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Space-like singularities of general relativity: A phantom menace?

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Abstract

The big bang and the Schwarzschild singularities are space-like. They are generally regarded as the ‘final frontiers’ at which space–time ends and general relativity breaks down. We review the status of such space-like singularities from three increasingly more general perspectives. They are provided by (i) A reformulation of classical general relativity motivated by the Belinskii, Khalatnikov, Lifshitz conjecture on the behavior of the gravitational field near space-like singularities; (ii) The use of test quantum fields to probe the nature of these singularities; and, (iii) An analysis of the fate of these singularities in loop quantum gravity due to quantum geometry effects. At all three levels singularities turn out to be less menacing than one might a priori expect from classical general relativity. Our goal is to present an overview of the emerging conceptual picture and suggest lines for further work. In line with the Introduction to Current Research theme, we have made an attempt to make it easily accessible to all researchers in gravitational physics.

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Notes

  1. The interesting case of a massless scalar field is discussed in [10] for FLRW cosmologies using the framework summarized here, and in [8] for Bianchi IX models, using a technically different approach that also leads to a well-defined evolution through the big-bang.

  2. This strategy is similar to that used in discussions of the BKL conjecture where one divides geometric fields by the trace K of the extrinsic curvature which is expected to diverge at the singularity to obtain the so-called “Hubble normalized fields" (see, e.g., [31]). A key difference is that whereas the focus in those treatments is on differential equations, here the focus is on a Hamiltonian framework that can serve as a point of departure for quantum theory.

  3. Details on results summarized below can be found in [7]. The operator \(D_i := E^a_i D_a\) is linear and satisfies the Leibnitz rule. Since \(D_a\) is determined by \(q_{ab}\), it ignores internal indices. Hence \(D_i\) also treats fields with internal indices as scalars. Note however that, given a function f on \(\mathbb {M}\), \(D_i f\) does not yield the exterior derivative df. Thus, \(D_i\) is not a connection on \(\mathbb {M}\). If we were to formally treat as a connection, it would have torsion determined by \(C_i{}^j\): \(D_{[i} D_{j]} f = -{T}^k_{\;\;ij} D_k f \) where \({T}^k_{\;\;ij} = \epsilon _{kl[i} C_{j]}^{\;\;l}\) [5, 7].

  4. By contrast, evolution equations of the ADM variables \((q_{ab}, P^{ab})\) as well as their triad analogs \((E^a_i, K_a^i)\) involve non-polynomial functions of these variables and, furthermore \(P^{ab}\) and \(K_a^i\) themselves diverge at the FLRW big bang.

  5. Sometime ago a similar result was obtained using Schrödinger picture for time evolution [44]. However, that reasoning relies on formal arguments that do not do full justice to the difficult quantum field theoretic issues associated with an infinite number of degrees of freedom. These are adequately handled in [3], along the lines of [2].

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Acknowledgements

This work was supported by the NSF Grants PHY-1806356, the Eberly Chair funds of Penn State, and the Alexander von Humboldt Foundation.

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Authors and Affiliations

  1. Institute for Gravitation and the Cosmos and Physics Department, The Pennsylvania State University, University Park, PA, 16802, USA

    Abhay Ashtekar, Adrián del Río & Marc Schneider

  2. Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea, 265, 34136, Trieste, Italy

    Marc Schneider

  3. INFN Sezione di Trieste, Via Valerio 2, 34127, Trieste, Italy

    Marc Schneider

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  1. Abhay Ashtekar
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Correspondence to Abhay Ashtekar.

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Ashtekar, A., del Río, A. & Schneider, M. Space-like singularities of general relativity: A phantom menace?. Gen Relativ Gravit 54, 45 (2022). https://doi.org/10.1007/s10714-022-02932-5

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