Inertia, gravity and the meaning of mass

Our concept of mass has evolved considerably over the centuries, most notably from Newton to Einstein, and then even more vigorously with the establishment of the standard model and the subsequent discovery of the Higgs boson. Mass is now invoked in various guises depending on the circumstance: it is used to represent inertia, or as a coupling constant in Newton’s law of universal gravitation, and even as a repository of a mysterious form of energy associated with a particle at rest. But recent developments in cosmology have demonstrated that rest-mass energy is most likely the gravitational binding energy of a particle in causal contact with that portion of the Universe within our gravitational horizon. In this paper, we examine how all these variations on the concept of mass are actually interrelated via this new development and the recognition that the source of gravity in general relativity is ultimately the total energy in the system.


Introduction
Already by Newton's time there were potentially two kinds of mass invoked in burgeoning physical laws.On the one hand, objects exhibited an acceleration in proportion to the force applied to them, implying they possessed a conserved 'inertial mass,' m i .And after Newton formulated his law of universal gravitation, it became apparent that a body also has 'gravitational mass,' m g , that today we would refer to as a gravitational coupling constant.Newton viewed these two quantities as being conserved, irreducible properties of matter, and simplified the description further by considering them to be indistinguishable [1].
Bondi [2] refined these definitions further, including also a possible dichotomy between 'passive' gravitational mass-that which responds to a gravitational field-and 'active' gravitational mass-that which creates the gravitational effect.He also allowed for the possibility of negative values for all these quantities.He argued, however, that the law of action and reaction in Newtonian physics implies the equality of active and passive gravitational masses.This concept has been tested experimentally many times since then, beginning with Kreuzer [3], who inferred an upper bound of ∼ 5 × 10 −5 for the fractional difference between the passive and active gravitational masses, to the latest measurement by Singh et al. [4], who lowered the limit considerably to ∼ 3.9 × 10 −14 .
All we can really say about inertial and gravitational masses is that the clues from nature point to a strict proportionality between them, since in principle Newton's gravitational constant (see Eq. 1 below) can always be adjusted to comply with any change in the ratio m i /m g .This proportionality is also the basis for Einstein's Principle of Equivalence, one of the most important founding tenets of general relativity [5].
A century later, we have a much more nuanced interpretation of mass, certainly with the establishment of the standard model [6,7,8] and the subsequent discovery of the Higgs boson [9,10,11].We now understand that inertial mass is better described as an emergent quality rather than an intrinsic property of matter, given that several fundamental particles, such as electrons, positrons and quarks, owe their inertia to a coupling with the Higgs field, while other composite particles, such as neutrons and protons, acquire inertia via the dynamical back reaction of accelerated quarks, which radiate gluons to conserve momentum (see ref. [12] for a recent review).In this context, m i ought to be viewed as a 'place holder' for something else, a notion we shall utilize liberally throughout this paper.
The gravitational mass has undergone quite an evolution as well.Following the broad acceptance of general relativity as the correct description of space and time, we now view the source of gravity to be the total energy, E, in the system.Coupled to another major discovery in relativity-the existence of rest-mass energyit now appears that Newton's gravitational mass may simply be a more primitive representation of E/c 2 .None of this necessarily gives us confidence, though, that m i and m g should be considered as representing the same thing.
But more recent work appears to have uncovered the origin of rest-mass energy [12], which in the end may tie all of these loose threads together.As we shall discuss later in this paper, the energy associated with a particle at rest appears to be its gravitational binding energy with that portion of the Universe contained within our gravitational horizon.Thus, in an odd twist of history, m i and m g appear to be linked after all-owing both of their existence to the energy of the particle.
To unravel this intricate quilt of physical attributes, we shall examine Newton's law of universal gravitation in the weak-field, static limit of general relativity, but go beyond its traditional application to particles 'with mass.'We now know that gravity accelerates 'massless' particles as well, as proven by the measured deflection of light in transit toward Earth through a cosmic medium with variable gravity.We shall therefore derive the analog of Newton's law for particles such as photons, which Newton would never have considered in his gravitational framework for want of any physical evidence of their existence.Were he alive today, however, he would no doubt have devised two versions of Equation (1): one for 'massive' particles, the other for 'massless.'This will be our principal task in § 2.2, with important foundational work in § 2.1.
Finally, in § 3, we shall unify the various concepts of inertial and gravitational mass via the rest-mass energy associated with them.We shall conclude with some closing thoughts in § 4.

Particles with Established Inertia
Newton's law of universal gravitation is an expression of the force experienced by a particle with established inertia m i and gravitational mass m g , due to a gravitating mass M g located at r = 0, which is understood relativistically in the limit of weak, static fields and low velocities.
Of necessity, the latter condition implies that particles coupling to the force must have non-zero inertia, m i = 0, for the acceleration g would otherwise be infinite.Equation (1) may also be written in terms of the gravitational potential, such that Equations ( 2) and ( 3) actually represent two distinct phenomena.The first accounts for the potential created by the hypothesized gravitating mass M g , while the second describes the response of a test particle m g to the presence of this potential.Thus, to ensure consistency between general relativity in the weak-field, static, lowvelocity limit and Newton's law of universal gravitation, two separate physical effects must be considered.First, the response of the particle given by Equation ( 3) is most naturally inferred from the geodesic equation, describing the trajectory of a free particle through the spacetime created by the gravitating mass M g (see Eq. 30 below).The affine parameter λ is often chosen to be the proper time τ in the particle's rest frame when m i = 0, though not for massless particles, such as a photon, for which τ is always zero.In this expression, Γ µ αβ are the Christoffel symbols containing information about the spacetime curvature, most directly calculated from the metric coefficients themselves: where the metric is formally written as The procedure for reducing Equation (4) to its Newtonian form is well known, so we won't dwell on the details, but mention only the key points, mostly in preparation for § 2.2 below.For a particle with inertia moving well below the speed of light, the second term is dominated by the α = β = 0 component, and therefore Then, to calculate Γ µ 00 for a weak gravitational field, we put where η αβ = diag(+1, −1, −1, −1) and |h αβ | ≪ 1.The inverse metric tensor is simply g αβ = η αβ − h αβ .If in addition the field is static, Substituting Equation ( 9) into (7), we thus find that so that d 2 x 0 /dτ 2 = 0, which means that dt/dτ is constant.That is, time progresses forward at a steady rate for a particle in Newtonian gravity, consistent with the prevailing view on the nature of time during Newton's era.
For the spatial coordinates (j = 1, 2, 3), we instead find that or, using the chain rule of differentiation, A comparison of Equations ( 3) and ( 12) therefore shows that, in order for Einstein's theory to correctly describe the motion of a particle with established inertia in a classical gravitational field, we must have m g → m i in the Newtonian limit (see § 3) which is, afterall, the basis for the Equivalence Principle that gave rise to the general relativistic description of particle trajectories in the first place.In addition, h 00 ≡ 2Φ/c 2 , so that In the next section, we shall learn that the gravitational coupling of particles believed to have zero inertia is very similar to this result, but differs from it in at least one very significant aspect.Before we begin to examine that situation, however, we must first understand the second phenomenon associated with Newtonian gravitythat giving rise to the gravitational potential itself (Eq.2).For this, we need to begin with the gravitational field equations in general relativity, which may be written G αβ is the Einstein tensor, written in terms of the Ricci tensor, the contracted Ricci tensor (also known as the curvature scalar) and the stress-energy tensor T αβ .In Equation (15), the quantity is known as the Riemann-Christoffel (or curvature) tensor.The appearance of the curvature scalar on the left-hand side of Equation ( 14) is sometimes inconvenient.Contracting this equation with α and β reduces it to the form Thus, an alternative representation of the field equations is It is beyond the scope of the present paper to describe how and why these equations are derived, but this topic is well covered in both the primary and secondary literature [5,13].We do point out, however, that the coefficient multiplying T αβ on the righthand side of Equations ( 14) and ( 19) was chosen in order for Einstein's equations to correctly reproduce the Newtonian potential in Equation ( 2) for a gravitating inertial source M g in the weak-field, static, low-velocity limit, which we now describe.
For simplicity, we adopt the so-called perfect fluid approximation, in which the stress-energy tensor excludes all possible shear forces associated with the transport of momentum components in directions other than those associated with the components themselves.The covariant form of this tensor may be written where u α is the local value of dx α /dτ for a comoving fluid element in the source, and p and ρ are the pressure and energy density, respectively, measured by an observer in a locally inertial frame comoving with the fluid at the instant of measurement.The trace of this stress-energy tensor is simply providing a clear, unequivocal affirmation that the source of spacetime curvature in general relativity is all forms of energy and momentum.This aspect of Einstein's theory cannot be overstated because it represents a clear departure from the Newtonian framework, in which gravity is due to an intrinsic 'mass' associated with the source, considered by Newton to be synonymous with inertia, and neither having anything to do with energy.Our continued examination of the meaning of m g , m i and M g below will be heavily based on this crucial distinction between Einstein's and Newton's theories, and their behavior in the classical limit.Equation ( 19) may thus also be written in the more suggestive form This expression is completely valid for all forms of energy, with or without inertia, and we shall use it also in the following section where we consider the gravitational coupling of particles believed to have no inertia.Here, however, we use the fact that matter (or its more common designation as 'dust') has essentially zero pressure, so T = ρ, and Equation ( 22) therefore implies that, to first order in the weak-field (Newtonian) limit (see Eq. 8), which relates derivatives of the metric coefficients (specifically h 00 and Φ) to the energy density, providing a direct link to the Poisson equation for Φ, from which Equation ( 2) is derived.
From Equation ( 15), we see that and R 0 000 = 0, so The second term on the right-hand side is zero for a static field, while the third and fourth terms are second order in h 00 .This leaves and, combining Equation ( 23) with ( 26), gives which is the relativistically derived Poisson's equation for the gravitational potential in the presence of an energy density ρ in the weak-field, static limit.It must be emphasized that this equation excludes any momentum (and therefore the pressure this would create) of the gravitating source specifically because we attributed inertia to the medium, allowing it to reside near the origin with a very low (or even zero) velocity.It is for this reason that T = ρ − 3p simply reduces to ρ in Equation ( 22), and we acknowledge the fact that the coefficient 4πG in Equation ( 27) was chosen to ensure that Einstein's and Newton's theories yield the same potential, Φ, when the gravitational mass, M g , is calculated solely from the energy density in the system, and nothing else.But there is another subtle, yet crucial, feature of this equation that we must fully understand, particularly when we begin to compare this result with its counterpart in the following section, addressing gravity in the presence of energy with what we believe to be zero inertia, i.e., in cases where T = ρ− 3p is not merely ρ.Equation ( 27) is fully consistent with Equation ( 2) only if we follow Newton in attributing the gravitational influence to a hypothesized 'gravitational mass,' M g , which necessarily would have to correspond to a gravitational mass density ρ g ≡ ρ/c 2 in Equation (27).
Those of us accustomed to the language of general relativity would be tempted to consider this as being self-evident.After all, isn't rest-mass energy simply given by this relation?Well yes, but not completely, as we shall soon find out.Later in this paper we shall better understand the distinction between gravitational and inertial mass, and realize that the interpretation of M g in Equation (2) as the 'rest-mass'-from the conversion of ρ to ρ g ≡ ρ/c 2 in Equation ( 27)-is valid only because Newton's law of universal gravitation was specifically formulated for particles with non-zero inertia in the low-velocity limit, where the gravitational and inertial masses are proportional to each other-or even equal, with an appropriate choice of units for the gravitational constant G.The situation is very different for photons, because m g for them is not zero.

Gravity with 'Zero Inertia'
Whether light has inertia and/or gravitational mass, and whether these two are equal, was something that could not easily be discussed or explored prior to the advent of relativity theory.But this did not prevent classical physicists from entertaining the idea that large heavenly bodies, such as the Sun, could in principle accelerate rays of light and cause them to deviate in discernible ways from straight-line trajectories.Very famously, Einstein himself used his Principle of Equivalence couched in Newtonian theory, essentially Equation (1) with m i and m g cancelled from both sides, to predict how much starlight would be bent upon grazing the surface of the Sun on its way toward Earth [14].
Without the full theory of general relativity to support this calculation, he partially relied on intuition, arguing that the rate of proper time (as seen by an observer fixed with respect to the source of gravity) varies with distance from the center of the gravitating body, thereby creating a speed of light varying with radius if one ignores the spatial variations.The latter assumption is key to understanding why he had to correct his prediction once general relativity was completed five years later.Applying Huygens's principle to a wave front passing through such a region, he could then calculate the degree of bending based on the time dilation produced by the central mass.This prediction amounted to about 0.875 seconds of arc, which turns out to be wrong by a factor 2. We shall see below why ignoring the spatial variations results in this not insignificant mistake.One may still predict the correct deflection angle using Newtonian theory, however, but only by using an alternative form of Equation ( 1) appropriate for light (see Eq. 52), which was not available to him at that time.
This type of speculation had already been carried out by others before him, even by Newton who, in his treatise on Opticks [15] published in 1704, asked the question: "Do not Bodies act upon Light at a distance, and by their action bend its Rays, and is not this action strongest at the least distance?"A century later, Johann Georg von Soldner had used Newton's Law of universal gravitation to calculate the deflection of starlight by the Sun treating "a light ray as a heavy body" and predicted a deflection angle of 0.84 arcseconds, virtually identical to Einstein's estimate based on his first attempt [16].
Einstein redid his calculation five years later once General Relativity was completed, taking into account all spacetime curvature effects and corrected his mistake, predicting a deflection angle of 1.745 arcseconds.As is well known by now, Sir Arthur Eddington subsequently led an expedition to an island off the coast of Africa, with a second group in Brazil, to measure the deflection of starlight grazing the edge of the Sun during the total eclipse of May 29, 1919.Their measurement provided a spectacular (if controversial) confirmation that General Relativity is the correct theory of gravity [17].
Questions have been raised about Eddington's analysis of their data because turbulence in Earth's atmosphere causes deflections of starlight comparable to those predicted by Einstein's theory.One must rely on the assumption that these are random in nature, so that they can be averaged away with the use of many images, leaving only the relativistic effect.By the end of their observations, Eddington and his coworkers had only two reliable images (with about five stars) at one site, and eight usable plates (with at least seven stars) at the Sobral location.Nineteen other plates taken with a second telescope had to be abandoned.Given this paucity of measurements, did Eddington really have the evidence to support Einstein's theory [18]?Some have argued that Eddington's enthusiasm for general relativity biased his approach.But many re-analyses between 1923 and 1956 of the plates from those expeditions yielded similar results within ten percent.A reanalysis in 1979 using the Zeiss Ascorecord and its data reduction software [19] yielded the same deflection as that calculated by Eddington, though with even smaller errors.The Sobral plates gave similar results, all consistent with general relativity.A modern assessment of Eddington's work has thus tended to show no credible evidence of bias in his conclusions [20].
Having said this, we now know, that photons do not have rest mass in the conventional form seen in the standard model of particle physics.So why are we justified in using General Relativity to calculate the gravitational acceleration of a photon when Einstein's theory is based on the equality of m i and m g (i.e., the Principle of Equivalence)?This question will become even more acute later in this paper, when we learn that Newton's approach of identifying gravitational mass suggests that m g strictly cannot be zero for light.To place this in context, we should ask ourselves whether we are missing a law of universal of gravitation, analogous to Equation (1), representing the weak-field, static limit of General Relativity for photons and other particles that do not have inertia in the conventional sense.
Such particles do not follow trajectories described by Equation (7).The magnitude of their velocity is always c, so one cannot adopt an asymptotically small speed to go along with the assumption of weak, static fields.As was the case in § 2.1, there are two effects we need to consider: the first arises from the impact of non-zero momentum on the source of gravity, modifying Newton's definition of M g , and the second provides the gravitational coupling of a particle we believe to have zero rest mass to the 'force' created by the former.
The first of these phenomena is quite straightforward to understand.Whereas the trace of the stress-energy tensor, T = ρ − 3p, reduces to ρ for 'dust,' the pressure cannot be ignored when the source is radiation since its momentum makes a significant contribution to the overall energy budget.For example, isotropic radiation exerts a pressure p = ρ/3, so T = 0. Thus, instead of Equation ( 23), we now get resulting in the modified Poisson's equation The factor 2 multiplying the density is directly due to the momentum within the source, which cannot be ignored when the gravitating particles cannot come to rest.When the source of gravity is radiation, Newton's 'gravitational mass' M g producing the potential in Equation (2) must therefore be twice the value one would naively have calculated from the energy density alone.(But beware that this is not the factor 2 associated with the deflection of starlight by the Sun.A correction such as this, in cases where the source of gravity is not completely inertial, is typically absorbed into the empirically determined value of M g .)Once the potential Φ and the Newtonian gravitational mass M g have been properly identified in Equation (2), we may then proceed with Equation (4) to derive the correct equations of motion for a photon in the vicinity of a spherically symmetric object, where the appropriate spacetime is described by the Schwarzschild metric: In this expression, is the Schwarzschild radius for an object with gravitational mass M g .Our goal is to uncover the radial acceleration experienced by the photon at a radius r ≫ r S , i.e., in the weak-field limit and, to facilitate the calculation, we shall assume that the photon's velocity is perpendicular to r = rr at that instant.The non-zero Christoffel symbols for this metric are simply Thus, letting overdot denote differentiation with respect to λ, we obtain the following expression from the µ = 0 component of Equation ( 4): Integrating this equation once gives where We align our coordinate system so that the photon's trajectory is restricted to the equatorial plane.Then, θ(λ) = π/2 and θ(λ) = 0, and Equation (35) becomes irrelevant.Equation (36) reduces to whose solution is simply where K 2 is a second constant of integration.Finally, each tangent vector along a null geodesic is light-like, so which provides us with the last equation we need: Now using Equations ( 34) and (38) with a change in variable to u ≡ 1/r, together with (which is valid because the motion has no θ dependence), we can modify Equation (40) to read as follows: Differentiating this equation once more with respect to φ gives us the equation of motion for a photon, which may be thought of as a relativistic version of the classical Binet orbit equation [21], best known for its use in calculating the deflection angle of starlight grazing the surface of the Sun on its way toward Earth.
In the absence of gravity (r S → 0), the solution to this equation would be a straight line perpendicular to r, given as where r 0 is the radius of closest approach to the origin, corresponding to the value of r at φ = 0.If we now make a weak-field approximation, consistent with r ≫ r S , the corresponding null trajectory will be a perturbation of this straight line, allowing us to write whose solution is Our interest here is not so much how the null geodesic is modified due to the accumulated effect of gravity over the ray's transit to Earth but, rather, the instantaneous acceleration experienced by a photon in the vicinity of φ = 0.The acceleration is obtained by differentiating this expression twice with respect to the observer's time, t, starting with We also have and so using also the relation in Equation ( 44), since gravity perturbs this trajectory only slightly in the weak-field limit.
A second differentiation results in the expression In this equation, however, the first term is simply the effect of seeing the straight line trajectory (Equation 44) in spherical coordinates.The radial acceleration due to gravity in Equation ( 50) is the rest of the righthand side.Thus, subtracting the first (geometric) term, substituting for dr/dt from Equation (49), and noting that r S ≪ r in the weak-field limit, we arrive at the Newtonian acceleration for a photon The dependence of this expression on the angle φ is simply due to the fact that the photon experiences zero acceleration in its longitudinal direction, so Equation (51) gives solely the component of acceleration in the radial direction.Be aware, however, that the approximations we have made in reaching this result are valid only near φ = 0, so this expression is not valid when sin φ → 1.On the other hand, given that φ → 0 when r → r 0 , this simply reduces to our final result, which is the maximal radial acceleration experienced by a photon transverse to its direction of motion.Equation ( 52) is the weak-field (r ≫ r S ), radial acceleration experienced by a photon moving perpendicular to r in the vicinity of a gravitational mass M g .It represents the Newtonian, static limit of General Relativity for particles we believe to have zero rest mass, the analog of Equation ( 1).Aside from their evident similarity, the other feature that stands out is the additional factor 2 emerging in the latter, which owes its appearance to the same physics responsible for the famous factor 2 in the calculation of the deflection angle of starlight grazing the surface of the Sun.This factor 2 appears as long as we use the same value of the gravitational constant, G, in both Equations ( 1) and (52).It is not due to a doubling of the source in Poisson's Equation (29), which arises from the contribution of momentum to the active gravitational mass, and would be absorbed into the overall 'measured' value of M g , independently of G.
It is instead due to the different geometries of particles with and without inertia along their longitudinal direction of motion.Simply put, particles with inertia satisfy Equation (1) in the low-velocity limit because, for them, the only metric coefficient in the Schwarzschild spacetime (Eq.30) that matters is g tt , i.e., the time dilation.As noted earlier in this section, Einstein himself used solely the effects of g tt to estimate the bending of light passing near the Sun, even though a lightwave was thought to be massless.These particles are moving too slowly compared to the speed of light for the distance covered during their acceleration to contribute significantly to ds 2 .For particles propagating at lightspeed, however, the impact of g rr cannot be ignored compared to g tt .In essence, the effects of spacetime curvature are doubled for photons compared to particles with established non-zero inertia.The inclusion of spatial variations resulting from g rr , once the full theory of general relativity was available to him, is the reason Einstein's recalculation of the deflection angle of starlight passing near the Sun doubled the effect he had anticipated in 1911.

Mass and Rest-mass Energy
The inference we draw from Equations (3), ( 27), (29) and ( 52) is that all particles experience a Newtonian-like gravitational attraction to each other in the nonrelativistic limit, whose coupling strength-according to general relativity-is the total energy in each of the gravitating objects.The Newtonian approach of assigning them 'masses' appears to be a way of representing these energies in terms of an inertial or gravitational context, a distinction that no doubt arises from the nature of rest-mass energy, as we shall further develop in this section.
In our previous, detailed examination of the origin of rest-mass energy [12], we showed that, of the four known forces, only gravity has all of the attributes required to satisfy a 'Principle of Equivalence'.Thus, the energy we commonly assign to a particle's rest mass is almost certainly gravitational in nature.Indeed, in the context of modern cosmology, the binding energy of a particle with gravitational mass m g in causal contact with that portion of the Universe within our gravitational horizon, R h ≡ c/H, where H is the Hubble constant, is exactly m g c 2 .If inertial mass is viewed as a surrogate for m g , we find in this result a natural explanation for the origin of rest-mass energy, though it still leaves open the question of whether m i and m g are truly different characteristics of the same object (or particle).
We should stress at this point that the proposal being discussed here, and introduced in ref. [12], is unique in a cosmological setting for the simple reason that it directly addresses the question of where rest-mass energy comes from, not simply mass.For example, other definitions of mass in cosmology, such a the Komar mass [22], or the Tolman mass [23], are statements concerning how much 'mass' is required to account for the (static) spacetime curvature in a closed volume, but none of these alternative approaches explains why the energy associated with m g must be m g c 2 .
In our proposal, on the other hand, any particle with gravitational mass m g has a binding energy m g c 2 due to its gravitational coupling to the energy contained within R h .
What we learn from Equation ( 52) is that particles we believe to have zero rest mass nevertheless also experience a gravitational force that accelerates them at a finite rate, albeit solely in a direction perpendicular to their velocity (with fixed magnitude c).Attempting to naively interpret this result in the context of Newton's original formulation of his law of universal gravitation (Eq.1), however, we would instead be compelled to assign them a non-zero inertial 'mass', for otherwise their acceleration would be infinite.Of course, this simple-minded approach does not comport very well with our conventional view that photons should be 'massless.'It appears, therefore, that our current definition of mass, inertial or otherwise, may be inaccurate, perhaps even defective.
Let us reconsider the Newtonian gravitational attraction between two particles.The clue we glean from general relativity, e.g., via Equation (21), is that the coupling constant for this interaction is the particle's total energy, E. And since the Newtonian gravitational force is symmetric between them, we write the force on particle 1 due to 2 as where r points from 1 to 2. Crucially, we now define where q here is the momentum of particle m i (distinguishing it from the symbol p we used earlier to denote the pressure).
According to our earlier study on the origin of rest-mass energy, [12], the binding energy of particle m g to that portion of the Universe within R h ≡ c/H, which also coincides with our apparent horizon [24] and the Hubble radius, equals its escape energy E esc = qc attained when its proper distance, R, approaches R h .Since q → m i c in this limit, we infer that E esc = m i c 2 .In contrast, a photon always has escape speed, even at R h , and is therefore unbound.For such particles, the total energy at any R is simply E = qc.
Consequently, Equation (54) may also be written though E esc → 0 for photons.With the definition in Equations ( 54) and (55), photons are therefore assigned a 'gravitational mass' while matter has the corresponding value Newton's law of universal gravitation is valid only in the low-velocity limit, however, for which qc ≪ m m i c 2 , and therefore fully consistent with the Principle of Equivalence.But what we have learned from Equation ( 52) is that a photon must also experience the force in Equation (53) in the Newtonian limit, albeit restricted to a direction perpendicular to its velocity.Newton's equation of motion for such a particle should therefore be written implying that for the photon as well, to be consistent with Equation (52).Not surprisingly, this is what we should have expected from the Principle of Equivalence applied to all particles, not just matter.Note, however, that unlike the inertial mass of matter, m γ i does not carry a 'rest energy' because the photon is always unbound.As such, m γ i does not appear in the photon's total energy budget analogous to Equation (54).

Conclusion
An important caveat for this work is that much of our discussion hinges on the viability of general relativity as the correct description of nature.In particular, all of the results we have discussed stem directly or indirectly from the fact that the source of gravity in Einstein's theory is the total energy of the system.No experimental test has ever provided evidence against this feature.A consideration of Einstein's theory in the weak-field, static limit then yields a workable interpretation of Newton's more empirical law of universal gravitation.We have shown in this paper that a direct comparison of these two approaches provides us with a possible explanation for the physical origin of gravitational and inertial mass, and perhaps also for the origin of rest-mass energy within the framework of one of the most famous solutions to Einstein's equations, i.e., the Friedmann-Lemaître-Robertson-Walker (FLRW) metric [13].
This work has also highlighted a tantalizing result that we should have anticipated all along.We have had an observational confirmation for several decades that astronomical sources of gravity bend null geodesics, fully confirming an important prediction of Einstein's theory.But the implied acceleration producing the deflection is finite, and when this is viewed in the weak-field, static limit (i.e., within a Newtonian framework), we cannot avoid the conclusion that light must also have a non-zero 'inertia' associated with the acceleration perpendicular to its velocity.
We should mention in passing that this notion of (at least some) particles possessing a velocity-dependent (which in our case translates into a directiondependent) inertia echoes the work of Lorentz and subsequent workers initiated in 1899 [25].The basis of his argument was the application of the nonrelativistic formula p = mv in the relativistic domain, which we now know is incorrect.Nevertheless, his influence at the turn of the century, prior to Einstein's introduction of special relativity, was significant enough to attract attention from the physics community to the broader question of the meaning of mass.In fact, his 1904 paper 'Electromagnetic Phenomena in a System Moving With Any Velocity Less Than That of Light' [26] introduced the "longitudinal" and "transverse" electromagnetic masses of the electron.This view was eventually supplanted by Einstein's moving observer's inertia, γm, however, so the notion of a direction-dependent inertia eventually subsided, though some of his concepts are still being considered today (see, e.g., ref. [27]).
In this paper, we have demonstrated that a direction-dependent inertia may not be out of the question after all, at least for photons.Without question, these particles have no inertia in the longitudinal direction, but they evidently do resist acceleration in the transverse direction.This inertia is apparently proportional (or even equal) to the gravitational mass inferred from the photon's energy but, at this stage, we have no idea what produces it.The speed of light is always c, however, so photons are always unbound within our gravitational horizon in the context of FLRW.This 'transverse' inertia thus carries no energy, and is therefore absent from the expression yielding the photon's total energy budget.
In summary, then, we have argued that gravitational mass, m g , is-in all cases--a surrogate for the particle's total energy which, in the context of Einstein's theory, is the actual source of spacetime curvature.Inertia no longer appears to be an intrinsic property but, rather, is an emergent feature due to several different mechanisms.For particles exhibiting a resistance to acceleration in their longitudinal direction of motion, this inertia appears to be proportional to m g .This conclusion is supported by the interpretation that rest-mass energy is a binding energy within our gravitational (or Hubble) horizon in the context of the FLRW cosmic spacetime.A consideration of particle motion near this horizon shows that the particle's energy approaches E esc = qc, with a momentum q = m i c, and Newton's law of universal gravitation in the low-velocity limit then shows that m g → m i (where the equality ensues with an appropriate choice of G).Photons exhibit an inertia transverse to their motion, and the implied inertial mass equals their gravitational mass, consistent with the Principle of Equivalence.But this inertia has no impact on their longitudinal motion, so it does not affect their energy budget, which is entirely kinetic.
We may have generated more questions than answers with this discussion, but hopefully in a manner that encourages further attempts at uncovering the true, physical relationship between a particle's inertial and gravitational mass(es).