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The Energy Density of a Gas of Photons

arXiv:hep-ph/9305302v1 24 May 1993IP-ASTP-12-93May, 1993The Energy Density of a Gas of PhotonsSurrounding a Spherical Mass M at a Non-Zero TemperatureAchilles D. SpeliotopoulosInstitute of PhysicsAcademia SinicaNankang, Taipei, Taiwan 11529AbstractThe equations determining the energy density ρ of a gas of photons in ther-modynamic equilibrium with a spherical mass M at a non-zero temperatureTs > 0 is derived from Einstein’s equations. It is found that for large r,ρ ∼1/r2 where the proportionality constant is a fundamental constant andis the same for all spherical masses at all temperatures.Bitnet address: PHADS@TWNAS886

§1. IntroductionIn the standard Schwarzchild solution [1] of Einstein’s equations fora static, spherical geometry the spacetime outside of a sphere of mass Mand radius R is taken to be a vacuum; empty and free of particles.

Thisis certainly true if the sphere is at absolute zero temperature. When, how-ever, the sphere has a non-zero temperature, then we would expect, on aphysically basis, a gas of thermal photons to be present.

If the system is inthermodynamic equilibrium, then the spacetime surrounding the sphere willnot be empty, but will instead be filled with blackbody radiation. As thesethermal photons also have a certain non-zero energy, we would expect thepresence of this blackbody radiation to also contribute towards determiningthe geometry of the spacetime.For any physically reasonable temperatures the energy density of thephotons is very small in comparison to the mass density of the sphere and thepresence of the photons is ignored in the usual Schwarzchild analysis.

Thisapproximation is certainly valid near the spherical body but what happenswhen one is very far away from the sphere? One should remember thatthe Schwarzchild solution is asymptotically flat.

Due to the mass M of thebody being confined within its radius R, as one goes further and furtheraway from the sphere the affect of its mass M on the curvature of spacetimebecomes less and less.The blackbody radiation, on the other hand, isunconfined. It extends from the surface of the sphere to fill the rest of thespacetime.

Although we would expect the energy density of the photonsto also decrease as one moves further and further away from the sphere,the important question is how fast it will do so. Let us, for the moment,consider a sphere of radius r > R and the total amount of energy withinit.

When r is near R, we would expect that the fraction ǫM of this energy2

which is due to the mass M will be much greater than ǫγ, the fraction of thetotal energy which is due to the photons. Consequently, we would expectthe mass M to be the dominant factor in determining the geometry of thespacetime within r and would expect the Schwarzchild solutions to be validin this region.

As, however, r increases, ǫM decreases since the mass M ofthe body is fixed, while ǫγ increases since the blackbody radiation extendsthroughout the spacetime. If the energy density of the photons decreasessufficiently rapidly so that ǫγ ≪ǫM for all r, then the mass M will always bethe dominant factor in determining the geometry of the spacetime.

Althoughthe Schwarzchild solution would be modified somewhat by the presence ofthe photons, we would not expect these modifications to be very drastic. Inparticular, the spacetime should still be asymptotically flat.

If, on the otherhand, the energy density does not decrease rapidly enough and ǫγ ≫ǫM forr larger than some r0, then we would expect that in this region of spacetimeit is the photons which will determine the spacetime geometry. In particular,we would expect the solutions to Einstein’s equations in this regime to bevery much different from the Schwarzchild solution.

The spacetime may noteven be asymptotically flat.In this paper we shall study some of the affects of non-zero temperatureson the static, spherically symmetric solutions of Einstein’s equations. Thesystem we shall be considering consists of a spherical body with a massM and a radius R which, due to the sphere being at a temperature Ts >0, is surrounded by blackbody radiation.

The system as a whole will beassumed to be in thermodynamic equilibrium with the body serving as theheat reservior for the system. In particular, this means that the sphericalbody is assumed to be in thermodynamic equilibrium with the photonssurrounding it.

It is moreover assumed that the body has not undergone3

complete gravitational collapse into a blackhole, is non-rotating, and is notelectrically charged. Nor shall there be any other massive objects present inthis spacetime.

The only difference between this system and the one studiedby Schwarzchild is the presence of a non-zero temperature for the sphere.Our aim in this paper is two fold. First, we shall derive a set of coupleddifferential equations which will determine the total energy density of thephotons.

As these are non-linear equations, we shall not be able to solvethem analytically. We shall, nonetheless, be able to obtain both asymptoticr →∞as well as r →R solutions to them.

Note, however, that we shallbe determining the photon’s total energy density in the gravitational field,and not its blackbody spectrum. Second, we shall show that the geometryof this spacetime differs drastically from the Schwarzchild geometry at larger.In fact with the presence of the photons the spacetime is no longerasymptotically flat.In order to avoid working with non-equilibrium systems, we have as-sumed that the mass M is in thermodynamic equilibrium with the blackbodyradiation surrounding it.

Unfortunately, even equilibrium quantum statisti-cal mechanics on curved spacetimes has yet to be satisfactorily formulated.The closest that we have come to a complete formalism is that given in [2].For various reasons, however, it will be difficult, if not impossible, to analyzethe system in the manner outlined therein and it is fortunate that all that weshall need is the Tolman-Oppenheimer-Volkoffequation for hydrostatic equi-librium. This, combined with the observation that the energy-momentumtensor of a gas of pure photons is traceless, shall be sufficient to determinethe energy density almost uniquely.

This method of deriving the energydensity has the added advantage of not only taking into account the affectsof the curvature of spacetime on the photons, but also the reciprocal affect4

of the photons on the spacetime curvature. Photons are not treated as testparticles in our analysis.

In this, and other, ways our method differs fromthat discribed in [2]. The only difficulty that we shall encounter is whenwe try to identify the temperature of the system.

Since we do not have anestablished formalism which will automatically do this for us, we shall, inthe end, have to rely on other physical arguments to do so.In the formalism given in [2], as indeed in most treatments of ther-modynamics in general relativity, the temperature of the system is takento be a constant throughout the spacetime. This is, it would seem to us,an oversimplification, for the following reason.

For massless particles thetemperature of the system at equilibrium may be interpreted physically asthe most probable energy that any one particle in the statistical ensemblemay have. (The case of massive particles is much more complicated andwill not be considered here.) In a gravitational field this should include notonly its kinetic energy, but also its gravitational energy as well.

The twocannot be seperated covariantly. Indeed, it is known that the frequency of aphoton, and thus its energy, when measured at different points in a gravita-tional field will either be “redshifted” or “blueshifted” with respect to oneanother.

We would on this basis expect that temperature too should varyfrom point to point on the manifold.§2. The Hydrostatic EquationWe begin with an N dimensional manifold M with a metric gµν whichhas a signiture of (−, +, +, +).

Greek indices shall run from 0 to N −1and the summation convention is used throughout. It is further assumedthat gµν is static, meaning that there exists a timelike Killing vector ξµ forthe system.

We shall also assume that the system contains only one heatreservior.5

Next, let Tµν be an energy momentum tensor operator defined on M.We shall, in a semi-classical approximation, treat gµν as a background, clas-sical field. We next denote the thermodynamic average of Tµν by ⟨Tµν⟩.We shall not need a specific definition of this average, but rather that itsatisfy a few basic properties that we would expect from any equilibriumthermodynamic average.

First, it should be “time independent”, meaningthat£ξ⟨Tµν⟩= 0,(1)where £ξ denotes the Lie derivative along the ξµ direction. It is for thisreason that we required M to have a timelike Killing vector.

Physically, itmeans that the background field gµν cannot change with respect to time thetotal energy contained in the matter fields so that the system as a wholecan be in equilibrium. Second, we require that the average be anomaly-free∇λ⟨Tµν⟩= ⟨∇λTµν⟩,(2)where ∇λ denotes the covariant derivative.

Third, after the average is takenthe energy momentum tensor should have the form ⟨Tµν⟩= ρuµuν +p(gµν +uµuν) were ρ and p are the proper energy density and pressure, respectively,and uµ is a unit velocity vector which must lie in the ξµ direction if (1) isto hold. We shall make this dependency explicit by writting⟨Tµν⟩= −ρξµξνξ2+ pgµν −ξµξνξ2.

(3)Then (1) requires thatξλ∇λρ = 0,ξλ∇λp = 0,(4)meaning that ρ and p are functions of vectors lying in the N −1 dimensionalhypersurface perpendicular to ξµ and/or ξ2 = ξµξµ.Finally, since the6

system is conserved, ∇µ⟨Tµν⟩= 0 and∇µp + (ρ + p)∇µ|ξ||ξ|= 0,(5)where |ξ| ≡p−ξ2 and we have used (3) and Killing’s equation∇µξν + ∇νξµ = 0,(6)in obtaining (5). This is the generalization of the Tolman-Oppenheimer-Volkoff[3] equation for hydrostatic equilibrium to general, static spacetimesand it reduces to the usual hydrostatic equation in the case of sphericallysymmetric spacetimes.We next consider a region of spacetime which contains only photons.In this region4πTµν = FµλF λν −14gµνFαβF αβ,(7)where Fµν is the field strength tensor.

This energy-momentum tensor op-erator is traceless T µµ = 0 in four spacetime dimensions.Consequently,⟨T µµ ⟩= 0 and for photons ρ = (N −1)p. The hydrostatic equation (5) isnow trivial to solve yieldingρ =σ|ξ|N ,(8)where σ is an arbitrary constant. The average energy momentum tensor forphotons is thus given by⟨Tµν⟩=1(N −1)σ|ξ|Ngµν −N ξµξνξ2.

(9)To determine ⟨Tµν⟩completely one must first determine ξµ for the manifold.As the presence of the photons will also affect the curvature of the spacetime,determining ξµ ultimately involves solving Einstein’s equations using (9) asthe source term1(N −1)σ|ξ|Ngµν −N ξµξνξ2= 18π Rµν,(10)7

where Rµν is the Ricci tensor and we are using geometrized units in whichG = c = kB = ¯h = 1. The Rµµ term is absent since ⟨Tµν⟩is traceless.The task now is to interpret (8) physically.

Let us consider, for themoment, the case of Minkowski spacetime and enclose the system in a very,very large box which is connected to a heat reservior at a fixed temperature.Killing’s equation is now a simple partial differential equation and we maychoose a coordinate system in which its solution for a timelike Killing vectoris ξfµ = (−βf, 0, 0, 0) where βf is a constant and the superscript f remindsus that this is the Minkowski spacetime.From (8) we find that in fourdimensions,ρf =σ(βf)4 ,(11)which, if we interpret 1/βf as the temperature of the heat reservior, is justBoltzmann’s law for photons. Then σ = π2k4B/(15¯h3c3) is identified withthe blackbody radiation constant.Using the Minkowski spacetime case as motivation, we shall tentativelyidentify the temperature T of the system asT = 1|ξ|,(12)in general, static spacetimes and see whether or not this will make physicalsense.

First, we note that although T does very with position, it is “timeindependent”, namely£ξT = T 32 ξµ∇µξ2 = 0,(13)as one would expect for a system in equilibrium.Second, we note thatvariations in T are due solely to the gravitational field. In fact, the temper-ature at various points of the manifold is related to one another by just theredshift factor,T(x)T(x′) =−ξ2(x′)−ξ2(x)1/2,(14)8

which is precisely what one would expect from time dilation and the fre-quency shift of photons in a gravitational field. Finally, we note that timelikeKilling vectors ξ in non-rotating systems are determined by the geometryof the spacetime only up to an overall constant, as can be seen explicitly inKilling’s equation (6).

Consequently, we have the freedom to attach an over-all constant to the Killing vector which we can then identify as the inversetemperature of the heat reservior. Indeed, from (14) we see that relativetemperatures between two points on the manifold are determined solely bygeometry and to determine an absolute temperature requires choosing a ref-erence point on the manifold from which we can measure all subsequenttemperatures with respect to.

The most natural reference point to chooseis the heat reservior of the system and to measure all other temperaturesbased on its value.The procedure for determining the energy density of thermal photons inany static geometry is now clear. First we solve Killing’s equations up to anoverall constant for a timelike Killing vector ξµ in terms of the componantsgµν of the metric of the spacetime.

To determine the overall constant, weuse as a boundary condition for ξµ the temperature Thr of the heat reserviorof the system by evaluating (12) at a point xhr on the surface of the heatreservior. It is required that the heat reservior have a constant temperaturethroughout its surface.

Finally, Einstein’s equations (10) are solved withthe appropriate boundary conditions to determine gµν.These boundaryconditions are usually also given at the heat reservior. The energy densityρ of the photons and the geometry of the spacetime are thus determined.§3 Spherical GeometryWe shall now attempt to solve (10) for a static, spherical geometry.Specifically, we shall consider a non-rotating spherical body of mass M, and9

radius R which at its surface has a temperature Ts. The spherical body willserve as the heat reservior for the system.

The most general metric whichis static and spherically symmetric has the form [3]ds2 = −fdt2 + hdr2 + r2dθ2 + r2 sin2 θdφ2,(15)where f and h are functions of r only. Killing’s equation is now straightfor-ward to solve giving ξµ = (−ckf(r), 0, 0, 0), and ξ2 = −c2kf(r) for a timelikeKilling vector.

ck is an arbitrary constant which is determined by evaluating(12) at the surface of the reservior: 1/ck = Tsf(R)1/2.As for the boundary conditions for f and h, we choose them in thefollowing manner. Take a point r > R and consider the amount of energycontained within a sphere of this radius which is due to the photons.

Clearly,because of the presence of the photons, the geometry of the spacetime withinr will be different from the Schwarzchild geometry. Let us, however, taker →R+ so that the amount of energy contained in the sphere due to thephotons gradually decreases.Their affect on the geometry of spacetimemust also do so correspondingly and just outside the body we would expectthe geometry of the manifold to be the same as that of the Schwarzchildgeometry.

Consequently, we choose as our boundary conditionslimr→R+ f(r) = 1 −2MR,limr→R+1h(r) = 1 −2MR . (16)Einstein’s equations given by (10) are8πρ0f 2= 1fhf ′′2 −f ′4f ′f + h′h+ f ′r,8πρ03f 2 = 1fh−f ′′2+ f ′4f ′f + h′h+ fh′r,8πρ03f 2 = −12rhf ′f + h′h+ h′rh2 + 1r21 −1h,(17)10

where ρ0 = σT 4s f 2(R) and the primes denote derivatives with respect to r.They may be reduced to two coupled, nonlinear differential equations32πρ03f 2= 1rhf ′f + h′h8πρ0f 2= h′rh2 + 1r21 −1h. (18)It is doubtful that these equations can be solved analytically.There is,nonetheless, one general feature that we can determine from these equationsalone.

Since we require h > 0 for all r > R, from (18) we find that f is amonotonically increasing function of r. Consequently, ρ = ρ0/f 2 is a mono-tonically decreasing function of r. The energy density of the photon gas isat its largest at the surface of the sphere and decreases monotonically as onegoes further and further away from it. This is exactly what we would haveexpected physically.

Notice also that since T = 1/|ξ| = Ts[f(R)/f(r)]1/2,the temperature of the photon gas also decreases monotonically with r.We shall now obtain approximate solutions to (18) in the small and larger limits. We first consider the near field solutions when r is near R and writef ≈1 −Γ, and h−1 ≈1 −Λ for Γ, Λ ≪1.

For the boundary conditions tobe consistant with this approximation, we shall require 2M/R ≪1. Thenignoring terms quadratic in Γ and Λ, we find that1h(r) ≈1 −2Mr(1 + 4πρ0r33M"1 −Rr4#),f(r) ≈1 −2Mr(1 −4πρ0r33M"1 −Rr2#),ρ(r) = ρ0(1 −2Mr+ 8π3 ρ0r2"1 −Rr2#)−2.

(19)Notice, however, that Γ and Λ increases quadratically with r and at somepoint rm they will no longer be valid.To estimate rm, we enforce the11

condition that Γ(r) ≪1 for r < rm. This gives4πρ03r3m ≈M,(20)as a determining equation for rm.

The near field solutions (19) are valid aslong as the total energy of the photons confined in a sphere of radius r ismuch less than the mass of the spherical body itself.As for the asymptotic, r →∞solutions, we obtain them in the followingmanner. First we defineρ =∆4πr2 ,1h = 1 −2K.

(21)Then (18) may be written asd∆dy = −2∆1 −2K23∆+ 4K −1dKdy = ∆−K ,(22)where y = log(r/r0) for some r0. These differential equations have a fixpoint at∆a = Ka = 314(23)where the derivatives of ∆and K vanish.

Then perturbing about this fixpoint,ddy∆−∆aK −Ka=−1/2−31−1 ∆−∆aK −Ka. (24)This matrix has eigenvaluesλ± = −34 ± i√474(25)so that the fix point (23) is stable.

Physically, this means that no matterwhat initial conditions are chosen for ∆and K, both functions will eventu-ally flow to the fix point (23) at large enough r. We can see this explicitly12

by solving (24)K(r) = 314(1 + Ar0r3/4sin √474log rr0!,)∆(r) = 314(1 + Ar0r3/4"√474cos √474log rr0!+ 14 sin √474log rr0! #).

(26)A, and r0 are constants which require matching boundary conditions that aregiven at small r to determine. As the solutions to (22) for intermediate r arenot known analytically, we are not able to do this explicitly.

Nevertheless,numerical calculations, and a formal perturbative solution of (18) treatingρ0 as the perturbation indicates that r0 ∼1/√ρ0 as long as 1/√ρ0 > rm.We would therefore expect (26) to hold whenever r ≫1/√ρ0. Note alsothat solutions to (22) approach the fix point (23) very slowly; basicly asr−3/4.In the very large r limit solutions to (22) asymptotically approachesfa = (56πρ0/3)1/2r, and ha = 7/4 where we have used the subscript a todenote the asymptotic solutions.

Thus the metric at large r isds2 = −56πρ03 12r dt2 + 74dr2 + r2dθ2 + r2 sin2 θdφ2. (27)and we can now see explicitly that this spacetime is not asymptotically flat.Next, we find that the asymptotic energy density isρa =3c456πGr2 ,(28)where we have replaced the correct factors of c and G. At very large r,the energy density decreases as 1/r2 with a proportionality constant whichis an universal number and is independent of either the mass M or thetemperature Ts of the sphere.

This is once again the consequence of (22)13

being a non-linear differential equation and having a stable, non-zero fixpoint. The temperature of the photon gas in the asymptotic limit is thenkBT(r) = 45¯h3c756π3Gr21/4= mplc2 4556π3l2plr2!1/4,(29)where mpl = (¯hc/G)1/2 is the Planck mass and lpl = mplG/c2 is the Plancklength and we have explicitly used σ = π2k4B/(15¯h3c3).

Although mpl isvery large, one should remember that (29) is valid only when r is also quitelarge. As lpl ∼10−33cm, this ensures that kBT(r) will always be very muchsmaller than the Planck energy.

In fact, (18) gaurentees that for r > R,T(r) ≤Ts, the temperaure at the surface of the sphere. We should alsomention that the asymptotic solutions are themselves solutions of (22) atany r, as can be seen explicitly.

They do not, unfortunately, satisfy thecorrect boundary conditions.We can also calculate the total average energy of the photonsE =Zd3x√hρ =Z ∞R√hdmdr dr,(30)from (18). Since m ∼r for large r, we do not expect this E to be finite.It should instead diverge linearly with r. This divergence is much milder,however, than in the case of flat spacetime where the total average energydiverges as the volume of the system.t§4.

DiscussionThe spacetime outside of a sphere with temperature Ts can thus bedivided into three regions. In the near field region, r ≪rm and the solutionsto Einstein’s equations are given by (19).

The geometry of the spacetimein this region is dominated by the mass M at r = 0 and the presence ofthe photons will not have an significant affect on it. In the intermediatefield region, rm < r < 1/ρo and the total energy contained in the thermal14

photons is now comparable to the mass of the sphere. Both the photonsand the mass M together will determine the geometry of the spacetime.

Inthe far field region, r ≫1/√ρ0 and the asymptotic solutions (26) are nowvalid. It is now the photons which are dominant over the mass M.We have in this paper considered only systems which are in thermo-dynamic equilirbium.

In particular, this means that the sphere must be inthermodynamic equilibrium with the gas of photons surrounding it. As thespacetime that we have been considering consisted of only the mass M andthe photons, this is equivilant to saying the sphere must be in thermody-namic equilibrium with the rest of its universe.

Since our universe is filledwith with the cosmic microwave background radiation which is at a temper-ature of ∼3o K, for a physical body to be in thermodynamic equilibriumwith the rest of our universe it must also be at a compareable temperature.There are very few actual physical bodies which are at such low tempera-tures. Moreover, because the mass M is in equilibrium with the photonssurrounding it, the amount of energy radiating away from the sphere mustexactly be balanced by the amount of energy impacting on the sphere bythe photon gas surrounding it.

It is for this reason that the energy density isdependent only on the geometry of the spacetime, and is why the definitionof the temperature (12) makes sense. It is also the reason why the intensityof the emitted radiation from the sphere does not have the charactoristic1/r2 behavior as one would naively expect.15

AcknowledgementsI would like to thank K.-W. Ng for many helpful discussions while thispaper was being written. This work is supported by the National ScienceCouncil of the Republic of China under contract number NSC 81-0208-M-001-78.PACS numbers: 04.20.-q, 04.20.Jb, 04.40.+c, 5.90.+mREFERENCES[1] K. Schwarzchild, Kl.

Math.-Phys. Tech., 189-196 (1916), Kl.

Math.-Phys. Tech., 424-434 (1916).

[2] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space,Chapters 1,2 (Cambridge University Press, Cambridge 1982). [3] R. M. Wald, General Relativity, Chapter 6, (The University of ChicagoPress, Chicago, 1984).16

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