the polymeric function, where ε denotes the product of the Immirzi parameter γ and the polymeric parameter δ (where typically $\epsilon = \gamma\delta \ll 1$): the former is an area parameter [1015–1017], whereas the latter was introduced to define the Hamiltonian constraint in terms of holonomies [1018], and was conservatively assumed to be constant in the approach of [1014]. Finally, $a_0 = A_{\min}/8\pi$ is related to the minimum area gap of LQG, i.e. the smallest non-zero eigenvalue of the kinematical area operator. Note that all three a₀, γ, and δ (and therefore ε and P) characterize universal hairs.

The above space-time depends upon the two parameters P (or equivalently ε) and a₀, and reduces to the Schwarzschild space-time in the limit P → 0, $a_0 \to 0$. However, since a₀ is expected to be extremely small in LQG, we do not expect it to play a relevant role as far as astrophysical BHs are concerned. Indeed, we have explicitly verified that a₀ plays a minor role in determining the shadow size. Therefore, for the purposes of this work, we set $a_0 = 0$, so that the only free parameter is P. We compute the shadow size numerically, and show its evolution as a function of P in figure 34. ⁵¹ We see that increasing the polymeric function decreases the shadow size, and the EHT observations set the upper limits $P \lesssim 0.05$ (1σ) and $P \lesssim 0.08$ (2σ). These can be translated into the upper limits $\epsilon \lesssim 0.5$ (1σ) and $\epsilon \lesssim 0.6$ (2σ), which are rather loose as on theoretical grounds we expect $\epsilon \ll 1$. However, this remains a valuable constraint on an interesting theoretical scenario, particularly given that observational constraints on LQG are not easy to obtain.

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The Kottler space-time [213], also known as Schwarzschild–de Sitter space-time, is the unique spherically symmetric solution to Einstein's equations sourced by a positive cosmological constant Λ. It is a particular case of the McVittie metric, which describes models interpolating between Schwarzschild and FLRW space-times in a continuous manner, and is part of a larger class of space-times which generically describe BHs embedded in an expanding Universe (see e.g. [1019–1043] for further relevant works). The Kottler BH is described by the following metric function [213]:

$$\begin{align} A(r)=1-\frac{2M}{r}-\frac{\Lambda}{3}r^{\,2} \,, \end{align} \tag{ 102 }$$

where Λ characterizes an universal hair. The Kottler space-time possesses two horizons for $0\lt\Lambda\lt\sqrt{3M}$, with the vector field $\partial_t$ only being time-like between these two horizons. We therefore restrict our discussion to static observers whose radial coordinate is bounded above by the coordinate of the outer horizon, which is anyhow always guaranteed to be the case since the latter corresponds to our de Sitter cosmological horizon in the physically interesting case where Λ is the cosmological constant responsible for cosmic acceleration.

As discussed at length in [43], the Kottler space-time is an excellent example of a BH whose shadow size depends explicitly on the radial coordinate of the observer, due to the metric described by equation (102) being manifestly non-asymptotically flat. Considering the importance of the Kottler space-time, we shall derive the exact expression for the shadow size, following the discussion of section 2.1. Explicitly, we find:

$$\begin{align} r_\mathrm{sh} = \frac{3\sqrt{3}M}{\sqrt{1-9\Lambda M^{\,2}}}\sqrt{1-\frac{2M}{r_\textrm O}-\frac{\Lambda}{3}r_\textrm O^2}\,, \end{align} \tag{ 103 }$$

which clearly reduces to $3\sqrt{3}M$ in the limit $\Lambda \to 0$ and $r_\textrm O \gg M$, the latter being clearly the case for the EHT observations of Sgr A$^*$. We further note that the requirement of the observer lying within the cosmological horizon ensures that the term in the square root to the right of equation (103) does not turn negative. As per our discussion in section 2.1, for the purposes of our discussion it suffices to consider a static observer, whereas considering an observer comoving with the expansion of the Universe would complicate the discussion considerably.

We show the evolution of the shadow size as a function of the cosmological constant Λ in equation (35). As a note of caution, we plot the shadow radius in units of the Newtonian mass M: since the Kottler space-time is not asymptotically flat, the ADM mass is not well-defined, however for the purposes of our discussion considering M will be sufficient. Figure 35 displays the features we could have expected from a qualitative assessment. For $\Lambda \lesssim 10^{-41}\,\mathrm{m}^2$, the product $\Lambda r_\textrm O^2 \ll 1$ considering $r_\textrm O \sim 8\,\mathrm{kpc} \simeq 2.5 \times 10^{20}\,\mathrm{m}$, and therefore the size of the Kottler BH is essentially that of a Schwarzschild BH of the same mass. Clearly, this includes the value of Λ indicated by cosmological observations, $\Lambda \sim {\cal O}(10^{-52})\,\mathrm{m}^2$ (corresponding to the dashed vertical line in figure 35), for which the product $\Lambda r_\textrm O^2 \sim {\cal O}(10^{-12})$, clearly too small to be appreciated by current EHT measurements. Only for larger values of Λ, when $\Lambda r^{\,2}$ becomes of order unity, does the effect of the cosmological constant become visible, acting to quickly decrease the shadow size. We therefore conclude that within a Kottler space-time where Λ is responsible for cosmic acceleration, there is no detectable imprint on the size of the BH shadow, as we could have expected from back-of-the-envelope calculations, but reassuringly confirmed by the full computation (see also [160] by two of us for different perspectives on the subject, as well as [1044, 1045]).

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One of the simplest extensions of the Einstein–Maxwell equations of GR and electromagnetism involves a direct coupling between the electromagnetic and Weyl tensors. The Weyl tensor $C_{\mu\nu\rho\sigma}$, invariant under conformal transformations of the metric, is a measure of the space-time shear, and is given by the following [1046]:

$$\begin{align} C_{\mu\nu\rho\sigma} = R_{\mu\nu\rho\sigma}- \left ( g_{\mu[\rho}R_{\sigma]\nu} - g_{\nu[\rho}R_{\sigma]\mu} \right )+\frac{1}{3}Rg_{\mu[\rho}g_{\sigma]\nu}\,, \end{align} \tag{ 104 }$$

where the square brackets indicate the antisymmetric part. A direct coupling between the electromagnetic and Weyl tensors is allowed, in which case the electromagnetic sector of the Lagrangian is given by [1047, 1048]:

$$\begin{align} {\cal L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\alpha C^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\,, \end{align} \tag{ 105 }$$

where α is a constant with dimensions of length squared. Assuming that the gravitational sector is still described by GR, Maxwell's equations are modified as follows:

$$\begin{align} \nabla_{\mu} \left ( F^{\mu\nu}-4\alpha C^{\mu\nu\rho\sigma}F_{\rho\sigma} \right ) = 0\,. \end{align} \tag{ 106 }$$

Clearly, the coupling to the Weyl tensor modifies the propagation of photons across the space-time, leading to a series of interesting phenomena including apparent superluminal propagation and birefringence [1049–1051]. This model is perhaps not a model of modified gravity in a strict sense, but is perhaps best interpreted as a model of modified electrodynamics, with such modifications actually expected when considering the effects of one-loop vacuum polarization on the photon effective action in QED as well as the effects of tidal gravitational forces (see [1047] for early discussions).

As in most modified electrodynamics models, here as well photons travel along null geodesics of an effective geometry. We follow [1052] who computed the effective geometry experienced by photons traveling in a Schwarzschild space-time in the presence of electromagnetic-Weyl coupling. As discussed in [1052], only two of the three possible polarizations turn out to be physical (likely as a consequence of gauge invariance and the Ward–Takahashi identities), which the authors refer to as $\ell$-polarization and m-polarization, or PPL and PPM respectively: we refer the reader to [1052] for further details on the derivation of the modified light-cone conditions and of the physical polarizations. The Weyl-corrected effective geometry corresponding to an underlying Schwarzschild space-time is then described by the following metric functions [1052] (see also [1053–1062] for further relevant works):

$$\begin{align} A(r)&= 1-\frac{2M}{r} \,,\nonumber \\ C(r)&= r^{\,2} \left ( \frac{r^3-8\alpha M}{r^3+16\alpha M} \right ) ^{-1}\quad \textrm{(PPL polarization)}\,,\\ C(r)&= r^{\,2} \left ( \frac{r^3+16\alpha M}{r^3-8\alpha M} \right ) ^{-1}\quad \textrm{(PPM polarization)}\,,\nonumber \end{align} \tag{ 107 }$$

where α characterizes an universal hair. The fact that only the angular metric function C(r) is modified is a direct reflection of the modifications to the light-cone structure due to the direct coupling between photons and the Weyl tensor.

We compute the shadow size numerically for both PPL- and PPM-polarized photons, and show its evolution as a function of the Weyl coupling α in figures 36 and 37 for PPL- and PPM-polarized photons respectively. We see that in the former case increasing α increases the shadow size, whereas the reverse happens in the latter case: this is not surprising, as the respective angular metric functions are the inverse of one another. For PPL-polarized photons, we find that the EHT observations set the upper limits $\alpha \lesssim 0.01M^{\,2}$ (1σ) and $\alpha \lesssim 0.2M^{\,2}$ (2σ). For PPM-polarized photons, the limits are instead looser and read $\alpha \lesssim 0.25M^{\,2}$ (1σ) and $\alpha \lesssim 0.3M^{\,2}$ (2σ). Taking $\alpha \lesssim {\cal O}(0.1)M^{\,2}$ as a representative limit, in SI units this translates to $\alpha \lesssim {\cal O}(10^{17})\,\mathrm{m}^2$, significantly weaker than some of the best constraints on this parameter arising from gravitational time delay/advancement, light deflection, and frequency shifts, with Cassini tracking setting the upper limit $\alpha \lesssim {\cal O}(10^{12})\,\mathrm{m}^2$ [1063, 1064].

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In [1065], Deser, Sarioglu, and Tekin (DST hereafter) introduced a class of gravitational models whose modifications to GR forbid Ricci-flat geometries, while preserving as many features of GR as possible, including the derivative order of the Einstein equations in Schwarzschild gauge. This class of models was introduced as a tool to better appreciate the nature of Ricci-flat geometries, and relies on the addition to the Einstein–Hilbert Lagrangian of terms which are non-polynomial in the Weyl tensor, as follows:

$$\begin{align} \Delta{\cal L} = \sqrt{-g}\,\beta_n \left \vert \textrm{tr}\,{\cal C}^n \right \vert ^{\frac{1}{n}}\,, \end{align} \tag{ 108 }$$

where ${\cal C}$ is the Weyl tensor already encountered earlier in equation (104), the index n labels the various models belonging to this class, and the trace operator acts upon the (mixed) Weyl tensor as follows:

$$\begin{align} \textrm{tr}\,{\cal C}^n \equiv C_{ab}^{\,\,\,\,\,cd}C_{cd}^{\,\,\,\,\,ef}{\ldots}C_{..}^{\,\,\,\,\,yz}C_{yz}^{\,\,\,\,\,ab}\,, \end{align} \tag{ 109 }$$

for n copies of the Weyl tensor, where n can be any arbitrary positive integer. To the best of our knowledge, [1065] was the first to introduce the non-polynomial gravity approach. Models of these type have later been considered in various works (particularly in the search for regular BHs), see for instance [1066–1075].

Without loss of generality, we shall follow [1065] and take n = 2, while defining $\sigma = \beta_2/\sqrt{3}$, so that equation (108) reduces to:

$$\begin{align} \Delta{\cal L}_{n=2} = \sqrt{-g}\,\sigma \sqrt{3 \left \vert C_{ab}^{\,\,\,\,\,cd}C_{cd}^{\,\,\,\,\,ef} \right \vert }\,. \end{align} \tag{ 110 }$$

This particular Lagrangian admits an explicit static spherically symmetric solution, the so-called DST BH, with metric function given by [1065]:

$$\begin{align} A(r)= \left [ \frac{1-\sigma}{1-4\sigma} - \left ( \frac{2M}{r} \right ) ^{\frac{1-4\sigma}{1-\sigma}} \right ] \left ( \frac{r}{2M} \right ) ^{\frac{6\sigma}{\sigma-1}} \,, \end{align} \tag{ 111 }$$

where σ characterizes an universal hair. It is easy to show that the above space-time is asymptotically flat for $0\lt\sigma\lt1$. In our subsequent analysis we have imposed σ > 0, whereas the upper limit guaranteeing asymptotic flatness σ < 1 turns out a posteriori to be satisfied given the very stringent limits on σ imposed by the EHT observations.

While slightly cumbersome, there exists a closed-form expression for the shadow radius, given by:

$$\begin{align} r_\mathrm{sh}=\frac{2^{\frac{3\sigma}{\sigma-1}}3^{\frac{\sigma-1}{4\sigma-1}}{\cal X}}{\sqrt{\left ( 3^{\frac{\sigma-1}{4\sigma-1}}{\cal X} \right ) ^{\frac{6\sigma}{\sigma-1}} \left ( \frac{\sigma-1}{4\sigma-1}-2^{\frac{4\sigma-1}{\sigma-1}} \left ( 3^{\frac{1-\sigma}{4\sigma-1}}{\cal X} \right ) ^{\frac{4\sigma-1}{\sigma-1}} \right ) }}\,, \end{align} \tag{ 112 }$$

where the quantity ${\cal X}$ is defined as:

$$\begin{align} {\cal X} \equiv \frac{8^{\frac{\sigma}{\sigma-1}}(4\sigma-1)}{(\sigma-1)(2\sigma+1)}\,, \end{align} \tag{ 113 }$$

which clearly reduces to ${\cal X} \to 1$ when σ → 0, from which it is relatively straightforward to see that the limit $r_\mathrm{sh} = 3\sqrt{3}$ is indeed recovered when σ → 0 in equation (112). We show the evolution of the shadow size as a function of σ in figure 38, where we see that increasing σ quickly decreases the shadow size. We note that the EHT observations set the very stringent limits $\sigma \lesssim 0.04$ (1σ) and $\sigma \lesssim 0.05$ (2σ), which directly translate into upper limits on the Lagrangian coupling β₂ in equation (110) of $\beta_2 \lesssim 0.07$ (1σ) and $\beta_2 \lesssim 0.09$ (2σ). To the best of our knowledge, we are not aware of any observational constraints on σ: a study on collisions of spinning particles near DST BHs was conducted by one of us in [1075], finding a behavior very close to that of GR, while the same work entertained the possibility that a very tiny discrepancy in the observed value of Mercury's perihelion precession might be explained by a value $\sigma \sim {\cal O}(10^{-9})$. However, our upper limits of order ${\cal O}(10^{-2})$ on σ represent the first ever robust observational constraints on a non-polynomial gravity model.

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In [1076] (see also [1077]), Grümiller constructed ab initio the most general theory for gravity in the IR (i.e. at large distances) consistent with an assumed set of symmetries and assumptions, including spherical symmetry, power counting renormalizability, analyticity, and so on. While the model was constructed based on well-founded theoretical assumptions, it also enjoys phenomenological motivation in its ability (at the time) to potentially mimic DM and/or dark energy, as well as explain a number of puzzles such as the flyby [1078] and Pioneer anomalies [1079]. While the theory is now understood to be unlikely to successfully mimic DM and dark energy (see e.g. [1080]), and interest in these anomalies has now partially faded [1081], here we shall simply consider it as being an interesting IR model for gravity.

In [1076], Grümiller constructed the Rindler gravity model from the following effective field theory for a scalar field Φ in 2 dimensions:

$$\begin{equation} S = \int \mathrm{d}^2x\sqrt{-g}\, \left [ \Phi^2R+2(\partial\Phi)^2-6\Lambda\Phi^2+8a\Phi+2 \right ]\,, \end{equation} \tag{ 114 }$$

where Λ will eventually play the role of dark energy, and a is a constant with SI units of acceleration. For a test particle located at large distance from a source (where the Newtonian acceleration is negligible), positive/negative a provides a constant acceleration towards/away from the source. The new term admits a simple interpretation: a Rindler acceleration, see [1082]. While the above theory is written in 2 dimensions and thus cannot be a realistic description of our world, given the interesting phenomenology associated to the Rindler acceleration outlined in [1076], here we shall adopt an equally phenomenological standpoint where we examine the consequences of a Rindler correction to the Schwarzschild metric (setting $\Lambda = 0$), taking the above model as theoretical motivation. That is, we consider the following metric function [1076] (see also [1083, 1084]):

$$\begin{align} A(r)=1-\frac{2M}{r}+2ar \,, \end{align} \tag{ 115 }$$

where a characterizes an universal hair, and which is manifestly not asymptotically flat ⁵² .

We compute the shadow size numerically, using the full computation outlined in section 2.1 for non-asymptotically flat metrics, and show its evolution against the Rindler acceleration a in figure 39, considering only positive values of a as required to explain the observational anomalies the model was constructed to account for [1076]. For convenience, rather than reporting the shadow evolution against aM, we report a in units of $\mathrm{pc}^{-1}$. We also note that a closed-form expression for the shadow radius can be found, given by:

$$\begin{align} r_\mathrm{sh} \simeq \frac{\sqrt{3} \left ( \sqrt{1+12a}-1 \right ) \sqrt{1+2ar_\textrm O}}{2a\sqrt{2\sqrt{1+12a}-1}} \end{align} \tag{ 116 }$$

which, although not obvious at first glance, does indeed reduce to $3\sqrt{3}$ as a → 0, as can be seen by Taylor expanding equation (116) in a (we have made the valid approximation that whenever the $2ar_\textrm O$ term is dominant over 1, it naturally also dominates over $2/r_\textrm O$). We have verified that this closed-form expression is in excellent agreement with our numerical results.

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As with the Kottler space-time considered previously in section 3.31, it is clear that the effects of Rindler gravity on the BH shadow only become significant when the term ar becomes sufficiently large. For smaller values of a required to mimic DM and/or address anomalous acceleration hints of the order of $a \sim 10^{-10}\,\mathrm{m}\,\mathrm{s}^{-2}$, ⁵³ corresponding to the dashed black box in figure 39, the effect on the shadow size is too small to be appreciated by the EHT observations. We conclude that, for phenomenologically interesting values of a, this model is in excellent agreement with the EHT observations but cannot be distinguished from its Schwarzschild counterpart, at least not based on the metric test we have considered here.

We have spent considerable time discussing BH solutions in modified theories of gravity, broadly defined. We shall now turn our attention to BHs surrounded by various forms of matter. The influence of external matter fields on BHs has been the subject of significant study, at least since the well-known Kiselev space-time was proposed [1092]. While particular emphasis has been placed on the possibility of quintessence surrounding BHs (see e.g. [1098–1117]), here we shall consider a much more strongly motivated scenario: that is, one where the BH is surrounded by DM. In fact, it is now an observationally established fact that DM is present in (almost) all galaxies, and is therefore expected to accumulate around SMBHs inhabiting galactic centers: therefore, it is worthwhile to consider how the surrounding DM back-reacts on the BH metric.

The above question has been considered in several works [1118–1120]. Specifically, consider a perfect fluid DM component, described by an equation of state w = 0. The metric function associated to such a configuration is then given by (see [1120]):

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{k}{r} \ln \left ( \frac{r}{\vert k \vert} \right ) \,, \end{align} \tag{ 117 }$$

where k is an integration constant, ultimately physically connected to the amount of DM surrounding the BH, and which characterizes a specific hair. We note that the above metric is not the metric one would obtain when setting w = 0 in the well-known Kiselev metric [1118]. The reason, as explained in [1121], is that the Kiselev metric does not describe a space-time surrounded by a perfect fluid in the sense we are using here: in fact, it is easy to show that the energy-momentum tensor sourcing the Kiselev metric inevitably possesses shear stress components, and therefore cannot describe a perfect fluid [1121]. This is not the case, instead, for the space-time described by equation (117), which has been explicitly derived by starting from a perfect fluid source term (see [1122–1141] for other relevant works).

While the size of the shadow cast by the BH described by equation (117) is easy to compute numerically, we also note that a closed-form expression exists and is given by:

$$\begin{align} r_\mathrm{sh} = \frac{9kW \left ( \frac{2}{3}e^{\frac{1}{3}+\frac{2}{k}} \right ) ^{\frac{3}{2}}}{2\sqrt{2+3W \left ( \frac{2}{3}e^{\frac{1}{3}+\frac{2}{k}} \right ) }} \end{align} \tag{ 118 }$$

where once more W denotes the principal branch of the Lambert W function. As a check, we find equation (118) to be in perfect agreement with our numerical results. We show the evolution of the shadow size as a function of the perfect fluid DM parameter k in figure 40, where we see that the shadow size quickly decreases as k increases. We find that the EHT observations set the upper limits $k \lesssim 0.07M$ (1σ) and $k \lesssim 0.15M$ (2σ). We are not aware of an explicit discussion of the physical meaning of k in the literature. On dimensional grounds, we expect the combination $c^2k/G$ to be related to the integrated amount of DM present in a shell surrounding the BH. Given the physical situation at play, we would expect the upper limits on $k/M$ to constrain the integrated amount of DM present in a shell size of size given by the photon sphere, although this does not allow us to estimate the (dimensionless) numerical factor relating these two quantities. At any rate, a more complete discussion requires a full physical interpretation of the parameter k, which is currently lacking in the literature: we plan to return to this issue in future work, to which we defer a more comprehensive discussion on the topic.

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Topological defects play an important role in several different context of physical interest, and likewise can be produced within a variety of different scenarios, including during phase transitions in the early Universe [1142], or associated to the spontaneous breaking of symmetries [1143–1146]. For example, global monopoles can arise in association to the breaking of a global O(3) symmetry down to a U(1) subgroup (this can be realized for instance with a triplet of scalar fields φⁱ ), and their stability has been studied in various works [1147–1149]. The gravitational field associated to a Schwarzschild BH carrying a global monopole charge was first studied in [1150, 1151] (see also [1152]), and the corresponding BH solution is described by the following metric function:

$$\begin{align} A(r)=1-k-\frac{2M}{r}\,, \end{align} \tag{ 119 }$$

where the fundamental interpretation of k depends on the model being examined. For instance, we can consider a triplet of scalar fields φⁱ satisfying a global O(3) symmetry, described by the following Lagrangian:

$$\begin{align} {\cal L} = \frac{1}{2} \left ( \partial\phi^i \right ) ^2-\frac{\lambda}{4} \left ( \phi^i\phi^i-\eta^2 \right ) ^2\,, \end{align} \tag{ 120 }$$

for which the associated global monopole is given by:

$$\begin{align} \phi^i = \eta f(r)\frac{x^a}{\vert \overline{\mathbf{x}} \vert}\,, \end{align} \tag{ 121 }$$

with $f(r) \to 1$ as $\vert \overline{\mathbf{x}} \vert \to 0$, and k in equation (119) is given by $k = 8\pi\eta^2$. In this case, k is therefore related to the amplitude of the monopole field. Interestingly, the same metric can describe a BH surrounded by a cloud of strings [1153], in which case k is associated to the gauge-invariant density of the cloud of strings $\sqrt{-\gamma}\rho = k/r^{\,2}$. In both cases, k characterizes a specific hair. See also [1154–1168] for further works on BHs carrying global monopole charges or surrounded by clouds of strings.

The above space-time is manifestly not asymptotically flat, itself a direct consequence of the presence of the global monopole charge: in fact, it is globally conical. We can easily find that the location of the photon sphere is given by:

$$\begin{align} r_\mathrm{ph}=\frac{3M}{1-k}\,. \end{align} \tag{ 122 }$$

To obtain the shadow size we can exploit the fact that $A(r_\textrm O) \to \sqrt{1-k}$ as $r_\textrm O \to \infty$ (and more precisely for $r_\textrm O \gg M$), although we need to pay attention to the mass parameter which sets the scale of the shadow radius. In fact, performing the coordinate transformations $t\to (1-k)^{-1/2} t^{^{\prime}}$, $r\to (1-k)^{1/2} r^{^{\prime}}$, $M\to (1-k)^{3/2} M^{^{\prime}}$, it is easy to see that the ADM mass is given by ${\cal M} = (1-k) M^{^{\prime}}$. We then find that the shadow size is given by:

$$\begin{align} r_\mathrm{sh}=\frac{3\sqrt{3}{\cal M}}{\sqrt{1-k}}\,. \end{align} \tag{ 123 }$$

The difference between whether we choose to report the shadow size in units of the ADM mass ${\cal M}$ or the parameter M appearing in equation (119), then, amounts to a factor of $(1-k)$. Here, we choose to report the shadow size in units of ${\cal M}$, and show its evolution against k in figure 41. We find that the EHT observations set the limits $k \lesssim 0.005$ (1σ) and $k \lesssim 0.1$ (2σ), which correspond to limits on the amplitude of the global monopole field of $\eta \lesssim 0.014$ and $\eta \lesssim 0.06$ respectively. Given the very tight limits on k, $1-k \approx 1$ and the ambiguity associated to reporting the shadow radius in units of ${\cal M}$ or M ends up making virtually no difference in our final results.

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Obtaining exact solutions to Einstein's equations in theories of gravity beyond GR is usually a difficult, if not impossible, task. While one case where this is possible is that where a perfect fluid acts as gravitational source, this ceases to be true as soon as the perfect fluid is coupled to more complex forms of matter-energy. In recent years, a very interesting technique has been proposed by Ovalle to overcome these difficulties: the minimal geometric deformation approach and its latter extension, the extended gravitational decoupling approach. We shall broadly refer to these approaches as 'gravitational decoupling'. This approach was initially proposed in the context of the Randall–Sundrum brane-world model [1169], but its power was quickly realized, to the extent that it has now been applied to an enormous variety of scenarios (see e.g. [1170–1219] for an inevitably incomplete list of examples). The strength of the gravitational decoupling approach is at least three-fold: (i) it allows to construct new viable BH solutions by extending simpler seed solutions; (ii) it can be used to systematically decouple a complex source energy-momentum tensor into more simple tractable components; and (iii) it can be utilized to find exact solutions in theories beyond GR. In the context of the present discussion, our interest in this approach rests upon its potential to describe BHs surrounded by various forms of matter. For complete details on the gravitational decoupling approach, we refer the reader to [1169, 1183, 1192].

Here, we shall base ourselves upon the work of [1220], where the authors applied the gravitational decoupling approach to obtain a minimal deformation of the seed Schwarzschild space-time describing a hairy BH sourced by spherically symmetric 'tensor-vacuum' $\theta_{\mu\nu}$ such that the existence of a well-defined event horizon is preserved, while the (primary) hair associated to the additional source does not violate the strong and dominant energy conditions. In short, [1220] obtains a general solution describing a spherically symmetric space-time which violates the NHT in a non-pathological way. In particular, the work finds that the charge associated to the primary hair increases the entropy of the space-time compared to its Schwarzschild value. Specifically, this BH solution is described by the following metric function [1220]:

$$\begin{align} A(r) = 1-\frac{2M}{r}+\alpha \exp \left ( -\frac{r}{M-\alpha\ell/2} \right ) \,, \end{align} \tag{ 124 }$$

where α and $\ell$ are the parameters which quantify deviations from the Schwarzschild metric, both characterizing specific hairs, with $0 \leqslant \ell \leqslant 2M$ representing the charge associated to the primary hair and the Schwarzschild space-time is recovered for $\alpha\ell = 2M$ (as well as in the physically less interesting case α = 0). In particular, the combination $\ell_0 = \alpha\ell$ quantifies the increase in the BH entropy from its Schwarzschild value $S = 4\pi M^{\,2}$. The fluid surrounding the BH which sources this space-time can be associated to DM.

For illustrative purposes, in order to focus our study on the charge parameter $\ell$, we fix α = 1.0, so that $\ell_0 = \ell$. We compute the resulting shadow size numerically, and show its evolution against $\ell$ in figure 42. We see that decreasing $\ell$ from $\ell = 2M$ decreases the shadow size, consistently with earlier findings of some of us in [142]. For this particular choice of α, we see that the EHT observations set the lower limits $\ell \gtrsim 0.35M$ (1σ) and $\ell \gtrsim 0.15M$ (2σ). These limits can be translated into upper limits on the entropy increase compared to the Schwarzschild value $S = 4\pi$ (in units where M = 1). In particular, we find that the associated hair cannot increase the entropy by more than $3\%$. Broadly speaking, this places constraints on the strength of NHT violation, in the presence of hair which does not violate the strong and dominant energy conditions. Finally, we stress that much as with a few cases studied earlier, these results only correspond to a slice of the full α-$\ell$ parameter space, and fixing α to smaller (larger) values would lead to weaker (tighter) constraints on $\ell$ (for more details see [142]).

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The final scenarios we study can generically be referred to as 'novel fundamental physics frameworks', which do not necessarily modify gravity (at least not directly, although as we shall see they lead to a sort of 'backreaction' on the metrics of BHs in GR). We begin by discussing modified UPs which go beyond the Heisenberg one.

While GR is strictly speaking not incompatible with QM, the introduction of gravity into quantum field theories appears to spoil their renormalizability, which is one of the main reasons why the construction of a full theory of quantum gravity has eluded us so far. Various proposals for rendering gravity UV-complete have been explored, ranging from the addition of higher order curvature terms (it is known that adding terms quadratic in the curvature tensor to the Einstein–Hilbert action renders the theory renormalizable [1221–1223], albeit at the price of a loss of unitarity), to the use of Lorentz violation as a field theory regulator [1224].

However, it has long been suspected that gravity should lead to an effective UV cut-off, in turn leading to a minimum observable length and/or a maximum possible momentum, based on gedanken experiments wherein the high energies used to resolve small distances eventually 'disturb' the space-time structure due to their gravitational effects, to the point that space-time itself cannot be resolved below some distance. Support for these ideas range from various general considerations, including within quantum gravity and QM [1225–1229], as well as string theory [1230–1234] (see also [1235–1244]). In the presence of a minimum length scale, gravity could therefore actually help rather than hamper achieving renormalizability.

These ideas have motivated approaches wherein the Heisenberg position-momentum commutator $[x,p] = i$, or equivalently the Heisenberg UP $\Delta x\Delta p \geqslant 1/2$, receives (quantum gravitational) non-linear corrections [1245–1247]. Such approaches are typically referred to as generalized uncertainty principle (GUP) or EUP, see e.g. [1248, 1249] for reviews. The notation and language when it comes to GUP and EUP is at times rather loose, but for consistency here we shall use the term 'GUP' to refer to the case where Heisenberg's UP receives corrections non-linear in momentum, typically associated to minimum length scenarios. On the other hand, we shall use the term 'EUP' to denote the case where the corrections are non-linear in position, as occurring in scenarios which introduce a (typically large) new fundamental scale (see e.g. [1250]). Here we shall first discuss GUP scenarios, before turning to EUP scenarios afterwards.

3.38.1. Generalized uncertainty principle.

Various GUP models non-linearly deforming the Heisenberg UP have been proposed in the literature, in most cases driven by phenomenological considerations themselves based on theoretical expectations. One of the earliest proposals is that of Kempf, Mangano, and Mann, wherein Heisenberg's position-momentum commutator receives corrections non-linear in momentum, as per the following [1251]:

$$\begin{align} [x_i, p_j] = i\delta_{ij} \left ( 1 + \beta_1\ell_\mathrm{Pl}^2 p^2 \right ) \,, \end{align} \tag{ 125 }$$

where $\ell_\mathrm{Pl}$ is the Planck length and β₁ is a dimensionless parameter. In turn, equation (125) implies the following corrected UP:

$$\begin{align} \Delta x\Delta p \geqslant \frac{1}{2} \left [ 1+ \beta_1\ell_\mathrm{Pl}^2(\Delta p)^2 \right ] \,, \end{align} \tag{ 126 }$$

with minimum measurable length given by $\Delta x \approx \ell_\mathrm{Pl}\sqrt{\beta_1}$.

A different GUP scenario which admits both a minimum length and a maximum momentum was instead presented in [1252, 1253] (see also [1254–1256]), and modifies Heisenberg's position-momentum commutator as follows:

$$\begin{align} [x_i, p_j] = i \left [ \delta_{ij}-\beta \left ( \delta_{ij}p+\frac{p_ip_j}{p} \right ) +\beta^2 \left ( \delta_{ij}p^2+3p_ip_j \right ) \right ] \,, \end{align} \tag{ 127 }$$

where $\beta = \beta_0\ell_\mathrm{Pl}$ and which in turn implies the following corrected UP:

$$\begin{align} \Delta x\Delta p \geqslant \frac{1}{2} \left ( 1-2\beta \langle p \rangle+4\beta^2 \langle p^2 \rangle \right ) \,. \end{align} \tag{ 128 }$$

The minimum measurable length implied by equation (128) is given by $\Delta x \approx \ell_\mathrm{Pl}\beta_0$, whereas the maximum momentum is given by $\Delta p \approx \ell_\mathrm{Pl}/\beta_0$. Notice that the deformed commutators of equations (125) and (127) have both been introduced on phenomenological grounds: however, in both cases the expected value of β is of order unity, and can be obtained using low-energy formulations of string theory [1257, 1258], by studying corrections to Hawking's theorem for the BH temperature obtained from two different approaches, or by computing the leading quantum corrections to Newton's gravitational potential [1259].

In the following, we shall consider the GUP characterized by equations (127) and (128). GUP-deformed BH solutions have been studied in several papers, and here we shall build upon the results of [1260–1262], where it has been shown that GUP effects physically lead to what one could consider quantum corrections to the BH ADM mass, which is modified to ${\cal M} = M+\beta M_\mathrm{Pl}^2/M^{\,2}$, where M can be interpreted as the 'bare' mass. Thus, the metric function for a GUP-corrected Schwarzschild BH reads [1260–1262]:

$$\begin{align} A(r) = 1-\frac{2M \left ( 1+\frac{\beta}{2M^{\,2}} \right ) }{r}\,, \end{align} \tag{ 129 }$$

where β characterizes an universal hair.

Despite the fact that various GUP tests have been performed in the literature using BHs and their environments (including their shadows), the form of the metric function given by equation (129) raises the question of whether one can actually discriminate a GUP-corrected BH from a 'genuine' Schwarzschild BH. In our case, we believe this might indeed be possible for Sgr A$^*$, for the following reason which echoes the argument presented previously for MOG in section 3.20. We interpret M as being the 'bare' mass of the system. Quantum effects which correct $M \to {\cal M}$ effectively change the total mass/energy of the system, as measured by an observer at infinity (given the definition of ADM mass). However, a measurement of the BH mass through dynamical probes sufficiently close to the event horizon will effectively probe M and not ${\cal M}$: this is indeed the case for the S-stars orbiting Sgr A$^*$ (and in particular the S2-star), whose motion has been used to obtain the exquisite mass-to-distance ratio measurements we adopt as priors. Moreover, the uncertainty on the mass (which does not disappear when we later fix M = 1, but is simply transferred to the mass-to-distance ratio) is part of what allows us to constrain the hair parameter β: we fix the bare mass, and ask how much of a change in the total mass/energy of the system we can tolerate when varying the hair parameter. While this physical interpretation has, to the best of our knowledge, never been explicitly discussed in the literature, it is the assumption which de facto underlies the many BH metric-based GUP tests (see e.g. [97, 1263–1266]).

Keeping in mind the assumptions and caveats discussed above, it is straightforward to show that the shadow radius associated to the GUP-corrected BH with metric function given by equation (129) is:

$$\begin{align} r_\mathrm{sh} = 3\sqrt{3}{\cal M} = 3\sqrt{3}M \left ( 1+\frac{\beta}{2} \right )\,. \end{align} \tag{ 130 }$$

We show the evolution of the shadow size as a function of β in figure 43. We see that the EHT observations set the upper limits $\beta \lesssim 0.01M^{\,2}$ (1σ) and $\beta \lesssim 0.14M^{\,2}$ (2σ). Restoring the Planck mass to obtain the dimensionless GUP parameter β, we require $\beta \lesssim 0.14(M/M_\mathrm{Pl})^2 \sim 2 \times 10^{88}$. Unsurprisingly this is approximately six orders of magnitude stronger than the limit obtained from the shadow of M87$^*$ in [97, 1264], as Sgr A$^*$'s mass is about three orders of magnitude smaller. However, this same limit is still weaker by several orders of magnitude compared to the limits obtained from independent tests such as perihelion precession, pulsar periastron shift, light deflection, gravitational waves, gravitational redshift, Shapiro time delay, geodetic precession, and quasiperiodic oscillations (see e.g. [1264, 1267, 1268]). Nonetheless, our limit remains a valuable complementary constraint on an fundamental physics scenario of great theoretical interest.

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3.38.2. Extended uncertainty principle.

In the GUP scenarios discussed so far, momentum appears to play a preferred role. However, other possibilities discussed in the literature envisage either a 'democratic' extension where both position and momentum play an equally important role (see e.g. [1250] by one of us), or scenarios where the corrections to Heisenberg's UP depend only on the position operator itself. It is worth reminding the reader once more that these deformations, while inspired by theoretical scenarios, are nonetheless introduced at the phenomenological level. With these caveats in mind, in the following we shall consider an EUP scenario, where Heisenberg's UP is corrected as follows:

$$\begin{align} \Delta x\Delta p \geqslant \frac{1}{2} \left [ 1+\alpha\frac{(\Delta x)^2}{L_{\star}^2} \right ] \,, \end{align} \tag{ 131 }$$

where $L_{\star}$ is a new large fundamental distance scale, and α is generically expected to be of order unity. It is fair to say that EUP scenarios have received much less attention than their GUP counterparts, the reason possibly being that the need for new macroscopic gravitational physics has only recently started to become appreciated. This is not unrelated to the recent realization that when dealing with BHs, quantum gravity effects may be relevant not only on microscopic scales, but perhaps more importantly on macroscopic horizon scales, and may well be tied to puzzling theoretical aspects such as the information paradox or singularity removal: see e.g. [1269–1271] for various proposals to address the information paradox which envisage new physics appearing at the gravitational radius of a system rather than at the Planck scale. Related recent works have put forward the possibility that the horizon itself may have a quantum origin, for example within the corpuscular framework, wherein BHs are described as Bose–Einstein condensates of N soft gravitons weakly confined to a region whose size is comparable to the BH gravitational radius [1272–1276]. See [182, 1277–1280] for further EUP-related works.

The effect of an EUP of the form given by equation (131) on BHs was studied in [1281]. The effect of the EUP is analogous to the GUP effect discussed in section 3.38.1, and amounts to (quantum) corrections to the BH ADM mass. As a result, the metric function of an EUP-corrected BH is given by [1281] (see also [1282, 1283]):

$$\begin{align} A(r) = 1-\frac{2M}{r} \left ( 1+\frac{4\alpha M^{\,2}}{L_{\star}^2} \right ) \,, \end{align} \tag{ 132 }$$

where α characterizes an universal hair. As is clear from equation (132), the EUP-inspired corrections are large only when $L_{\star}$ is sufficiently small (this makes sense if we think of $L_{\star}$ as corresponding to an inverse energy scale). We show the evolution of the shadow size as a function of $\log_{10}(L_{\star}/\sqrt{\alpha})$, with $L_{\star}$ measured in pc, in figure 44. We find it convenient to express $L_{\star}$ in these units since the possibility of partially mimicking the galactic effects of DM has been discussed in the literature [1281], and relies upon $L_{\star}/\sqrt{\alpha}$ lying on ${\cal O}(0.1$–$10)\,\mathrm{kpc}$ scales, i.e. those relevant to dwarf and spiral galaxies. From figure 44 we see that EUP effects do not lead to visible changes in the shadow radius as long as $L/\sqrt{\alpha} \gtrsim 10^{-5}\,\mathrm{pc}$. For smaller values of $L/\sqrt{\alpha}$, the shadow radius increases drastically. Values of $L/\sqrt{\alpha} \simeq 10^{-6}\,\mathrm{pc}$ are completely excluded by the EHT observations, as the resulting shadow is much larger than what these observations allow. In addition, the black shaded band in figure 44 denotes the range corresponding to galactic and sub-galactic scales $L_{\star} \sim {\cal O}(0.1$–$10)\,\mathrm{kpc}$ (with α = 1) discussed previously, where EUP-related effects have been argued to potentially mimic the effects of DM. On these scales, EUP effects leave no visible imprint on the size of Sgr A$^*$'s shadow: this is to be expected, given that such scales are several orders of magnitude above the scale of Sgr A$^*$'s gravitational radius, which is in the ${\sim} {\cal O}(10^{-7})\,\mathrm{pc}$ range ⁵⁴ .

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Another approach (arguably enjoying stronger theoretical motivation compared to the ones discussed previously) which naturally introduces a minimum length scale occurs within NCGs. The term NCG encompasses a rather large ensemble of approaches: while we shall here shall use it in the sense of a space-time with non-zero commutation relations among the position coordinates [1286], of enormous importance is the spectral approach to NCG pioneered by Connes [1287]. The non-commutativity of space-time coordinates naturally encapsulates the discrete nature of space-time expected within several high-energy frameworks, including attempts towards constructing a theory of quantum gravity. For instance, this is a natural outcome of string theory, where target space-time coordinates become non-commuting operators on a D-brane [1288]. It is worth noting, however, that motivation for NCG covers a wide range of frameworks going well beyond string theory.

One of the earliest approaches towards constructing a non-commutative field theory was based on the Weyl–Wigner–Moyal $\ast$-product [1289–1293]. However, this approach left open several important questions, concerning among others the issues of Lorentz invariance and UV finiteness. This prompted the development of a different approach, the so-called coherent state formalism, wherein physical coordinates are commuting c-numbers, built as expectation values on coherent states [1294, 1295]: within this approach, one can construct an explicitly UV-finite non-commutative field theory (see also [1296]). The literature on NCG is extremely vast and it is not possible to do it justice here: for representative examples of important works on NCG and its applications to various scenarios of interest, as well as related ideas, we refer the reader to [1297–1346].

In the approach we shall follow, the non-commutativity of space-time will be encoded in the commutation relation between space-time coordinates:

$$\begin{align} [x^{\mu},x^{\nu}]=i\vartheta^{\mu\nu}\,, \end{align} \tag{ 133 }$$

where $\vartheta^{\mu\nu}$ is an anti-symmetric real matrix. Within the coherent state formalism, the issues of covariance and unitarity are solved if one assumes the following block-diagonal form for $\vartheta^{\mu\nu}$:

$$\begin{align} \vartheta^{\mu\nu} = \vartheta \textrm{diag} \left ( \epsilon_{ij}\,,\epsilon_{ij}\,,{\ldots}\epsilon_{ij} \right ) \,, \end{align} \tag{ 134 }$$

where ε_ij is the two-dimensional Levi-Civita tensor, and ϑ; is a fundamental constant with dimensions of length squared, and reflects the fact that within NCG the best possible localization of a particle is within a fundamental cell of area ϑ. It has been shown in [1294, 1295] that the main outcome of this approach is to replace point-like objects with smeared structures, a fact which can be seen explicitly by computing the NCG-induced modifications to the Feynman path integral. This basically amounts to a substitution rule, where Dirac delta functions are replaced by a distribution of 'typical' (in a loose sense) width $\sqrt{\vartheta}$. The same of course can and will occur for finite mass distributions, including the mass density of a particle-like gravitational source: rather than being localized at the origin, this will be smeared and diffuse throughout a region of typical width $\sqrt{\vartheta}$, as a result of the uncertainty encoded in the non-commutativity of the space-time coordinates in equations (133) and (134). Applying this 'substitution rule', [1347] was the first to study NCG-inspired BHs: we use the term 'inspired' since, in this approach, the choice of the function with which to replace the Dirac delta essentially amounts to a phenomenological, but theory-inspired, choice. In the following, we shall consider two different choices for such a function describing the smearing of the mass distribution of a point-like gravitational source: a Gaussian in section 3.39.1 and a Lorentzian in section 3.39.2.

3.39.1. Gaussian mass distribution.

We begin by choosing a Gaussian mass distribution, as first considered in [1347]. We note that this is the choice which enjoys strongest theoretical motivation. In this case, the point-like mass distribution $\rho(r) = M\delta(r)$ is replaced by the following:

$$\begin{align} \rho_{\vartheta}(r) = \frac{M}{\left ( 4\pi\vartheta \right )^{\frac{3}{2}}}\exp \left ( -\frac{r^{\,2}}{4\vartheta} \right )\,, \end{align} \tag{ 135 }$$

so that the point-like gravitational source diffuses throughout a region of size $\sqrt{\vartheta}$. Given this choice of mass distribution, the corresponding BH solution was studied in [1347] by constructing the (covariantly conserved) energy-momentum tensor corresponding to the energy density of equation (135). The final result is that the metric function describing NCG-inspired BHs within this scenario is given by [1347] (see also [1348–1362] for further works on NCG-inspired BHs):

$$\begin{align} A(r) = 1-\frac{4M}{r\sqrt{\pi}} \gamma \left ( \frac{3}{2},\frac{r^{\,2}}{4\vartheta} \right ) \,, \end{align} \tag{ 136 }$$

where ϑ characterizes an universal hair, γ denotes the lower incomplete gamma function, and the Schwarzschild metric is recovered in the limit $r/\sqrt{\vartheta} \to \infty$. For $\sqrt{\vartheta}\lt0.5252$ the above space-time possesses two horizons, whereas an extremal BH with degenerate horizons at r = 1.5873 is recovered for $\sqrt{\vartheta} = 0.5252$. In the following, we shall focus on the $\sqrt{\vartheta} \leqslant 0.5252$ regime.

We compute the shadow size numerically, and show its evolution as a function of the NCG parameter ϑ in figure 45. We find that the shadow size depends extremely weakly on ϑ, so much so that the evolution cannot be appreciated by the naked eye. This is consistent with the earlier findings of [1363]. We find that for the extremal value $\sqrt{\vartheta} = 0.5252$, the relative change in the shadow size compared to that of a Schwarzschild BH with the same mass is of order ${\cal O}(10^{-3})$. We therefore conclude that this particular NCG-inspired BH is in perfect agreement with the EHT observations, but distinguishing it from its Schwarzschild counterpart appears extremely unlikely, unless unrealistic angular resolutions of ${\cal O}(\mathrm{nas})$ can be reached (which is extremely far from feasible).

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3.39.2. Lorentzian mass distribution.

Remaining in the realm of NCG-inspired phenomenology, another possibility which has been considered in the literature is that where a point-like gravitational source is smeared across a non-Gaussian distribution. An example which has been studied in this sense envisages a Lorentzian distribution (see e.g. [1364–1366]), so that equation (135) is modified to the following:

$$\begin{align} \rho_{\vartheta}(r) = \frac{M\sqrt{\vartheta}}{\pi^{\frac{3}{2}}(r^{\,2}+\pi\vartheta)^2}\,, \end{align} \tag{ 137 }$$

where again the source diffuses across a region of linear size $\sqrt{\vartheta}$. The resulting NCG-inspired BHs were studied in [1365], and shown to be described by the following metric function (to lowest order in $\sqrt{\vartheta}$):

$$\begin{align} A(r) = 1-\frac{2M}{r}+\frac{8M\sqrt{\vartheta}}{\sqrt{\pi}r^{\,2}} \,, \end{align} \tag{ 138 }$$

where ϑ characterizes an universal hair, and which is clearly of the RN-like form, so we can write down the following closed-form expression for the shadow radius:

$$\begin{align} r_\mathrm{sh} = \frac{\sqrt{2\sqrt{\pi}} \left ( 3+\sqrt{9-64\sqrt{\frac{\theta}{\pi}}} \right ) }{\sqrt{\frac{32\sqrt{\frac{\theta}{\pi}}+\sqrt{9-64\sqrt{\frac{\theta}{\pi}}}-3}{8\sqrt{\theta}}}}\,. \end{align} \tag{ 139 }$$

We show the evolution of the shadow size as a function of the NCG parameter ϑ in figure 46. Compared to the previous Gaussian mass distribution case, we now find a stronger effect on the BH shadow size, which decreases with increasing ϑ. This is likely due to the fact that a Lorentzian distribution has heavier tails compared to a Gaussian one with the same value of ϑ. We find that the EHT observations set the upper limits $\vartheta \lesssim 0.02M^{\,2}$ (1σ) and $\vartheta \lesssim 0.04M^{\,2}$ (2σ), both comparatively weak as one would typically expect NCG effects to show up at much smaller scales (see e.g. [1367]).

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Since the discovery of Hawking evaporation and the fact that a temperature can be associated to BHs [219, 1368], the intriguing connections between thermodynamics and gravity have flourished and given birth to the vibrant field of BH thermodynamics [1369–1371]. The Bekenstein–Hawking entropy relation envisages a simple proportionality relation between the entropy and area of a BH [1368, 1372], and is unquestionably the most studied entropy relation in the BH thermodynamics field. However, over the past decades various alternative entropy relations have been explored, with a wide range of different theoretical motivations, including the fact that the non-extensive nature of gravitational systems such as BHs calls for a non-extensive entropy-area relation (unlike the Bekenstein–Hawking entropy): see e.g. [1373–1382] for important examples thereof.

As first explicitly noted by one of us in [155], in the context of a BH the presence of a modified entropy-area relation effectively back-reacts on other thermodynamical quantities, including the BH's mass. If M is the BH 'bare' mass and $S(A) = S(M)$ is the entropy-area/mass relation in question, the 'dressed' mass $\widetilde{M}$ is then given by $\widetilde{M} = (\partial S/\partial M)/8\pi$. The necessity of constructing a consistent thermodynamical system in the presence of a modified entropy relation is indeed an issue which has started receiving significant attention recently, see e.g. discussions in [1383–1386], as well as [1387].

A particularly interesting modified entropy relation has been considered by Barrow [1388], motivated by possible space-time foam effects. Barrow entropy leads to a fractal structure for the horizon with a volume which is finite (but larger than in the standard case), and area which can be either finite or infinite. In the toy model considered by Barrow, a three-dimensional fractal is built around the BH event horizon, modifying its total surface and volume ⁵⁵ . In this picture, the entropy-area law of a BH is modified to:

$$\begin{align} S = \left ( \frac{A}{4} \right ) ^{1+\frac{\Delta}{2}}\,, \end{align} \tag{ 140 }$$

where Δ characterizes an universal hair, $0\leqslant \Delta \leqslant 1$ is related to the dimension of the fractal, with $\Delta = 0$ resulting in the usual Bekenstein–Hawking entropy $S = A/4$, and $\Delta = 1$ leading the largest deviation of the BH entropy from the Bekenstein–Hawking value. Since its proposal, Barrow entropy has been at the center of several interesting applications, among which it is worth mentioning its use in driving a model of holographic dark energy [1389–1391] (see also [1392–1415]).

As shown in [155] by one of us, the back-reaction of Barrow entropy on the Schwarzschild metric modifies the 'bare' mass to ${\cal M} = (\Delta/2+1)(4\pi)^{\Delta/2}M^{\Delta+1}$, so that the metric function of the Barrow entropy-corrected BH is given by [155]:

$$\begin{align} A(r) = 1-\frac{(\Delta+2)M^{\Delta+1}(4\pi)^{\Delta/2}}{r} \,. \end{align} \tag{ 141 }$$

It is therefore trivial to show that the shadow size is given by ⁵⁶ :

$$\begin{align} r_\mathrm{sh} = 3\sqrt{3}{\cal M}=3\sqrt{3} \left ( \frac{\Delta}{2}+1 \right ) M^{1+\Delta}\pi^{\Delta/2}\,. \end{align} \tag{ 142 }$$

The shadow size increases with increasing Δ, reflecting the fact that the BH surface area increases due to the quantum gravity effects envisaged in [1388].

We show the evolution of the shadow size as a function of Δ in figure 47. With the same caveats already discussed previously for MOG in section 3.20, as well as GUP and EUP scenarios in section 3.38 (see also [1385] for a different point of view), we clearly see that the EHT observations set extremely strong constraints on the parameter Δ. In particular, we find the upper limit $\Delta \lesssim 0.001$ at 1σ, tighter than cosmological constraints requiring $\Delta \lesssim 0.1888$ [1416, 1417], but weaker than Big Bang Nucleosynthesis constraints requiring $\Delta \lesssim 1.4 \times 10^{-4}$ [1418]. At 2σ, we have the weaker constraint $\Delta \lesssim 0.035$, which is nonetheless still stronger than the cosmological one. One particularly interesting consequence of these limits is that the behavior of a holographic DE component based on Barrow entropy is prevented from deviating significantly from the cosmological constant Λ, particularly in the phantom regime, and for this reason we expect that Barrow entropy cannot play a significant role in the context of the Hubble tension [1419–1423]: see e.g. [1424–1438] for relevant discussions in relation to phantom dark energy. This highlights an interesting complementarity between BH shadows and cosmology, already hinted to in some of our earlier results, and which we plan to explore more in future works.

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Although in this work we have covered a large number of scenarios, our limited resources meant that we were inevitably forced to make a choice as to which models to study, leaving out an even larger number of potentially interesting ones. Here, we very briefly comment on a few interesting models which have been left out.

3.41.1. Vector-tensor Horndeski gravity.

While Horndeski's scalar-tensor theory [610] is probably among the best known modified gravity frameworks, much less known is the fact that Horndeski also wrote down the most general 4D vector-tensor theory with second order equations of motion [1439]. While not sharing the same success as its scalar-tensor counterpart, the theory has started re-gaining interest recently (see e.g. [1440–1459]). The theory allows for the coupling of vector fields to gravity through their field-strength tensors, and this can be done for any vector field, including the gauge fields of the Standard Model. With $F^{\mu\nu}$ (possibly carrying additional gauge and/or family indices) denoting the field-strength tensor of a gauge field, the gauge-gravity coupling is of the form:

$$\begin{align} {\cal L} \supset \frac{\lambda}{4M_\mathrm{Pl}^2} \left ( RF_{\mu\nu}F^{\mu\nu}-4R_{\mu\nu}F^{\mu\alpha}F_{\alpha}^{\nu}+R_{\mu\nu\alpha\beta}F^{\mu\nu}F^{\alpha\beta} \right ) \,, \end{align} \tag{ 143 }$$

with λ the dimensionless gauge-gravity coupling, which of course can take different values for each gauge field.

In [1460], one of us derived upper limits on λ by estimating the order-of-magnitude of the gauge-gravity coupling as ${\sim} \lambda{\cal R}{\cal F}^2/M_\mathrm{Pl}^2$, with ${\cal R}$ and ${\cal F}$ the typical curvature and field-strength tensor strengths respectively, and requiring that the Horndeski interaction term be weaker than both the Einstein–Hilbert and Maxwell Lagrangian terms, in order for the associated effects to be subdominant. This leads to upper limits of the order of:

$$\begin{align} \vert \lambda \vert \lesssim \left ( \frac{\ell_{\cal R}}{\ell_\mathrm{Pl}} \right ) ^2\,, \end{align} \tag{ 144 }$$

where $\ell_{\cal R}$ is the length scale of the system under consideration. For instance, in the case of BH shadows one expects the correction to the shadow size due to the gauge-gravity interaction to be of the order of:

$$\begin{align} \frac{\delta r_\mathrm{sh}}{r_\mathrm{sh}} \sim \lambda \left ( \frac{\ell_\mathrm{Pl}}{\ell_{\cal R}} \right ) ^2\,, \end{align} \tag{ 145 }$$

where $\ell_{\cal R} \sim M_\mathrm{bh}/M_\mathrm{Pl}^2$, with $M_\mathrm{bh}$ the BH mass. Requiring that the fractional correction to the shadow size be much smaller than unity results in the upper limit:

$$\begin{align} \vert \lambda \vert \lesssim \left ( \frac{M_\mathrm{bh}}{M_\mathrm{Pl}} \right ) ^2\,. \end{align} \tag{ 146 }$$

For M87$^*$, equation (146) gives the limit $\vert \lambda \vert \lesssim 10^{88}$. Of course, with Sgr A$^*$ we can improve this limit by about 6 orders of magnitude as a direct consequence of the fact that we are probing a regime of much higher curvature (see figure 1), obtaining $\vert \lambda \vert \lesssim 10^{82}$. This is still 7 order of magnitude short of the strongest constraint obtained in [1460] from neutron stars, and is overall weak in a broad sense. It is plausible that cosmological or GW constraints on this theory may yield much stronger bounds but such constraints, to the best of our knowledge, are not yet available.

3.41.2. Bumblebee gravity.

Bumblebee gravity, first proposed by Kostelecký and Samuel in [761] (see also [1461, 1462]), is a class of effective field theories of gravity exhibiting spontaneous Lorentz symmetry breaking driven by the dynamics of a vector field acquiring a VEV. As such, the model constitutes an interesting playground for studying effects associated to spontaneous breaking of Lorentz symmetry, and has been the subject of a large number of studies (see e.g. [395, 1058, 1162, 1463–1487]). An exact non-rotating BH solution in bumblebee gravity was constructed in [1488], and is described by a single additional parameter $\ell$, which is related to the VEV of the bumblebee field, with the full metric given by:

$$\begin{align} \mathrm{d}s^2 = - \left ( 1 - \frac{2M}{r} \right ) \mathrm{d}t^2+ \left ( 1+\ell \right ) \frac{\mathrm{d}r^{\,2}}{1-\frac{2M}{r}}+r^{\,2}\mathrm{d}\Omega^2 \,, \end{align} \tag{ 147 }$$

However, in this metric the only modifications to the Schwarzschild space-time appear in the g_rr term, and do not therefore change the size of the shadow, which remains $3\sqrt{3}M$. Therefore, for the purposes of the metric test presented in this work, BHs in bumblebee gravity are not distinguishable from a Schwarzschild BH with the same mass. A rotating version of this solution was derived in [1489] (see also [1490]): for what concerns the shadow size computation, this BH has the same form of the Kerr metric, with an effective spin given by $a_{\star}\sqrt{1+\ell}$, and therefore can only be sensibly constrained if one has an independent spin measurement. We note, however, that the correctness of this BH solution was criticized in [1491] (see also [1492, 1493]).

3.41.3. Dynamical Chern–Simons gravity.

Dynamical Chern–Simons gravity modifies the Einstein–Hilbert action by adding a scalar field ϑ with a standard kinetic term and a potential, and coupling this field to the Pontryagin density obtained by contracting the Riemann tensor with its dual, as follows [1494, 1495]:

$$\begin{align} {\cal L} \supset \frac{\alpha}{4}\vartheta R_{\mu\nu\rho\sigma} {^{*}R^{\mu\nu\rho\sigma}}\,, \end{align} \tag{ 148 }$$

where the dual of the Riemann tensor is given by:

$$\begin{align} ^{*}R^{\mu\nu\rho\sigma} = \frac{1}{2}\epsilon^{\rho\sigma\alpha\beta}R^{\mu\nu}_{\alpha\beta}\,. \end{align} \tag{ 149 }$$

Such a coupling enjoys strong motivation from various theoretical scenarios, including string theory, LQG, effective field theories of inflation, and so on, and has been extensively studied in the literature [1496–1510].

As shown in [1494], non-rotating BHs in dynamical Chern–Simons gravity are still described by the Schwarzschild space-time. Physically speaking, this is a consequence of the fact that the stationary Cotton tensor vanishes on a spherically symmetric background. Therefore, at the level of non-rotating solutions as considered in the present work, the EHT observations cannot be used to test dynamical Chern–Simons gravity. Rotating BH solutions in this theory have been considered in various works. While exact solutions valid for all values of the spin are not known, a number of slowly-rotating solutions have been derived in the literature [1511–1515].

3.41.4. Casimir wormhole.

The Casimir effect, i.e. the force exerted between two plane parallel, close, uncharged, metallic plates in vacuum, was predicted theoretically almost a century ago [1516], and later verified experimentally. The associated Casimir energy has long been known to be a potential candidate source to generate and support a traversable WH (see e.g. [369, 1517], as well as [1518–1527] for later works). In [1528], Garattini explicitly constructed various interesting traversable WH solutions supported by Casimir energy. One example is described by the following metric:

$$\begin{align} \mathrm{d}s^2 = - \left ( \frac{3r}{3r+a} \right ) ^2 \mathrm{d}t^2+\frac{\mathrm{d}r^{\,2}}{1-\frac{2a}{3r}+\frac{a^2}{3r^{\,2}}}+r^{\,2}\mathrm{d}\Omega^2 \,, \end{align} \tag{ 150 }$$

where a is the throat radius. From the above we can immediately read off that the WH's ADM mass is $M = 3a$, so that the metric function can be rewritten as:

$$\begin{align} A(r) = \left ( \frac{r}{r+M} \right ) ^2\,. \end{align} \tag{ 151 }$$

It is easy to see that the above metric function is identical to the metric function of the null NS considered in section 3.15. The key difference between the two, rendering one a WH and the other a NS, lies in the g_rr metric element. However, this does not impact the computation of the photon sphere and associated shadow size. Just as for the null NS, we find that the Casimir WH does not have a photon sphere (formally the solution to the equation determining the photon sphere is $r_\mathrm{ph} = 0$). We would nonetheless expect it to cast a shadow of size $r_\mathrm{sh} = M$, which is immediately ruled out at very high significance by the EHT observations. However, a definitive answer as to whether or not the Casimir WH casts a shadow would require a light trajectory analysis similar to the one performed by some of us in [74] for the null NS (see figure 1(f) thereof), given the presence of the WH throat at $r = M/3$, which may prevent the results of [74] from fully carrying over here. We defer this analysis to a future work.

3.41.5. T-duality.