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The Asymptotic Form of the Energy Density of

arXiv:hep-ph/9305303v1 24 May 1993IP-ASTP-19-93May, 1993The Asymptotic Form of the Energy Density ofWeakly Interacting Particlesin a Static, Spherical GeometryAchilles D. SpeliotopoulosInstitute of PhysicsAcademia SinicaNankang, Taipei, Taiwan 11529AbstractThe asymptotic form of the energy density for a gas of particles surroundinga sphere of mass M and radius R is studied using Einstein’s equations. Itis shown that if the pressure of the gas p varies linearly with the energydensity ρ for small ρ, then ρ ∼1/r2 for large r.PACS numbers: 04.40.+c, 04.20.Jb, 98.62.Gq, 04.90.+eBitnet address: PHADS@TWNAS886

In the Schwarzchild solution of Einstein’s equations for a static, spheri-cal geometry [1] the spacetime surrounding the sphere of mass M and radiusR is taken to be empty and free of particles. The real world is certainly notso clean and simple, however and a more realistic model would also includea gas of particles which surround the sphere.

With the addition of theseparticles the question then becomes what affect, if any, they will have onthe geometry of the spacetime. At first glance this would seem to be animpossible question to answer.

Not only must an equation of state for theparticles be known, this equation of state must also have been calculated inthe presence of the very gravitational field that it itself determines. Whatwe shall find, however, is that due to the restrictions that Einstein’s equa-tions themselves put on the energy density of the particles, the asymptoticbehavior of this spacetime will be surprisingly simple to determine.In this paper we shall study some of the asymptotic properties of astatic, spherical geometry filled with a gas of particles.

To be specific, thesystem we shall be considering consists of a sphere with mass M and radiusR which is surrounded by an arbitrary gas of particles. We shall not needto specify the type of particles surrounding the sphere, nor shall we needto fix their temperature or density.

All that we shall require is that theyhave an equation of state which has certain properties. For convenience wehave also assumed that the sphere has not undergone complete gravitationalcollapse into a black hole and that the system as a whole is in thermodynamicequilibrium.Moreover, unlike the usual treatment of matter in generalrelativity, we shall not a prior`ı confine the particles to be within a sphereof any fixed radius, but will instead let the system itself determine how farinto the spacetime the particles will spread.Although we have used the word “gas” to describe the particles, this isonly a matter of convenience.

The particles are not only allowed to interact2

with gravity, thereby determining the geometry of the spacetime, but alsowith each other. Since the most stringent constraint on the energy density,and thus the geometry of the spacetime, comes not from the equation ofstate of the particles, but rather from Einstein’s equations themselves, allthat we need require is that the equation of state satisfy a few physicallyreasonable requirements.

In the end we shall find that if the particles in-teract among themselves to any significant extent, then the spacetime willbe asymptotically flat. The energy density of the particles will die offfasterthan 1/r2 and we may charactorize the particles as being confined within asphere of a certain radius which will require a specific choice of the equationof state to determine precisely.

If the particles are very weakly interacting,on the other hand, then we shall find that the spacetime is not asymptoti-cally flat. The energy density will die offas 1/r2 for large r and we cannotcharactorize the particles as being confined within a sphere of any fixed ra-dius.

The affect of the particles on the geometry of the spacetime is insteadall pervasive.We begin with the most general form of the metric for a static, sphericalgeometry [2]ds2 = −fdt2 + hdr2 + r2dθ2 + r2 sin2 θdφ2 ,where f and h are unknown functions of r only. We are only interested inthe structure of the spacetime for r > R and shall always assume that thisholds.

As usual, we shall take the average energy momentum tensor for theparticles to be of the form ⟨Tµν⟩= ρuµuν +p(gµν +uµuν) where uµ is a unitvelocity vector which lies in the direction of the timelike Killing vector forthe system, while ρ and p are the energy density and pressure, respectively,of the particles surrounding the sphere. The two Einstein’s equations we3

shall find of use are [2]8πρ = h′rh2 + 1r21 −1h,(1a)8πp = f ′rfh −1r21 −1h,(1b)where the primes denote derivatives with respect to r and we are using unitsin which G = c = 1. From the conservation equation ∇µ⟨Tµν⟩= 0 we alsohave the Tolman-Oppenheimer-Volkoff(TOV) equation [2] for hydrostaticequilibriumdpdr + 12 (p + ρ) 1fdfdr = 0 .

(2)At this point we shall assume that we are given an equation of state forthe particles which we shall write asp = w(ρ)ρ ,(3)where w(ρ) is a functional of ρ. It is further assumed that this equationof state includes the rest mass of the particles and was calculated in thepresence of the gravitational field.As such both p and ρ must also befunctions of r. We next make the anzatz that w(ρ) itself has no explicitr dependence, but is instead dependent on r only through its dependenceon ρ.

We justify this anzatz with the following. First, we know that inMinkowski spacetime the equation of state may always be written in theform given in eq.

(3). As the system is in thermodynamic equilibrium, andas there are no external fields present, any r dependence in w(ρ) must thenbe due to the curvature of the spacetime.

Second, log f may, under certaincircumstances, be interpreted as being twice the gravitation potential andis therefore a measure of the “gravitational force” on the particles. Third,we note that all explicit r dependence in the TOV equation can be canceledand one can instead consider it as a differential equation determining p in4

terms of f. In fact, any dependence of p on r is due to f. Finally, thegeometry of the spacetime, namely f and h, is ultimately determined bythe energy density and pressure of the particles. Since the equation of stateon curved spacetimes must also reduce to the equation of state calculatedin Minkowski spacetimes in the limit where f →1 and h →1, we wouldtherefore expect w(ρ) to depend on r only implicitly through ρ.The equation of state will in general be very complicated, if it can becalculated at all.

Fortunately, we shall not need any specific form of eq. (3),but rather that it have a couple of physically reasonable properties whichwe would expect from any physical system.

First, the pressure p must befinite as ρ →0, meaning that the gas of particles cannot become infinitelystiffas the energy density of the particles is reduced to zero. Actually, weshall need the more stringent requirement thatlimρ→0 w(ρ) ≡w0 ,exists and is a finite.

This means, in particular, that for small ρ, p ∼ρ1+n forn > 0. We justify this requirement with the observation that if the particleswere consisted only of photons, then the pressure must be one third theenergy density.

This holds even in curved spacetimes. Since the photonsdo not interact among themselves, we have at least one example of a gasof non-interacting particles for which the pressure varies linearly with theenergy density.

We would therefore expect that if the particles do interactwith one another, then the pressure should vary at least linearly with theenergy density, if not stronger. We would not expect the pressure to have aweaker power law dependence on the energy density than a linear one.With the Einstein’s equations, and the TOV equation, we have nowthree differential equations determining ρ, f, and h. (Since the equation ofstate is given, p is determined as soon as ρ is.) We shall then, presumeably,5

need a set of three initial conditions to determine the system completely.The initial condition for ρ is straightforward; we need only take take its valueat the surface of the sphere ρR ≡ρ(R). The initial conditions for f and h,on the other hand, are much more difficult to determine.

Fortunately, weare only interested in the asymptotic nature of the spacetime and we shalllater find that the asymptotic forms of both ρ and h are independent oftheir initial conditions. Whatever initial condition we choose for h is thusimmaterial for our purposes.

f, on the other hand, does depend on initialconditions; not only on its own, but also on ρR as well. Since, however, wemay always rescale the time coordinate, its dependence on its own initialcondition is not extremely relevant and we may formally take it to be f(R).Because the equation of state may be quite complicated, the solutionof eqs.

(1) for all r is not trivial to find. In, however, the asymptotic limitwhere r →∞the situation simplifies dramatically.

To see how this happens,we integrate eq. (1a) to formally give1h = 1 −2m0r−8πrZ rRρ(¯r)¯r2d¯r ,for r > R where m0 is an integration constant.Since ρ is not knownexplicitly, this would seem to be of little use.

Note, however, that h > 0for all r > R. Consequently, ρ →0 as r →∞. In fact, ρ must die offatleast as fast as 1/r2.

Thus, because we are only interested in the asymptoticbehavior of the spacetime, only the behavior of the equation of state whenρ →0, namely w0, will be relevant in our analysis. Keeping this in mind,we next make use of the equation of state in eq.

(2) to obtain1fdfdr = −21 + w(ρ)ρdwdρ + w(ρ) 1ρdρdr . (4)We then defineρ =∆4πr2 ,6

and note that ∆either vanishes as r →∞or else approaches a constantvalue.The important point is that ∆must be finite as r →∞.Thendefining h−1 = 1 −2K, eqs. (1) become∆=dKdy + K ,(5a)∆w0 = −w0(1 −2K)1 + w0 1∆d∆dy −2−K ,(5b)for large y where y ≡log(r/r0) for some r0.

Solutions to these equationshave two different behaviors depending upon whether or not w0 vanishes.Case 1: w0 = 0From eq. (5b), K →0 as r →∞.

Consequently from eq. (5a), ∆→0as well.

This means that the energy density, and consequently the pressure,must decrease faster than 1/r2 for large r.Then from eq. (4), f →ckwhere ck is a constant which can be set to 1 by suitably rescaling the timecoordinate.

Consequently, if w0 = 0, then f →1 and h →1 as r →∞andthe spacetime is asymptotically flat.Case 2: w0 ̸= 0This case is much more interesting. We write eqs.

(5) as a set of twocoupled, non-linear differential equationsdKdy =∆−K ,(6a)d∆dy = −1 + w0w0∆1 −2K∆w0 +1 + 5w01 + w0K −2w01 + w0. (6b)Fortunately, these two equations are autonomous, meaning they have noexplicit y dependence.

Obtaining asymptotic solutions to eqs. (6) is thenstraightforward and involves looking for fixed points (∆a, Ka) of eqs.

(6)where the derivatives of both ∆and K vanish. (See [3].

We caution thereader that, depending on the literature, the term “critical point” is oftenused instead of “fixed point”.) From eqs.

(6) one of these fixed points occurs7

at ∆a = 0 = Ka, which is the w0 = 0 case once again. A second fixed pointoccurs at∆a = Ka =2w0(1 + w0)2 + 4w0.If we next expand eqs.

(6) about this fixed point, thenddyδ∆δK=−2w01+w0−2(1+5w0)(1+w0)21−1 δ∆δK,(7)where δ∆= ∆−∆a and δK = K −Ka. The asymptotic behavior of thesolutions to eqs.

(6) depends on the eigenvalues λ± of this matrix. Writtingλ± = −η ± iϕ, we find thatη = 121 +2w01 + w0,ϕ = 12(7 + 42w0 −w20)1/21 + w0.Since ρ ≥0 for all r, ∆a must be positive.

This condition, combined withthe requirement that h−1 = 1 −2Ka > 0 for all r, means that w0 > 0.Moreover, because p ≤ρ/3, w0 ≤1/3, and it is then straightforward to seethat ϕ will always be a real number. Consequently, this fixed point is stableand all solutions to eqs.

(6) will eventually spiral counterclockwise into thisfixed point in the ∆-K plane no matter what their initial conditions wereoriginally. We can see this explicitly by solving eq.

(7) to giveK(r) ≈∆a(1 + A rr0−ηsinϕ log rr0),∆(r) ≈∆a(1 + A rr0−η "ϕ cosϕ log rr0+ (1 −η) sinϕ log rr0 #),for large r. A, and r0 are constants which depend on the specific equationof state and initial condition for ρ and p. Notice also that because 0

From eq. (4) we can solve for f in terms of ρ.

In the asymptotic limitwe find that f ≈krq where q = 4w0/(1 + w0),k =f(R)4πρR∆aq/2exp(2 ρR dwdρ |ρR + wR1 + wR!−q2+ 2Z 10x log x d2dx2xw′(x) + w(x)1 + w(x)dx)and wR ≡w(ρR). Since 0 < w0 ≤1/3, 0 < q ≤1.

Consequently, for large rthe metric becomesds2 = −krqdt2 +11 −2Kadr2 + r2dθ2 + r2 sin2 θdφ2 ,and we can see explicitly that this spacetime is not asymptotically flat.To summerize, we have shown that if the particles interact among them-selves to such an extent that w0 = 0, then a static, spherical spacetime con-taining these particles is asymptotically flat, as expected. Moreover, we seethat the energy density of these particles must necessarily decrease fasterthan 1/r2; most probably very much faster.

As the Schwarzchild geometryis also asymptotically flat and involves a mass density which is confinedwithin a sphere of definite radius, we may therefore charactorize these par-ticles as effectively being confined within a sphere of a certain radius whichwill require a specific equation of state to determine. The affect of theseparticles on the curvature of the spacetime, like the affect of the mass M inthe Schwarzchild solution, eventually dies away and the spacetime asymp-totically approaches the Minkowski spacetime.

When, on the other hand,the particles interact with each other so weakly that w0 ̸= 0, then theirenergy density decreases as 1/r2 and their affect on the curvature of thespacetime is correspondingly long range. They cannot be charactorized asbeing contained within a sphere of any definite radius and are instead spreadthroughout the spacetime.

In fact, we find that f ∼rq while h approaches9

a constant as r →∞and a spacetime filled with these weakly interactingparticles is not asymptotically flat.What is of even greater interest is the universal nature of the energydensity in the asymptotic limitρa = ∆ac44πr2G ,(8)where we have replaced the correct factors of c and G.(The subscripta denotes the asymptotic limit.) Notice that the form of ρa is the sameirrespective of the mass and radius of the sphere, irrespective of any “initialcondition” for ρ at r = R, and irrespective of any form that the equation ofstate may take so long as w0 ̸= 0.

In fact, the only property of the particlesthat it does depend upon is w0, a dimensionless number. From dimensionalarguments, if the particles surrounding the sphere are massless, then w0must simply be a number.

It cannot even depend on the temperature of thegas, since, aside from the Planck mass, there is no other length scale onecan use to construct a w0. If, on the other hand, the particles have a massm, then w0 is either once again a number, or else is a function of the ratioT/m, where T is the temperature of the gas.Notice that although this universality extends to h, it does not extendto include f. For large r, f ≈krq and although q depends only on w0,k depends not only on f(R) but also on ρR.

Consequently, the motion oftest particles in this spacetime will always be dependent the specific choiceof initial conditions for f and ρ, and thus on the detail properties of theparticles and of the spherical mass.The question remains as to how one may go about calculating w0.For certain cases this is trivial.Suppose Tµν is the energy momentumoperator for the particles such that its equilibrium average is ⟨Tµν⟩=ρuµuν + p(gµν + uµuν).If this energy momentum operator is traceless,10

as it is for a gas of pure photons, then ⟨T µµ ⟩= 0 and p = ρ/3. Conse-quently, any system which has a traceless energy momentum operator hasa w(ρ) = w0 = 1/3.

Calculating w0 for other systems, on the other hand,is a much more formidable task, although at first glance not an impossibleone. Because w0 is determined when the energy density of the particles isvanishingly small, this suggests that when calculating w0 one may, as a firstapproximation, neglect the affect of the particles on the curvature of thespacetime and may instead treat them as test particles.

One can then inprinciple use the vacuum solutions of Einstein’s equations as a backgroundfield and calculate the equation of state for the system using methods de-scribed in [4] .We end this paper with a couple of observations. First, the 1/r2 be-havior in eq.

(8) is precisely what one would expect for the energy densityof “dark matter” based on Newtonian gravity [5]−[7]. Unfortunately, whenone has gone to such a large r such that eq.

(8) holds, one also finds thatthe spacetime no longer close to being Minkowskian. It is instead extremelycurved (see [8]).

Second, the form of ρa is quite peculiar in that it containsno explicit length scale for r. The only length scale which can be constructedsolely from universal constants, however, is the Planck length l2pl = ¯hG/c3.If we use it, then we can write eq. (8) asρa = ρpl∆a4πlplr2(9)where ρpl = c7/(¯hG2) is the Planck energy density.

Although ρpl is verylarge, eq. (9) is valid only at very large r. Since lpl ∼10−33cm, this ensuresthat the size of ρ will always be physically reasonable.AcknowledgementsThis work is supported by the National Science Council of the Republicof China under contract number NSC 82-0208-M-001-086.11

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

Tech., 424-434 (1916). [2] R. M. Wald, General Relativity, Chapter 6, (The University of ChicagoPress, Chicago, 1984).

[3] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods forScientists and Engineers, Chapter 4, section 4.4, (McGraw-Hill BookCompany, New York, 1978). [4] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space,Chapters 1,2 (Cambridge University Press, Cambridge 1982).

[5] V. C. Rubin, W. K. Ford, and N. Thonnard, Ap. J.

Letters, 225 L107(1978). [6] V. Trimble, Ann.

Rev. Astron.

and Astrophys., 25 425 (1987). [7] E. W. Kolb and M. S. Turner, The Early Universe, Chapter 1 (Addison-Wesley Publishing Company, Inc., New York, 1990).

E. W. Kolb and M.S. Turner, The Early Universe: Reprints, Chapter 1 (Addison-WesleyPublishing Company, Inc., New York, 1988).

[8] A. D. Speliotopoulos, submitted to Phys. Rev.

Lett., April 1993.12

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