Abstract
In 1916, shortly after Einstein published his general relativity [1], Schwarzschild published a solution of the Einstein equation [4] that was later understood, mostly due to the works of Finkelstein [5] and Kruskal [6], as describing a region of spacetime whose curvature is so great that nothing—even light—can escape from it.
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Notes
- 1.
According to different accounts, the term “black hole” had been introduced in the sixties of the last century either by Wheeler or Dicke (Dicke compared that spacetime region to a prison in India called the Black Hole, because no one who entered it left it alive).
- 2.
Non-singular (or singularity-free) black holes are called regular black holes.
- 3.
Papapetrou particularly emphasizes the serious anomaly on the Schwarzschild sphere, whose physical meaning, I think, has not been thoroughly examined [13]:
But these geodesics are space-like for \( r > 2m \) and time-like for \( r < 2m \). The tangent vector of a geodesic undergoes parallel transport along the geodesic and consequently it cannot change from a time-like to a space-like vector. It follows that the two regions \( r > 2m \) and \( r < 2m \) do not join smoothly on the surface \( r = 2m \).
- 4.
The only comments after my talk “On the asymptomatic formation of black holes” at the Third Minkowski Meeting (11–14 September 2023, Albena, Bulgaria) and during the panel discussion on the nature of black holes just stated that there was no contradiction between the two predictions of the Schwarzschild solution. This is self-evident when those predictions are regarded as mathematics. But they become problematic when it is clamed that they apply to the physical world.
- 5.
In addition, taking the term “asymptotically” seriously requires a number of explicit definitions such as “asymptotically existing” (“asymptotic existence”), “asymptotically non-existing,” “asymptotically real” or even more confusing definitions such as “asymptotically alive” and “asymptotically dead”.
- 6.
I believe that the use of double standards has not been intentional, but rather following “an inner voice” which whispers that there is a contradiction (when the Schwarzschild solution is regarded as describing a real physical situation) that should be avoided or at least addressed.
- 7.
The other (relevant) one was mentioned in the previous chapter—the Nobel Prize in Physics for 1993 whose reason for awarding the Prize was also carefully and correctly worded: “for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation.”
- 8.
There was a dramatic development in black hole physics—on 1 December 2023 Roy Kerr posted a very important paper “Do Black Holes have Singularities?” (arXiv:2312.00841) where he demonstrated that “There is no proof that black holes contain singularities when they are generated by real physical bodies.”
Therefore, the assertion “black hole formation is a robust prediction of the general theory of relativity” does not seem to be indisputably true if the two defining features of black holes are a singularity and an event horizon, which appears to be the accepted understanding (moreover, Penrose himself in the quote below states it explicitly that a black hole is formed when “physical singularity is encountered”).
References
A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 49, (1916). New publication of the original English translation ([2]) in [3]
A. Einstein, The foundation of the general theory of relativity, in The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, ed. by H.A. Lorentz, A. Einstein, H. Minkowski, H. Weyl. With Notes by A. Sommerfeld. Translated by W. Perrett and G.B. Jeffery (Methuen and Company, Ltd., 1923; reprinted by Dover Publications Inc., 1952)
V. Petkov (ed.), The Origin of Spacetime Physics, 2nd edn. Foreword by A. Ashtekar. (Minkowski Institute Press, Montreal, 2023)
K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. K. Preuss. Akad. Wiss. 1, 189 (1916)
D. Finkelstein, Past-future asymmetry of the gravitational field of a point particle. Phys. Rev. 110, 965 (1958)
M.D. Kruskal, Maximal extension of Schwarzschild metric. Phys. Rev. 119, 1743 (1960)
S. Murk, Nomen non est omen: why it is too soon to identify ultra-compact objects as black holes. Int. J. Mod. Phys. D. https://doi.org/10.1142/S0218271823420129
R. Geroch, What is a singularity in general relativity? Ann. Phys. 48, 526–540 (1968)
A. Einstein, On a stationary system with spherical symmetry consisting of many gravitating masses. Ann. Math., Second Series 40(4), 922–936 (1939)
P.O. Hess, M. Schäfer, W. Greiner, Pseudo-Complex General Relativity (Springer, Heidelberg, 2016)
C. Bambi (ed.), Regular Black Holes: Towards a New Paradigm of Gravitational Collapse (Springer, 2023), p. 5
A. Ashtekar, J. Olmedo, P. Singh, Regular Black Holes from Loop Quantum Gravity in [11], pp. 235–236
A. Papapetrou, Lectures on General Relativity (Reidel, Dordrecht, 1974), pp.85–86
P.A.M. Dirac, General Theory of Relativity (Princeton, Princeton University Press, 1996), pp.35–36
J.R. Oppenheimer, H. Snyder, On continued gravitational contraction. Phys. Rev. 56(455), 456 (1939)
R. Penrose, Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 18 (1965)
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Petkov, V. (2024). Black Holes. In: Seven Fundamental Concepts in Spacetime Physics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-49730-8_7
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