Deformations of GR and BH thermodynamics

Even though a 'pure connection' description of non-zero scalar curvature Einstein metrics is possible, it appears that a description with a certain set of auxiliary fields is more useful.

Let Aⁱ be an ${\rm{SO}}(3)$ connection on a four-manifold, and ${F}^{i}={\rm{d}}{A}^{i}+(1/2){\epsilon }^{{ijk}}{A}^{j}\wedge {A}^{k}$ be its curvature two-forms Let ${M}^{{ij}}$ be a symmetric $3\times 3$ matrix of auxiliary fields that is required to satisfy a certain condition, see below. Consider the following action

$$\begin{eqnarray}&&S[A,M]=\int {\rm{Tr}}({MF}\wedge F).\end{eqnarray} \tag{ 4 }$$

The matrix M is restricted to lie on a co-dimension one surface

$$\begin{eqnarray}&&g(M)=0\end{eqnarray} \tag{ 5 }$$

in the space of symmetric $3\times 3$ matrices, where $g(M)$ is a scalar-valued function. The surface that is relevant to Einstein connections/metrics is

$$\begin{eqnarray}&&{g}_{{\rm{E}}}(M)={\rm{Tr}}({M}^{-1})-{\rm{\Lambda }},\end{eqnarray} \tag{ 6 }$$

where $4{\rm{\Lambda }}$ is the scalar curvature. The claim is that the critical points of (4), with M restricted to lie on (5) with (6) give rise to Einstein metrics.

Varying the action (4) with respect to the connection we get

$$\begin{eqnarray}&&{{\rm{d}}}_{A}({M}^{{ij}}{F}^{j})=0.\end{eqnarray} \tag{ 7 }$$

Here we assumed that the manifold is compact, so that the integration by parts necessary to derive this field equation is justified. In the asymptotically hyperbolic case of relevance for us below, the bulk term in the action needs to be supplemented with a multiple of the Chern–Simons term on the boundary, see [10] and below.

To get the equation for M, it is useful to impose the constraint (5) with the help of a Lagrange multiplier. To do this, we introduce an auxiliary four-form field ${ \mathcal V }$, to which we shall refer as the auxiliary volume form. We can then write

$$\begin{eqnarray}&&{F}^{i}\wedge {F}^{j}={X}^{{ij}}\;{ \mathcal V }.\end{eqnarray} \tag{ 8 }$$

The matrix ${X}^{{ij}}$ of curvature wedge products is defined, as ${ \mathcal V }$ modulo multiplication by a function. We can then write the action as

$$\begin{eqnarray}&&S[A,M,\mu ]={\int }_{M}({\rm{Tr}}({MX})-\mu \;g(M)){ \mathcal V }.\end{eqnarray} \tag{ 9 }$$

The Euler–Lagrange equations for M are then

$$\begin{eqnarray}&&X=\mu \displaystyle \frac{\partial g}{\partial M},\end{eqnarray} \tag{ 10 }$$

which says that the matrix X of curvature wedge products must lie in the direction orthogonal to the surface $g(M)=0$. When this equation can be solved, together with $g(M)=0$, for $M(X)$, the solution can be substituted to (7) to obtain the 'pure connection' formulation of field equations, in which we obtain second-order PDE's for the connection. However, in many circumstances a more useful alternative appears to be to interpret (7) as a first-order PDE on the matrix M. Solving it one can then find X from (10) and thus obtain a set of first-order PDE's on the connection components. It is this strategy that will be followed below in solving concrete problems in this setting.

For the case of GR (6) the equation relating X and M takes the form $X=-\mu {M}^{-2}$. We can write it as ${MX}=-\mu {M}^{-1}$. Then, taking the trace and using the condition (6) we get

$$\begin{eqnarray}&&X=\displaystyle \frac{{\rm{Tr}}({MX})}{{\rm{\Lambda }}}{M}^{-2},\end{eqnarray} \tag{ 11 }$$

which gives X up to scale.

We now describe the metric, constructed from the connection. This metric is Einstein when the connection satisfies the Euler–Lagrange equations (7) and (11).

First, one constructs the conformal metric. This is the unique conformal metric that makes the triple of curvature two-forms Fⁱ self-dual. Explicitly, this is the conformal metric

$$\begin{eqnarray}&&\sqrt{{\rm{det(g)}}}\;{g}_{\mu \nu }\sim {\tilde{\epsilon }}^{\alpha \beta \gamma \delta }{\epsilon }^{{ijk}}{F}_{\mu \alpha }^{i}{F}_{\nu \beta }^{j}{F}_{\gamma \delta }^{k}.\end{eqnarray} \tag{ 12 }$$

Second, one fixes the volume form ${{ \mathcal V }}_{m}$ of the metric to be

$$\begin{eqnarray}&&-2{\rm{\Lambda }}\;{{ \mathcal V }}_{m}={\rm{Tr}}({MF}\wedge F).\end{eqnarray} \tag{ 13 }$$

The numerical factor in this formula is adjusted to agree with some conventions introduced later.

An interesting class of deformations of the Einstein condition is obtained by simply changing the co-dimension one surface that restricts the matrix M in the action (4). This corresponds to changing the function $g(M)$ in the constraint equation $g(M)=0$. As we have reviewed above, the function that gives rise to Einstein connections is (6). Replacing M by ${M}^{-1}$ in this function, i.e. ${g}_{\mathrm{inst}}={\rm{Tr}}(M)-{\rm{\Lambda }}$ turns out to give self-dual General Relativity, with solutions being half-flat Einstein metrics, see [7] for a discussion of this. More generally, deformations of Einstein connections can always be parametrised by making the cosmological constant in (6) a function of the trace-free part of the matrix ${M}^{-1}$:

$$\begin{eqnarray}&&g(M)={\rm{Tr}}({M}^{-1})-{\rm{\Lambda }}({M}_{\mathrm{tf}}^{-1}),\qquad {M}_{\mathrm{tf}}^{-1}\equiv {\rm{\Psi }}:= {M}^{-1}-\displaystyle \frac{1}{3}{\mathbb{I}}{\rm{Tr}}({M}^{-1}),\end{eqnarray} \tag{ 14 }$$

so that

$$\begin{eqnarray}&&{M}^{-1}={\rm{\Psi }}+\displaystyle \frac{{\rm{\Lambda }}({\rm{\Psi }})}{3}{\mathbb{I}}.\end{eqnarray} \tag{ 15 }$$

A particularly simple one-parameter family of modifications is obtained by taking

$$\begin{eqnarray}&&{\rm{\Lambda }}({\rm{\Psi }})={{\rm{\Lambda }}}_{0}-\displaystyle \frac{\alpha }{2}{\rm{Tr}}({{\rm{\Psi }}}^{2}),\end{eqnarray} \tag{ 16 }$$

with α being a parameter of dimensions of inverse curvature. The modification of the Einstein condition that ensues becomes important in regions with large self-dual part of the Weyl curvature.

In the deformed case, one continues to define the metric by the condition that the curvature two-forms are self-dual. The volume form is then fixed to be a constant multiple of the form ${\rm{Tr}}({MF}\wedge F)$, so that the action (4) continues to receive the interpretation of the total volume of the space.

We take the following spherically symmetric ansatz

$$\begin{eqnarray}&&{A}^{1}=a(R){\rm{d}}t+\mathrm{cos}(\theta ){\rm{d}}\phi ,\qquad {A}^{2}=-b(R)\mathrm{sin}(\theta ){\rm{d}}\phi ,\qquad {A}^{3}=b(R){\rm{d}}\theta .\end{eqnarray} \tag{ 17 }$$

The easiest way to motivate this ansatz is to consider a spherically symmetric metric ${\rm{d}}{s}^{2}={f}^{2}(R){\rm{d}}{t}^{2}+{g}^{2}(r){\rm{d}}{r}^{2}+{R}^{2}({\rm{d}}{\theta }^{2}+{\mathrm{sin}}^{2}(\theta ){\rm{d}}{\phi }^{2})$, and compute the self-dual part of the associated Levi-Civita connection. One obtains a connection of the form above, with functions $a,b$ related to the functions $f,g$ and their first derivatives. One then abstracts from the metric considerations and takes (17) as the appropriate ansatz. A more direct way to establish (17) as a good spherically symmetric ansatz is to explicitly check that this connection is spherically symmetric. Thus, one assumes that there is the group ${\rm{SO}}(3)$ acting in the $t=\mathrm{const}$ planes, with orbits of this action being copies of S². One can then check that the Lie derivative of the Lie algebra valued connection one-form Aⁱ in the direction of vector fields generating the action of ${\rm{SO}}(3)$ is a gauge transformation. Here we make no claim as to the generality of the ansatz (17). But it is certainly sufficient for our purposes of describing a large class of spherically symmetric solutions.

The curvatures of (17) are

$$\begin{eqnarray}{F}^{1} & = & -a^{\prime} {\rm{d}}t\wedge {\rm{d}}R+({b}^{2}-1)\mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi ,\\ {F}^{2} & = & {ab}\;{\rm{d}}\theta \wedge {\rm{d}}t-b^{\prime} \mathrm{sin}(\theta ){\rm{d}}R\wedge {\rm{d}}\phi ,\\ {F}^{3} & = & -{ab}\mathrm{sin}(\theta ){\rm{d}}t\wedge {\rm{d}}\phi +b^{\prime} {\rm{d}}R\wedge {\rm{d}}\theta ,\end{eqnarray} \tag{ 18 }$$

where the primes denote the derivative with respect to the radial coordinate, at this stage arbitrary. The diagonal X matrix can then be taken to be

$$\begin{eqnarray}&&{X}^{11}=\displaystyle \frac{a^{\prime} }{a},\qquad {X}^{22}={X}^{33}=\displaystyle \frac{{bb}^{\prime} }{{b}^{2}-1}.\end{eqnarray} \tag{ 19 }$$

The field equations are obtained by substituting the above ansatz into the action (4). We have

$$\begin{eqnarray}&&S[a,b,M]=-8\pi \beta \int {\rm{d}}R({M}_{1}({b}^{2}-1)a^{\prime} +2{M}_{2}{abb}^{\prime} ),\end{eqnarray} \tag{ 20 }$$

where we integrated over the angular and time coordinates, assuming that t is periodic with period β. By varying this action with respect to functions $a(R),{b}^{2}(R)-1$ we easily get the following system

$$\begin{eqnarray}&&({M}_{1}a({b}^{2}-1))^{\prime} =f(X)a({b}^{2}-1),\qquad ({M}_{2}a({b}^{2}-1))^{\prime} =f(X)a({b}^{2}-1),\end{eqnarray} \tag{ 21 }$$

where we manipulated the equations to map them into a form in which the right-hand side is the same. Here

$$\begin{eqnarray}&&f(X)={\rm{Tr}}({MX})={M}_{1}\displaystyle \frac{a^{\prime} }{a}+{M}_{2}\displaystyle \frac{({b}^{2}-1)^{\prime} }{{b}^{2}-1}.\end{eqnarray} \tag{ 22 }$$

Of course the same equations can also be obtained from the equations ${d}_{A}({MF})=0$, but the above derivation from a simple action is more economic.

We now choose the radial coordinate so that

$$\begin{eqnarray}&&f(X)a({b}^{2}-1)=1.\end{eqnarray} \tag{ 23 }$$

The solution for the matrix M is then immediately written down

$$\begin{eqnarray}&&{M}_{1}=f(X)(R+{R}_{1}),\qquad {M}_{2}=f(X)(R+{R}_{2}),\end{eqnarray} \tag{ 24 }$$

where ${R}_{\mathrm{1,2}}$ are integration constants. Alternatively, we can write

$$\begin{eqnarray}&&{M}_{1}^{-1}=\displaystyle \frac{1}{f(X)(R+{R}_{1})},\qquad {M}_{2}^{-1}=\displaystyle \frac{1}{f(X)(R+{R}_{2})}.\end{eqnarray} \tag{ 25 }$$

The metric is computed from the requirement that the curvature two-forms (18) are self-dual, and that the metric volume form is a constant multiple of the coordinate volume form. This last condition follows from our choice of the radial coordinate (23). We get the following metric

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}=\sqrt{\displaystyle \frac{f(X)}{| {\rm{\Lambda }}| {X}_{1}{X}_{2}^{2}}}\left({X}_{1}\displaystyle \frac{{a}^{2}{b}^{2}}{{b}^{2}-1}{\rm{d}}{t}^{2}+{X}_{1}{X}_{2}^{2}\displaystyle \frac{{b}^{2}-1}{{b}^{2}}{\rm{d}}{R}^{2}+{X}_{2}({b}^{2}-1){\rm{d}}{{\rm{\Omega }}}^{2}\right),\end{eqnarray} \tag{ 26 }$$

where as usual ${\rm{d}}{{\rm{\Omega }}}^{2}$ is the metric on the unit sphere. If desired, one can easily go the Lorentzian signature by simply changing $t\to \sqrt{-1}\;t$. But for most of our purposes the Euclidean metric above is sufficient.

Anticipating what will happen, let us assume that the function b² has a simple zero for some value $R={R}_{+}$. Below we shall see that this is what happens in GR at the location of the Euclidean event horizon. This continues to be true in modified theories, and the purpose of this subsection is to establish the value of the angle deficit at the tip of the arising conical singularity. We can do this in full generality, without specifying the theory we work with.

Thus, let us write

$$\begin{eqnarray}&&{b}^{2}=({b}^{2}{)}_{+}^{\prime }(R-{R}_{+})+\cdots .\end{eqnarray} \tag{ 27 }$$

Substituting this into (26) we see that the correct radial coordinate near $R={R}_{+}$ is given by

$$\begin{eqnarray}&&\rho ={\left(\displaystyle \frac{f(X)}{| {\rm{\Lambda }}| {X}_{1}{X}_{2}^{2}}\right)}^{1/4}{({X}_{1}{X}_{2}^{2}({b}^{2}-1))}^{1/2}2\sqrt{\displaystyle \frac{R-{R}_{+}}{({b}^{2}{)}_{+}^{\prime }}}.\end{eqnarray} \tag{ 28 }$$

Here all the terms are evaluated at $R={R}_{+}$. We then substitute the $R-{R}_{+}$ found from this into the expression for the coefficient of the ${\rm{d}}{t}^{2}$ term. We use the relation

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}({b}^{2})^{\prime} =({bb}^{\prime} )={X}_{2}({b}^{2}-1)\end{eqnarray} \tag{ 29 }$$

to observe that all terms cancel leaving

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={a}_{+}^{2}{\rho }^{2}{\rm{d}}{t}^{2}+{\rm{d}}{\rho }^{2}\end{eqnarray} \tag{ 30 }$$

as the metric near the tip of the cone. We learn that the correct period for the identification of the t coordinate is

$$\begin{eqnarray}&&\beta =\displaystyle \frac{2\pi }{{a}_{+}}.\end{eqnarray} \tag{ 31 }$$

It is very interesting that the components of the connection seem to have direct physical meaning. Thus, at the horizon the component b vanishes, while the component a just measures the BH temperature!

The work [9] identified entropy with the boundary term of the Einstein–Hilbert action evaluated at the Euclidean horizon. Here we develop a similar interpretation for our theories.

Thus, we first consider (20) and notice that in obtaining the associated Euler–Lagrange equations we have neglected the boundary terms

$$\begin{eqnarray}&&-8\pi \beta ({M}_{1}({b}^{2}-1)a+2{M}_{2}{{ab}}^{2})\end{eqnarray} \tag{ 32 }$$

evaluated at both ends of the interval of integration over the radial coordinate. These terms at $R={R}_{\infty }$ are taken care of by the Chern–Simons invariant that we will add at the boundary at infinity, see below.

We now need to consider what happens at the point $R={R}_{+}$. As we have seen, at this value of the radial coordinate ${b}^{2}=0$, and so the second term in (32) drops out. However, as we have also shown above, at this point we have (31) and so the first term reduces just to

$$\begin{eqnarray}&&16{\pi }^{2}{M}_{1}^{+},\end{eqnarray} \tag{ 33 }$$

where ${M}_{1}^{+}\equiv {M}_{1}{| }_{+}$. If our expectation about the entropy being the boundary term of the action is correct, this quantity divided by $16\pi G$ must be the entropy

$$\begin{eqnarray}&&{ \mathcal S }=\displaystyle \frac{\pi {M}_{1}^{+}}{G}.\end{eqnarray} \tag{ 34 }$$

This quantity can be written in a more invariant way as follows. Let us consider the two-form ${M}^{{ij}}{F}^{j}$ restricted to the horizon two-sphere. By inspection of (18) we see that the only term that survives at the horizon is ${F}^{1}{| }_{+}=-\mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi$. This means that $\sqrt{{| {MF}| }^{2}}={M}_{1}^{+}\mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi$, and the entropy formula (2) given in the Introduction is a covariant expression for our result (34). Below we will confirm that (34) is the correct expression for the entropy by explicitly computing the Legendre transform of the free energy for the one-parameter family of modified theories (16).

We now specialise to the case of GR. We write the equation relating M and X as ${MX}=-\mu {M}^{-1}$ and take the trace of this equation. We get $-\mu =f(X)/{\rm{\Lambda }}$, where we have used the definition (22) and the condition ${\rm{Tr}}({M}^{-1})={\rm{\Lambda }}$. But we can also take the trace of (25) to get

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{1}{f(X)}\left(\displaystyle \frac{1}{R+{R}_{1}}+\displaystyle \frac{2}{R+{R}_{2}}\right)\equiv \displaystyle \frac{3R+2{R}_{1}+{R}_{2}}{f(X)(R+{R}_{1})(R+{R}_{2})}.\end{eqnarray} \tag{ 35 }$$

It is now convenient to shift the radial coordinate so as to impose

$$\begin{eqnarray}&&2{R}_{1}+{R}_{2}=0.\end{eqnarray} \tag{ 36 }$$

We now take ${R}_{1}\equiv \bar{R}$ to be the single parameter of the solution. This gives

$$\begin{eqnarray}&&f(X)=\displaystyle \frac{3R}{{\rm{\Lambda }}(R+\bar{R})(R-2\bar{R})}.\end{eqnarray} \tag{ 37 }$$

We now have $X=-\mu {M}^{-2}$, with μ expressed in terms of $f(X)$ that we just solved for. So, the final expressions for the quantities ${X}_{1},{X}_{2}$ are

$$\begin{eqnarray}&&{X}_{1}=\displaystyle \frac{(R-2\bar{R})}{3R(R+\bar{R})},\qquad {X}_{2}=\displaystyle \frac{(R+\bar{R})}{3R(R-2\bar{R})}.\end{eqnarray} \tag{ 38 }$$

We now integrate the relations (19) to obtain the components of the connection

$$\begin{eqnarray}&&a={K}_{1}\displaystyle \frac{R+\bar{R}}{{(3R)}^{2/3}},\qquad {b}^{2}-1={K}_{2}\displaystyle \frac{R-2\bar{R}}{{(3R)}^{1/3}},\end{eqnarray} \tag{ 39 }$$

where ${K}_{\mathrm{1,2}}$ are integration constants. This gives a complete solution to the problem, modulo the issue of fixing (or interpreting) the integration constants ${K}_{\mathrm{1,2}}$ and $\bar{R}$.

There is a relation between the integration constants ${K}_{\mathrm{1,2}}$ that follows from our gauge-fixing condition $f(X)a({b}^{2}-1)=1$. Indeed, substituting the $f(X)$ that we have found above, as well as the connection components, we get

$$\begin{eqnarray}&&{K}_{1}{K}_{2}={\rm{\Lambda }}.\end{eqnarray} \tag{ 40 }$$

Let us fix ${K}_{\mathrm{1,2}}$ from the large R asymptotics of the metric (26). We have ${X}_{1},{X}_{2}=1/3R$ asymptotically, and $f(X)=3/{\rm{\Lambda }}R$. So, the conformal factor in (26) behaves as

$$\begin{eqnarray}&&\sqrt{\displaystyle \frac{f(X)}{| {\rm{\Lambda }}| {X}_{1}{X}_{2}^{2}}}=\displaystyle \frac{9R}{| {\rm{\Lambda }}| }=3{l}^{2}R.\end{eqnarray} \tag{ 41 }$$

So, overall, for large R the metric reads

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={l}^{2}\left({K}_{1}^{2}\displaystyle \frac{{R}^{2/3}}{{3}^{4/3}}{\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{R}^{2}}{{(3R)}^{2}}+{K}_{2}\displaystyle \frac{{R}^{2/3}}{{3}^{1/3}}{\rm{d}}{{\rm{\Omega }}}^{2}\right).\end{eqnarray} \tag{ 42 }$$

We would like this to be the usual (asymptotically) hyperbolic metric

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={l}^{2}\left({r}^{2}{\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{{r}^{2}}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2}\right).\end{eqnarray} \tag{ 43 }$$

This means that asymptotically at least

$$\begin{eqnarray}&&{K}_{1}^{2}\displaystyle \frac{{R}^{2/3}}{{3}^{4/3}}={r}^{2},\qquad {K}_{2}\displaystyle \frac{{R}^{2/3}}{{3}^{1/3}}={r}^{2}\end{eqnarray} \tag{ 44 }$$

combining this relation with (40) we get

$$\begin{eqnarray}&&R={r}^{3}{l}^{2}\end{eqnarray} \tag{ 45 }$$

and

$$\begin{eqnarray}&&{K}_{1}={\left(\displaystyle \frac{3}{l}\right)}^{2/3},\qquad {K}_{2}={\left(\displaystyle \frac{3}{{l}^{4}}\right)}^{1/3}.\end{eqnarray} \tag{ 46 }$$

Thus, finally, with these integration constants fixed from the asymptotics the connection components are

$$\begin{eqnarray}&&a=\displaystyle \frac{R+\bar{R}}{{({Rl})}^{2/3}},\qquad {b}^{2}-1=\displaystyle \frac{R-2\bar{R}}{l{({Rl})}^{1/3}}.\end{eqnarray} \tag{ 47 }$$

We can now write down the final expression for the metric. For the conformal factor we get

$$\begin{eqnarray}&&\sqrt{\displaystyle \frac{f(X)}{| {\rm{\Lambda }}| {X}_{1}{X}_{2}^{2}}}=\displaystyle \frac{3{l}^{2}{R}^{2}}{R+\bar{R}}.\end{eqnarray} \tag{ 48 }$$

After numerous cancellations we get the following metric

$$\begin{eqnarray}&&{\rm{d}}{s}^{2}={l}^{2}\left({b}^{2}{\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{{b}^{2}}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2}\right),\end{eqnarray} \tag{ 49 }$$

where (45) is the relation that is valid everywhere. In terms of the radial coordinate r the function b² takes the form

$$\begin{eqnarray}&&{b}^{2}=1+\displaystyle \frac{{r}^{3}{l}^{2}-2\bar{R}}{{{rl}}^{2}}=1-\displaystyle \frac{2\bar{R}}{{{rl}}^{2}}+{r}^{2}.\end{eqnarray} \tag{ 50 }$$

This becomes the usual Euclidean Schwarzschild AdS metric if we further replace $r\to r/l$ and identify

$$\begin{eqnarray}&&\bar{R}={Ml},\end{eqnarray} \tag{ 51 }$$

where M is the black hole mass.

At the value ${r}_{+}$ of the radial coordinate satisfying

$$\begin{eqnarray}&&1-\displaystyle \frac{2M}{{{lr}}_{+}}+{r}_{+}^{2}=0,\end{eqnarray} \tag{ 52 }$$

which is where the event horizon would be located in the Lorentzian signature solution, the metric has a conical singularity unless we identify time with the correct period.

We already worked out the general formula for the inverse temperature β in (31), so we just need to apply the formula and see how the familiar AdS-Schwazschild solution temperature arises. The formula (31) tell us that we just need to evaluate the value of connection component a at the location of the event horizon. In terms of the r coordinate the function a (47) is given by

$$\begin{eqnarray}&&a=\displaystyle \frac{1}{r}\left({r}^{2}+\displaystyle \frac{M}{{rl}}\right).\end{eqnarray} \tag{ 53 }$$

Using (52) to express everything in terms of ${r}_{+}$ and substituting into (31) we immediately get the familiar expression

$$\begin{eqnarray}&&\beta =\displaystyle \frac{4\pi {r}_{+}}{1+3{r}_{+}^{2}}.\end{eqnarray} \tag{ 54 }$$

It is possible to parametrise the solution by ${r}_{+}$ rather than M/l because, as it is easy to check, any positive value of ${r}_{+}$ can arise as M/l varies between zero and infinity. Indeed, for $M\ll l$ ${r}_{+}=2M/l$, while for $M\gg l$ ${r}_{+}={(2M/l)}^{1/3}$.

The solution is essentially the same as in [7], and so we simply list the relevant formulas. We define the matrix $R={\rm{diag}}(R+{R}_{1},R+{R}_{2},R+{R}_{3})$. Then the matrix X is given by formula (79) of [7]

$$\begin{eqnarray}&&X=\displaystyle \frac{{R}^{-2}\left({\mathbb{I}}-\tfrac{\partial {\rm{\Lambda }}({\rm{\Psi }})}{\partial {\rm{\Psi }}}\right)}{{\rm{Tr}}\left[{R}^{-1}\left({\mathbb{I}}-\tfrac{\partial {\rm{\Lambda }}({\rm{\Psi }})}{\partial {\rm{\Psi }}}\right)\right]}.\end{eqnarray} \tag{ 55 }$$

The tracefree matrix Ψ is given by

$$\begin{eqnarray}&&{\rm{\Psi }}={\rm{\Lambda }}({\rm{\Psi }})\left[\displaystyle \frac{{R}^{-1}}{{\rm{Tr}}({R}^{-1})}-\displaystyle \frac{1}{3}{\mathbb{I}}\right].\end{eqnarray} \tag{ 56 }$$

The only difference with the case considered in [7] is that now the second and third eigenvalues of all matrices are equal.

As in [7], we find the function ${\rm{\Lambda }}(R)\equiv {\rm{\Lambda }}({\rm{\Psi }}(R))$ by solving a certain quadratic equation. We introduce the function

$$\begin{eqnarray}&&{s}_{2}(R)=\displaystyle \frac{1}{{(R+{R}_{1})}^{2}}+\displaystyle \frac{2}{{(R+{R}_{2})}^{2}}=\displaystyle \frac{3{R}^{2}+2R({R}_{2}+2{R}_{1})+{R}_{2}^{2}+2{R}_{1}^{2}}{{(R+{R}_{1})}^{2}{(R+{R}_{2})}^{2}}.\end{eqnarray} \tag{ 57 }$$

We make the same choice of the origin of the R coordinate as in the GR case, namely (36), and set ${R}_{1}\equiv \bar{R}$. We then have

$$\begin{eqnarray}&&{s}_{2}(R)=3\displaystyle \frac{{R}^{2}+2{\bar{R}}^{2}}{{(R+\bar{R})}^{2}{(R-2\bar{R})}^{2}}.\end{eqnarray} \tag{ 58 }$$

We then introduce another function

$$\begin{eqnarray}&&q(R):= \left(\displaystyle \frac{{s}_{2}}{{s}_{1}^{2}}-\displaystyle \frac{1}{3}\right)=\displaystyle \frac{2{\bar{R}}^{2}}{3{R}^{2}},\end{eqnarray} \tag{ 59 }$$

which is non-negative and diverges at the singularity at $R=0$. The function ${\rm{\Lambda }}(R)$ is then given as the solution of the quadratic equation

$$\begin{eqnarray}&&-\displaystyle \frac{\alpha }{2}q(R){{\rm{\Lambda }}}^{2}={\rm{\Lambda }}-{{\rm{\Lambda }}}_{0}.\end{eqnarray} \tag{ 60 }$$

This gives

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{1}{\alpha q}(\sqrt{1+2\alpha {{\rm{\Lambda }}}_{0}q}-1)=\displaystyle \frac{3{R}^{2}}{2\alpha {M}^{2}{l}^{2}}\left(\sqrt{1-\displaystyle \frac{4\alpha {M}^{2}}{{R}^{2}}}-1\right),\end{eqnarray} \tag{ 61 }$$

where we have substituted (51). As we shall see, this relation will still be valid because asymptotically we will have the solution behaving as Schwarzschild, with modifications becoming important only deep inside the manifold. We see that one must be careful here and for positive α make sure that α is not large enough so that the expression under the square root stays positive.

Having determined ${\rm{\Lambda }}(R)$, we can write down the solution for the matrix X explicitly, this is formula (97) of [7]

$$\begin{eqnarray}&&X=\displaystyle \frac{{R}^{-2}}{{s}_{1}(1+\alpha x{\rm{\Lambda }})}\left(1+\alpha {\rm{\Lambda }}\left(\displaystyle \frac{{R}^{-1}}{{s}_{1}}-\displaystyle \frac{1}{3}\right)\right).\end{eqnarray} \tag{ 62 }$$

Spelling it out explicitly

$$\begin{eqnarray}{X}_{1} & = & \displaystyle \frac{R-2{Ml}}{3{Ml}}\left(\displaystyle \frac{1}{\sqrt{{R}^{2}-4\alpha {M}^{2}}}-\displaystyle \frac{1}{R+{Ml}}\right),\\ {X}_{2} & = & -\displaystyle \frac{R+{Ml}}{6{Ml}}\left(\displaystyle \frac{1}{\sqrt{{R}^{2}-4\alpha {M}^{2}}}-\displaystyle \frac{1}{R-2{Ml}}\right).\end{eqnarray} \tag{ 63 }$$

We note that for $\alpha \lt 0$ the singularity appearing in the case of GR at $R=0$ is absent in the modified theory. The components of the matrix X stay finite at the would be singularity at $R=0$. There is a simple pole in X₁ located at $R=-{Ml}$, and a simple pole in X₂ located at $R=2{Ml}$, but these are present already in GR solution. These are coordinate singularities that do not correspond to any singularity in the connection components.

We can now integrate the relations (19) and obtain the connection components. Somewhat surprisingly, this can be done in a closed form. We get

$$\begin{eqnarray}a & = & \displaystyle \frac{R+{Ml}}{{(l/2)}^{2/3}{(R+\sqrt{{R}^{2}-4\alpha {M}^{2}})}^{2/3}}\mathrm{exp}\displaystyle \frac{1}{3{Ml}}(\sqrt{{R}^{2}-4\alpha {M}^{2}}-R),\\ {b}^{2}-1 & = & \displaystyle \frac{R-2{Ml}}{l{(l/2)}^{1/3}{(R+\sqrt{{R}^{2}-4\alpha {M}^{2}})}^{1/3}}\mathrm{exp}-\displaystyle \frac{1}{3{Ml}}(\sqrt{{R}^{2}-4\alpha {M}^{2}}-R).\end{eqnarray} \tag{ 64 }$$

We chose the integration constants so that we get the same asymptotics as the GR solutions (47). Alternatively, the above solutions go to their GR counterparts as one takes the deformation parameter $\alpha \to 0$. For $\alpha \lt 0$ there is no longer a singularity at $R=0$, with the connection components staying finite there. There is another region for negative R, which we do not analyse here as our main interest is thermodynamics.

The fact that the Schwarzschild $r=0$ singularity gets resolved in modified theories of this type is not new, and has been studied (in a different formulation from the one considered here) in [12]. This reference also contains more information about the causal structure of the arising space times. We do not consider all these issues in the present paper, remaining in the Euclidean signature where the thermodynamic partition function can be computed.

The equation ${b}^{2}=0$ gives the following equation to solve for ${R}_{+}$ in terms of M

$$\begin{eqnarray}&&\mathrm{exp}\displaystyle \frac{1}{3{Ml}}(\sqrt{{R}_{+}^{2}-4\alpha {M}^{2}}-{R}_{+})=\displaystyle \frac{2{Ml}-{R}_{+}}{l{(l/2)}^{1/3}{({R}_{+}+\sqrt{{R}_{+}^{2}-4\alpha {M}^{2}})}^{1/3}}.\end{eqnarray} \tag{ 65 }$$

Unfortunately, it is no longer possible to solve for $M({R}_{+})$ explicitly. But we can use this equation to get for ${a}_{+}$

$$\begin{eqnarray}&&{a}_{+}=\displaystyle \frac{2({Ml}+{R}_{+})(2{Ml}-{R}_{+})}{{l}^{2}({R}_{+}+\sqrt{{R}_{+}^{2}-4\alpha {M}^{2}})}.\end{eqnarray} \tag{ 66 }$$

The auxiliary matrix M that appears in our Lagrangian is asymptotically a multiple of the identity matrix for any asymptotically hyperbolic solution, see (25). Because of this, the divergences appearing in the bulk part of the action (4) are the same as those in a multiple of the Chern–Simons functional of the boundary connection. This means that the Chern–Simons invariant for the restriction of the connection to the boundary can be used to renormalise the bulk part of the action.

We now compute the renormalised volume of the Euclidean BH metric, using the Chern–Simons invariant as the counter term necessary for the renormalisation. First, the bulk part is trivially computed thanks to our radial coordinate fixing condition. We have

$$\begin{eqnarray}&&\int {\rm{Tr}}({MF}\wedge F)=-2\int f(X)a({b}^{2}-1){\rm{d}}R{\rm{d}}t\mathrm{sin}(\theta ){\rm{d}}\theta {\rm{d}}\phi =-8\pi \beta ({R}_{\infty }-{R}_{+}),\end{eqnarray} \tag{ 67 }$$

where ${R}_{\infty }$ is the upper limit of the R integration, and we have used the radial coordinate fixing condition (23).

We now compute the CS functional for the connection (17). We write

$$\begin{eqnarray}&&{\rm{d}}{A}^{1}={\rm{d}}a\wedge {\rm{d}}t-\mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi \end{eqnarray} \tag{ 68 }$$

and so then

$$\begin{eqnarray}&&{A}^{1}\wedge {\rm{d}}{A}^{1}=-a{\rm{d}}t\wedge \mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi +\mathrm{cos}(\theta ){\rm{d}}\phi \wedge {\rm{d}}a\wedge {\rm{d}}t.\end{eqnarray} \tag{ 69 }$$

The object $\mathrm{cos}(\theta ){\rm{d}}\phi$ is not a globally defined one-form, and so we cannot simply restrict ${A}^{1}{\rm{d}}{A}^{1}$ to the three-boundary and integrate. Instead, let us rewrite the troublesome term as

$$\begin{eqnarray}&&\mathrm{cos}(\theta ){\rm{d}}\phi \wedge {\rm{d}}a\wedge {\rm{d}}t=-{\rm{d}}(a{\rm{d}}t\wedge \mathrm{cos}(\theta ){\rm{d}}\phi )-a{\rm{d}}t\wedge \mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi .\end{eqnarray} \tag{ 70 }$$

We can now restrict to the three-boundary and integrate. The integration gets rid of the total derivative term. Overall we get twice the contribution from the first term in (69), and not once as one might naively assume. Adding the contribution from the $(1/3){\epsilon }^{{ijk}}{A}^{i}\wedge {A}^{j}\wedge {A}^{k}=2{A}^{1}\wedge {A}^{2}\wedge {A}^{3}$ we get

$$\begin{eqnarray}&&{S}^{\mathrm{CS}}=2a({b}^{2}-1)\int {\rm{d}}t\wedge \mathrm{sin}(\theta ){\rm{d}}\theta \wedge {\rm{d}}\phi =8\pi \beta a({b}^{2}-1).\end{eqnarray} \tag{ 71 }$$

We can evaluate the quantity $a({b}^{2}-1)$ e.g. from (64). We have

$$\begin{eqnarray}&&a({b}^{2}-1)=\displaystyle \frac{2(R+{Ml})(R-2{Ml})}{{l}^{2}(R+\sqrt{{R}^{2}-4\alpha {M}^{2}})}.\end{eqnarray} \tag{ 72 }$$

We can now expand this evaluated at $R={R}_{\infty }$. We get

$$\begin{eqnarray}&&-\displaystyle \frac{3}{{\rm{\Lambda }}}a({b}^{2}-1){| }_{{R}_{\infty }}={R}_{\infty }-{Ml}+O(1/{R}_{\infty }),\end{eqnarray} \tag{ 73 }$$

so that

$$\begin{eqnarray}&&-2{\rm{\Lambda }}{RV}=\underset{{R}_{\infty }\to \infty }{\mathrm{lim}}\left(\int {\rm{Tr}}({M}^{-1}F\wedge F)+\displaystyle \frac{3}{{\rm{\Lambda }}}{S}_{{R}_{\infty }}^{\mathrm{CS}}\right)=8\pi \beta ({Ml}-{R}_{+}).\end{eqnarray} \tag{ 74 }$$

Note that the CS functional does not simply remove the divergent term ${R}_{\infty }$ in (67), but also adds a finite piece to the answer. This is precisely the same finite piece that appears in the literature, see e.g. [13], formula (2.18). This finite piece is, however, usually obtained by considering the trace of the extrinsic curvature of the boundary. Here our Chern–Simons prescription gives it automatically.

The Euclidean partition function is the Euclidean Einstein–Hilbert action, which is a multiple of the volume. The Euclidean action is

$$\begin{eqnarray}&&I=-\displaystyle \frac{1}{16\pi G}\int \sqrt{g}\;(R-2{\rm{\Lambda }})=-\displaystyle \frac{2{\rm{\Lambda }}}{16\pi G}V,\end{eqnarray} \tag{ 75 }$$

where V is the total volume. In the case of asymptotically hyperbolic manifolds the renormalised volume must be used. Substituting from (74) we get

$$\begin{eqnarray}&&I=\displaystyle \frac{\beta ({Ml}-{R}_{+})}{2G}.\end{eqnarray} \tag{ 76 }$$

It is not possible to solve the equation (65) for either ${R}_{+}$ or M. However, we can rewrite it as a relation between

$$\begin{eqnarray}&&x=\displaystyle \frac{{R}_{+}}{{Ml}}\end{eqnarray} \tag{ 77 }$$

and M/l. We get

$$\begin{eqnarray}&&M/l=\displaystyle \frac{{(x+\sqrt{{x}^{2}-4\tilde{\alpha }})}^{1/2}}{\sqrt{2}{(2-x)}^{3/2}}{{\rm{e}}}^{(\sqrt{{x}^{2}-4\tilde{\alpha }}-x)/2},\end{eqnarray} \tag{ 78 }$$

where

$$\begin{eqnarray}&&\tilde{\alpha }=\alpha /{l}^{2}.\end{eqnarray} \tag{ 79 }$$

For the case of GR the quantity x changes $x\in (0,2)$, with small values corresponding to small black holes $M\ll l$ and values close to two corresponding to large black holes $M\gg l$. One is not allowed to consider negative x in the case of GR because of the square root in the numerator in (78). However, the modified theory with $\tilde{\alpha }\lt 0$ allows for negative x as well, with large negative x giving exponentially large masses.

Using this parametrisation we can rewrite all the quantities as functions of x. We get for the inverse temperature

$$\begin{eqnarray}&&\beta =\displaystyle \frac{\pi (x+\sqrt{{x}^{2}-4\tilde{\alpha }})}{(M/l)(1+x)(2-x)}=\displaystyle \frac{\pi \sqrt{2}{(2-x)}^{1/2}{(x+\sqrt{{x}^{2}-4\tilde{\alpha }})}^{1/2}}{1+x}{{\rm{e}}}^{-(\sqrt{{x}^{2}-4\tilde{\alpha }}-x)/2}\end{eqnarray} \tag{ 80 }$$

and for the partition function

$$\begin{eqnarray}&&I=\left(\displaystyle \frac{{l}^{2}}{2G}\right)\beta (M/l)(1-x)=\left(\displaystyle \frac{{l}^{2}}{2G}\right)\displaystyle \frac{\pi (1-x)(x+\sqrt{{x}^{2}-4\tilde{\alpha }})}{(1+x)(2-x)}.\end{eqnarray} \tag{ 81 }$$

It is now straightforward (but tedious) to check that

$$\begin{eqnarray}&&\displaystyle \frac{\partial I}{\partial \beta }=\displaystyle \frac{\partial I/\partial x}{\partial \beta /\partial x}=\displaystyle \frac{{Ml}}{G}.\end{eqnarray} \tag{ 82 }$$

Thus, the mass parameter of our solution is indeed the thermodynamic energy (multiple of).

We can also compute the entropy

$$\begin{eqnarray}&&{ \mathcal S }=\displaystyle \frac{\partial I}{\partial \beta }\beta -I=\left(\displaystyle \frac{{l}^{2}}{2G}\right)\displaystyle \frac{\pi (x+\sqrt{{x}^{2}-4\tilde{\alpha }})}{2-x}.\end{eqnarray} \tag{ 83 }$$

This result for the entropy is rather simple. It is definitely not equal to the area of the sphere at ${b}^{2}=0$, as can be easily checked from the metric (26).

The entropy obtained is no longer the horizon area, as it is in GR, but this is as expected for it is known that the entropy-area relation is characteristic of General Relativity. However, there is another way to characterise the gravitational entropy. Thus, as is shown in [9], the entropy arises from the boundary term needed to make the Einstein–Hilbert action variational principle well-defined. This boundary term, computed around the would-be conical defect point of the Euclidean solution, is a multiple of the entropy.

We have already computed the boundary term in (34). We will now show that (83) coincides with (34). Indeed, from (25) we have ${M}_{1}=f(X)(R+\bar{R})$ with $f(X)={s}_{1}(R)/{\rm{\Lambda }}({\rm{\Psi }})$. Thus, overall we have

$$\begin{eqnarray}&&{M}_{1}=\displaystyle \frac{3R}{{\rm{\Lambda }}(R-2\bar{R})},\end{eqnarray} \tag{ 84 }$$

with Λ given by (61). Re-writing everything in terms of the quantity $x={R}_{+}/{Ml}$ we have

$$\begin{eqnarray}&&{M}_{1}^{+}=\displaystyle \frac{{l}^{2}(x+\sqrt{{x}^{2}-4\tilde{\alpha }})}{2(2-x)}.\end{eqnarray} \tag{ 85 }$$

Comparing with (83) we see that (34) indeed holds.

We have obtained the expression (83), interpreted as (34) for a particular one-parameter family of modifications of GR. However, the reasoning that led to (34) is general, and relies just on the fact that ${b}^{2}=0$ at the horizon. Thus, it can be expected that the entropy formula (34) and its covariant version (2) hold for an arbitrary member of our modified family of gravity theories.