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Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

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Abstract

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein–Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approach that of a subextremal Reissner–Nordström-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon (“near” \(i^+\)), in such a way that \(r=+\infty \) is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the cosmic no-hair conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote to those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.

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Notes

  1. Or in Reissner–Nordström-de Sitter [10, 25] or Kerr-de Sitter [25].

  2. See point 4 of Theorem 1.1 for the formulation used here.

  3. See below the clarification of this terminology.

  4. To set the convention, whenever we refer to an RNdS solution \(({{\mathcal {M}}}, g)\) we mean the maximal domain of dependence \(D(\Sigma ) = {{\mathcal {M}}}\) of a complete Cauchy hypersurface \(\Sigma = \mathbb {R}\times \mathbb {S}^2\).

  5. It is also standard to refer to these as expanding regions, but we prefer the designation “cosmological”, since the local regions (regions I and III in Fig. 1) are also expanding [3].

  6. These stability results concern the Einstein vacuum and Einstein–Maxwell equations, and not the Einstein–Maxwell-scalar field system.

  7. Note that \((1-\mu )\) depends on (u, v) only through \((r,\varpi )\). In what follows, in a slight abuse of notation, we will interchangeably regard \((1-\mu )\) as a function of either pair of variables, with the meaning being clear from the context.

  8. Note that in this setting Birkhoff’s theorem is less restrictive and allows solutions which do not belong to the RNdS family, see [14][Theorem 1.1] and [20].

  9. Non-degenerate in the terminology of [4].

  10. From now on, all topological statements will refer to the topology of \([0,+\infty [ \times \left[ 0,V\right] \) induced by the standard topology of \(\mathbb {R}^2\).

  11. There is a difference in a factor of \(4\pi \) due to our different convention for the value of Newton’s gravitational constant.

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Acknowledgements

We thank Anne Franzen for sharing and allowing us to use Fig. 1. This work was partially supported by FCT/Portugal through UID/MAT/04459/2013 and Grant (GPSEinstein) PTDC/MAT-ANA/1275/2014. Pedro Oliveira was supported by FCT/Portugal through the LisMath scholarship PD/BD/52640/2014.

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Authors and Affiliations

  1. Mathematics Department, Lisbon University Institute – ISCTE, Lisbon, Portugal

    João L. Costa

  2. Mathematics Department, Instituto Superior Técnico ULisboa, Lisbon, Portugal

    José Natário & Pedro Oliveira

  3. CAMGSD, Instituto Superior Técnico ULisboa, Lisbon, Portugal

    João L. Costa, José Natário & Pedro Oliveira

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  1. João L. Costa
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  2. José Natário
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  3. Pedro Oliveira
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Correspondence to José Natário.

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Communicated by Mihalis Dafermos.

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Costa, J.L., Natário, J. & Oliveira, P. Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes. Ann. Henri Poincaré 20, 3059–3090 (2019). https://doi.org/10.1007/s00023-019-00825-z

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