A gravitational collapse singularity theorem consistent with black hole evaporation
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A gravitational collapse singularity theorem consistent with black hole evaporation E. Minguzzi1 Received: 10 July 2019 / Revised: 4 May 2020 / Accepted: 28 May 2020 © Springer Nature B.V. 2020
Abstract The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work, I show that the causality conditions in Penrose’s theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in the absence of predictability and hence compatibly with quantum field theory and black hole evaporation. Keywords Singularity theorems · Black holes · Causality Mathematics Subject Classification 83C57 · 83C75 · 53C50
1 Introduction The celebrated Penrose singularity theorem [1,2] establishes that in a globally hyperbolic spacetime admitting a non-compact Cauchy hypersurface H , under the null energy condition, any trapped surface S leads to the formation of a spacetime singularity in its causal future. If the weak cosmic censorship conjecture is correct, then the spacetime singularity will be hidden behind a horizon so leading to the formation of a black hole. These objects, particularly stationary ones, have then been investigated with the theory of quantum fields over curved spacetimes. A rather robust prediction states that the black hole will eventually evaporate in a finite, though huge, time [3–5]. It was observed by Kodama [6] and Wald [7] that the phenomenon of black hole evaporation is in tension with global hyperbolicity and hence with the possibility of predicting the evolution of the spacetime manifold by means of Einstein’s equations.
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E. Minguzzi [email protected] Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Via S. Marta 3, 50139 Florence, Italy
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Recently, Lesourd [8] has given more precise arguments which show that even causal continuity, a causality condition much weaker than global hyperbolicity, would be violated. We see that we are in the presence of a loophole argument. In order to infer the formation of singularities, we need to assume global hyperbolicity. Unfortunately, one would expect the formation of a black hole and hence its very evaporation. If so, the global assumption of global hyperbolicity could not have been satisfied in the first place. In other words, the hypotheses of Penrose’s theorem seem too strong for a realistic universe, as taking into account the quantum effects it appears that the existence of a black hole implies a certain amount of unpredictability. These observations open the possibility that trapped surfaces do not necessarily imply the formation of singularities, at least when quantum effects are taken into account. The situation would be clarified and the problem solved if we could generalize Penrose’s theorem by removing the assumption of global hyperbolicity. Much of the discussion about the information loss paradox might have been i
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