Abhas Mitra

Why gravitational contraction must be accompanied by emission of radiation in both Newtonian and Einstein gravity

(and not just the Newtonian gravitational potential energy E N g ) decreases for such contraction. In the era of General Relativity (GR) too, it is justifiably believed that gravitational contraction must release radiation energy. However no GR version of (Newtonian) Helmholtz- Kelvin (HK) process has ever been derived. Here, for the first time, we derive the GR version of the appropriate virial theorem and Helmholtz Kelvin mechanism by simply equating the well known expressions for the gravitational mass and the Inertial Mass of a spherically symmetric static fluid. Simultaneously, we show that the GR counterparts of global “internal energy”, “gravitational potential energy” and “binding energy” are actually different from what have been used so far. Existence of this GR HK process asserts that, in Einstein gravity too, gravitational collapse must be accompanied by emission of radiation irrespective of the details of the collapse process.

Radiation pressure supported stars in Einstein gravity: eternally collapsing objects

Even when we consider Newtonian stars, that is, stars with surface gravitational redshift z « 1, it is well known that, theoretically, it is possible to have stars supported against self-gravity almost entirely by radiation pressure. However, such Newtonian stars must necessarily be supermassive. We point out that this requirement for excessively large M in the Newtonian case is a consequence of the occurrence of low z « 1. However, if we remove such restrictions, and allow for the possible occurrence of a highly general relativistic regime, z >>1, we show that it is possible to have radiation pressure supported stars (RPSSs) at an arbitrary value of M. Since RPSSs necessarily radiate at the Eddington limit, in Einstein gravity, they are never in strict hydrodynamical equilibrium. Further, it is believed that sufficiently massive or dense objects undergo continued gravitational collapse to the black hole (BH) stage characterized by z = oo. Thus, late stages of BH formation, by definition, would have z >>1, and hence would be examples of quasi-stable general relativistic RPSSs. It is shown that the observed duration of such Eddington limited radiation pressure dominated states is t ≈ 5 x 10 8 (1 + z) yr. Thus, t → ∞ as BH formation (z → ∞) takes place. Consequently, such radiation pressure dominated extreme general relativistic stars become eternally collapsing objects (ECOs) and the BH state is preceded by such an ECO phase. This result is also supported by our previous finding that trapped surfaces are not formed in gravitational collapse and the value of the integration constant in the vacuum Schwarzschild solution is zero. Hence the supposed observed BHs are actually ECOs.

Quantum information paradox: Real or fictitious?

One of the outstanding puzzles of theoretical physics is whether quantum information indeed gets lost in the case of black hole (BH) evaporation or accretion. Let us recall that quantum mechanics (QM) demands an upper limit on the acceleration of a test particle. On the other hand, it is pointed out here that, if a Schwarzschild BH exists, the acceleration of the test particle would blow up at the event horizon in violation of QM. Thus the concept of an exact BH is in contradiction with QM and quantum gravity (QG). It is also reminded that the mass of a BH actually appears as an integration constant of Einstein equations. And it has been shown that the value of this integration constant is actually zero! Thus even classically, there cannot be finite mass BHs though zero mass BH is allowed. It has been further shown that during continued gravitational collapse, radiation emanating from the contracting object gets trapped within it by the runaway gravitational field. As a consequence, the contracting body attains a quasi-static state where outward trapped radiation pressure gets balanced by inward gravitational pull and the ideal classical BH state is never formed in a finite proper time. In other words, continued gravitational collapse results in an ‘eternally collapsing object’ which is a ball of hot plasma and which is asymptotically approaching the true BH state with M = 0 after radiating away its entire mass energy. And if we include QM, this contraction must halt at a radius suggested by the highest QM acceleration. In any case no event horizon (EH) is ever formed and in reality, there is no quantum information paradox.

Likely formation of general relativistic radiation pressure supported stars or ‘eternally collapsing objects’

Hoyle & Folwler showed that there could be Radiation Pressure Supported Stars (RPSS) even in Newtonian gravity. Much later, Mitra found that one could also conceive of their General Relativistic (GR) version, “Relativistic Radiation Pressure Supported Stars” (RRPSSs). While RPSSs have z � 1, RRPSSs have z � 1, where z is the surface gravitational redshift. Here we elaborate on the formation of RRPSSs during continued gravitational collapse by recalling that a contracting massive star must start trapping radiation as it would enter its photon sphere. It is found that, irrespective of the details of the contraction process, the trapped radiation flux should attain the corresponding Eddington value at sufficiently largez � 1. This means that continued GR collapse may generate an intermediate RRPSS with z � 1 before a true BH state with z = 1 is formed asymptotically. An exciting consequence of this is that the stellar mass black hole candidates, at present epoch, should be hot balls of quark gluon plasma, as has been discussed by Royzen in a recent article entitled “QCD against black holes?”.

Comments on “The Euclidean gravitational action as black hole entropy, singularities, and space-time voids” [J. Math. Phys. 49, 042501 (2008)]

We point out that the space-time void inferred by Castro [J. Math. Phys. 49, 042501 (2008)] results from his choice of a discontinuous radial gauge. Further since the integration constant α0=2M0 (G=c=1) occurring in the vacuum Hilbert/Schwarzschild solution of a neutral “point mass” is zero [Arnowitt et al., in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962), Chap. 7, p. 227; also Phys. Rev. Lett. 4, 375 (1960). A. Mitra, Adv. Space Res. 38, 2917 (2006); Proceedings of the XIth Marcel-Grossmann Conference on General Relativity (World Scientific, Singapore, 2008), Vol. 3, p. 1968], Castro’s gauge reduces to the well behaved and physical Hilbert gauge. Physically this means that true Hilbert/Schwarzschild black holes have unique gravitational mass M=0. Accordingly, the unphysical space-time void inferred by Castro is actually nonexistent.

A generic relation between baryonic and radiative energy densities of stars

By using elementary astrophysical concepts, we show that for any self-luminous astrophysical object the ratio of radiation energy density inside the body (ρr) and the baryonic energy density (ρ0) may be crudely approximated, in the Newtonian limit, as ρr/ρ0∝GM/Rc2, where G is constant of gravitation, c is the speed of light, M is gravitational mass and R is the radius of the body. The key idea is that radiation quanta must move out in a diffusive manner rather than stream freely inside the body of the star. When one would move to the extreme general relativistic case, i.e. if the surface gravitational redshift z≫ 1, it is found that ρr/ρ0∝ (1 +z). Earlier treatments of gravitational collapse, in contrast, generally assumed ρr/ρ0≪ 1. Thus, actually, during continued general relativistic gravitational collapse to the black hole state (z∞), the collapsing matter may essentially become an extremely hot fireball with ρr/ρ0≫ 1, a la the very early Universe, even though the observed luminosity of the body as seen by a faraway observer L∞∝ (1 +z)−1 0 as z∞, and the collapse might appear as ‘adiabatic’.

Einstein energy associated with the Friedmann–Robertson–Walker metric

Following Einstein’s definition of Lagrangian density and gravitational field energy density (Einstein in Ann Phys Lpz 49:806, 1916, Einstein in Phys Z 19:115, 1918, Pauli in Theory of Relativity, B.I. Publications, Mumbai, 1963), Tolman derived a general formula for the total matter plus gravitational field energy (P0) of an arbitrary system (Tolman in Phys Rev 35:875, 1930, Tolman in Relativity, Thermodynamics & Cosmology, Clarendon Press, Oxford, 1962, Xulu in hep-th/0308070, 2003). For a static isolated system, in quasi-Cartesian coordinates, this formula leads to the well known result

Friedmann-Robertson-Walker metric in curvature coordinates and its applications

{P_0 = \int \sqrt{-g} (T_0^0 - T_1^1 - T_2^2 - T_3^3) d^3 x,}