Relativistic Stars with a Linear Equation of State: Analogy with Classical Isothermal Spheres and Black Holes

A&A 483, 673–698 (2008) Astronomy DOI: 10.1051/0004-6361:20078287 & c ESO 2008 Astrophysics

Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes

P. H. Chavanis

Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France e-mail: [email protected] Received 16 July 2007 / Accepted 10 February 2008

ABSTRACT

We complete our previous investigations concerning the structure and the stability of “isothermal” spheres in general relativity. This concerns objects that are described by a linear equation of state, P = q, so that the pressure is proportional to the energy density. In the Newtonian limit q → 0, this returns the classical isothermal equation of state. We specifically consider a self-gravitating radiation (q = 1/3), the core of neutron stars (q = 1/3), and a gas of baryons interacting through a vector meson field (q = 1). Inspired by recent works, we study how the thermodynamical parameters (entropy, temperature, baryon number, mass-energy, etc.) scale with the size of the object and find unusual behaviours due to the non-extensivity of the system. We compare these scaling laws with the area scaling of the black hole entropy. We also determine the domain of validity of these scaling laws by calculating the critical radius (for a given central density) above which relativistic stars described by a linear equation of state become dynamically unstable. For photon stars (self-gravitating radiation), we show that the criteria of dynamical and thermodynamical stability coincide. Considering finite spheres, we find that the mass and entropy present damped oscillations as a function of the central density. We obtain an upper bound for the entropy S and the mass-energy M above which there is no equilibrium state. We give the critical value of the central density corresponding to the first mass peak, above which the series of equilibria becomes unstable. We also determine the deviation from the Stefan-Boltzmann law due to self-gravity and plot the corresponding caloric curve. It presents a striking spiraling behaviour like the caloric curve of isothermal spheres in Newtonian gravity. We extend our results to d-dimensional spheres and show that the oscillations of mass-versus-central density disappear above a critical dimension dcrit(q). For Newtonian isothermal stars (q → 0), we recover the critical dimension dcrit = 10. For the stiffest stars (q = 1), we find dcrit = 9 and for a self-gravitating radiation (q = 1/d)wefinddcrit = 9.96404372... very close to 10. Finally, we give simple analytical solutions of relativistic isothermal spheres 2 in two-dimensional gravity. Interestingly, unbounded configurations exist for a unique mass Mc = c /(8G). Key words. hydrodynamics – instabilities – relativity – stars: neutron – gravitation – black hole physics

1. Introduction and the entropy scale as T ∼ N4/3, E ∼ N7/3,andS ∼ N (Hertel & Thirring 1971; Chavanis 2002d). These exotic scalings (with Self-gravitating systems have a very strange thermodynamics respect to conventional thermodynamics) come from the long- due to the attractive long-range, unshielded nature of the gravi- range action of gravity, which makes the system spatially inho- tational potential (Padmanabhan 1990; Chavanis 2006c). In par- mogeneous and renders the energy and the entropy non additive. ticular, they can display negative specific heats, inequivalence On the other hand, in the context of black hole physics, of statistical ensembles, and phase transitions associated with Bekenstein (1973) and Hawking (1975) have shown that the en- gravitational collapse. Furthermore, since these systems are spa- tropy of a black hole scales as its area S ∼ R2 and that its temper- tially inhomogeneous and their energy non-additive, the usual ature (measured from an observer at infinity) scales as T ∼ 1/R. thermodynamic limit (N → +∞ with N/V fixed) is clearly irrel- These results largely remain mysterious and are usually con- evant and must be reconsidered. As a result, the thermodynami- sidered to reflect a fundamental description of spacetime at the cal parameters have unusual scalings with respect to the number quantum level. In order to explain the area scaling of the entropy, of particles or system size (see Sect. 7.1 of Chavanis 2006c). researchers have invoked holography (Bousso 2002), quantum For classical self-gravitating isothermal spheres, the mass scales gravity, string theory, entanglement entropy (Srednicki 1993), like the size of the object M ∼ R, which is the exact scaling and brick-wall models (’t Hooft 1985). of the singular isothermal sphere. As a result, the classical ther- In recent works, Banks et al. (2002)andPesci(2007) con- modynamic limit (CTL) for self-gravitating systems corresponds sidered perfect fluids in general relativity with a linear equation to N → +∞ in such a way that N/V1/3 is fixed (de Vega & of state P = q and found that for the stiffest case compatible Sanchez 2002). It is then found that the temperature, the en- with causality (q = 1), the entropy scales as the area S ∼ R2 and ergy, and the entropy scale like T ∼ 1, E ∼ N,andS ∼ N the temperature as T ∼ 1/R. Here, these unusual scalings are (Chavanis & Rieutord 2003). When quantum effects are taken a consequence of the non-extensivity of the system due to the into account, the mass scales with the radius as M ∼ R−3,which action of gravity. As mentioned above, exotic scalings are also is the exact mass-radius relation for nonrelativistic white dwarf encountered in Newtonian gravity. However, that purely classi- stars. As a result, the quantum thermodynamic limit (QTL) for cal systems can exhibit scaling laws analogous to black holes is self-gravitating fermions corresponds to N → +∞ in such a way intriguing and deserves further investigation. The study of these that NV is fixed. It is then found that the temperature, the energy, systems should evidence what, in the area scaling of the black

Article published by EDP Sciences 674 P. H. Chavanis: Relativistic stars with a linear equation of state hole entropy, is the reflect of a fundamental theory of quantum regular spheres with very large radii, are dynamically unstable. gravity and what mainly stems from the non-extensive nature of Secondly, we consider general relativistic systems of astrophys- the system. ical interest described by a linear equation of state for which In a preceding paper (Chavanis 2002b), denoted Paper I, we all the parameters entering into the scaling laws (including the studied the structure and the stability of spherically symmetric multiplicative factor) can be calculated explicitly. In Sect. 3,we relativistic stars with a linear equation of state P = q.These consider a self-gravitating radiation (photon star) corresponding = systems are sometimes called “isothermal” (by an abuse of lan- to q 1/3. This problem was first studied by Sorkin et al. (1981) guage) because in the Newtonian limit q → 0, they reduce to in a seminal paper. For this system, we note that the entropy is = classical isothermal spheres described by the Emden equation proportional to the particle number (S λN) and the energy = 2 (Chandrasekhar 1972). Just as for their Newtonian counterparts, is proportional to the mass (E Mc ). Therefore, the maxi- they must be enclosed within a box (of radius R)soastoprevent mization of N at fixed M (dynamical stability) is equivalent to their evaporation and make their total mass finite. For a given the maximization of S at fixed E (thermodynamical stability). We immediately conclude that the conditions of dynamical and volume, we showed the existence of a critical mass-energy Mc above which there is no equilibrium state. In that case the sys- thermodynamical stability coincide. We also note that the dy- = tem is expected to collapse and form a black hole. Furthermore, namical variational problem δN µδM can be rewritten in the = −1 considering the series of equilibria, we showed that the mass vs. form of a thermodynamical variational problem δS T δE where T = c2/(λµ) is to be interpreted as a temperature (we central density relation M(0) presents a series of damped oscillations. The configurations of hydrostatic equilibrium become shall see that T corresponds to the temperature at infinity T0 given by the Tolman relation). We can therefore apply the results unstable above a certain value of the central density 0,c corresponding to the maximum mass (first peak). Secondary extrema of Paper I to that system. For a fixed box radius, we find that = in the series of equilibria correspond to new modes of instability. there exists a maximum mass Mc 0.2465...MPR/LP (where We also numerically observed that the baryon number vs. cen- MP and LP are the Planck mass and the Planck length) and a maximum entropy S = 0.6217...k (R/L )3/2 above which there tral density N(0) presents extrema at the same locations as the c B P is no equilibrium state. Moreover, the series of equilibria M( ) mass M(0), but we were not able to explain this observation. 0 and S ( ) present damped oscillations and become unstable (dy- A first motivation of the present paper is to clarify this re- 0 − namically and thermodynamically) above a critical central den- sult. In Sects. 2.1 2.3 and in Appendix D.1, applying the gen- sity = 0.439...c4/(GR2) corresponding to the first mass peak. eral argument of Weinberg (1972) to a system described by a 0,c ff New modes of instability appear at each secondary peaks. These linear equation of state, we show that the Oppenheimer-Volko stability results, which have been proved analytically in Paper I equation of hydrostatic equilibrium in general relativity can be and which are developed in the present paper, complete and sim- obtained by extremizing the baryon number N[] at fixed mass- = plify the early analysis of Sorkin et al. (1981). For a fixed central energy M[]. From the condition δN µδM (where µ is a density, the photon star becomes unstable above a critical ra- Lagrange multiplier), it becomes clear that extrema of M(0)in 4 1/2 dius Rc = 0.663...(c /G0) . For large radii, the entropy scales the series of equilibria correspond to extrema of N(0), as we ob- as S ∼ R3/2.Thisdiffers from the black hole scaling, but this served numerically. We also argue in Sect. 2.1 and Appendix D.2 is consistent with the Bekenstein inequality (Bekenstein 1981). that only maxima of baryon number N[]atfixedmassM[]are We also determine the deviation from the Stefan-Boltzmann law dynamically stable. More precisely, the maximization of N[]at due to self-gravity and plot the corresponding caloric curve. In fixed M[] forms a criterion of formal nonlinear dynamical sta- Sect. 4.1, we find similar results for a gas of completely degener- bility for a perfect fluid with respect to the Einstein equations. ate ultra-relativistic fermions at T = 0 modelling the core of neu- In Paper I, we considered the linear dynamical stability prob- tron stars. In particular, the baryon number scales as N ∼ R3/2 lem and showed that the system becomes unstable after the first for R → +∞. We also consider, in Sect. 4.2, a gas of baryons mass peak in the series of equilibria M(0). In Appendix D.2 of interacting through a vector meson field. This model, introduced the present paper, we show that the first mass peak also corre- by Zel’dovich (1962), is described by a linear equation of state sponds to the point at which the hydrostatic configurations cease with q = 1. For this system the baryon number scales as the area to be maxima of the baryon number N[] (at fixed mass) and be- N ∼ R2, analogously to the black hole entropy. come saddle points. Therefore, the conditions of linear and nonlinear dynamical stability coincide. Similar results are obtained A last motivation of our paper is to extend our results to a for barotropic spheres described by the Euler-Poisson system in space of dimension d. Such generalization is quite common in Newtonian gravity (see Chavanis 2006a). black hole physics and quantum gravity, where is it advocated that extra-dimensions can appear at the micro-scales, an idea Another motivation of our paper, inspired by the works of stemming from the Kaluza-Klein theory. In Newtonian grav- Banks et al. (2002)andPesci(2007), is to investigate the scaling ity, the influence of the dimension of space on the laws of behaviour of the thermodynamical parameters (mass-energy M, physics has only been considered recently (Sire & Chavanis baryon number N,entropyS , temperature T, etc.) with the sys- 2002; Chavanis & Sire 2004; Chavanis 2004, 2006a,b, 2007a). → +∞ tem size R. In particular, for R , it is found in Sects. 2.4 We have shown that the structure of the system is highly depen- 3q+1 − 2q and 2.5 that M ∼ R, N ∼ S ∼ R q+1 ,andT ∼ R q+1 .These dent on the dimensionality of space and that the problem is very scaling laws were implicit in our preceding paper, but they de- rich because it involves several critical dimensions. Therefore, it serve to be emphasised. We complete previous studies in two is natural to complete this type of investigations. In continuity respects. In Sects. 2.6 and 2.7, we use the stability criteria ob- with our study of classical isothermal spheres (Sire & Chavanis tained in Paper I and Appendix D to determine the domain of 2002), we show in Sect. 5 that the oscillations in the mass-central validity of these scaling laws precisely. We show that, for a fixed density profile disappear above a critical dimension dcrit(q)de- central density 0, the system becomes unstable above a criti- pending on the index q. Above this dimension, the configura- cal radius Rc so that the scaling laws only hold approximately tions are stable for any central density, contrary to the case close to this maximum radius. Pure scaling-law profiles, corre- d < dcrit. This implies that the pure scaling law profiles corresponding to singular spheres with infinite central energy or to sponding to the singular solution are now stable. For Newtonian P. H. Chavanis: Relativistic stars with a linear equation of state 675 isothermal stars (q → 0) we recover the critical dimension steady states of the Einstein equations. However, these first- dcrit = 10 (Sire & Chavanis 2002). For the stiffest stars (q = 1), order variations tell nothing about the stability of the system. we find dcrit = 9 and for a self-gravitating radiation (q = 1/d), Only maxima of N at fixed mass M are dynamically stable, so we find dcrit = 9.96404372..., very close to 10. The oscillations we must consider the sign of the second-order variations of N to exist for any q ∈ [0, 1] when d < 9, and they cease to exist for settle the stability of the system. In Appendix D, we consider the any q ∈ [0, 1] when d ≥ 10. We also note that the dimension case of a linear equation of state P = q (see below) so that the d = 2 is critical so that the results obtained for d > 2 do not baryon number is a functional N[] of the energy density alone. pass to the limit d → 2. In two-dimensional gravity, we obtain In that case, we can evaluate the second-order variations of N in Sect. 6 analytical expressions for the density profile of rela- and study the stability of the system. tivistic isothermal spheres for any q. Unexpectedly, they have a The maximization problem (4)or(D.1) is similar to the mini- finite radius for q 1 (and the density profile decreases like mization of the energy functional W[ρ]atfixedmassM[ρ] = M a Gaussian for q = 1) contrary to their Newtonian analogue for a self-gravitating barotropic gas in Newtonian gravity. It is where the density decreases as r−4 (see, e.g., Sire & Chavanis known that this minimization problem provides a criterion of 2 2002). Furthermore, they exist at a unique mass Mc = c /8G. formal nonlinear dynamical stability for the Euler-Poisson sys- Similarly, classical isothermal spheres in d = 2 exist at a unique tem (see Chavanis 2006a). Similarly, we expect that the max- mass Mc = 4kBT/Gm for a given temperature, or equivalently imization problem (4)or(D.1) provides a criterion of formal at a unique temperature kBTc = GMm/4 for a given mass (see, nonlinear dynamical stability for the Einstein equations. e.g., Chavanis 2007b). Because of that, we may expect that these structures are only marginally stable. 2.2. The equation of state To close the system of Eqs. (1)−(2), we need to specify an equa- 2. Relativistic stars tion of state relating the pressure P to the energy density .The In this section, we complete the results obtained in Paper I con- first law of thermodynamics can be expressed as cerning the structure and the stability of relativistic stars with a 1 s linear equation of state. d = −Pd + Td , (6) n n n 2.1. The equations governing equilibrium where n is the baryon number density and s the entropy den- The condition of hydrostatic equilibrium for a spherically sym- sity in the rest frame. We assume in the following that the metric perfect fluid in general relativity is described by the term Td(s/n) can be neglected. In that case, the first law of ther- Oppenheimer-Volkoff (1939) equation modynamics reduces to P + 2GM(r) dP + P GM(r) 4πG d = dn. (7) 1 − = − + Pr , (1) n c2r dr c2 r2 c2 We now assume a “gamma law” equation of state of the form where P(r) is the pressure, (r) is the energy density, and P = q with q = γ − 1. (8) 1 r M r = r πr2 r ( ) 2 ( )4 d (2) c 0 In that case, Eq. (7) can be integrated at once and we find the polytropic relation is the mass-energy contained within the sphere of radius r.IfR denotes the radius of the configuration, the total mass-energy is P = Knγ, (9) M = M(R). We also need the baryon number N. It is obtained by multiplying the baryon number density n(r) by the proper vol- where K is a constant. Combining Eqs. (8)and(9), we find that ume element eλ(r)/24πr2dr and integrating over the whole con- the baryon density is related to the energy density by figuration. This leads to an expression of the form q 1/γ − = 1/γ R 2GM(r) 1/2 n . (10) N = n(r) 1 − 4πr2 dr. (3) K rc2 0 The velocity of sound is given by (dP/d)1/2c = q1/2c so that If we restrict ourselves to spherically symmetric configurations the principle of causality requires q ≤ 1. In the following, we and isentropic perturbations, it can be shown that the maximiza- consider 0 ≤ q ≤ 1(i.e.1≤ γ ≤ 2). tion problem There are two situations where the term Td(s/n) can be neglected. The first situation is when T = 0. This is the situation { | = } Max N[,n] M[] M fixed , (4) that prevails in the core of neutron stars where the thermal en- determines stationary solutions of the Einstein equations that are ergy is much smaller than the Fermi energy so that the neutrons are completely degenerate. The second situation is when the en- dynamically stable (Weinberg 1972). The critical points of the = baryon number N at fixed mass M for isentropic perturbations tropy per baryon s/n λ is a constant. This is the case in su- solve the variational problem permassive stars where convection keeps the star stirred up and produces a uniform entropy distribution. This is also the case for δN − µδM = 0, (5) a gas of self-gravitating photons where the pressure is entirely due to radiation (see Sect. 3). For a self-gravitating radiation, the where µ is a Lagrange multiplier enforcing the conservation Gibbs-Duhem relation of mass. This variational principle leads to the Oppenheimer- Vo l ko ff equation (Weinberg 1972); therefore, it determines = −P + Ts+ µn, (11) 676 P. H. Chavanis: Relativistic stars with a linear equation of state simplifies itself since the chemical potential for photons vanishes of Paper I for different values of q. It has to be noted that the (µ = 0). In that case, we obtain asymptotic behaviour for ξ → +∞ of the regular solutions behave like the singular sphere (20). = − + − P Ts. (12) Since the energy density decreases as r 2 for r → +∞, rela- Combining the foregoing relations, we find that tivistic stars with a linear equation of state have an infinite mass like their Newtonian analogues (see, e.g., Chavanis 2002a). This 1 s(r) λq T(r) γ−1 reflects the tendency of the system to evaporate. To circumvent = = ffi n(r) + , (13) this di culty, we shall enclose the system within a box of ra- λ 1 q K dius R. Then, the solution of Eqs. (17)−(18) must be terminated γ at the normalized box radius λq T(r) γ−1 P(r) = q(r) = K · (14) 1/2 + 4πG0(1 + q) 1 q K α = R. (21) c4q These relations are valid more generally for any system with s/n = λ constant and µ = 0. This is the case for example in It can be noted at that point that α is a measure of the central the cosmology developed by Banks & Fischler (2001) based on density 0 for a given box radius R, or a measure of the radius R the holographic principle. Central to their discussion is a perfect for a given central density 0. The energy density contrast is a fluid with equation of state P = and T ∝ s ∝ 1/2. monotonic function of α given by R≡ 0 = eψ(α). (22) 2.3. The general relativistic Emden equation (R) Considering the equation of state (8), we introduce the dimen- Finally, it is convenient in the analysis to introduce the Milne sionless variables ξ, ψ and M(ξ) by the relations (Chandrasekhar variables (Chandrasekhar 1942) 1972; Chavanis 2002b) −ψ = ξe = u ,vξψ . (23) 4 1/2 ψ −ψ c q = 0e , r = ξ, (15) 4πG0(1 + q) The description of the solutions of the general relativistic Emden equation in the Milne plane is given in Paper I. and 4π c4q 3/2 M(r) = 0 M(ξ). (16) 2.4. Singular solution c2 4πG (1 + q) 0 We first give the values of the thermodynamical parameters In terms of these variables, the Oppenheimer-Volkoff corresponding to the singular solution of the general relativis- Eqs. (1)−(2) can be reduced to the following dimension- tic Emden equation. They present exact scaling laws that share less forms some analogies with the scaling laws for the entropy and temperature of black holes. We stress from the beginning, however, that 2q M(ξ) dψ M(ξ) 1 − = + qξe−ψ, (17) the singular solutions are unstable. The question of the stability 1 + q ξ dξ ξ2 of regular isothermal spheres is considered in Sect. 2.6. dM The singular energy and pressure profiles are = ξ2e−ψ. (18) dξ 4 = P(r) = qQc −2 (r) + r . (24) In the Newtonian limit q → 0, these equations reduce q 4πG(1 q) to the Emden equation (Chandrasekhar 1942). Therefore, Using relation (10), we find that the baryon number density and − Eqs. (17) (18) represent the general relativistic equivalent of the entropy profile are given by the Emden equation. It is in this sense that relativistic stars de- 1 2 4 q+1 scribed by a linear equation of state resemble classical isother- s(r) q Qc − 2 mal spheres. Just as in Newtonian gravity, there exists a singular n(r) = = r q+1 . (25) λ 4πGK(1 + q) solution

− Q 2(1 + q) From Eq. (14), the temperature profile is e ψs = , where Q = · (19) 2 + 2 + q ξ (1 q) 4q 2 4 q+1 1 + q q Qc − 2q T(r) = K r q+1 . (26) The singular energy density profile is λq 4πGK(1 + q) qQc4 Finally, using Eqs. (106)−(107) of Paper I, we find that the func- = r−2. (20) s 4πG(1 + q) tions ν(r)andλ(r) determining the metric are given by

−λ r ν r 4q This solution was first found by Klein (1947) and re-discovered e ( ) = 1 − pQ, e ( ) = Ar q+1 , (27) by numerous researchers including Misner & Zapolsky (1964), Chandrasekhar (1972), etc. Considering now the regular solu- where A is a constant and − tions of Eqs. (17) (18), we can always suppose that 0 repre- 2q p = · (28) sents the energy density at the centre of the configuration. Then, 1 + q Eqs. (17)−(18) must be solved with the boundary conditions ψ(0) = ψ (0) = 0. The corresponding solutions must be com- Comparing Eqs. (26)and(27), we check explicitly that the puted numerically and some density profiles are given in Fig. 8 Tolman relation (see Appendix B) is satisfied. P. H. Chavanis: Relativistic stars with a linear equation of state 677

Suppose now that the singular solution is terminated by a box Newtonian gravity, isothermal spheres also display negative spe- with radius R. According to Eqs. (2)and(24), the total mass is cific heats but for a completely different reason (Lynden-Bell & Wood 1968). According to the Virial theorem 2K + W = 0, we pQc2 ≡ + ∼− ∼−3 M = R. (29) have E K W K 2 NkBT (where K is the kinetic energy 2G and W the potential energy) leading to negative specific heats = = − 3 Therefore, we find that the mass M scales linearly with the ra- C dE/dT 2 NkB < 0. In that case, the energy becomes dius R. This is the same scaling as for the singular isothermal more and more negative as temperature increases. Therefore, the origin of negative specific heats for classical isothermal spheres sphere in Newtonian gravity (see, e.g., Chavanis 2002a,b). This ff is also the scaling entering in the Schwarzschild relation (A.3). and black holes is di erent. On the other hand, according to Eqs. (3)and(25), the baryon number and the total entropy can be written 2.6. The mass-energy + S 1 q 1 −1/2 We now consider regular isothermal spheres and discuss the sta- N = = Q 1+q (1 − pQ) λ 3q + 1 bility of the system along the series of equilibria and the domain 2 4 1/γ of validity of the scaling laws. It is shown in Paper I that the nor- q c 3q+1 ×4π R q+1 . (30) malized mass-energy is related to the parameter α by the relation 4πGK(1 + q) 2GM pv0(1 − qu0) Finally, using the Tolman relation (B.7)andEq.(26), the tem- χ ≡ = , (32) Rc2 1 + pv perature measured by an observer at infinity is given by 0

q where u0 = u(α)andv0 = v(α) are the values of the Milne vari- + + 2 4 q 1 2q 1 q q Qc / − ables at the box radius R. The foregoing relation can be rewritten T = K (1 − pQ)1 2R q+1 . (31) 0 λq 4πGK(1 + q) 2GM = χ(α). (33) The surface temperature T(R) has the same scaling with R as T0, Rc2 differing only in the factor (1 − pQ)1/2 according to the Tolman relation. It defines the series of equilibria parametrized by α going from 0to+∞. For a fixed box radius R, we can write 1/2 2.5. Analogy with the black hole entropy 4πG(1 + q) α = a1/2, a = R. 0 with 4 (34) Some interesting analogies between relativistic stars described c q by a linear equation of state and black hole thermodynamics 1/2 Therefore, α = a is a measure of the central density so that have been discussed previously by Banks et al. (2002)andPesci 0 Eq. (33) determines the relation between the total mass M and (2007). Let us briefly develop their arguments in connection to the central density . The corresponding curve M( ) is plot- the present study. 0 0 tedinFigs.1 and 2. It presents damped oscillations around an When P = , the velocity of sound (dP/d)1/2c is equal to asymptote at M = M ≡ χ Rc2/(2G) corresponding to the sin- the velocity of light so that Eq. (8) with q = 1isthestiffest s s gular solution (29). For α → 0, the density is almost uniform equation of state compatible with the principle of causality. In and that case, we see from Eqs. (30)and(31) that the entropy scales like the area S ∼ R2 and that the temperature scales like the in- 2q χ(α) ∼ α2. (35) verse of the radius T ∼ 1/R. As noticed by Banks et al. (2002), 3(1 + q) this is similar to the scaling of the Bekenstein-Hawking entropy and temperature for black holes (see Appendix A). We also note For α → +∞, the density profile tends to the singular sphere (20) from Eq. (25) that the entropy profile scales as s(r) ∼ r−1.Asdis- and cussed by Pesci (2007), this shows that the total entropy is due to → = the contribution of the whole volume. Therefore, for relativistic χ(α) χs pQ. (36) stars with a linear equation of state, the area scaling of the en- The mass is maximum for a certain value of the central density ff tropy is a volume e ect resulting from the inhomogeneity of the corresponding to α = α (first peak). The maximum mass is ff c system. This di ers from the result of Oppenheim (2002)who given by considered self-gravitating classical systems reaching the above scaling laws for entropy and temperature while approaching the 2GM c = χ . (37) Schwarzschild radius. In this limit, he showed that all entropy Rc2 c lies on the surface and the area scaling of the entropy is due to The values of αc, χc,andχs depend on q.Forq = 1/3, we have external layers. ======More generally, for a linear equation of state of the form (8), αc 4.7, χc 0.493, p 1/2, Q 6/7, χs 3/7andforq 1, 3q+1 we have αc = 4.05, χc = 0.544, p = 1, Q = 1/2, χs = 1/2. There the entropy scales as S ∼ R q+1 .For0 ≤ q ≤ 1, the expo- is no equilibrium state with M > Mc.ForM < Mc,thestable nent is always smaller than 2 (the area scaling law) obtained configurations correspond to α<α,i.e.tosufficiently small = c for q 1. This is in agreement with the Bekenstein inequality central densities. The configurations with α>α in the series of ≤ = 2 c S S BH kBπ(R/LP) (where LP is the Planck length). Finally, equilibria (i.e. those located after the first mass peak) are dynam- ∼ ∼ we note that the energy scales like E M R and that the tem- ically unstable. New modes of instability appear at each mass − 2q − q+1 perature scales like T ∼ R q+1 . This leads to E ∼ T 2q implying peak. These results are proved analytically in Paper I where a negative specific heats C = dE/dT < 0 (see Appendix A for linear dynamical stability analysis of box-confined systems with black holes and Sect. 3.5 for the self-gravitating radiation). The a linear equation of state is performed. This study is completed mass-energy is positive but it decreases with the temperature. In in Appendix D where it is shown that the first mass peak is also 678 P. H. Chavanis: Relativistic stars with a linear equation of state

0.6 10

Fixed R M q=1 Non−extensive scaling c M q=1 s ε (M=R) Fixed 0 α Unstable 5 c 0.4 Unstable 2 Stable 0 M ,R Extensive scaling c c 3 ln(M)+Cst 2GM/Rc (M=R ) 0.2 −5 Stable

0 −10 −5 0 5 10 −5 0 5 10 1/2 ln(bR) ln(aε0 ) Fig. 1. Mass as a function of the central density for a fixed box radius. Fig. 3. Mass as a function of the box radius for a fixed central density. This corresponds to the curve χ(α). For a system evolving at fixed vol- This corresponds to the curve αχ(α). ume, the system becomes unstable at the first mass peak. More generally, the mass-radius relation for a fixed central density is given by q=1 0.54 First peak Fixed R 2GbM = χ(bR)bR. (42) c2

Unstable This relation is plotted in Fig. 3. The scaling law M ∼ R is 2 0.52 only valid in the limit R → +∞. However, as we have seen, the system becomes unstable for R > Rc ≡ αc/b,i.e.foramass

2GM/Rc 2 M > Mc ≡ c Rcχc/2G. Therefore, the solutions exhibiting a pure scaling law profile (like the singular isothermal sphere or 0.5 Stable the regular isothermal spheres with R → +∞) are dynamically unstable. However, for stable configurations close to the critical Secondary peak radius Rc, we observe in Fig. 3 that the linear scaling holds ap- 0.48 proximately, so that the results given in Sect. 2.4 are correct in −202468 ε 1/2 that sense. ln(a 0 ) Fig. 2. Mass as a function of the central density for a fixed box radius showing the secondary peaks. 2.7. The baryon number Using Eq. (3) and introducing the dimensionless variables de- the point at which the steady states pass from maxima of N[](at fined previously, we find that the baryon number is given by fixed mass) to saddle points of N[] (at fixed mass). Therefore, α −1/2 N 1 − ψ(ξ) M(ξ) the system loses its stability at the first mass peak in agreement ∆ ≡ = 1+q − 2 3q+1 e 1 p ξ dξ, (43) with the Poincaré turning-point argument (Katz 2003; Chavanis N∗ α 1+q 0 ξ 2006c) applied to the series of equilibria M( ). 0 with For a fixed central density 0, we can write 1 1/2 2 4 q+1 + 3 q c 4πG0(1 q) N∗ = 4πR · (44) α = bR, with b = · (38) 2 + c4q 4πGKR (1 q) Therefore, α = bR is a measure of the system size R so that the The foregoing relation can be rewritten equation N =∆(α). (45) c2R N∗ M = χ(α) (39) 2G = 1/2 For a fixed box radius, using α a0 , this equation determines determines the relation between the total mass and the radius. In the relation between the baryon number and the central density. the limit α → 0, the system is homogeneous and, using Eq. (35), The corresponding curve is plotted in Fig. 4. It presents damped we obtain the usual scaling for an extensive system oscillations around an asymptote at N = Ns ≡ ∆sN∗ correspond- 4π ing to the singular solution (30). According to Eq. (5)theex- M = 0 R3, R → . 2 ( 0) (40) trema of N(α) occur for the same values of α as the extrema of 3 c M(α) in the series of equilibria. This leads to angular points in If we now consider the limit α → +∞ corresponding to the sin- the curve N(M), as shown in Fig. 5.Forα → 0, corresponding gular sphere, using Eq. (36), we get the non-extensive scaling to an almost uniform distribution, we have 2 c R 1 2 M = χ , (R → +∞). (41) ∆(α) ∼ α q+1 , (46) s 2G 3 P. H. Chavanis: Relativistic stars with a linear equation of state 679

0.6

Nc q=1 Ns ε 10 Fixed 0 Non−extensive α Unstable 2 c scaling (N=R ) 0.4

Unstable * Stable N/N ln(N)+Cst 0 Extensive scaling 0.2 3 q=1 (N=R ) Fixed R Stable

−10 0 −5 0 5 10 −5 0 5 10 ln(bR) ε 1/2 ln(a 0 ) Fig. 6. Baryon number as a function of the box radius for a ﬁxed central Fig. 4. Baryon number as a function of the central density for a ﬁxed 3q+1 box radius. We have plotted ∆(α). density. We have plotted α q+1 ∆(α).

0.6 For a fixed central density, using α = bR, the equation Fixed R q=1 1 2 4 q+ Unstable q c 1 3q+1 =∆ q+1 N (α)4π + R , (49) 0.4 4πGK(1 q) Stable determines the relation between the baryon number and the ra- * dius. In the limit α → 0, the system is homogeneous and, using N/N Eq. (46), we have the usual scaling for an extensive system 0.2 4π N = n R3, (R → 0). (50) 3 0 If we now consider the limit α → +∞, corresponding to the 0 singular sphere, using Eq. (47) we find the non-extensive scaling 0 0.2 0.4 0.6 2GM/Rc2 1 2 4 q+1 q c 3q+1 N =∆4π R q+1 , (R → +∞). (51) Fig. 5. Baryon number as a function of the mass for a fixed box ra- s 4πGK(1 + q) dius. We have plotted ∆(α) as a function of χ(α) so that the curve is parametrized by α.Forq = 1, we note that, asymptotically, ∆(α) χ(α) ∼ = More generally, for a fixed central density, the baryon number is so that the relation M N is linear. We have drawn a line y x for expressed as a function of the radius by comparison. This special behaviour arises only for q = 1. 1 + 1 4πGK(1 + q) q 1 3q+1 3q+1 b q+1 N = (bR) q+1 ∆(bR). (52) and for α → +∞ corresponding to the singular sphere, we have 4π q2c4

3q+1 + q+1 1 q 1 − / This relation is plotted in Fig. 6. The scaling law N ∼ R is ∆(α) → ∆ = Q 1+q (1 − χ ) 1 2. (47) s 3q + 1 s only valid in the limit R → +∞. However, as we have seen, the system becomes unstable for R > Rc ≡ αc/b, i.e. for a baryon = The baryon number is maximum for α αc and its value is number N > Nc ≡ ∆cN∗(Rc). Therefore, the solutions exhibiting a pure scaling law profile (like the singular isothermal sphere Nc or the regular isothermal spheres with R → +∞) are dynami- =∆c. (48) N∗ cally unstable. However, for the stable configurations close to 3q+1 + the critical radius Rc, we note that the scaling N ∼ R q 1 holds The values of αc, ∆c and ∆s depend on q.Forq = 1/3, we have 1/4 approximately, so that the results given in Sect. 2.4 are correct αc = 4.7, ∆c = 0.925, ∆s = (8/21) = 0.7856... and for q = 1, in that sense. Finally, in Fig. 7, we have plotted the curve N(M) we have αc = 4.05, ∆c = 0.546, ∆s = 1/2. There is no equilib- for a fixed central density. For small mass, we have the scaling rium state with N > Nc.ForN < Nc, the stable configurations 3q+1 + correspond to α<αc,i.e.tosufficiently small central densities. N ∼ M and for large mass, we have N ∼ M q 1 . Configurations with α>αc in the series of equilibria are dy- As a final remark, we note that the density contrast (22)isa namically unstable. We can note that, for q = 1, the asymptotic monotonic function of α (behaving like the inverse of the den- values χs = 1/2and∆s = 1/2 coincide. More precisely, the sity profile reported in Fig. 8 of Paper I) that could be used to curves χ(α)and∆(α) turn out to superimpose for α 1. This parametrize the series of equilibria M(α)andN(α) in place of implies that the curve N(M) becomes linear in the region where α. In particular, the critical energy density contrast correspond- α 1, as can be seen in Fig. 5. This behaviour is, however, not ing to αc is Rc = 30.1forq = 1andRc = 22.4forq = 1/3. general and, for q 1, the curves χ(α)and∆(α)aredifferent Configurations with a density contrast greater than these values (see, e.g., Figs. 18 and 19 of Paper I for q = 1/3). are dynamically unstable. 680 P. H. Chavanis: Relativistic stars with a linear equation of state

20 3.2. The Stefan-Boltzmann law q=1 Non−extensive Fixed ε 2 0 scaling (N=M ) We first recall some basic elements of thermodynamics that ap- 10 ply to a pure radiation modeled as a gas of photons. We first ig- ff Unstable nore gravitational e ects. The distribution function of a perfect 0 gas of bosons without interaction is given by the Bose-Einstein statistics

Extensive scaling ln(N)+Cst 1 1 −10 (N=M) f = , (61) 3 β−µ − Stable h e 1

−20 where µ is the chemical potential. For ultra-relativistic particles, we have = pc. On the other hand, for particles without rest mass, like photons, the chemical potential µ = 0. Therefore, the −30 −30 −20 −10 0 10 distribution function of a gas of photons is simply ln(M)+Cst 1 1 Fig. 7. Baryon number as a function of the mass for a fixed central den- = · f βpc (62) 3q+1 h3 e − 1 sity. We have plotted α q+1 ∆(α) as a function of αχ(α) so that the curve is parametrized by α. The particle number, the kinetic energy, and the pressure can be computed from Eqs. (53)−(55). Writing these relations lo- 3. Self-gravitating radiation: photon stars cally and using the properties of the Bose integrals, we obtain the standard results In this section, we apply the preceding results to a radiation that is so intense that self-gravity (in the sense of general relativ- 1 8π π4 P = = (k T)4 , (63) ity) must be taken into account. This is sometimes referred to 3 h3c3 B 90 as “photon stars” (Schmidt & Homann 2000). This problem was first considered by Sorkin et al. (1981). Our work confirms and 8π 3 extends the results of these authors. n = (kBT) ζ(3). (64) h3c3

3.1. Equation of state for ultra-relativistic particles The relation (63) between the energy and the temperature is the Stefan-Boltzmann law. Using the Gibbs-Duhem relation (11) Let us consider a perfect gas of non-interacting particles that can with µ = 0, we directly obtain the entropy be relativistic. We call f (p) the numerical density of particles with impulse p. The particle number, the kinetic energy, and the 5 32π 3 pressure are s = kB (kBT) . (65) 90h3c3 N = V f dp, (53) For a pure radiation, the pressure is related to the photon density by E = V f (p)dp, (54) kin P = Knγ, (66) 1 d P = fp dp, (55) with 3 dp 4 π4 hc where the kinetic energy of a particle is γ = , K = · (67) ⎧ ⎫ 3 90 4 1/3 ⎪ 1/2 ⎪ 8πζ(3) ⎨⎪ p2 ⎬⎪ (p) = mc2 ⎪ 1 + − 1⎪ . (56) ⎩ m2c2 ⎭ On the other hand, the entropy is related to the photon density by

In the ultra-relativistic limit, the kinetic energy of a particle with 4π4 s = λn, with λ = k . (68) impulse p reduces to 90ζ(3) B (p) = pc. (57) If we now consider a gas of self-gravitating photons in general Therefore, the pressure is related to the kinetic energy by relativity (photon stars), the preceding relations still hold lo- 1 E cally. They have the form considered in Sect. 2.2 with q = 1/3. P = kin · (58) 3 V Therefore, a self-gravitating radiation can be studied with the theory developed in Sect. 2. We note that the singular proﬁles of In the local form, this leads to the linear equation of state Sect. 2.4 scale like 1 P = q with q = · (59) − − − 3 P ∝ ∝ r 2, s ∝ n ∝ r 3/2, T ∝ r 1/2. (69) Note that, in a d-dimensional universe (see Sect. 5), the corresponding value of this parameter would be These are also the asymptotic behaviours of the regular proﬁles for r → +∞. The numerical constants in Eqs. (24)−(26) can 1 q = · (60) be obtained explicitly by using the values of K, γ,andλ given d above. P. H. Chavanis: Relativistic stars with a linear equation of state 681

0.6 10

Fixed R q=1/3 Non−extensive scaling 0.5 ε (M=R) q=1/3 M Fixed 0 c Unstable 5

0.4 α Unstable c 2

0.3 0 Extensive scaling Stable 3 ln(M)+Cst 2GM/Rc (M=R ) 0.2

−5 Stable 0.1

0 −10 −5 0 5 10 −5 0 5 10 1/2 ln(aε0 ) ln(bR) Fig. 8. Mass as a function of the central density for a ﬁxed box radius. Fig. 9. Mass as a function of the box radius for a ﬁxed central density.

However, configurations with 3.3. The mass-energy of self-gravitating photons 4 1/2 We now apply the general results of Sect. 2.6 to the specific c R > Rc ≡ αc , (77) context of a self-gravitating radiation. The consideration of an 16πG0 explicit physical example allows us to determine the numeri- α>α = . cal value of the multiplicative constants appearing in the scal- corresponding to c 4 7, are dynamically and thermody- ing laws. It is convenient to introduce the Planck length and the namically unstable. This corresponds to Planck mass, 1 c8 1/2 M > M ≡ χ α · c c c 3 (78) G 1/2 c 1/2 2 16πG 0 L = , M = · (70) P c3 P G 3.4. The entropy of self-gravitating photons In terms of these parameters, the relation (33) can be rewritten The total entropy of the gas of photons is M 1 R = −1/2 χ(α) , (71) R 2GM(r) MP 2 LP S = s r − πr2 r. ( ) 1 2 4 d (79) 0 rc with Using Eq. (68), we find that 16πG 1/2 α = 0 R. (72) S = λN, (80) c4 where N is the number of photons (3). Using the results of For a given box radius, the mass-central density relation M(0) Sect. 2.7, we obtain presents damped oscillations around an asymptote R 3/2 1 R S = A∆(α)kB , (81) Ms = χs MP , (73) LP 2 LP with the numerical constant χ = / with s 3 7 corresponding to the singular solution (see Fig. 8). 1/4 There is no equilibrium state with a mass larger than 1 8π3 A = = 0.672188... (82) 3 15 = 1 R Mc χc MP , (74) For a given box radius, the entropy vs. central density S ( ) 2 LP 0 presents damped oscillations around an asymptote where χ = 0.493 corresponding to α = 4.7. On the other hand, c c 3/2 the configurations with central density R S s = A∆skB (83) LP 4 2 ≡ c αc 0 >0,c , (75) ∆ = 1/4 16πG R with s (8/21) , corresponding to the singular solution (see Fig. 10). There is no equilibrium state with an entropy greater corresponding to α>αc are dynamically and thermodynami- than cally unstable (see Sect. 3.5). 3/2 For a given central density, the mass-radius relation M(R) R S c = A∆ckB , (84) is represented in Fig. 9. For small radii, we have the extensive LP scaling M ∼ R3 and for large radii, we get the scaling law where ∆c = 0.925 corresponding to αc = 4.7. The configurations with central density > are dynamically and thermody- M = 1 R → +∞ 0 0,c χs , (R ). (76) namically unstable. Eliminating between Eqs. (71)and(81), MP 2 LP 0 682 P. H. Chavanis: Relativistic stars with a linear equation of state

1 Non−extensive S 3/2 c q=1/3 scaling (S=R ) 10 Fixed ε 0.8 0 α c Unstable

0.6 Unstable *

S/S Stable ln(S)+Cst 0 0.4 Extensive scaling 3 q=1/3 (S=R ) Stable 0.2 Fixed R

−10 0 −5 0 5 10 −5 0 5 10 ln(bR) ε 1/2 ln(a 0 ) Fig. 12. Entropy as a function of the box radius for a ﬁxed central Fig. 10. Entropy as a function of the central density for a ﬁxed box density. radius.

0.94 20

Fixed R Fixed ε 0 Non−extensive scaling 0.89 q=1/3 3/2 10 q=1/3 (S=M )

0.84 Unstable 0 Unstable * 0.79 Stable Extensive scaling S/S ln(S) (S=M) −10 0.74 Stable

−20 0.69

0.64 −30 0.35 0.4 0.45 0.5 −30 −20 −10 0 10 2 2GM/Rc ln(M) Fig. 11. Entropy as a function of mass for a fixed box radius. Fig. 13. Entropy as a function of mass for a fixed central density. we obtain the entropy vs mass curve S (M). This curve is represented in Fig. 11. Since the peaks of entropy coincide with the 3.5. Thermodynamical stability and temperature mass peaks in Figs. 8 and 10, this implies that the entropy vs. mass curve presents angular points at each peak1. However, only We have seen in Sect. 2.1 that the maximization of the particle number N at fixed mass-energy M provides a criterion of the upper part of the curve, corresponding to α<αc,isstable. For a given central density, the entropy vs. radius rela- formal nonlinear dynamical stability. Since the entropy of the tion S (R) is represented in Fig. 12. For small radii, we have the self-gravitating radiation is proportional to the particle num- = extensive scaling S ∼ R3 and, for large radii, we get the scaling ber (S λN) and since the energy is proportional to mass = 2 law (E Mc ), we conclude that the criterion of formal nonlin- 3/2 ear dynamical stability is equivalent to the maximization of the R entropy at fixed energy: S = A∆skB , (R → +∞). (85) LP Max {S [] | E[] = E fixed}, (88) However, configurations with R > Rc are dynamically and thermodynamically unstable. This corresponds to that is to say, to the criterion of thermodynamical stability in c7 3/4 the microcanonical ensemble. This proves very simply that the 3/2 2 S > S c = A∆cα kB · (86) criteria of dynamical and thermodynamical stability coincide . c 16πG2 0 Now, using S = λN and E = Mc2, the variational principle (5) Finally, eliminating R between Eqs. (71)and(81), we obtain the determining the critical points, can be rewritten as entropy vs. mass curve S (M) represented in Fig. 13.Forsmall mass, we have the extensive scaling S ∼ M and for large mass, 1 δS = δE, (89) we get T ∆ 3/2 3/2 s M 2 For isothermal spheres in Newtonian gravity (Chavanis 2006a), we S = 2 A kB · (87) 3/2 have found that the minimization of the energy functional W at fixed χs MP mass (formal nonlinear dynamical stability) is equivalent to the min- 1 This is similar to the entropy vs. energy curve S (E) in Newtonian imization of the Boltzmann free energy FB at fixed mass (thermody- gravity (see Fig. 4 of Chavanis 2002d). namical stability in the canonical ensemble). P. H. Chavanis: Relativistic stars with a linear equation of state 683

1.5 1.5

q=1/3 Fixed R Fixed R q=1/3 T(R)

T(R) 1 1

α T * T T T α c Unstable 0.5 0.5 Stefan−Boltzmann law 4 Stable (M=T )

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 −7 −2 3 8 ε 1/2 M ln(a 0 ) Fig. 15. Temperature as a function of the energy for a fixed box radius. Fig. 14. Temperature as a function of the central density for a fixed box We have plotted θ(α) as a function of χ(α). radius. This curve represents actually the function θ(α). 1 ≡ 2 with T c /(λµ). This can be interpreted as the first principle of Fixed R thermodynamics where T is identified with the thermodynamical 0.8 q=1/3 temperature. Now, using Eqs. (D.7)and(63), we find that T(R) 2GM 1/2 T = T R − , 0.6 T ( ) 1 2 (90) Rc * where T(R) is the surface temperature. Comparing Eq. (90) with S/S 0.4 the Tolman relation (B.7), we observe that T is equal to the temperature measured by an observer at infinity: 0.2 T = T0. (91) Stefan−Boltzmann law (S=T3) Using Eqs. (63), (22), (21)and(33), we find that the temper- 0 ature T is related to the parameter α by the relation 0 0.5 1 1.5 T 1 L 1/2 k T = θ(α)M c2 P , (92) B 3A p R Fig. 16. Entropy as a function of the temperature for a fixed box radius. We have plotted ∆(α) as a function of θ(α). where α1/2 θ(α) = [1 − χ(α)]1/2. (93) central density between Eqs. (71), (81)and(92), we obtain the R(α)1/4 temperature vs. energy curve T(M) and the entropy vs. temper- For a given box radius, Eq. (92) determines the relation T( ) ature curve S (T). For small values of mass or temperature, self- 0 gravity is negligible and we recover the Stefan-Boltzmann law. between the temperature and the central density. This curve ff (see Fig. 14) presents damped oscillations around the temper- For larger values, general relativistic e ects must be taken into 1/4 account and the curves form a spiral. Figures 15 and 16 show ature of the singular solution corresponding to θs = Q (1 − 1/2 1/4 the deviation from the Stefan-Boltzmann law due to general rel- χs) = (96/343) . There is no equilibrium state above a critical temperature ativity. For a system evolving at fixed radius, the series of equilibria is stable (dynamically and thermodynamically in the mi- 1/2 1 2 LP crocanonical ensemble) until the first turning point of mass or k T∗ = θ∗ M c , (94) B 3A p R entropy. The caloric curve of Fig. 15 is strikingly similar to the caloric curves T(E) of Newtonian isothermal spheres (see, e.g., with θ∗ = 0.897 corresponding to the first peak at α∗ = 1.47. Chavanis 2002a). Since α∗ <α, the dynamical and thermodynamical instabil- c For a given central density, the temperature vs. radius ity occurs after the first peak of temperature3. Eliminating the plot T(R) is represented in Fig. 17. For small radii, the temper- 3 = We can wonder whether the turning point of mass-energy M(0)is ature is an intensive variable T const., and for large radii, we associated with a loss of microcanonical stability and the turning point have the scaling law of temperature T( ) is associated with a loss of canonical stability as 0 θ L 1/2 for Newtonian isothermal spheres; see Antonov (1962), Lynden-Bell & k T = s M c2 P , (95) Wood (1968), Katz (1978), Padmanabhan (1990)andChavanis(2002a). B 3A P R Note, however, that the mass-energy M[]isalinear functional of the = 1/4 density , like the mass M[ρ] in Newtonian gravity and contrary to the with θs (96/343) . However, configurations with R > Rc energy E[ρ] in Newtonian gravity. Furthermore, we note that a self- are dynamically and thermodynamically unstable (in the micro- gravitating radiation becomes unstable above a critical temperature or canonical ensemble). This corresponds to mass-energy, while a classical self-gravitating isothermal gas becomes 3/4 1 θ G5/4 unstable below a critical temperature or energy. These are important T < T = (16π)3/4 c 0 · (96) differences that need to be discussed further. c 3/2 11/4 3A αc c 684 P. H. Chavanis: Relativistic stars with a linear equation of state

1 20

Intensive variable Fixed ε T=Cst Unstable 0 0 q=1/3 10 T(R) Stable S=1/T3 −1 T 0

−2 −1/2 T(R)=R ln(S)

ln(T)+Cst Unstable −10 Stable −3 T=R−1/2 q=1/3 −20 Fixed ε −4 0

−30 −5 −5 −4 −3 −2 −1 0 1 −10 −5 0 5 10 ln(T) ln(bR) Fig. 19. Entropy as a function of the temperature for a ﬁxed central den- Fig. 17. Temperature as a function of the box radius for a ﬁxed central ∆ 3/2 1/2 density. This corresponds to the function θ(α)/α1/2. sity. We have plotted (α)α as a function of θ(α)/α .

1 3.6. Comparison with the black hole entropy

0 The scaling laws (76), (85), and (95) that we obtained for the Stable mass, the entropy, and the temperature of a self-gravitating radiation can be compared with the corresponding scaling laws −1 T(R) (A.3), (A.6), and (A.7) for black holes. For the self-gravitating radiation, we have T Unstable −2 1/4 ln(T) 3 R 2 4π3 R 3/2 M = MP , S rad = kB , (100) 2 14 LP 3 315 LP −3 M=1/T Fixed ε 1/4 1/2 0 3 140 L k T = M c2 P · (101) −4 q=1/3 B rad 7 9π3 P R ∼ ∼ 3/2 ∼ −1/2 −5 They behave as M R, S rad R ,andTrad R .Forthe −25 −15 −5 5 black holes, we have ln(M) 1 R R 2 Fig. 18. Temperature as a function of the energy for a fixed central den- M = MP , S BH = πkB , (102) sity. We have plotted θ(α)/α1/2 as a function of αχ(α). 2 LP LP 1 L k T = M c2 P · (103) Eliminating the radius between Eqs. (71)and(92), we can ex- B BH 4π P R press the temperature as a function of the mass-energy. For small 2 −1 masses we have T = const. and for large masses we obtain a They behave as M ∼ R, S BH ∼ R ,andTBH ∼ R .Wehave T −2 law: calculated the exact values of the numerical constants appearing in the expressions of the mass, entropy, and temperature of a self- M M c2 2 = 1 2 P · gravitating radiation. Comparing with black holes, the form of 2 χsθs (97) MP 18A kBT the laws are similar but the exponents are different. In particular, ∝ 3/4 However, the configurations are unstable for M > Mc or T < Tc. we note that S rad S BH . For large enough radii R where this 2 Using E = Mc , the specific heat of the self-gravitating radiation scaling law applies we have S rad S BH in agreement with the (for sufficiently large masses or sufficiently low temperatures) is Bekenstein (1973) inequality. given by dE χ θ2 (M c2)3 4. Other explicit examples C = = − s s k P < . 2 B 3 0 (98) dT 9A (kBT) In this section, we give other physical examples where the gen- As for black holes (see Appendix A), the specific heat is nega- eral theory developed in Sect. 2 applies. tive. Finally, eliminating the radius between Eqs. (81)and(92), we can express the entropy as a function of the temperature. For 4.1. Neutron stars small entropies we have T = const., and for large entropies, we − obtain a T 3 law: In the simplest models, a neutron star can be viewed as an ideal ∆ θ3 M c2 3 gas of self-gravitating fermions. Their density is so large that S = s s k P · gravitation must be described in terms of general relativity. The 2 B (99) 27A kBT distribution function of the neutrons is given by the Fermi-Dirac However, the configurations are unstable for S > S c or T < statistics Tc. The curves T(M)andS (T) for a fixed central density 0 are 2 1 f = , (104) plotted in Figs. 18 and 19. h3 eβ−µ + 1 P. H. Chavanis: Relativistic stars with a linear equation of state 685 where µ is the chemical potential and (p) the energy per par- Let us first consider a gas of charged baryons without gravi- ticle. In the completely degenerate limit (T = 0), the neutrons tational interaction. In the Zel’dovich model, the (repulsive) po- have momenta less than a threshold value p0 (Fermi momen- tential of interaction between two charges is given by = 3 tum) and their distribution is f 2/h . There can only be 2 3 g 2 neutrons in a phase space element of size h on account of u(ξ) = e−µξ, (112) the Pauli exclusion principle. Furthermore, in the core of neu- ξ = tron stars, the neutrons are ultra-relativistic so that (p) pc. where ξ = |r − r |, g is the charge of the baryons and µ the According to Sect. 3.1, they are described by a linear equation = = quantum of mass of the meson (the mass of the meson is m of state of the form (8) with q 1/3. On the other hand, us- µ/c). Since T = 0, the energy simply represents the rest mass ing Eqs. (53)and(55), the density and the pressure are given by 2 = 3 3 = 4 3 energy of the baryons nmbc plus the energy of interaction n 8πp0/(3h )andP 2πcp0/(3h ). Eliminating p0 between these two expressions, we obtain 1 +∞ 2πn2g2 = nm c2 + n2 u ξ πξ2 ξ = nm c2 + · b ( )4 d b 2 (113) P = Knγ, (105) 2 0 µ The pressure is then given by with 2 2 1/3 ∂(/n) 2πn g 4 1 3 P = − = · (114) γ = , K = hc. (106) ∂(1/n) µ2 3 8 π It can be seen from Eqs. (113)and(114) that, in the limit of large Therefore, the core of neutron stars can be studied with the the- n,wehave ory developed in Sect. 2 with q = 1/3. The numerical values of K and γ are explicitly given by Eq. (106). The singular pro- P = , (q = 1). (115) files are This represents the most rigid equation of state compatible with 3c4 c4 relativity theory since the velocity of sound (dP/d)1/2c is equal = , P = , (107) 56πGr2 56πGr2 to the velocity of light for this value of q.Thevalueq = 1is clearly an upper bound4. On the other hand, Eq. (114) can be c9 1/4 rewritten n = r−3/2. 3 3 (108) 1029G h P = Knγ, (116) These are also the asymptotic behaviours of the regular profiles with for r → +∞. On the other hand, for the singular solution, the 2πg22 total mass and the total baryon number within a sphere of ra- γ = 2, K = · (117) dius R are m2c2 M 1 R Therefore, a gas of baryons interacting through a vector meson = χ , (109) M 2 s L field can be studied with the theory developed in Sect. 2 with P P q = K γ 1. The numerical values of and are explicitly given by R 3/2 Eq. (117). The singular profiles are N = B∆ , (110) s 4 LP c = P = r−2, (118) = ∆ = 1/4 16πG with χs 3/7, s (8/21) and m2c6 1/2 1 4 1/4 n = r−1. (119) B = = 0.35408... (111) 32π2g2G2 3 π These are also the asymptotic behaviours of the regular profiles The description of an ultra-relativistic and completely degener- for r → +∞. On the other hand, for the singular solution, the ate gas of neutrons in a box (modelling the core of a neutron star) total mass and total baryon number within a sphere of radius R is similar to the description of a self-gravitating radiation given can be written as in Sect. 3. We just need to replace the entropy (proportional to the photon number) by the neutron number. M 1 R = χs , (120) MP 2 LP 4.2. Baryons interacting via a vector meson field 2 3 2 =∆ m c R N s 2 , (121) Finally, we consider a model introduced by Zel’dovich (1962) g LP that provides the stiffest equation of state P = compatible with the requirements of relativity theory. It consists in a gas 4 Note that the above formulae are valid only for µ>1/R where R is of baryons interacting through a vector meson field (in addi- the system size. For µ = 0 (Coulomb interaction), the potential is not tion to the gravitational force). In the case of neutron stars, the screened and the baryons repel each other at large distances so that a pressure must be imposed to retain the charges. Now, using the Virial gravitational contraction is balanced by the quantum pressure + = = = = = 4/3 theorem 2K W 3PV and considering K T 0, we obtain P /3 P Kn due to the Pauli exclusion principle. In the Zel’dovich for the electromagnetic field. If we take the contribution of the rest mass model, the gravitational contraction is balanced by the pressure in into account, we get P ≤ /3 so that the upper bound is q = 1/3 2 P = Kn due to the electrostatic repulsion of the baryons. For as advocated by Landau & Lifshitz (1960). The model of Zel’dovich dense objects (n 1), the form of pressure considered by (1962) shows that this bound can be exceeded for a classical vector Zel’dovich is expected to prevail over the quantum pressure. field with a mass µ. 686 P. H. Chavanis: Relativistic stars with a linear equation of state with χs = 1/2and∆s = 1/2. Interestingly, these are the same and for large radii, we get the scaling law (121). However, con- scalings as for the mass and the entropy of a black hole (see figurations with R > Rc are dynamically unstable. Eliminating R Appendix A). We have obtained the exact value of the prefactor. between Eqs. (122)and(128), we obtain the baryon number vs. A more complete description of this system is provided by mass curve N(M) represented in Fig. 7. For small mass, we have the analysis of Sect. 2. The different curves have been plotted for the extensive scaling N ∼ M and for large mass, we get q = 1 so that they are directly relevant to the Zel’dovich model provided that the numerical values Eq. (117) are incorporated in ∆ m2c3 M 2 N = 4 s · (130) the discussion. The relation (33) can be written 2 2 χs g MP M 1 R = χ(α) , (122) MP 2 LP 5. Relativistic stars in d dimensions with In this section, we extend the theory of relativistic stars with a linear equation of state to d-dimensional spheres. This study 8πG 1/2 α = 0 R. (123) completes our investigations (Sire & Chavanis 2002; Chavanis c4 &Sire2004; Chavanis 2004, 2006a,b, 2007a) of the dependence of the laws of physics, regarding Newtonian gravity, on the di- For a given box radius, the mass-central density relation M( ) 0 mension of space. It can also have applications in the theory of presents damped oscillations around the mass M ,givenby s compact objects where extra dimensions can appear on the mi- Eq. (120), corresponding to the singular solution (see Figs. 1 croscale, an idea originating from the Kaluza-Klein theory. and 2). There is no equilibrium state with a mass larger than 1 R Mc = χc MP , (124) 5.1. The equations governing equilibrium 2 LP Let us consider the (d + 1)-dimensional, spherically symmetric = = where χc 0.544 corresponding to αc 4.05. On the other metric given by hand, the configurations with central density ds2 = eν(r)d(ct)2 − eλ(r)dr2 − r2dΩ. (131) c4 α 2 > ≡ c , (125) 0 0,c 8πG R For a perfect fluid, the Einstein equations of general relativity governing the hydrostatic equilibrium of a spherical distribution corresponding to α>αc are dynamically unstable. For a given of matter are central density, the mass-radius relation M(R) is represented in ∼ 3 d 16πG Fig. 3. For small radii, we have the extensive scaling M R , (re−λ) − (3 − d)e−λ = (d − 2) − r2, (132) and for large radii, we get the scaling law (120). However, con- dr (d − 1)c4 figurations with 1/2 dP 1 dν c4 = − (P + ) , (133) R > Rc ≡ αc , (126) dr 2 dr 8πG0

α>α −λ corresponding to c, are dynamically unstable. This corre- e dν 1 − 16πG sponds to = (d − 2)(1 − e λ) + P. (134) r dr r2 (d − 1)c4 8 1/2 ≡ 1 c · Combining Eqs. (132)−(134), we obtain the d-dimensional gen- M > Mc χcαc 3 (127) 2 8πG 0 eralization of the Oppenheimer-Volkoff equation On the other hand, the relation (45) can be written aGM(r) dP 1 − = c2rd−2 dr m2c3 R 2 N =∆α · ( ) 2 (128) 1 aGM(r) 8πG g LP − ( + P) (d − 2) + Pr , (135) c2 2rd−1 (d − 1)c2 For a given box radius, the baryon number vs. central density N( ) presents damped oscillations (see Fig. 4). There is no with 0 equilibrium state with N larger than r S d − M(r) = rd 1dr, (136) 2 2 m2c3 R c 0 N =∆ , c c 2 (129) g LP and d/2 where ∆c = 0.546, corresponding to αc = 4.05. The config- 2π 16π S = , a = · (137) urations with central density 0 >0,c are dynamically unsta- d Γ(d/2) (d − 1)S d ble. Eliminating 0 between Eqs. (122)and(128), we obtain the baryon number vs. mass curve N(M). Since the peaks of baryon These equations are only defined for d > 1. For d = 2, we have number coincide with the mass peaks in Figs. 1 and 4,thisim- S 2 = 2π and a = 8. For d = 3, we have S 3 = 4π and a = 2. plies that the baryon number vs. mass curve presents angular We also recall that the value of the gravity constant G and of d 2 points at each peak αn (see Fig. 5). For a given central den- its dimension G ∼ R /(MT ) changes with the dimension of sity, the baryon number vs. radius relation N(R) is represented space d. The dimension d = 2 is critical and will be treated in Fig. 6. For small radii, we have the extensive scaling S ∼ R3, separately in Sect. 6. In this section, we consider that d > 2. P. H. Chavanis: Relativistic stars with a linear equation of state 687

Using Eqs. (132)and(134), we find that the metric func- The Newtonian limit corresponds to q → 0 (see Appendix C). tions λ(r)andν(r) satisfy the relations Taking the derivative of Eq. (148), we find that aGM(r) − −λ(r) = − d 1 M e 1 − , (138) ψ = − · (149) c2rd 2 ξ M + 16πPrd dν 1 d− d− aM r c2 = ( 1)( 2) ( ) · (139) Substituting Eqs. (148)and(149)inEq.(147), we obtain a dif- dr r rd−2c2 − ferential equation for the mass profile d−2 aGM(r) 1 In the empty space surrounding the star, P = = 0. Therefore, − (d − 2) M(ξ) + qM = d 1 − M = ξ · if M M(R) denotes the total mass-energy of the star, (150) ξ M d−2 − 2q Eqs. (138)−(139) become for r > R ξ 1+q M(ξ) aGM dν d − 2 e−λ(r) = 1 − , = · (140) In addition, the metric functions determined by Eqs. (138) c2rd−2 dr rd−2c2 − and (133) can be expressed as r aGM 1 2q M(ξ) 2q The second equation is readily integrated in e−λ = 1 − ,ν= ψ + const., (151) 1 + q ξd−2 1 + q aGM ν(r) = ln 1 − · (141) rd−2c2 where the constant is determined by the matching with the outer Schwarzschild solution (141)atr = R. Substituting the foregoing expressions for λ and ν in Eq. (131), we obtain the (d + 1)-dimensional generalization of the Schwarzschild form of the metric outside a star 5.3. Singular solution aGM dr2 ds2 = 1 − d(ct)2 − − r2dΩ. (142) For d > 2, Eqs. (147)and(148) admit a singular solution of the rd−2c2 − aGM form 1 c2rd−2 − Q 2(d − 2)(1 + q) This metric is singular at e ψs = , where Q = · (152) 2 − + 2 + 1 ξ (d 2)(1 q) 4q aGM d−2 r = ≡ R (d). (143) c2 S For this singular solution, the metric is of the form

This does not mean that spacetime is singular at that radius λ 4q ν 4q e = 1 + , e = Aξ 1+q . (153) but only that this particular metric is. Indeed, the singularity (d − 2)(1 + q)2 can be removed by a judicious change of coordinates (see, e.g., Weinberg 1972). Furthermore, it can be shown that, for a spher- Coming back to original variables, the singular energy density ical system in hydrostatic equilibrium, the radius of the star profile is satisfies − 4 1 Ps(r) (d 1)qQc 1 − (r) = = · (154) d2 aGM d 2 s + 2 R ≥ , (144) q 8πG(1 q) r 4(d − 1) c2 Considering the thermodynamical variables of Sect. 2.4,wefind which is the generalization of the Buchdahl theorem in d di- the scalings5 mensions (Ponce de Leon & Cruz 2000). Therefore, for d > 2, s(r) the points outside the star always satisfy r > RS(d) so that the n(r) = ∝ r−2/(q+1), T(r) ∝ r−2q/(q+1), (155) Schwarzschild metric is never singular for these stars. λ

− dq+d−2 − + ∝ d 2 ∝ ∝ q+1 ∝ 2q/(q 1) 5.2. The general relativistic Emden equation M R , S N R , T R . (156) Considering the linear equation of state (8), we introduce the 5 In particular, the entropy scales with the energy E = Mc2 like S ∼ Eb dimensionless variables ξ, ψ,andM(ξ) by the relations with b = (dq + d − 2)/[(q + 1)(d − 2)]. Interestingly, this is the same scaling as the one obtained by Kalyana Rama (2007) in a cosmological 1/2 − 4 context once the physical size LH of the horizon in his paper is identified = −ψ = (d 1)c q 0e , r + ξ, (145) with the box size R (K.R., private communication). This author studies 8πG0(1 q) the phase transition, below a critical temperature, from a universe dom- and inated by highly excited strings to a FRW universe. He then argues that the final spacetime configuration (q, d) that emerges should maximize S (d − 1)c4q d/2 the entropy at fixed energy (with the additional condition d ≥ 3). This M(r) = d 0 M(ξ). (146) = = 2 + leads to q 1andd 3, providing a possible explanation of why c 8πG0(1 q) our universe is three-dimensional. According to this approach, we have In terms of these variables, Eqs. (135)and(136) can be reduced passed from a universe with d = 9 dimensions dominated by strings to the following dimensionless forms to a three-dimensional universe dominated by radiation, then particles. Using completely different arguments, we have also found that the di- 2q M(ξ) dψ M(ξ) mension d = 3 of our universe is special. For example, compact objects 1 − = (d − 2) + qξe−ψ, (147) ≥ + d−2 d−1 like white dwarf stars would be unstable in a universe with d 4dimen- 1 q ξ dξ ξ sions (Chavanis 2007a). The modifications of the laws of gravity with dM − − the dimension of space and the special role played by the dimension = ξd 1e ψ. (148) dξ d = 3 are very intriguing. 688 P. H. Chavanis: Relativistic stars with a linear equation of state

The constants of proportionality can be easily obtained from the so that z → z0 = ln Q for ξ → +∞. In terms of this new variable, expressions in Sect. 2.4.Forq = 0, corresponding to Newtonian Eqs. (147)−(148) become isothermal stars, we obtain the classical scalings E ∼ N, S ∼ N and T ∼ 1 with N ∼ Rd−2 (Chavanis 2004). For q = 1, cor- − 2q M(ξ) dz 1 − responding to the stiffest stars, we find that M ∼ Rd−2 and 1 + q ξd 2 dξ S ∝ N ∝ Rd−1 so that the entropy scales like the area in any d − 2 + (d + 2)q M(ξ) ez 2 + + q − = 0, (167) dimension of space (Banks et al. 2002). For q < 1, it scales ac- 1 + q ξd−1 ξ ξ cording to a power less than the area. On the other hand, the ∝ = dM − temperature scales like T 1/R in any dimension. For q 1/d, = ξd 3ez. (168) corresponding to a self-gravitating radiation or to a neutron star dξ in d dimensions, we have the scaling laws We set z = z0 + f with f 1 and linearize the equations. We −2 − 2d − 2 then find that f satisfies the equation P ∝ ∝ r , n ∝ s ∝ r d+1 , T ∝ r d+1 , (157) 2 + + − d−1 d f (d 1)q d 1 d−2 d f − + d−2 d(d 1) − 2 ξ ξ M ∝ R , S ∝ N ∝ R d+1 , T ∝ R d+1 . (158) dξ2 q + 1 dξ 2 2(d − 2 + 2dq + (d − 2)q ) − Finally, for the sake of completeness, we give the expression + ξd 3 f = 0. (169) (1 + q)2 of the Stefan-Boltzmann law in d dimensions. From the Bose- Einstein distribution with a chemical potential µ = 0, we find The further change of variables ξ = et transforms Eq. (169)toa that the pressure, the density of photons, and the entropy density linear equation with constant coefficients. We find are given by d2 f dq + d − 2 d f 2(d − 2 + 2dq + (d − 2)q2) + + f = 0. (170) 1 (d − 1)!S d + dt2 q + 1 dt (1 + q)2 P = = (k T)d 1ζ(d + 1), (159) d hdcd B Looking for solutions of the form f ∝ ekt, we obtain (d − 1)!S n = d (k T)dζ(d), (160) −(d − 2 + dq) ± ∆(q) hdcd B k = , (171) 2(q + 1) (1 + d)(d − 1)!S ∆ s = k d (k T)dζ(d + 1). (161) where (q) is the discriminant B hdcd B ∆(q) = (d − 4)2q2 + 2d(d − 10)q + (d − 2)(d − 10). (172) The equivalent expressions for neutron stars in d dimensions are This function is itself quadratic and its discriminant is δ = given in Chavanis (2007a). −128(d − 1)(d − 10). We have ∆(q) → +∞ for q →±∞. Therefore, for d > 10, ∆(q) > 0forallq.Ford < 10, ∆(q) = 0 5.4. Asymptotic behaviour has two roots. Noting that ∆(0) = (d − 2)(d − 10), we conclude that one root is negative and the other is positive. On the other Considering now the regular solutions of Eqs. (147)−(148), we hand, noting that ∆(1) = 4(d − 9)(d − 1), we conclude that the can always suppose that 0 represents the energy density at the positive root is in the range ]1, +∞[ford < 9 and in the range centre of the configuration. Then, Eqs. (147)−(148)mustbe [0, 1] for d ≥ 9. Summarising: (i) for d < 9, ∆(q) < 0forall solved with the boundary conditions q ∈ [0, 1]; (ii) for d > 10, ∆(q) > 0forallq ∈ [0, 1] (iii) for 9 ≤ d ≤ 10, then ∆(q) ≤ 0for0≤ q ≤ q∗ and ∆(q) ≥ 0for ψ(0) = ψ (0) = 0. (162) q∗ ≤ q ≤ 1 (see Figs. 20 and 21)whereq∗(d) is the positive root of ∆(q) = 0, i.e. The corresponding solutions must be computed numerically. √ However, it is possible to determine the asymptotic behaviours d(10 − d) + 32(d − 1)(10 − d) ξ → q∗(d) = · (173) explicitly. For 0, (d − 4)2 2 4 ψ = aξ + bξ + ... (163) For d = 9, q∗ = 1andford = 10, q∗ = 0. When ∆ ≤ 0, the asymptotic behaviour of Eqs. (147)−(148)forξ → +∞ is with ⎡ ⎛ √ ⎞⎤ ⎢ ⎜ ⎟⎥ − Q ⎢ A ⎜ −∆ ⎟⎥ d − 2 + qd ψ ∼ ⎢ + ⎝⎜ ξ + δ⎠⎟⎥ , = e 2 ⎣1 d−2+dq cos ln ⎦ (174) a , (164) ξ + 2(q + 1) 2d ξ 2(q 1) and and it presents damped oscillations around the singular sphere. The curve (174) intersects the singular solution (152) infinitely [d(d+2)q2+(2d2−4d−8)q+d(d−2)](2−d−qd) often at points that asymptotically increase geometrically in b = · (165) 8d2(d+2)(1+q) the ratio 1: exp[2π(1 + q)/ −∆(q)]. As shown in Paper I for d = 3(seealsoSect.5.6), this property is responsible for the → +∞ For ξ , the asymptotic behaviour of the solution of damped oscillations of the mass-central density relation M( ). − 0 Eqs. (147) (148) can be obtained by extending the procedure When ∆ ≥ 0, the asymptotic behaviour of Eqs. (147)−(148)for = developed by Chandrasekhar (1972)ind 3. We introduce a ξ → +∞ is new variable z defined by ⎡ ⎤ ⎢ ⎥ z −ψ Q ⎢ A ⎥ −ψ e e ∼ ⎢1 + √ ⎥ , (175) e = , (166) ξ2 ⎣ d−2+dq− ∆ ⎦ ξ2 ξ 2(q+1) P. H. Chavanis: Relativistic stars with a linear equation of state 689

30 such that ∆(1) = 0(orq∗(d) = 1) corresponding to dcrit = 9. Interestingly, this coincides with the critical dimension arising 20 in superstring theory that may have some connection to the limit d=11 case q = 1 (see, e.g., Kalyana Rama 2006). (iii) Finally, for neu- 10 tron stars or for a self-gravitating radiation (q = 1/d), the critical d=10 dimension above which the mass-central density relation M(0) d=9.96404372 becomes monotonic is such that ∆(1/d) = 0(orq∗(d) = 1/d). It (q) 0 ∆ q*(d) q*=1/9.96404372 is solution of the fourth degree equation d=9 −10 d4 − 10d3 + d2 − 8d + 16 = 0, (177) = = −20 leading to dcrit 9.96404372... very close to d 10. As a conse- d=8 quence, for 2 < d < dcrit = 9.96404372... the mass-radius rela- d=3 tion of neutron stars M(R) should present a spiraling behaviour −30 0 0.2 0.4 0.6 0.8 1 around the point corresponding to the singular sphere (see Fig. 2 q of Meltzer & Thorne 1966, and Fig. 15 of Paper I), while for = Fig. 20. Discriminant ∆(q) as a function of q for diﬀerent values of the d > dcrit 9.96404372..., it should tend to the point correspond- dimension of space d (characteristic dimensions are described in the ing to the singular sphere without spiraling. A similar property text). The mass-central density relation M(0) presents damped oscilla- holds for the caloric curve of the self-gravitating radiation (see tions when ∆(q) < 0 and a monotonic behaviour when ∆(q) ≥ 0. Sect. 5.7).

1 5.5. The Milne variables d=9 (stiffest stars) As in the Newtonian theory of isothermal spheres 0.8 (Chandrasekhar 1942), it is convenient to introduce the Milne variables No oscillation 0.6 ξe−ψ u = ,v= ξψ . (178) q Damped oscillations ψ q (d) 0.4 * In terms of these variables, the system of Eqs. (147)−(148) ff q=1/d can be reduced to a single first-order di erential equation (see 0.2 d=9.96404372 Paper I for d = 3). For ξ → 0, one has (radiation) (Newtonian stars) d=10 1 b 1 u = − + ξ2 + ..., v = 2aξ2 + ... (179) 0 2 23456789101112131415 2a a 2 d and for ξ → +∞ Fig. 21. Phase diagram in the (d, q) plane. The line q∗(d) separates Q the region where the mass-central density relation M(0)presents u → u = ,v→ v = 2. (180) damped oscillations from the region where the mass-central density re- s 2 s lation M(0) is monotonic. The critical dimension corresponding to a = ff self-gravitating radiation is obtained by taking the intersection between The solution curves in the (u,v) plane in d 3 for di erent the line q∗(d) and the curve q = 1/d. values of q are given in Fig. 9 of Paper I. According to the discussion in Sect. 5.4, they tend to the point (us,vs) corresponding to the singular sphere by forming a spiral. By contrast, for and it tends to the singular sphere without oscillating. In that d ≥ dcrit(q) (defined in Sect. 5.4), they reach the singular solu- case, the mass-central density relation M(0) is monotonic (see tion (us,vs) directly, without spiraling (see Fig. 22). Sect. 5.6). For a given value of q, the critical dimension above which the oscillations disappear is such that ∆(q) = 0 leading to 5.6. The mass-density profile = 4 + 3 + + dcrit(q) + q 3q 1 . (176) If the system is enclosed within a box, the solution of q 1 2 Eqs. (147)−(148) must be terminated at a radius α given by Let us consider specific systems. (i) For classical isothermal 1/2 → 8πG (1 + q) spheres (q 0), the critical dimension above which the mass- α = 0 R. 4 (181) central density relation M(0) becomes monotonic is such that (d − 1)c q ∆(0) = 0(orq∗(d) = 0) corresponding to dcrit = 10. This returns the result obtained by Sire & Chavanis (2002) who stud- According to Eq. (146) the relation between the total mass M of ied the d-dimensional Emden equation. As a consequence, for the configuration and the parameter α is = 2 < d < dcrit 10, the caloric curve β(E) of Newtonian isother- 2q M(α) Rd−2c2 mal spheres presents a spiraling behaviour around the point cor- M = · (182) 1 + q αd−2 aG responding to the singular sphere, while for d ≥ dcrit = 10, it tends to the point corresponding to the singular sphere without Solving for M(ξ)inEq.(147) and taking ξ = α,weget spiraling (see Sect. 5.7). (ii) For relativistic stars with the stiffest equation of state (q = 1), the critical dimension above which M(α) αψ (α) − qα2e−ψ(α) = , (183) the mass-central density relation M(0) becomes monotonic is αd−2 d − 2 + pαψ (α) 690 P. H. Chavanis: Relativistic stars with a linear equation of state

4 1.2 q=1/d d=2 q=1/d d=2 1

3 0.8 2 d=5 c d=2.5 χ

d−2 s 0.6

v 2 d=dcrit=9.96404372 d=3

d=3 =aGM/R d=2.5 χ 0.4

1 0.2 d=5

d=dcrit=9.96404372 0 0 −5 0 5 10 0 0.5 1 1.5 2 ln(α) u Fig. 23. Mass vs. central density for a fixed box radius. We have con- Fig. 22. The (u,v) plane for isothermal spheres in general relativity. We sidered the case q = 1/d of a self-gravitating radiation. For 2 < d < = have considered the case q 1/d of a self-gravitating radiation. For 2 < d = 9.96404372, the curve presents damped oscillations around the = crit d < dcrit 9.96404372, the curve forms a spiral around the point (us,vs) mass χ corresponding to the singular sphere. The series of equilibria ≥ s corresponding to the singular sphere. For d dcrit it reaches that point becomes unstable after the first mass peak at (α ,χ ). For d ≥ d ,the = c c crit directly, without spiraling. The case d 2 is studied in Sect. 6. curve is monotonic. In that case, all the configurations of the series of equilibria (with arbitrary central density) are stable up to the singular solution with the maximum mass χ = χ . The case d = 2 is studied in = = s where p is defined by Eq. (28). Writing u0 u(α)andv0 v(α), Sect. 6. we obtain

aGM pv0(1 − qu0) and Appendix D) in a space of d dimensions, we find that the ≡ χ α = · d−2 2 ( ) (184) R c d − 2 + pv0 point of marginal stability in the series of equilibria precisely corresponds to the criterion (187). Therefore, dynamical insta- The curve χ(α) gives the mass M(0) as a function of the central bility sets it at the turning point of mass, as expected. When = = density for a fixed box radius. It starts from χ 0forα 0and ∆(q) < 0, the series of equilibria is stable until the first mass peak = − = − + tends to an asymptotic value χs pQ/(d 2) 4q/[(d 2)(1 at (α , M ), and when ∆(q) ≥ 0, the series of equilibria is stable 2 c c q) + 4q], corresponding to the singular sphere, as α → +∞.The for all values of the central density (including the singular sphere mass associated to the singular sphere is that is marginally stable). This implies that, for d ≥ dcrit(q), the pure scaling laws (156) correspond to stable configurations con- Rd−2c2 M = χ · (185) trary to the case d = 3. s s aG

The properties of the mass-central density curve χ(α) can be ob- 5.7. Self-gravitating radiation tained by a graphical construction in the Milne plane. Let us look for the presence of oscillations by finding possible extrema In this section, we briefly describe the behaviour of the curves of χ(α). Taking the derivative with respect to α of Eq. (182), plotted in Sect. 3 as a function of the dimension of space. For using Eqs. (148)and(183), and finally introducing the Milne sake of generality we give the formulae for any q ∈ [0, 1], but in variables, we get the figures, we focus on the self-gravitating radiation q = 1/d. Let us first consider a fixed box radius. In that case, the 1/2 dχ p v0(1 − qu0) parameter α ∝ is a measure of the central density. The = u0v0 − (d − 2) · (186) 0 dα α d − 2 + pv0 mass-central density relation M(0) is plotted in Fig. 23.For 2 < d < dcrit(q), it presents damped oscillations around the mass Therefore, the extrema of the curve χ(α), determined by the con- of the singular sphere, and for d ≥ dcrit the convergence to the dition dχ/dα = 0, satisfy mass of the singular sphere is monotonic. The entropy S = λN, where N is the number of particles, 1 pv = (d − 2) − q − 1 . (187) − 0 u R aGM(r) 1/2 0 N = n(r) 1 − S rd−1dr, (188) c2rd−2 d The intersections between this curve and the solution curve (u,v) 0 in the Milne plane determine the values of α for which χ(α)is is given as a function of the central density by the relation extremum. We easily check that the curve (187) passes by the α −1/2 point (u ,v ) corresponding to the singular solution. Therefore, 1 − ψ(ξ) M(ξ) d−1 s s S ∝ ∆(α) = e 1+q 1 − p ξ dξ. (189) when ∆(q) < 0, there is an infinity of intersections, and the dq+d−2 d−2 α 1+q 0 ξ curve χ(α) presents an infinity of damped oscillations around → +∞ the singular sphere χs. This is the case extensively described in For α (singular sphere), we have = ∆ ≥ d 3 (see Paper I). When 0, there is only one intersec- 1 + q 1 + −1/2 tion (corresponding to the singular sphere), and the curve χ(α) ∆(α) → ∆s = Q 1 q (1 − χs) . (190) dq + d − 2 increases monotonically up to the singular sphere χs.Thecor- responding curves for a self-gravitating radiation are plotted in For d < dcrit, the entropy-central density relation S (0)presents Fig. 23. If we extend the dynamical stability analysis (see Paper I damped oscillations at the same locations as M(0). This implies P. H. Chavanis: Relativistic stars with a linear equation of state 691

1 5

Fixed R q=1/d Newtonian isothermal spheres 0.8 4 d=2 d=3

0.6 3 d=2.5 S v d=3 d =10 0.4 d=5 crit 2 d=5

0.2 1

d=dcrit=9.96404372 0 0 0.1 0.2 0.3 0.4 0.5 0 M 0246810 u Fig. 24. Entropy vs. mass-energy for a ﬁxed box radius. The peaks disappear for d ≥ dcrit. Fig. 26. The solution of the classical Emden equation in the (u,v)plane as a function of the dimension of space.

1.4 q=1/d 5 Fixed R Newtonian isothermal 1.2 spheres d=d =9.96404372 crit d=2 4 1 d=5

0.8 T 3 d−2

0.6 d=3 GMm/R 0.4 2 η=β

0.2 1 0 dcrit=10 0 0.1 0.2 0.3 0.4 0.5 M 0 Fig. 25. Temperature vs. mass-energy for a fixed box radius. The spiral −5 5 15 ln(α) shrinks to a point for d ≥ dcrit. Fig. 27. Mass-central density profile for a fixed temperature and box ≥ radius (or temperature vs. density contrast for a fixed mass and box ra- that the curve S (M) presents some peaks at these points. For d dius) as a function of the dimension of space for Newtonian isothermal dcrit, the oscillations in S (0) and the peaks in S (M) disappear spheres. (see Fig. 24). The temperature aGM T = T R − ( ) 1 d−2 2 (191) R c the scaling laws are given by Eqs. (155)−(156). In particular, the d−2 2 is given as a function of the central density by the relation mass behaves like M = χsR c /(aG)forR → +∞.Weagain emphasise that for d ≥ dcrit the solutions are stable for any ra- α2q/(q+1) dius R, contrary to the case d < d where they become unstable T ∝ θ(α) = [1 − χ(α)]1/2, (192) crit R(α)q/(q+1) for R > Rc (see Sect. 2). Finally, we compare the results obtained previously for a where R(α) = eψ(α) is the density contrast. For α → +∞ (singu- q self-gravitating radiation in general relativity with the results + 1/2 lar sphere), we have θ(α) → θs = Q q 1 (1 − χs) .Ford < dcrit, obtained for Newtonian isothermal spheres (Sire & Chavanis the temperature-central density relation T(0) presents damped 2002). In that case, the critical dimension at which the oscilla- oscillations at locations different from M(0). This implies that tions disappear is dcrit = 10. In Fig. 26, we plot the solution curve the curve T(M) forms a spiral. For d ≥ dcrit, the oscillations of the Emden equation in the (u,v) plane. In Figs. 27 and 28,we in T(0) and the spiral in the caloric curve T(M) disappear (see plot the inverse temperature η and the energy Λ as a function 1/2 Fig. 25). This is similar to what happens to the caloric curve β(E) of the parameter α = (S dGβmρ0) R, which is a measure of of a Newtonian isothermal gas for d ≥ dcrit = 10 (see Sire & the density contrast. In Fig. 29, we plot the caloric curve β(E). Chavanis 2002). For d = 2, the caloric curve tends to a plateau with temperature Alternatively, if we fix the central density, the parameter Tc = GMm/4kB and energy E →−∞. This corresponds to the α ∝ R is a measure of the system size. The curves M(R) ∝ formation of a Dirac peak as T → Tc.For2 < d < 10, the αd−2χ(α), S (R) ∝ α(dq+d−2)/(q+1)∆(α), and T(R) ∝ α−2q/(q+1)θ(α) caloric curve forms a spiral. For d ≥ 10, the spiral shrinks to a behave like the ones reported in Figs. 9, 12,and17 (note that, for point. These results are strikingly similar to those obtained for a d ≥ dcrit(q), the small oscillations are suppressed). For large R, self-gravitating radiation in general relativity. 692 P. H. Chavanis: Relativistic stars with a linear equation of state

2 6. Two dimensional gravity d=2.2 d=2.5 In this section, we focus on the dimension d = 2, which presents 1 d=2 peculiar features and where analytical results can be obtained. d=3

2 0 6.1. Self-conﬁned solutions /GM

d−2 For d = 2, the Oppenheimer-Volkoﬀ Eqs. (135)−(136) reduce to d=5

−ER −1

Λ= 8GM(r) dP 8πG Newtonian isothermal 1 − = − ( + P)Pr, (193) spheres c2 dr c4 −2 d =10 crit with 2π r −3 M(r) = rdr. (194) −2 0 2 4 6 8 10 12 14 c2 ln(α) 0 Fig. 28. Energy vs. density contrast for a fixed mass and box radius as a From Eqs. (138)and(139), we find that the functions defining function of the dimension of space for Newtonian isothermal spheres. the metric are given by 16πGPr −λ = − 8GM(r) dν = c4 · 5 e 1 , (195) c2 dr 1 − 8GM(r) Newtonian isothermal c2 spheres d=2 In the empty space surrounding the star, P = = 0. Therefore, 4 for r > R,wehavee−λ = 1 − 8GM/c2 and ν = cst. The first rela- 2 tion requires M ≤ Mc ≡ c /8G. In fact, we find in the following d=2.5 3 examples (valid for a linear equation of state) that steady solu- d−2 2 d=3 tions exist only for M = Mc ≡ c /8G. We can wonder whether

dcrit=10 this result is general. GMm/R 2 d=2.2 For a linear equation of state, defining η=β d=5 c4q 1/2 = −ψ, r = ξ, 1 0e (196) 8πG0(1 + q) and 0 −3 −1 1 3 5 7 c2q Λ=−ERd−2/GM2 M(r) = M(ξ), (197) 4G(1 + q) Fig. 29. The caloric curve as a function of the dimension of space for Newtonian isothermal spheres. the generalized Emden equation takes the form 2q dψ 1 − M(ξ) = qξe−ψ, (198) 1 + q dξ dM = ξe−ψ. (199) 5.8. Particular values of the parameters dξ In this section, we regroup the values of the different parameters We note that there is no Newtonian limit (q → 0) in d = 2(see defined in the text for different values of q and different dimen- Appendix C). The differential equation for the mass profile is sions of space. 1 M qM For q = 1(stiffest stars), we have p = 1, Q = θ = d−2 ,and − = · (200) s d−1 ξ M − 2q =∆ = 1 = = = 1 1+q M χs s d−1 . In particular, for d 3, we have p 1, Q = 1 =∆ = 1 = = θs 2 and χs s 2 .Furthermore,αc 4.05, χc 0.544, This equation can be easily integrated once to yield ∆c = 0.546, α∗ = 1.30, and θ∗ = 0.676. On the other hand, for = = = = 7 =∆ = 1 1+q dcrit 9, we have p 1, Q θs ,andχs s . 2 8 8 M = − 2q = = 2 A0 1 + M , (201) For q 1/d (self-gravitating radiation), we have p d+1 , ξ 1 q 2 Q = 2d(d −2)(d +1)/[(d −2)(d +1) +4d], χs = pQ/(d −2), θs = Q1/(d+1)(1 − χ )1/2,and∆ = d+1 Qd/(d+1)(1 − χ )−1/2. In partic- where A0 is a positive constant. This can again be integrated eas- s s d(d−1) s ily to yield = = 1 = 6 = 3 = 96 1/4 ular, for d 3, we have p 2 , Q 7 , χs 7 , θs ( 343 ) ,and 8 1/4 1−q ∆s = ( ) .Furthermore,αc = 4.7, χc = 0.493, ∆c = 0.925, 2 21 2q 2 2 α∗ = 1.47 and θ∗ = 0.897. On the other hand, for P(d ) = 10 1 − M = −K ξ + B, (202) crit 1 + q (closest upper integer value of the critical dimension), we have = 2 = 110 = 55 = 110 1/11 1331 1/2 p 11 , Q 63 , χs 1386 , θs ( 63 ) ( 1386 ) ,and where B and K (related to A0) are some constants. Here, we as- ∆ = 11 110 10/11 1386 1/2 = s 90 ( 63 ) ( 1331 ) . sume q 1 (the case q 1 will be treated separately). Because P. H. Chavanis: Relativistic stars with a linear equation of state 693

M = 0atξ = 0, we find that B = 1. Therefore, the mass profile This can be solved easily to yield is given by 2 2 M(ξ) = 1 − e−K ξ , (215) + 1 q 2 2 2 M(ξ) = 1 − (1 − K ξ ) 1−q . (203) 2q where we have used M(0) = 0. The density profile is the Gaussian Using Eq. (199), we find that the normalized density profile is − − 2 2 given by e ψ = 2K2e K ξ . (216)

2 − 1 + q 1+q Using ψ(0) = 0, we get K = 1/2sothat ψ = 2 − 2 2 1−q e 2K − (1 K ξ ) . (204) q(1 q) 2 e−ψ = e−ξ /2, (217) Thus, we find that the density vanishes at a finite distance ξ0 = = − 2 1/K. Using furthermore the fact that ψ(0) 0, we obtain M(ξ) = 1 − e ξ /2. (218) 2(1 + q) 1/2 ξ = · (205) Returning to original variables, we find that 0 q(1 − q) −(r/L)2 (r) = 0e , (219) Therefore, the mass and density profiles can be written6 2 + 2 c −(r/L)2 1 q 2 1−q M(r) = 1 − e , (220) M(ξ) = 1 − 1 − (ξ/ξ0) , (206) 8G 2q where 1+q −ψ 2 1−q e = 1 − (ξ/ξ ) . (207) 1/2 0 c4 L = (221) Returning to original variables, we find that 8πG0 2 2 is a typical lengthscale. The metric is explicitly given by c − M(r) = 1 − 1 − (r/r )2 1 q , (208) 0 2 2 8G e−λ = e−(r/L) , e−ν = e−(r/L) . (222)

1+q 2 1−q Again, we find a family of solutions parametrized by the central (r) = 0 1 − (r/r0) , (209) density. It is noteworthy that the total mass M = M(+∞) = Mc 4 1/2 is the same for all these configurations and is again given by c Eq. (213). For → +∞, the density profile tends to a Dirac r0 = · (210) 0 4πG0(1 − q) peak. These results have to be contrasted from their Newtonian The metric is explicitly given by counterpart where the density decreases as ξ−4 for ξ → +∞

2 (see, e.g., Sire & Chavanis 2002). However, the universal mass −λ 2 1−q = 2 e = 1 − (r/r0) , (211) Mc c /(8G) seems to be the general relativistic equivalent of the critical temperature Tc = GMm/(4kB) or the critical = 2q mass Mc 4kBT/(Gm) in 2D Newtonian gravity for isother- ν = − ln 1 − (r/r )2 . (212) 1 − q 0 mal spheres. Indeed, for a fixed temperature, unbounded two- dimensional, self-gravitating isothermal spheres have a unique This defines a family of solutions parametrized by the central mass Mc (see, e.g., Chavanis 2007b). density. These solutions can have different radii r ,buttheyall Using Eq. (10), the baryon number is given by 0 have the same mass M = M(r0) = Mc given by +∞ N = n(r)eλ(r)/22πrdr, (223) 2 c 0 Mc = · (213) G + + 8 with n(r) = (q/K)1/(q 1)(r)1/(q 1). From the analytical expres- λ(r)/2 For r0 → 0, the density profile tends to a Dirac peak. In two sions Eqs. (209)(211)or(219)(222), we note that n(r)e = dimensions, the case of photon stars (self-gravitating radiation) n0 in the star. For q 1, using Eq. (210) we explicitly obtain and neutron stars corresponds to q = 1/2. On the other hand, for 1 4 q q+1 − ff q = q c q+1 the sti est equation of state corresponding to 1, Eq. (201) N = . (224) simplifies in K 4G(1 − q) 0 M The baryon number diverges for → 0, i.e. r → +∞, suggest- = A (1 − M). (214) 0 0 ξ 0 ing that the system tends to evaporate (recall that stable states tend to maximize N at fixed mass M)7.Forq = 1, the baryon 6 It is amusing to note that the form of the density profile is similar to number diverges, whatever the central density. a “Tsallis distribution” with index p = 2q/(1 + q). For q = p = 1, we obtain a “Boltzmann distribution” (217). These analogies with general- 7 This result is different from the case of 2D Newtonian gravity where ized thermodynamics (Tsallis 1988) are, of course, purely coincidental. the Boltzmann free energy FB[ρ] happens to be independent of the They show that the “Tsallis distribution” can arise in very different con- central density ρ0 parametrizing the family of isothermal solutions at texts that are not necessarily related to thermodynamics. M = Mc or T = Tc (see Chavanis 2007b). 694 P. H. Chavanis: Relativistic stars with a linear equation of state

1.2 3

d=2 R/r0=10 1 2.5 q=1/2 Self−confined profiles

0.8 2 c

0.6 ε 1.5 R/r0=3/2 M/M Box−confined profiles 0.4 1 d=2 R/r =1 0.2 q=1/2 0.5 0

R/r0=2/3

0 0 0123 0 0.2 0.4 0.6 0.8 1 1.2 1.4

R/r0 r/R Fig. 30. For box-confined systems, this figure represents the mass as a Fig. 31. Density energy profile (in units of c4/2πGR2) for different val- 2 2 function of the central density (proportional to (R/r0) )forq = 1/2. ues of the central density (proportional to (R/r0) )forq = 1/2. The pro- 4 2 file is self-confined if r0 ≤ R corresponding to 0 > c /[4πG(1 − q)R ]. For r0 → 0or0 → +∞, the density profile tends to a Dirac peak. 6.2. Box-confined solutions 1.2 If the system is confined within a box of radius R, the previous results remain valid for r ≤ R.Ifr0 > R, the system is confined 1 by the wall (ρ(R) 0), and if r0 < R, the system is self-confined (ρ(r0) = 0). Let us consider here box-confined configurations (r0 > R). For q 1, the mass-central energy relation for fixed R 0.8 is given by c 0.6 2 2 c − M/M M = 1 − 1 − (R/r )2 1 q , (225) 8G 0 0.4 4 1/2 d=2 = c · r0 (226) 0.2 q=1 4πG0(1 − q)

The mass-central density (for a fixed box radius) is plotted in 0 = 0123 Fig. 30 and the density profile is plotted in Fig. 31.Forq 1, R/L we have Fig. 32. For box-confined systems, this figure represents the mass as a 2 c 2 2 = M = 1 − e−(R/L) , (227) function of the central density (proportional to (R/L) )forq 1. 8G where Eliminating ξ between these two expressions, we obtain c4 1/2 v = −2lnu. (232) L = · (228) 8πG0 The (u,v) curve is parametrized by ξ.Forξ = 0, we have (u,v) = The mass-central density (for a fixed box radius) is plotted in (1, 0) and for ξ → +∞,wehave(u,v) = (0, +∞). The solution Fig. 32 and the density profile is plotted in Fig. 33. curve in the (u,v) plane is represented in Fig. 34. If the system is confined within a box of radius R, the previous results remain valid for ξ ≤ α ≤ ξ0 where 6.3. The Milne variables 8πG (1 + q) 1/2 The Milne variables are defined by Eqs. (178). Using the analyt- = 0 α 4 R. (233) ical solution (207), we find for q 1that c q

2 From the criterion (187), we can extrapolate that the condition 1 2 2 qξ u = (1 − (ξ/ξ ) ) 1−q ,v= · (229) of marginal stability in d = 2 corresponds to u0 = 0. Therefore, q 0 1 − (ξ/ξ )2 0 the solutions that are confined by a box are stable, since u0 = Eliminating ξ between these two expressions, we obtain u(α) > 0, while the self-confined solutions are marginally stable since u0 = u(ξ0) = 0. q−1 = 2 2 − v qξ0 (qu) 1 . (230) 7. Conclusion The (u,v) curve is parametrized by ξ.Forξ = 0, we have (u,v) = (1/q, 0) and for ξ → ξ0,wehave(u,v) = (0, +∞). For q = 1, In this paper, we have carried out a thorough analysis of the using the analytical solution (217), we find that structure and stability of relativistic stars (and self-gravitating radiation) described by a linear equation of state. In order to pre- 2 u = e−ξ /2,v= ξ2. (231) vent evaporation, we have placed these objects in a cavity. We P. H. Chavanis: Relativistic stars with a linear equation of state 695

3 a Dirac peak (black hole) with mass Mc (collapse). On the other R/L=10 ≥ d=2 hand, for d dcrit(q)(wheredcrit(q) is a nontrivial dimension 2.5 q=1 depending on q), the mass vs central density proﬁle no longer displays oscillations, and all the conﬁgurations of the series of 2 equilibria are stable whatever their central density. In that case, R/L=3/2 the pure scaling laws are meaningful. The physical implication of this result needs further investigation. For the self-gravitating

ε 1.5 radiation, we have found that the critical dimension has the noninteger value d = 9.96404372... This is an interesting (and R/L=1 crit 1 intriguing) result because it arises solely from the combination of the Einstein equations and the Stefan-Boltzmann law, so it has 0.5 R/L=2/3 a relatively fundamental origin. We have also shown that the structure of relativistic stars 0 with a linear equation of state in general relativity is strikingly 0 0.2 0.4 0.6 0.8 1 1.2 1.4 similar to the structure of isothermal stars in Newtonian grav- r/R ity. Basically, the analogy stems from the resemblence between Fig. 33. Density energy profile (in units of c4/8πGR2) for different val- the general relativistic Emden equation and its Newtonian coun- ues of the central density (proportional to (R/L)2)forq = 1. For L → 0 terpart and the fact that they coincide for q → 0. On the other → +∞ or 0 , the density profile tends to a Dirac peak. hand, stable relativistic stars with a linear equation of state maximize the baryon number N[] at fixed mass-energy M[] (for- 5 mal nonlinear dynamical stability for the Einstein equations).

ξ ξ d=2 Similarly, isothermal stars in Newtonian gravity minimize the = 0 W 4 energy functional [ρ]atfixedmassM[ρ] (formal nonlinear ξ→∞ dynamical stability for the Euler-Poisson system) or minimize the Boltzmann free energy functional FB[ρ]atfixedmassM[ρ] 3 (thermodynamical stability in the canonical ensemble). As a result, the curves of Figs. 1 and 4 are respectively similar to (i) the v mass-versus-central density M(ρ0) at fixed temperature and vol- q=1/2 2 ume; and (ii) the energy functional W(ρ0) or the Boltzmann free energy FB(ρ0)-versus-central density at fixed temperature and q=1 volume for Newtonian isothermal spheres (Chavanis 2002d). On 1 the other hand, by interpreting the mass M[] as an energy and

ξ=0 the baryon number N[] as an entropy, the criterion of formal ξ=0 nonlinear dynamical stability in general relativity is equivalent 0 0 0.5 1 1.5 2 2.5 3 to a criterion of thermodynamical stability in the microcanoni- u cal ensemble. Therefore, the curves of Figs. 8, 10, 11,and15 are R R Fig. 34. The (u,v) curve of “isothermal” spheres in two-dimensional respectively similar to (i) the energy E( ) and the entropy S ( )- gravity for q = 1/2andq = 1. versus-density contrast at fixed mass and volume; (ii) the entropy S (E)-versus-energy at fixed mass and volume; and (iii) the caloric curve β(E) at fixed mass and volume for Newtonian have studied the stability of the steady states (i) by linearizing isothermal spheres (Chavanis 2002d). Therefore, depending on the Einstein equations around the stationary solution and study- the interpretation, relativistic stars with a linear equation of state ing the sign of the squared pulsation σ2 (see Paper I); and (ii) by share analogies with classical isothermal spheres in both micro- considering the sign of the second-order variations of the baryon canonical and canonical ensembles. These interesting analogies number (see Appendix D). We have found analytically that the are intriguing and deserve further investigation. instability occurs precisely at the first mass peak when we vary the central density for a fixed box radius. We have applied these Note added in proof Coincidentally, two other authors V. Vaganov results to explicit examples: self-gravitating radiation, the core arXiv:0707.0864 arXiv:0707.0961 of neutron stars, stiffest stars, Zel’dovich model etc. We have de- [ ] and J. Hammersley [ ]have, in two recent papers, independently carried out studies of the self- termined their domains of stability precisely and found some up- gravitating radiation in d dimensions (in an asymptotically anti-de Sitter per bounds on the thermodynamical parameters like the entropy. space Λ ≤ 0) related to the one developed in Sect. 5. These authors We have obtained scaling laws (with a precise determination of find that the oscillations in the mass-central density relation disappear the prefactor) that we compared with the scaling laws of black above a critical dimension. On the basis of numerical calculations, J. holes. We have stressed, however, that pure scaling laws corre- Hammersley obtains a simple relation between the critical density 0,c spond to unstable configurations and that they can hold only ap- and the dimension d and argues that the critical dimension is dcrit = 10 + proximately close to the critical radius Rc. We have generalized ( 1 if we include time). Our analytical approach (for a cosmological our results in d dimensions and found two critical dimensions. In constant Λ=0) is more precise and shows that the critical dimension = corresponding to the self-gravitating radiation has the noninteger value d 2, steady state solutions exist for a unique value of the mass = M = c2/(8G) and they can be obtained analytically. For this dc 9.96404372... very close to 10 (a minor mistake in calculation was c made in the first version of this manuscript and led to a slightly dif- mass, we get an infinite family of solutions parametrized by the ferent value for the critical dimension. This mistake was pointed out to central density. The density profiles vanish at a finite radius for me by V. Vaganov, who performed, in a new Appendix B of his paper q 1 and decrease as a Gaussian for q = 1. These configurations [arXiv:0707.0864v4], a phase plane analysis of the Oppenheimer- are probably marginally stable. They may evolve dynamically ei- Vol koff equation with Λ=0. This provides an alternative derivation ther toward a completely spread profile (evaporation) or toward of the asymptotic results obtained in our Sect. 5.4). Our approach also 696 P. H. Chavanis: Relativistic stars with a linear equation of state provides a thorough description of the structure and stability of rela- Appendix B: The Tolman relation tivistic stars with a linear equation of state p = q for any q ∈ [0, 1] and d ≥ 2. Note that the existence of a critical dimension above which Global thermodynamic equilibrium requires that the redshifted the oscillations of the mass-central density relation disappear had been temperature eν(r)/2T(r) is uniform throughout the medium. This noted previously in Sire & Chavanis (2002) for Newtonian isothermal is called the Tolman (1934) relation spheres (corresponding to q → 0). In that case, dcrit = 10 exactly. The ν(r)/2 present study extends this work in general relativity. e T(r) = T0, (B.1) Acknowledgements. I am grateful to Kalyana Rama for pointing out his work where T is a constant. Since ν(r) → 0forr → +∞, we conclude after a first version of this paper was placed on [arXiv:0707.2292]. 0 that T0 is the temperature measured by an observer at infinity. Let us check that this relation is satisfied by our equations. From Appendix A: Some elements of black hole Eq. (14), we find that thermodynamics T(r) ∝ (r)q/(q+1). (B.2) In this appendix, we recall elementary notions of black holes thermodynamics in order to facilitate the comparison with the On the other hand, according to Eq. (102) of Paper I, the Einstein results obtained in this paper. equations for a spherically distribution of matter give In the seventies, different works (Christodoulou 1970; dP 1 dν Penrose & Floyd 1971; Hawking 1971) have shown that the area = − ( + P) · (B.3) A = 4πR2 of a black hole (more precisely the area of its event dr 2 dr horizon) cannot decrease. Noting the analogy with the second For the linear equation of state (8), this relation can be easily law of thermodynamics, these results led Bekenstein (1973)to integrated into conjecture that black holes have an entropy proportional to their 2 area A = 4πR . On the basis of dimensional analysis, he obtained − q+1 ν(r) (r) = Ae q 2 , (B.4) kBA S BH = λ , (A.1) L2 where A is a constant. Comparing Eqs. (B.2)and(B.4), we ob- P tain the Tolman relation (B.1). Now, applying this relation at the where λ is a dimensionless constant. If black holes have entropy boundary r = R of the system, we obtain and energy, they must possess a temperature. Hawking (1975) ν(R)/2 showed that black holes emit thermal radiation at a temperature e T(R) = T0. (B.5) c3 k T = · (A.2) On the other hand, according to Eq. (109) of Paper I, we have B 8πGM Considering a Schwarzschild black hole for which ν(R) = ln(1 − 2GM/Rc2). (B.6) Rc2 M = , (A.3) Therefore, the temperature at infinity is related to the tempera- 2G ture at the surface of the star by and writing the first law of thermodynamics in the form 2 2GM d(Mc ) = TdS BH, (A.4) T = T(R) 1 − · (B.7) 0 Rc2 one gets This relation shows that the thermodynamical temperature de- 3 c fined by Eq. (89) coincides with the temperature measured by an kBT = · (A.5) 32πλGM observer at infinity. Comparing this expression with the expression of the Hawking temperature (A.2) it is found that λ = 1/4. This leads to the following expression of the Black Hole (or Bekenstein-Hawking) Appendix C: The Newtonian limit entropy8 in a d-dimensional universe k A S = B · (A.6) In this Appendix, we briefly discuss the Newtonian limit BH 2 + 4LP of the Einstein equations in a (d 1)-dimensional spacetime. Considering first the general relativistic Emden On the other hand, using Eq. (A.3), the black hole tempera- Eqs. (147)−(148), the Newtonian limit corresponds to q → 0. In ture (A.5) can be written that limit Eqs. (147)−(148) reduce to c kBT = , (A.7) 4πR dψ = − M(ξ) dM = d−1 −ψ (d 2) − , and ξ e , (C.1) emphasising the scaling T ∼ 1/R with the radius. Finally, the dξ ξd 1 dξ energy E = Mc2 of the black hole is related to its temperature by and they combine to give c5 E = · (A.8) 1 d d−1 dψ −ψ 8πGkBT ξ = (d − 2)e . (C.2) ξd−1 dξ dξ Therefore, black holes have negative specific heats dE c5 This is the familiar Emden equation (Chandrasekhar 1942) with C = = − < 0. (A.9) an additional factor d − 2. For d > 2, we can rescale the param- dT 8πGk T 2 B eters so as to recover the Emden equation exactly. For d = 2, 8 Coincidentally, the initials are the same. we see that there is no Newtonian limit in d = 2. This can also P. H. Chavanis: Relativistic stars with a linear equation of state 697 r δM r = 4π δr 2 r be seen, more fundamentally, at the level of the Oppenheimer- Substituting ( ) c2 0 d in Eq. (D.3), interchanging Vo l ko ff Eqs. (135)−(136). The Newtonian limit corresponds to the order of the integrals, and writing the first-order condition as c → +∞. In that limit, Eq. (135) reduces to Eq. (5), we obtain R −1/2 dP 8π(d − 2) GM(r) − q 1 2GM(r) = − ρ · 2 q+1 − d−1 (C.3) 4πr drδ 1 dr (d − 1)S d r q + 1 rc2 0 R −3/2 This is to be compared with the classical condition of hydrostatic G 1 2GM(r ) + (r ) q+1 1 − 4πr dr equilibrium c4 r c2 r K 1/γ R 4π dP GNewtonM(r) −µ δr2 r = . = −ρ · (C.4) 2 d 0 (D.5) dr rd−1 q 0 c We find that the gravitational constants are related to each This condition must be true for all variations, implying other by 1/γ −1/2 K 1 1 − q 2GM(r) µ = q+1 1 − (d − 2)8π q c2 q + 1 rc2 G = G . (C.5) Newton Einstein − (d − 1)S d R 3/2 G 1 2GM(r ) + r q+1 − πr r . 4 ( ) 1 2 4 d (D.6) They only coincide in d = 3. In d = 1, the Newtonian c r r c = gravitational constant is infinite and it vanishes in d 2. = Therefore, there is apparently no Newtonian limit in one- and If we take r R, we find that the value of µ is given by two-dimensional gravity. 1/γ −1/2 K 1 1 − q 2GM µ = (R) q+1 1 − · (D.7) q c2 q + 1 Rc2 Appendix D: Dynamical stability analysis On the other hand, if we take the derivative of Eq. (D.6) with In this Appendix, we consider the formal nonlinear dynamical respect to r, we obtain, after simplification, stability of a box-confined relativistic star with a linear equation 2GM(r) d 1 q + 1 GM(r) 4πGq of state (8). Specifically, we study the maximization problem 1 − = − + r , (D.8) c2r dr c2 q r2 c2 Max {N[] | M[] = M fixed}, (D.1) which is the Oppenheimer-Volkoff Eq. (1) with a linear equation and show that it provides the same condition of stability as the of state P = q. This is a particular case of the general result condition of linear dynamical stability studied in Paper I by con- given by Weinberg (1972) which is valid for an arbitrary equa- sidering the growth rate of a solution of the linearized Einstein tion of state provided that the perturbations are adiabatic. equations. Therefore, linear and nonlinear dynamical stability coincide. Stability is lost when the mass-central density pro- D.2. The second-order variations file M(0) presents an extremum. The solutions on the series of equilibra M(0) are nonlinearly dynamically stable before the We now turn to the more complicated second-order variations. turning point of mass, and they become linearly dynamically Writing unstable after the turning point of mass. This is similar to the d f 4π case of barotropic stars described by the Euler-Poisson system f = δM(r), = δr2, (D.9) in Newtonian gravity (Chavanis 2002a,c, 2006a). dr c2 the second variations of the baryon number are given by D.1. The first-order variations R −3/2 G 1 2GM(r) − q d f δ2N = dr 1 − q+1 f For a linear equation of state, using Eqs. (3)and(10), the baryon q + 1 r rc2 dr number can be expressed in terms of the energy density in the 0 2 R −5/2 form 3G 2GM(r) 1 2 + r π − q+1 f 4 d 4 1 2 − 2c 0 rc 1/γ R 1/2 q 1 2GM(r) − + 2 R 4 1/2 2 N = q 1 1 − 4πr dr. (D.2) c q 2GM(r) − 2q+1 d f K rc2 − r − q+1 , 0 d 2 2 1 2 (D.10) 0 4πr 2(q + 1) rc dr

Therefore, the ﬁrst-order variations of baryon number and mass 1/γ are where we have omitted, for brevity, the positive term (q/K) in factor on the r.h.s. This is the sum of three integrals that will − 1/γ R 1/2 q 1 2GM(r) be denoted I1, I2,andI3. Integrating by parts the ﬁrst integral, δN = q+1 − 1 2 we get K 0 rc ⎛ ⎞ ⎜ GδM(r) ⎟ R 2 −5/2 ⎜ 1 δ 2 ⎟ 3G 2GM(r) 1 × ⎜ + rc ⎟ πr2 r, I + I = 4π 1 − q+1 ⎝ 2GM(r) ⎠ 4 d (D.3) 1 2 4 2 q + 1 1 − 0 2c rc rc2 ⎛ ⎞ ⎜ −3/2 ⎟ R G d ⎜1 2GM(r) q ⎟ 4π ⎜ − + ⎟ 2 δM = δr2dr. (D.4) − ⎝⎜ 1 − q 1 ⎠⎟ f dr. (D.11) 2 2(q + 1) dr r rc2 c 0 698 P. H. Chavanis: Relativistic stars with a linear equation of state

Expanding the derivative, we obtain the point of marginal stability Λ=0 in the series of equilibria. It is obtained by solving the differential equation R −3/2 G GM r q 1 2 ( ) − + 2 I + I = 1 − q 1 d f 2 dψ − d f 1 2 4(q + 1) r2 rc2 + − + + qξeλ ψ 0 dξ2 ξ dξ dξ −1 24πG 2GM(r) 2 2qr 1 d × 1 − qr + λ−ψ 2qξ dψ 4 2 + +e 1 + f = 0, (D.18) c rc q 1 dr q + 1 dξ − 2GM(r) 1 = = + 3 1 − − 1 f 2dr. (D.12) with f (0) f (α) 0. The same equation was obtained in rc2 Paper I (see Eq. (167)) by determining the condition of marginal ff linear dynamical stability. Therefore, the thresholds of linear Using the Oppenheimer-Volko Eq. (D.8), we find after and nonlinear dynamical stability coincide. Equation (D.18)was simplification solved in Paper I and led to the relation (I-171) identical to R −3/2 (I-148). Therefore, the points where Λ=0 correspond to ex- G 1 2GM(r) − q I + I = 1 − q+1 trema of mass in the series of equilibria M(α). A first eigen- 1 2 4(q + 1) r2 rc2 0 value Λ1 becomes positive at the first mass peak (implying in- 4qr 1 d 2 stability), and new modes of stability are lost at subsequent × 2 − f dr. (D.13) Λ Λ Λ = q + 1 dr extrema. At the ith extremum, we have 1 > 2 > ... > i 0 > Λi+1 > Λi+2 > .... On the other hand, integrating by parts the third integral in Eq. (D.10) and using the Oppenheimer-Volkoff Eq. (D.8), we References obtain after simplification Antonov, V. A. 1962, Vest. Leningr. Gos. Univ., 7, 135 − Banks, T., & Fischler, W. 2001 [hep-th/0111142] c4 q R 1 2GM(r) 1/2 I = rf − Banks, T., Fischler, W., Kashani-Poor, A., McNees, R., & Paban, S. 2002, Class. 3 2 d 2 1 2 8π (q + 1) 0 r rc Quantum Grav., 19, 4717 Bekenstein, J. D. 1973, PRD, 7, 2333 2 − 2q+1 d f d f 2 1 d Bekenstein, J. D. 1981, PRD, 23, 287 × q+1 + − − dr2 dr r dr Bousso, R. 2002, Rev. Mod. Phys., 74, 825 Chandrasekhar, S. 1942, An Introduction to the Theory of Stellar Structure − 2GM(r) 1 4πG (Dover) + − q + r . 1 2 4 ( 1) (D.14) Chandrasekhar, S. 1972, A limiting case of relativistic equilibrium, in General rc c Relativity, papers in honour of J.L. Synge, ed. L. O’ Raifeartaigh (Oxford) Chavanis, P. H. 2002a, A&A, 381, 340 In conclusion, Chavanis, P. H. 2002b, A&A, 381, 709 (Paper I) R Chavanis, P. H. 2002c, A&A, 386, 732 G 1 − q 2qr 1 d 2 = 3λ/2 q+1 − 2 Chavanis, P. H. 2002d, PRE, 65, 056123 δ N 2 e 1 f dr 2(q + 1) 0 r q + 1 dr Chavanis, P. H. 2004, PRE, 69, 066126 Chavanis, P. H. 2006a, A&A, 451, 109 4 R + c q 1 λ/ − 2q 1 Chavanis, P. H. 2006b, C. R. Physique, 7, 331 + drf e 2 q+1 2 2 Chavanis, P. H. 2006c, Int J. Mod. Phys. B, 20, 3113 8π (q + 1) 0 r Chavanis, P. H. 2007a, PRD, 76, 023004 d2 f d f 2 1 d 4πG × + − − + eλ (q + 1)r . (D.15) Chavanis, P. H. 2007b, Physica A, 384, 392 dr2 dr r dr c4 Chavanis, P. H., & Rieutord, M. 2003, A&A, 412, 1 Chavanis, P. H., & Sire, C. 2004, PRE, 69, 016116 To determine the sign of δ2N, we are led to consider the eigen- de Vega, H. J., & Sanchez, N. 2002, Nucl. Phys. B, 625, 409 Christodoulou, D. 1970, PRL, 25, 1596 value equation Hawking, S. 1971, PRL, 26, 1344 Hawking, S. 1975, Commun. Math. Phys., 43, 199 4 + c q 1 λ/2 − 2q 1 e q+1 Hertel, P., & Thirring, W. 1971, Commun. Math. Phys., 24, 22 8π (q + 1)2 r2 ’t Hooft, G. 1985, Nucl. Phys. B, 256, 727 Kalyana Rama, S. 2007, Phys. Lett. B, 645, 365 d2 f d f 2 1 d 4πG × + − − + eλ (q + 1)r Katz, J. 1978, MNRAS, 183, 765 dr2 dr r dr c4 Katz, J. 2003, Found. Phys., 33, 223 Klein, O. 1947, Ark. Mat. Astr. Fys., 34, 19 G 1 3λ/2 − q 2qr 1 d Landau, L. D., & Lifshitz, E. M. 1960, The classical theory of fields (Fizmatgiz) + e q+1 1 − f =Λf, (D.16) 2(q + 1) r2 q + 1 dr Lynden-Bell, D., & Wood, R. 1968, MNRAS, 138, 495 Meltzer, D. W., & Thorne, K. S. 1966, ApJ, 145, 514 with the boundary conditions f (0) = f (α) = 0 (see Paper I). Misner, C. W., & Zapolsky, H. S. 1964, Phys. Rev. Lett., 12, 635 Introducing the variables defined in Sect. 2.3, the foregoing Oppenheim, J. 2002, Phys. Rev. D, 65, 024020 Oppenheimer, J. R., & Volkoff, G. M. 1939, Phys. Rev., 55, 374 equation can be rewritten as Padmanabhan, T. 1990, Phys. Rep., 188, 285 Penrose, R., & Floyd, R. M. 1971, Nature, 229, 177 d2 f 2 dψ d f 2qξ dψ + − + + qξeλ−ψ + eλ−ψ 1 + f = Pesci, A. 2007, Class. Quantum Grav., 24, 2283 dξ2 ξ dξ dξ q + 1 dξ Ponce de Leon, J., & Cruz, N. 2000, Gen. Relativ. Gravit., 32, 1207 Schmidt, H. J., & Homann, F. 2000, Gen. Relativ. Gravit., 32, 919 4 − 1 2q+1 qc q+1 2 −λ/2 − ψ Sire, C., & Chavanis, P. H. 2002, PRE, 66, 046133 ξ e e q+1 Λ f, (D.17) 2πG2 0 Sorkin, R. D., Wald, R. M., & Jiu, Z. Z. 1981, Gen. Relativ. Gravit., 13, 1127 Srednicki, M. 1993, Phys. Rev. Lett., 71, 666 with f (0) = f (α) = 0. This equation determines a discrete set of Tolman, R. C. 1934, Relativity, Thermodynamics and Cosmology (Oxford: eigenvalues Λ1 > Λ2 > Λ3, ... If all the eigenvalues Λ are nega- Clarendon) tive, then δ2N ≤ 0 and the configuration is a maximum of N at Tsallis, C. 1988, J. Stat. Phys., 52, 479 Weinberg, S. 1972, Gravitation and Cosmology (John Wiley) fixed mass. If at least one eigenvalue is positive, the configura- Zel’dovich, Ya. B. 1962, Soviet Phys. J.E.T.P., 14, 1143 tion is an unstable saddle point. We therefore need to determine