arXiv:gr-qc/0605066v3 10 Jul 2006 ievra ec negative a Hence vice-versa. alta h pcfi eti endthrough defined is be heat re- us must specific let the time, this, that appreciate long To call a too. gravity for Einstein in gravity valid heat”, Newtonian specific for “negative known occurrence of whether ask phenomenon fluid. can the the of one up perspective, heating thermodynamics and From accompanied radiation perspective, of be general emission must both a by collapse from gravitational actual demand that can simpli- and one collapse assumptions fications, actual associated another of their Put with solution equations of collapse. irrespective whether, gravitational way, whether during phys- such wonder effects of would occurrence ical for one reason fundamental sight, a is first there at en- effects obvious such radiated though, seem of Surprisingly, amount them[1]. increases. net steadily while hot- of ergy becomes collapse few fluid the the during that a nev- ter indicate details, only them their of in here all differ ertheless, vastly recall col- studies such GR may all spherical Though we There radiative and on neutrinos. studies lapse and many photons been as have such emis- radiation the of by accompanied sion be must collapse gravitational xetdt emr rnucdfrafli ujc to subject fluid a is for effect pronounced an more for such be only Intuitively to known expected is gravity. heat” Newtonian specific weak “negative of nomenon vice-versa. and system the from diation) T> that so dT temperature the raises compression Gravitational where and h rvttoa otato utb copne yEmiss by Accompanied be Must Contraction Gravitational Why ti eeal eivdta eea eaiitc(GR) Relativistic General that believed generally is It eepaiehr ht nasrc es,ti phe- this sense, strict a in that, here emphasize We dT .Anegative A 0. dQ stecrepnigiceeto temperature. of increment corresponding the is steaon fha netdit h system the into injected heat of amount the is ftecnrcinpoes hshpesbcuetettlN total the because happens This process. contraction the of rvttn ytmms eacmaidb ees fradiat of release by accompanied be must system gravitating adntjs h etna rvttoa oeta energy potential gravitational Newtonian the just not (and ewrs rvttoa olpe eaiitcAstrophy Relativistic Collapse, 04.20.Cv Gravitational 95.30.Lz, Keywords: 95.30.Sf, 04.40.Dg, numbers: PACS the of gravitation u irrespective too, been radiation gravity have of Einstein what in that, from energy” asserts different “internal process actually global are of energy” counterparts GR G “binding the the t that derive equating show we we simply time, by first mass mechanism the gravitational Kelvin for Helmholtz (Newtoni Here, and of theorem derived. version beli GR been justifiably no ever is However has it energy. too, (GR) radiation Relativity release General of era the yuigvra hoe,HlhlzadKli hwdta th that showed Kelvin and Helmholtz theorem, virial using By INTRODUCTION C ol hndemand then would P u enhsk apeceke ,D617Heidelberg D-69117 1, Saupfercheckweg Kernphysik, fur MPI dQ n the and en oso et(ra- heat of loss means nNwoinadEnti Gravity Einstein and Newtonian in nrilMass Inertial eal fteclas process collapse the of details Q< dQ C Dtd eray1,2018) 15, February (Dated: = dQ/dT and 0 ba Mitra Abhas fashrclysmercsai ud Simultaneously, fluid. static symmetric spherically a of udi yrdnmcleulbim tflosthat follows it equilibrium, hydrodynamical in rela- fluid a from below 5]: reviewed physics perspective[4, is basic modern process tively the H-K Without account, the process. historical behind (HK) may further Kelvin which into - generation going Helmholtz energy as elab- of called Kelvin[3] process be 1861, this in on later, orated years Few should bodies radiation. self-gravitating emit of contraction that 1854 in utb copne yeiso frdainadan and radiation collapse of GR temperature. emission fluid details, by increasing the accompanied exercise of be entire irrespective must The why, a show of fluid. conservation will symmetric energy spherically global of static Energy perspective Binding the and from energy Self-Gravitational defini- global GR at pro- arrive tions Newtonian automatically shall then this would we We of counterpart accordingly cess. 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Note that for the above mentioned ideal gas having an EOS p = nkT , Eq.(7) reduces to the more familiar form N 3 N Eg +3 p dV =4πR pb (1) Eg +2U = 0 because here γ = 5/3. However Eqs.(1-9) Z are valid for any fluid obeying relation (5) and not nec- where, EN is the Newtonian gravitational potential en- essarily by an “ideal gas” alone, as long as the fluid may g be assumed to obey an EOS of the form (5). Thus, in ergy, p = p(r) is the isotropic pressure, pb is the pres- sure at the boundary r = R, and the volume element principle, γ is arbitrary here subject to general thermo- dV =4πr2dr. For a laboratory gas sphere, it is possible dynamical constraints. to have p(r) uniform pb. In such a case, Eq.(1) In Eq.(9), EN is the Newtonian binding energy of the would reduce≈ to ≈ fluid and must be negative for a fluid which is already as- sumed to be bound. For attractive gravity, for any case, EN +4πR3 p 4πR3 p (2) bound or unbound, one must have EN < 0. In the present g b ≈ b g case, as soon as we set pb = 0, we imply the system to and which would express the obvious fact that for a lab- be self-bound, and one must have EN < 0. Thus from oratory gas EN 0. Note that it is necessary to have g ≈ Eq.(9), it transpires that, one must have γ > 4/3. A pb > 0 in such a case. limiting case of γ = 4/3 would signify a transition to In contrast, if the fluid is assumed to be self-contained, unbound systems. For “unbound systems” one expects i.e., bound by its own gravity then one expects to have, EN > 0. If the system would be unbound, one would have pb > 0 and further the system would not be pb = 0 (3) in hydrostatic equilibrium. In such a case, one needs to use dynamical form of virial theorem to study it. Also, and obtains the better known form of Eq.(1): by noting Eq.(7), one might think that for unbound sys- tems, one might have γ < 4/3. But this would be an incorrect conclusion, because as explained above, for un- EN +3 p dV = 0 (4) g Z bound systems, Eq.(7) would cease to be valid. It may be borne in mind that γ is an inherent thermodynam- This is known as static and scalar virial theorem. If the ical parameter and cannot be dictated by gross global adiabatic index of the fluid is γ, then hydrodynamical behavior of the fluid. p = (γ 1)e (5) A limiting value of γ =4/3 corresponds to a situation − when the momenta of the constituent particles of the where e is the internal energy density. For some ideal fluid, P = , and thus, cannot be strictly realized ex- ∞ fluids, γ may also be considered as the “ratio of specific cept for singular situations[6]. For instance note that the heats”. For example, for a monoatomic ideal gas having critical ultrarelativistic White Dwarf of Chandrasekhar an equation of state (EOS) p = nkT , where n is num- has R = 0 because it strictly corresponds to a fluid hav- ber density of the monoatomic molecules and k is the ing γ =4/3[2]. Boltzmann constant γ is the ratio of specific heats with If, additionally, the fluid obeys a polytropic equation a unique value γ 5/3. For an ultrarelativistic gas with of state particle momenta≡ or for a pure photon gas, γ is → ∞ again the ratio of specific heats having the unique value p = Kργp (10) γ 4/3[2]. The internal energy for the entire fluid is ≡ where, K and γp are uniform over the fluid, it follows U = e dV (6) that Z 3 GM 2 Using Eqs.(5) and (6) in (4), and assuming γ to be uni- EN = − (11) g 5 n R form, we have −

N where γp 1+1/n and G is the gravitational constant. Eg + 3(γ 1) U = 0 (7) ≡ − Note that, in general, γ = γp and only if the fluid is 6 so that considered to undergo adiabatic change, one would have γ = γp. Further, as we would see, all contraction pro- 1 N cesses are expected to be accompanied by emission of ra- U = − Eg (8) 3(γ 1) diation, and they must be non-adiabatic in a strict sense. − It may be also mentioned that, in the Newtonian case, The total Newtonian energy of the fluid is the fluid density appearing in Eq.(10) essentially means 3γ 4 rest mass density: ρ = ρ0. E = EN + U = − EN (9) N g 3(γ 1) g If one differentiates Eq.(8), for slow contraction, one − 3 will have that this result follows from the Newtonian HK process and does not depend on the details of either the physical dU 1 dEN = − g (12) properties of the fluid or the collapse process. dt 3(γ 1) dt − Since the above result is of generic nature, it is ex- pected to be qualitatively valid even in case of strong Also, from Eq.(11), we see that gravity. In fact, even after the introduction of General dEN 2 Relativity (GR) into astrophysics, the idea that gravi- g 3 GM ˙ = 2 R (13) tational contraction must result into radiation output is dt 5 n R − naturally and justifiably used[1]. While considering the ˙ N Since R < 0 for contraction, while the value of Eg de- configuration of static fluid spheres in GR, Buchdahl[7] N posed the question whether the amount total radiation creases during contraction its absolute value Eg in- N | | emitted by gravitational contraction can even exceed the creases. From Eq.(12) , we find that, as Eg increases 2 during such contraction, so does U. However,| the| amount initial value of E = Mc itself. The answer was in the of total gravitational energy released by the contraction, negative. Yet, for Einstein gravity no exact counterpart N of Eqs.(12-17) exists. Thus, we cannot assert, as a prin- dEg , is not fully accounted for by the gain in the value of| U: | ciple, that self-gravitating matter has “negative specific heat” in Einstein gravity too. And we want to address 1 dU = dEN < dEN (14) this precise aspect in the present paper. 3(γ 1)| g | | g | − For overall energy conservation, it is therefore necessary STATIC FLUIDS IN GENERAL RELATIVITY that the rest of the energy gain 1 3γ 1 Let us consider a static self-gravitating fluid sphere 1 dEN = − dEN = dQ (15)  − 3(γ 1) | g | 3(γ 1)| g | described by the metric − − 2 is radiated away by the system. This could have been ds2 = A2(r)dt2 B2(r)dr2 r2(dθ2 + sin θdφ2) (18) found directly by differentiating Eq.(9): − − Here we have taken G = c = 1. Recall that one often N 2 ν 2 λ dEN 3γ 4 dEg uses the symbols A = e = g00 and B = e = g . = − (16) − rr dt 3(γ 1) dt The radial coordinate r is the luminosity distance and as − before, the coordinate volume element is dV = 4πr2dr. Using Eq.(13) into above Eq., we find that The proper volume element however is 2 dE 3γ 4 3 GM 2 N = − R<˙ 0 (17) d = B(r) dV =4πr B(r) dr (19) dt 3(γ 1) 5 n R2 V − − Note that since B(r) > 1, d > dV , and this might be In case of a gas confined in a laboratory by physical seen as the effect of “stretching”V of space by gravity. The inclosure, p > 0. One can also imagine the physical b energy momentum tensor of the body in mixed tensor inclosure to be a perfect insulator and one may con- form is ceive of a radiationless adiabatic contraction for arbi- trary γ 4/3. But in an astrophysical context, there T i = (p + ρ)uiu + pgi (20) is neither≥ any physical inclosure nor any perfect insulat- k k k ing surrounding. Hence, it appears from Eq.(15) that a where ui is the fluid 4-velocity, ρ is the total mass energy strictly adiabatic contraction (dQ = 0) would be possi- density (excluding contribution due to self-gravitation), ble only for the idealized case of γ = 4/3. And Eq.(9) and p is the isotropic pressure. would show that, in such a case, one would already have The total mass energy density of a cold fluid (excluding EN = 0, i.e., they system would be unbound. In reality, negative self-gravitational energy) is one has γ =4/3 only for pure radiation or when the en- ∗ ergy of the particles per unit rest mass E = , which ρ = ρ0 + e (21) is possible only for a singular situation in case∞ the “gas” is not already a pure radiation[6]. where ρ0 = mN n is the proper rest mass energy density, Thus, Eq.(17) shows that the total (Newtonian) energy mN is nucleon rest mass, n is nucleon proper number of the system decreases for contraction and it could be density (not to be confused with polytropic index), e is i so only if the system radiates appropriate amount of en- the internal energy density, and gk is the metric tensor. ergy. Since U increases, the fluid become hotter while it The explicit form of B(r) is[8] radiates (dQ < 0). Therefore a self-gravitating fluid has − − a negative specific heat and this fact is well known. Note B(r) = [1 2M(r)/r] 1/2 = (1 2m/r) 1/2 (22) − − 4 where As before, note that, neither M0 nor U is defined with r respect to S∞. By using Eqs.(20-28), it can be verified M(r)= m = 4πρ r2dr (23) that Z0 EG + U = M M0 (30) The total energy of the system as perceived by a distant − inertial observer, S∞, i.e., the gravitational mass of the From the above equation, it may appear that the GR fluid is equivalent of EN , the binding energy, is

R R EGR = M M0 = EG + U (31) M = ρ dV = (ρ/B) d (24) − Z0 Z0 V Suppose the gas was initially infinitely dispersed to in- This total energy is also known as gravitational mass or finity. Further, let the gas molecules be rest with respect Schwarzschild mass of the fluid although actually this to S∞. Under such a case, the initial mass energy of the should have been named by the name of Hilbert. Note cloud is just the rest mass energy: that M(t =0)= M0 (32) R R M = (ρ/B) d = ρ d (25) And if the contraction of the cloud into a finite size would Z0 V 6 Z0 V indeed release energy, one must have M M0 < 0. But unlike the Newtonian case, as noted by− Tooper[9], it as one might have expected. The reason for this is that is not apparent from the definitions of E and U that total mass energy includes not only local contributions G E = E + U < 0! This is so because while in the from ρ but also the negative global contribution of self- GR G Newtonian case, EN and U are related through Eq.(7), gravitational energy. It is this negative latter contribu- g there is no known relationship between E and U in the tion which reduces the effective net proper energy density G GR case. Further there might be examples when the from ρ to ρ/B(r). occurrence of a negative E “is not a sufficient condi- On the other hand, the total proper energy content of GR tion for instability of the system against an expansion the sphere, by excluding any negative self energy con- to infinity”[9]. This means that E may not be the tribution , i.e., the energy obtained by merely adding GR correct GR equivalent of a “binding energy”. Had E individual local energy packets, is GR been the true GR equivalent of EN , probably, one would R have had equations similar to (8-9) involving EG, U and Mproper = ρ d (26) EGR. But no such equations exist. Thus, although, intu- Z0 V itively, one expects GR contraction to release radiation The conventional definition GR self- gravitational en- energy, one really cannot show it unlike the Newtonian ergy of the body is[7, 8] case developed in previous section. In Eq.(30), we may note that, while, EG, U and M0 R are not defined with respect to (w.r.t.) S∞, M, on the EG = M Mproper = (1 √ grr) ρ dV (27) − Z0 − − other hand is defined w.r.t. S∞. And this may be the fundamental reason that EGR defined in Eq.(31) may not Since B(r) > 1 in the presence of mass energy, EG is be the true “binding energy” of the fluid. a -ve quantity, as is expected. Note however that the EG defined by Eq.(27) is the sum of appropriate lo- cal (proper) quantities somewhat like the definition of ANOTHER DEFINITION OF FLUID MASS Mproper in Eq.(26); and is not defined with respect to the inertial observer S∞. Note also that, in contrast, gravi- For any stationary gravitational field, total four mo- tational mass M is indeed the mass-energy measured by mentum of matter plus gravitational field is conserved S∞. and independent of the coordinate system used[10, 11]: The proper rest mass energy of the fluid is P i = (T i0 + ti0) dV (33) M0 = ρ0 d (28) Z Z V where tik is the energy momentum pseudo-tensor asso- If there are no initial anti-baryons or anti-leptons, M0 = ciated with the gravitational field. Further, the inertial mN N, where N is the total number of baryons, and is mass (same as gravitational mass), i.e, the time compo- therefore a conserved quantity. nent of the 4-momentum of any given body in GR can The proper internal energy of the fluid is be expressed as[10–12] ∞ 0 1 2 3 3 U = e d (29) M = (T0 T1 T2 T3 ) √ g d x (34) Z V Z0 − − − − 5 where Accordingly, the total mass energy content of the fluid,

4 2 excluding any self-energy, as measured by an inertial g = r A(r)B(r) sin θ (35) − frame such as the distant observer, S∞ is different from 3 Mproper and is given by[8] (Eq. 11.1.19): is the determinant of the metric tensor gik and d x = dr dθ dφ. Since R 1 2 3 0 T1 = T2 = T3 = p; T0 = ρ (36) Mmatter = A(r) ρ d = A(r) B(r) ρ dV (41) − Z0 V Z it follows that[11, 12] Physically this means that local energy content in a given ∞ ∞ cell ρd is measured (redshifted) as A(r)ρd by the iner- V V M = (ρ +3p)A(r)B(r)dV = (ρ +3p)A(r)d tial observer S∞. Z0 Z0 V (37) In fact, the inertial observer S∞ would see the rest mass energy, i.e., the proper energy, of a nucleon too When the body is bound and pb = ρb = 0 for r R, then, the foregoing integrals shrink to[11, 12] ≥ to be reduced by the same factor √g00. Hence, when the body is finite and not dispersed to infinity, the total R R rest mass energy of the body, as reckoned by the inertial M = (ρ +3p)A(r)B(r)dV = (ρ +3p)A(r)d observer S∞, is Z0 Z0 V (38) R Although our result would not depend on splitting of M˜ 0 = ρ0 A(r) d (42) the foregoing equation, (since we would simply equate Z0 V the “total” expression of “inertial” and “gravitational” Similarly, the redshifted global internal energy of the fluid mass), we might nevertheless do so as measured by S∞ is R R R M = ρ A(r) d + 3 p A(r) d (39) ˜ Z0 V Z0 V U = e(r)A(r) d (43) Z0 V for the sake of obtaining physical insight. We can, now, quickly identify the 1st term on the RHS of Eq.(39) as Mmatter. Again bear in mind the fact that Global Definitions our eventual result would not depend on such identifica- tion or splitting because it would be obtained by equat- ing the “total” expressions for inertial and gravitational The Newtonian virial theorem (4) is essentially a masses. statement of energy conservation involving negative self- Further, we see that, the 2nd term on the RHS of gravitational energy and positive thermodynamic energy Eq.(39) is the global energy associated with pressure as of the fluid. In the Newtonian case, there exists global perceived by S∞. Accordingly, we rewrite Eq.(39) as: inertial frames and a statement of energy conservation can be made in a trivial way. But in GR, even for this M = Mmatter + Mpressure (44) simplest case of a static fluid sphere, there is no global inertial frame. Thus the exercise of having global defini- Again recall that M too is defined only with respect to tions of related energies and to enact their conservation S∞. Having done this splitting, we are in a position to is a highly non-trivial task. As is well known, for sta- obtain the GR Virial Theorem, which is essentially an tionary systems, however global energy can be defined accounting of global energies involved in the problem. in a meaningful way for asymptotically flat spacetimes. And since global energies, in GR, can be defined only Further, when the energy is defined with reference to an w.r.t. the inertial observer S∞, all relevant integrals must observer at a spatial infinity (S∞), we obtain the so-called be defined w.r.t. the same observer. And this is what we ADM Mass[13]. Also global energy conservation can be have just done. meaningfully defined only with reference to S∞. Note that EG and U are essentially summation over local appropriate values and not over the corresponding GR HELMHOLTZ KELVIN PROCESS quantities measured by S∞. It may be mentioned that, if any locally measured energy is ǫ, then the energy mea- Let us simply transpose Eq.(39) (irrespective of its sured by the far away inertial observer is the redshifted splitting) as quantity

R R ρ A(r) d M + 3p A(r)d = 0 (45) ǫ∞ = √g00 ǫ = A(r) ǫ (40) Z0 V− Z0 V 6 to reexpress as When we carry out this above integration with ρ = constant, we obtain R ˜ Eg + 3p A(r) d = 0 (46) 3 M 2 Z0 V E˜ = − (55) g 5 R Or, ˜ N Therefore, in the Newtonian limit, Eg = EG = Eg , ˜ E˜g + 3p √ g00 grr dV = 0 (47) though in general Eg and Eg are different. Z − Now, using the thermodynamical relation (5) and where Eq.(43) in Eq.(46), as before, we will have

E˜g + 3(γ 1)U˜ = 0 (56) E˜g = M Mmatter = (AB 1) ρ dV (48) − − Z − By direct comparison with Eqs.(4) and (7), we can easily Or else, identify Eqs.(48) and (56) as the appropriate GR version of static scalar virial theorem. Note that Eqs.(46) and

E˜g = (√ g00grr 1) ρ dV (49) (56) naturally reduce to their Newtonian forms, Eqs.(1) Z − − and (7) for sufficiently weak gravity with g00 grr 1. Thus we may interpret the existence of the≈− Newtonian≈ Since AB < 1 in the presence of mass-energy, we have virial theorem too as due to equivalence of “gravitational E˜ < 0 as is expected. Clearly, we have, obtained, now g mass” and “inertial mass”. an equation similar to (4). It appears then that the above From Eq.(56), we obtain defined E˜g rather than the previously defined EG is the true measure of self-gravitational energy as perceived by 1 U˜ = − E˜ (57) an inertial observer S∞. This is so because EG is not 3(γ 1) g defined with respect to S∞, the accountant for global − energy. In contrast, both the components of E˜g, namely, If the fluid undergoes quasistatic contraction and ∆ de- M and Mmatter are defined w.r.t. S∞. notes the associated changes in relevant quantities, then Further, recall that, in GR, the effect of “gravitational we will have potential” is conveyed by g00. But EG (Eq.[27]) does not contain g00 at all. In contrast, E˜g indeed involves 2 1 +1 gravitational potential term g00 = A (Eq.[49]). ∆U˜ = − ∆E˜ = ∆E˜ (58) 3(γ 1) g 3(γ 1)| g| What would be the value of true global GR self- − − ˜ gravitational energy Eg in the Newtonian limit? ˜ To see this we consider a sphere with ρ = constant for Here we have used the fact that since Eg < 0 it must decrease for contraction. Thus as is expected, the in- which[8] ternal energy of the fluid must increase for gravita- 1 − tional contraction. If the appropriately averaged value A(r)= [3(1 2M/R)1/2 B(r) 1] (50) 2 − − of A(r)=g ¯00 during this contraction, the increase in proper internal energy would be Using Eq.(22) in (50), we further see that ∆U˜ 1 ∆Eg 1 1 2 ∆U = = | | > 0 (59) A(r)B(r) = [3(1 2M/R) / B(r) 1] (51) g¯00 3(γ 1) g¯00 2 − − − Since g00 < 1 in the presence of gravity, this means that, Again using Eq.(22) in (51), we obtain the rate of increase in proper internal energy would be

1 − higher than in the corresponding Newtonian case. A(r)B(r)= [3(1 2M/R)1/2(1 2m/r) 1/2 1] (52) 2 − − − As we look back at Eq.(58), the increase in the value of E˜ , namely ∆E˜ is not fully accounted for by the | g| | g| Now if we proceed to linearized gravity limit with increase in the value of U˜. Note that in the absence of M/R 1 and m/r 1, we will have ≪ ≪ initial antiparticles, the contribution of rest mass-energy 3 is unaffected during the process. Therefore, for the sake A(r)B(r)=1 (m/r M/R)] (53) of global energy conservation, as reckoned by the inertial − 2 − observer S∞, the fluid must radiate out an amount of Using Eq.(52) in (48), we see that energy +∆Q given by

3 1 ˜ 3γ 4 ˜ E˜g = (M/R m/r)ρ dV (54) ∆Q = 1 ∆Eg = − ∆Eg (60) −2 Z −  − 3(γ 1) | | 3(γ 1)| | − − 7 in order to be able to contract. Note, as before, that We may see that, unlike in Eq.(30), all the quantities γ > 4/3 in a strict sense, as long as particle momenta are involved in Eq.(63) are defined w.r.t. S∞ and which sug- finite. To see that in the GR context too, that γ = 4/3 gests that E˜ is indeed the true binding energy of the implies singular situation, look at the Ist row of Table I fluid. The reader is again requested here to appreciate of [14] which shows that in this case again Rmax = 0, the subtle point why the true binding energy of the fluid where Rmax is the maximum possible radius of the con- is given by Eq.(63) rather than by Eq.(31).: figuration. Further, for γ = 4/3, the next entry in the Though M0 is a conserved quantity, once the fluid is 2 same row shows that R0 = 0 where R0 = 2GM/c is contracted into a finite size, it gets dissociated from the the Schwarzschild radius. This implies that for γ =4/3, inertial frame S∞. And S∞, who is doing the global en- one has R0 = 0. This latter fact implies that even in ergy accounting, sees the locally defined rest mass energy 2 GR, the total mass energy Mc = 0 just as EN = 0 for to be redshifted to M˜ 0 rather than as M0. On the other γ =4/3 (Eq.[9]). Occurrence of Rmax = 0 implies a fluid hand, the total mass-energy of the fluid, again defined sphere that has collapsed to a singular point. And as w.r.t. S∞ is M. Therefore, the global binding energy per Ref.(14), the singular configuration then would have of the fluid, i.e., the difference between the total mass zero mass energy. Incidentally, in the classic paper[13], energy and rest mass energy as seen by the same inertial Arnowitt, Deser and Misner too found that a neutral observer S∞ is M M˜ 0. Note that the existence of equa- “point particle” has zero “clothed mass”. Then Chan- tions (62-63) does− not depend on such interpretations drasekhar’s exercise[14], in addition, suggests that if the because they, in any case, crept up spontaneously. fluid would attain such a singular state, the value of Although, in a Newtonian case, the notion of a bind- γ 4/3. This is also in perfect agreement with the ing energy always existed, to the best of our knowledge, → notion that a singular state should be infinitely hot with such a notion was never before properly derived in the complete domination of radiation energy over rest mass GR context. A relativistic “bound system” may thus be energy[6]. defined as one having E˜ < 0, and an “unbound system” In fact, in one would misconceive of a situation with will have E˜ > 0. γ < 4/3, irrespective of whether it is a Newtonian or a While, in the Newtonian case, “Total Energy” is the GR case, one would have to ensure injection of energy binding energy EN , in GR, total global energy, as mea- 2 into the system to let it collapse. This would mean that sured by S∞, always is E = Mc . in the absence of external injection of energy, the system would not contract/evolve at all in defiance of basic tenet of global gravitation. DISCUSSIONS Thus, one must indeed have γ > 4/3 and, in a strict sense, there cannot be any adiabatic gravitational contrac- The important idea of Helmholtz and Kelvin, devel- tion. Consequently, all strictly adiabatic gravitational oped in the 19th century, that gravitational contraction collapse studies are of only academic interest. Thermo- should both raise the internal energy and cause the fluid dynamically, the global specific heat of the contracting to radiate was always expected to be valid irrespective of fluid is negative because dQ < 0 while the temperature the strength of the gravity. However, the original deriva- and internal energy of the fluid increase. tion to this effect was made in the framework of extremely weak, i.e., Newtonian gravity. We showed here that even for arbitrary, strong gravity, this process indeed remains GENERAL RELATIVISTIC BINDING ENERGY valid. In fact, as shown by Eq.(59), the process becomes even more effective compared to the Newtonian case as Eq.(60) suggests that, we might isolate a quantity gravity increases. Pictorially, we may think that stronger 3γ 4 gravity churns out more radiation from matter even in E˜ = − E˜ (61) 3(γ 1) g the absence of chemical or thermonuclear energy sources. − A similar conclusion is supported by a recent work which as the total energy of the system excluding any rest mass shows that the ratio of radiation energy density to rest contribution. This is actually the “Binding Energy” of mass energy density of a self-luminous contracting ob- the gravitating system, defined by a given distribution of ject is proportional to its surface redshift z [6]. When mass-energy. z 1, the self-gravitating contracting object is “matter ≪ By using Eq.(57), it is seen that dominated”, i.e., ρ0 ρ , but when, z 1, the object ≫ r ≫ becomes radiation dominated : ρ ρ0 like the very ˜ ˜ ˜ r E = Eg + U (62) early Universe[6]. The present study≫ provided an addi- Further using Eqs.(39), (42), (43) and (48), it also tran- tional explanation for this result. These studies show spires that that the actual fate of radiative physical gravitational collapse could be radically different from traditional pic- E˜ = E˜ + U˜ = M M˜ 0 (63) tures of continued gravitational collapse inspired by the g − 8 pressureless dust collapse where a Black Hole (BH) or a “global energy” from any preferred theoretical perspec- Naked Singularity is catastrophically formed in a finite tive, correct or incorrect. This is so because, unlike a comoving proper time. Traditional GR collapse studies generic case, the definition of “global mass energy” of are usually done by (i) assuming dust models with p 0 a chargeless static spherically symmetric fluid (measured ≡ even when the fluid is supposed to have collapsed to sin- by S∞) is very well known since long[10, 12]. And we just gularity, or (ii) considering pressure but neglecting all appealed to the Principle of Equivalence to equate the heat transport, dQ 0. But as shown by Eqs. (58) and already well known expressions for “gravitational mass” ≡ (60), dQ = 0 implies (a) dU˜ = 0 and (b) dE˜g = 0. The (Schwarzshild mass) and “inertial mass” - the time com- condition (a) is satisfied only for dust and thus despite a ponent of linear 4-momentum. It is this simple opera- formal consideration of existence of pressure, in the con- tion which yielded the GR virial theorem and GR HK text of collapse, a fluid satisfying condition (a) becomes mechanism (Eqns. [12-17]). It is the same principle of similar to a pressureless, internal energyless dust. The equivalence which demanded that, from energy conser- condition (b) is not satisfied even for a dust unless it vation considerations, all the relevant globally defined has M = U = Eg = fixed = 0 at the beginning of the energies are defined w.r.t. the unique inertial frame S∞ collapse. But no isolated fluid with finite size can have rather than w.r.t. a series of (infinite) proper frames. To M = 0 (the Universe may, however, have M = 0 even the best of our knowledge, this is the maiden derivation being of finite or infinite extent). In several numerical of HK process using GR. And this is also the maiden studies of supposed radiative collapse, one implicitly or proper GR explanation for the intuitive notion that a 2 explicitly assumes Q M0c . At the advanced stage self-gravitating object has effective global “negative spe- of collapse, this assumption≪ fails[6] and such cases effec- cific heat” in Einstein gravity too. tively become similar to case (ii) of adiabatic continued For further appreciation of the “global quantities” in- collapse valid for M = 0. volving √g00 in our study which arose spontaneously and In contrast, in a breakthrough research on physical not imposed from new theoretical perspective, we recall gravitational collapse, Herrera and Santos[1] have shown that the “Poisson’s Equation” in GR has the form[16] that the force exerted on the collapsing fluid by the out- 2 ward propagating heat/radiation may stall the continued √g00 =4πG√g00 (ρ +3p) (64) ∇ collapse process and formation of either a finite mass BH or Naked Singularity may be averted. Herrera, Di And only when one moves to weak gravity with Prisco and Barreto[1] have successfully made a numeri- GM/r 1 and √g00 1 GM/r 1, one obtains ≪ ≈ − ≈ cal model of continued collapse to substantiate this path- the more familiar form breaking idea. Such ideas are consistent with the model independent generic studies[6] which show that contin- 2 ued catastrophic collapse indeed degenerates into a ra- φ =4πG (ρ +3p) (65) ∇ diation pressure supported hot quasistatic state called where the weak “gravitational potential φ GM/r. Of eternally collapsing objects because of outward force due ∼ to collapse generated radiation at extremely deep grav- course, Eq.(64) would also degenerate into Eq.(65) in a itational potential wells, z 1 where z is the surface local free falling frame where g00 = 1. But for fluids hav- gravitational redshift of the≫ collapsing object. It is be- ing finite pressure there cannot be any such global free cause of the resultant reduction in the value of M due to falling frame and therefore Eq.(65), in the present con- continuous radiation outpour that no apparent horizon text, can be recovered only for weak gravity (in any case or event horizon is formed until M = R = 0[6]. we are dealing with a static fluid. Formally a strict adi- Finally, the GR definition of “binding energy” of a abatic collapse is possible only for a pressureless “dust” static fluid is M M˜ 0 rather than M M0. with p = U 0, though, physically, M = 0 in such a − − ≡ Newtonian HK process is a direct sequel of static New- case. tonian virial theorem. Similarly, we needed to derive Since virial theorem is important for the study of com- the exact GR version of the static virial theorem. This pact objects and gravitational contraction, the exact rel- GR virial theorem involved globally defined quantities ativistic virial theorem obtained here could be useful for measured w.r.t. the same inertial observer S∞. In gen- relativistic astrophysics, either now or in future. eral, the notion of global energy is far from transparent and unique in GR. For example, for non-static and non- spherically symmetric systems or charged systems, there ACKNOWLEDGEMENTS could be various notions of “energy” and “mass”; to ap- preciate this one may have a look at a recent long review The author thanks Felix Aharonian for encouragement. paper [15]. However the present paper must not be con- MPI fur Kernphysik, Heidelberg is thanked for the kind fused with such studies. The aim of this paper was not invitation and hospitality. Both the anonymous referees at all to define any new definition of either “mass” or are also thanked for unprejudiced fair assessment of the 9 manuscript and some minor suggestions. Finally, the Ed- Soc., 216, 1001, (1985); L. Herrera, J. Jimenez, and G.J. itorial Staff members of Phys. Rev. D. are thanked for a Ruggeri, Phys. Rev. D., 22(10), 2305 (1980); P.C. Vaidya, speedy processing of this manuscript. Astrophys. J., 144, 943 (1966) [2] S. Chandrasekhar, An Introduction to the Study of Stellar Structure (Dover, 1967) [3] Lord Kelvin, 1861, Brit. Assoc. Repts., Part II, 26 (1861) [4] R.L. Bowers and T. Deeming, Astrophysics I, Stars (Jones and Bartlett, Boston, 1984) ∗ [email protected] Electronic address: [5] R. Kippenhahn & A. Weigert, 1990, Stellar Structure and † Also at Theoretical Astrophysics Section, Bhabha Evolution, (Springer-Verlag, Berlin, 1990) Atomic Research Center, Mumbai-400085, India [6] A. Mitra, Mon. Not. R. Astron. Soc. Lett., 367, L66, [1] L. Herrera, A. Di Prisco, and W. Barreto, Phy. Rev. (2006) (gr-qc/0601025); also, Mon. Not. R. Astron. Soc, D73, 024008 (2006); L. Herrera & N.O. Santos, Phys. 369, 492, (2006), (gr-qc/0603055); Found. Phys. Lett., Rev. D70, 084004, (2004); S. Maharaj and M. Goven- 13, 543 (2000), Found. Phys. Lett., 15, 439 (2002), astro- der, Int. J. Modern Phys. D, 14, 667 (2005); L. Herrera, ph/0408323, gr-qc/0512006 Phys. Lett. A, 300, 157 (2002); S. Wagh, M. Govender, [7] H.A. Buchdahl, Phys. Rev., 116, 1027 (1959) K. Govinder, S. Maharaj, P. Muktibodh and M. Moodly, [8] S. Weinberg, Gravitation, (John Wiley, NY, 1972) Class. Quantum Grav., 18 , 2147 (2001); M. Govender [9] R.F. Tooper, ApJ., 140, 434 (1964) and K. Govinder, Phys. Lett. A, 283, 71 (2001); M. [10] R.C. Tolman, Phys. Rev., 35, 875 (1930) Govender, R. Maartens and S. Maharaj, Month. Not. [11] L.D. Landau & E.M. Lifshitz, The Classical Theory of R. Astron. Soc., 310, 557 (1999) M. Govender, S. Ma- Fields, (Pergamon Press, Oxford, 4th ed., 1975) haraj and R. Maartens, Class. Quantum Grav.,15, 323 [12] E. Whittaker, Proc. Roy. Soc. A, 149, 384 (1935) (1998); L. Herrera and J. Mart´ınez, Gen. Rel. Grav., 30, [13] R. Arnowitt, S. Deser, and C. Misner, Phys. Rev., 122, 445 (1998); L. Herrera and J. Mart´ınez, Astrophysics and 997 (1961). Also see, gr-qc/0405109 Space Science, 259, 235 (1998); A. Di Pris’co, L. Herrera [14] S. Chandrasekhar, Phys. Rev. Lett., 12, 114 (1964) and M. Esculpi, and N.O. Santos, Gen. Rel. Grav., 29, [15] L.B. Szabados, 2002, Living Reviews of Relativity, 7, 4 1391 (1997); A. Di Pris’co, L. Herrera and M. Esculpi, (2002) Class Quantum Grav., 13, 1053 (1996); B.C. Tewari, As- [16] C.S. Unnikrishnan & G.T. Gillies, Phys. Rev. D, 73, trophys. Sp. Sc., 149(2), 233 (1988); A.K.G. de Oliveria, 101101(R), (2006) N.O. Santos, and C.A. Kolassis, Mon. Not. R. Astron.