On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordström or Reissner-Nordström-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the "Cauchy horizon." It is located inside the black hole in these spacetimes. The strong cosmic censorship conjecture asserts that the Cauchy horizon does not, in fact, exist in practice because the slightest perturbation (of the metric itself or the matter fields) will become singular there in a sufficiently catastrophic way that solutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmic censorship holds, the Cauchy horizon will be converted into a "final singularity," and determinism will hold. Recently, however, it has been found that, classically this is not the case in RNdS spacetimes in a certain range of mass, charge, and cosmological constant. In this paper, we consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship. We find that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, V , along a radial null geodesic transverse to the Cauchy horizon as T V V ∼ C/V 2 with C independent of the state and C = 0 generically in RNdS spacetimes. This divergence is stronger than in the classical theory and should be sufficient to convert the Cauchy horizon into a singularity through which the spacetime cannot be extended as a (weak) solution of the semiclassical Einstein equation. This behavior is expected to be quite general, although it is possible to have C = 0 in certain special cases, such as the BTZ black hole. arXiv:1912.06047v1 [gr-qc] 12 Dec 2019As we shall show in sec. 3, the behavior of the quantum stress tensor is determined by the trace anomaly and the continuity equation alone. We find that eq. (5) holds, with C ∼ κ 2 c − κ 2 − . Our arguments are technically rather 3 Our arguments rest on the assumption of regularity across the cosmological horizon, or the weaker condition that Tvv vanishes on the cosmological horizon, with v the "Killing parameter". This condition has a natural limit in RN, that of absence of radiation infalling from spatial infinity. It seems natural to discuss the behavior at the Cauchy horizon in terms of such asymptotic conditions, as without asymptotic conditions also the classical consideration of strong cosmic censorship becomes meaningless, as discussed for example in the Introduction to [12]: The domain of dependence of a compact achronal set in any globally hyperbolic spacetime has a Cauchy horizon, across which it is of course smoothly ex...

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“…We transform to regular coordinates V = V − and V = V c near CH R and H L c , respectively, (see eqs. (17) and (18)). We obtain…”

Section: Behavior Of the Classical And Quantum Stress-energy Tensor Nmentioning

confidence: 94%

“…The tip of the lightcone only needs to be examined a little more carefully for the proof that the commutator property holds, and plays a role similar to i − in the subsequent construction in RNdS 17. This is to be understood in the sense of wave packets, cf.…”

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confidence: 99%

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

Hollands

Wald

Zahn

2020

Class. Quantum Grav.

125

134

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“…In Part 1 [7] of this trilogy, we showed the equivalence (under appropriate regularity conditions for the initial data) between the Einstein-Maxwellscalar field equations (1)−(4) under spherical symmetry and the system of first order PDEs (19)−(28). We established existence, uniqueness and identified a breakdown criterion for solutions of this system.…”

Section: Introductionmentioning

confidence: 99%

“…In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon.…”

Section: Introductionmentioning

confidence: 99%

“…

AbstractThis paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon.In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols.

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confidence: 99%

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On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

et al. 2017

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This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon.In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when Λ > 0.

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Fate of strong cosmic censorship conjecture in presence of higher spacetime dimensions

Rahman

Chakraborty

SenGupta

et al. 2019

J. High Energ. Phys.

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A well posed theory of nature is expected to determine the future of an observer uniquely from a given set of appropriate initial data. In the context of general relativity, this is ensured by Penrose's strong cosmic censorship conjecture. But in recent years, several examples are found which suggest breakdown of the deterministic nature of the theory in Reissner-Nordström-de Sitter black holes under the influence of different fundamental fields. Nevertheless, the situation has been reassuring for the case of astrophysically meaningful Kerr-de Sitter black hole solutions which seems to respect the conjecture. However, the previous analyses were done considering only the effect of scalar fields. In this paper, we extend the study by considering Dirac fields in Kerr-de Sitter background and show that there exist a parameter space which does not respect the conjecture.

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On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

Cited by 44 publications

References 46 publications

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime

On the Global Uniqueness for the Einstein–Maxwell-Scalar Field System with a Cosmological Constant: Part 3. Mass Inflation and Extendibility of the Solutions

Fate of strong cosmic censorship conjecture in presence of higher spacetime dimensions

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