Relativistic Gravitational Collapse by Thermal Mass
aa r X i v : . [ g r- q c ] S e p Relativistic Gravitational Collapse by Thermal Mass
Zacharias Roupas ∗ Centre for Theoretical Physics, The British University in Egypt, Sherouk City 11837, Cairo,Egypt
Abstract
Gravity and thermal energy are universal phenomena which compete over the stabilization ofastrophysical systems. The former induces an inward pressure driving collapse and the latter astabilizing outward pressure generated by random motion and energy dispersion. Since a contract-ing self-gravitating system is heated up one may wonder why is gravitational collapse not halted inall cases at a sufficient high temperature establishing either a gravo-thermal equilibrium or explo-sion. Here, based on the equivalence between mass and energy, we show that there always existsa temperature threshold beyond which the gravitation of thermal energy overcomes its stabilizingpressure and the system collapses under the weight of its own heat.
Mass and energy are equivalent and gravitate irrespective from their origins. Likewise, motion ofmaterial particles and transmission of energy by field waves are phenomena that manifest the universalnature of heat. Thus, gravity and thermal energy are the two universal properties of physical realityknown as yet, while they are ultimately tightly connected. Gravity causes motion and vice versa.Nevertheless, they have counter primary effects on the stability of systems. Gravity drives contraction,while thermal pressure expansion.When a system contracts adiabatically it is heated. Since thermal energy gravitates there mightexist a threshold upper temperature beyond which gravity of heat overcomes its outward pushingpressure. This is the question we address. The answer may be emphasized, by using heuristicallyonly three basic equations of physics; Einstein’s equivalence between mass and energy m = E/c ,Newton’s law of gravitational energy E gravity ∼ − GM /R and the equation of internal thermal energy E thermal ∼ N kT . Assuming that thermal energy gravitates and since gravitational energy grows tothe square of mass it follows that gravitational energy grows to the square of temperature. However,thermal energy grows only linearly with temperature. Thus, there should exist a temperature at whichgravitational energy induced by thermal mass overcomes thermal energy. Consider in particular aspherical self-gravitating system with radius R and mass M = N m + E thermal /c consisting of thetotal rest mass of N constituents with mass m and the thermal mass M thermal = E thermal /c . Theself-gravitational energy of the system to first order in ( GM/Rc ) is proportional to E gravity ∼ − G M R = − N G m R (cid:18) kTmc (cid:19) . (1)The gravitational energy overcomes the thermal pressure and the system gets destabilized if | E gravity | > E thermal ⇒ (cid:18) kTmc (cid:19) > Rc GM rest kTmc . (2)The smaller root of this inequality, call it T , suggests that for sufficiently low temperature T < T the gravitational energy overcomes thermal energy as expected (gravothermal catastrophe). The new ∗ Essay written for the Gravity Research Foundation 2020 Awards for Essays on [email protected] Weinberg, as in section 11.1 of Ref. [1] has shown that the relativistic self-gravitational energy E gravity may beexpanded in terms of G M ( r ) /rc as E gravity = − R R πr (cid:26) G M ( r ) rc + (cid:16) G M ( r ) rc (cid:17) + O (cid:0) ( G M /rc ) (cid:1)(cid:27) ρ ( r ) c dr, where ρ ( r ) denotes the mass density at radius r and M ( r ) denotes the total mass-energy included in radius r . Since ρ ∼ M ,the relativistic self-gravitational energy of a static body is of order E gravity ∼ O (cid:0) M (cid:1) . T c , steaming from thebigger root of (2) that is generated by the gravitation of thermal mass. For any T > T c (3)the outward pointing pressure of thermal energy is overcome by its own gravitation and the systemgets destabilized. The characteristic temperature for kT /mc ≫ kT c ≃ mc Rc GM rest . (4)Therefore, by this simple analysis it is in addition predicted that this characteristic temperaturedepends on the compactness of rest mass GM rest /Rc .The result (4) is astonishingly close to the exact one, obtained by a precise relativistic treatment[2–6]. We consider the equation of state of the relativistic ideal gas P ( ρ ( r ) , T ( r )) = ρ ( r ) c b ( r )(1 + F ( b ( r ))) , F ( b ) = K ( b ) K ( b ) + 3 b − b = mc kT , (5)where K ν ( b ) are the modified Bessel functions, P is the pressure and ρ is the total mass density.The equation of state may also be written as P = ( ρ rest /m ) kT where ρ rest is the rest mass density.Keeping the rest mass compactness fixed at several prescribed values, we solve the relativistic equationof hydrostatic equilibrium and Tolman equation, respectively,d P ( r )d r = − (cid:18) ρ ( r ) + P ( r ) c (cid:19) (cid:18) Gµ ( r ) r + 4 πG P ( r ) c r (cid:19) (cid:18) − Gµ ( r ) rc (cid:19) − , (6)d T ( r )d r = T ( r ) P ( r ) + ρ ( r ) c d P ( r )d r , (7)where µ ( r ) is the total mass-energy contained within radius r . Indeed, we find that the characteristictemperature depends on the rest mass compactness of the system as predicted by our previous heuristicanalysis. In Figure 1(a) is plotted the caloric curve for GM rest /Rc = 1 /
4, a typical rest masscompactness value of cool neutron stars. Each point corresponds to a thermodynamic equilibriumconfiguration. All points in the branch LH are stable, while all other branches (the two spirals) areunstable and the empty regions signify the complete absence of any kind of equilibrium. In y-axisis denoted the Tolman temperature T Tolman , which is the global quantity that remains constant inequilibrium [7], while local temperature T ( r ) in Gravity attains a radial gradient (7) according to theTolman-Ehrenfest effect [8].The caloric curve has the form of a double spiral, discovered firstly in [2]. As we increase the restmass, the spiral shrinks and becomes a point for GM rest /Rc = 0 .
35. No equilibria may be attainedabove this value.The double spiral manifests two kinds of gravothermal instabilities. The low-energy one desig-nated with point L is the relativistic generalization of Antonov instability, called also ‘gravothermalcatatrophe’ [9, 10]. This instability is of Newtonian origin and is caused due to the decoupling of aself-gravitating core with negative specific heat capacity from a halo with positive specific heat ca-pacity. A heat transfer from the core to the halo cannnot be reversed. This low-energy instability inadiabatic conditions occurs as the system gets expanded and not contracted. The subsequent coolingresults in weakening heat’s ability to sustain the gravity of rest mass.The upper spiral on the other hand designates a purely relativistic instability caused by theweight of heat during contraction and heating [2, 6]. In Figure 1(b) it is evident that starting fromthe equilibrium L and heating up the system a little, the density contrast, namely the edge over thecentral density of the sphere, increases. The system becomes more homogeneous as expected becauseof the increase of thermal energy over gravitation. However, at a certain temperature, designatedby point Σ, a peculiar reverse occurs. The system becomes less homogeneous while heated! Thegravity of thermal energy dominates over its outward pointing pressure. Heating the system furtherresults only in condensing the system and the appearance of the relativistic gravothermal instability,designated by point H , that is caused by the weight of heat.2 (a) Double spiral of the carolic curve (b) Density contrast Figure 1:
The series of hydrostatic, thermal equilibria of the relativistic self-gravitating ideal gas for GM rest /Rc = 1 /
4. The branch LH is stable, while all points not in this branch are unstable and the emptyregions outside points L , H signify the absence of equilibria. At point Σ the gravitation of thermal mass takesover its outward pushing thermal pressure and any further temperature increase results in decreasing the edgeto central density instead of homogenizing the system. Point L designates the relativistic generalization of theNewtonian gravothermal catastrophe and point H the relativistic gravothermal instability caused by thermalmass. This result suggests that the gravitational collapse to a black hole cannot be halted by thermalenergy alone. When all fuels and other stabilizing mechanisms are exhausted, thermal energy cannotcome to the rescue no matter how hot the collapsing system becomes.This phenomenon especially applies to supernova. Equation (4) and Figure 1(a) suggest that kT c is expected to be of the same order with mc for a supernova core, where m should be identifiedwith the rest mass of the primary gas constituent. If the remnant core attains a temperature anddensity that allows QCD deconfinent [11–14], m can be identified with the rest mass of up quarkor even strange quark at higher densities [15, 16] giving kT c ∼ − ∼ M eV [17]. If the core attainshigher temperature, it will continue to collapse under the weight of its own heat. Because of the core’snegative specific heat capacity, even if significant amount of radiation is emitted from the core it willonly result in further increasing its temperature and thus its gravitational thermal mass.The relativistic gravothermal instability shall in addition have implications for cosmology. In anopen Universe, an early region which becomes the observable Universe cannot be arbitrarily denseand hot in the past. For a certain compactness there always exists some temperature above which thesystem cannot expand because of the extra gravity of heat. The same is true for density fluctuationsin an open Universe and for the whole patch of a closed Universe. Nevertheless, the expansion addsan additional inertial force, pushing outwards, which should be taken into account.We highlighted here some basic underlying physics of gravitational instability that has been, asyet, evading attention. This is that random motion and energy dispersion can generate such intensegravity that overcomes its own stabilizing, against gravitational collapse, pressure. This effect impliesthe inevitability of black hole formation and shall dominate its very late stages.
Acknowledgements
The author thanks Amr El-Zant for discussions on cosmology.
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