McVittie's Legacy: Black Holes in an Expanding Universe

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Abstract

We prove that a class of solutions to Einstein's equations---originally discovered by G. C. McVittie in 1933---includes regular black holes embedded in Friedman-Robertson-Walker cosmologies. If the cosmology is dominated at late times by a positive cosmological constant, the metric is regular everywhere on and outside the black hole horizon and away from the big bang singularity, and the solutions asymptote in the future and near the horizon to the Schwarzschild-de Sitter geometry. For solutions without a positive cosmological constant the would-be horizon is a weak null singularity.

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Quasi-Local Gravitational Energy

Sean A Hayward (1993)

A dynamically preferred quasi-local definition of gravitational energy is given in terms of the Hamiltonian of a `2+2' formulation of general relativity. The energy is well-defined for any compact orientable spatial 2-surface, and depends on the fundamental forms only. The energy is zero for any surface in flat spacetime, and reduces to the Hawking mass in the absence of shear and twist. For asymptotically flat spacetimes, the energy tends to the Bondi mass at null infinity and the \ADM mass at spatial infinity, taking the limit along a foliation parametrised by area radius. The energy is calculated for the Schwarzschild, Reissner-Nordstr\"om and Robertson-Walker solutions, and for plane waves and colliding plane waves. Energy inequalities are discussed, and for static black holes the irreducible mass is obtained on the horizon. Criteria for an adequate definition of quasi-local energy are discussed.

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Cosmological Multi-Black Hole Solutions

Jennie Traschen, David Kastor (1992)

We present simple, analytic solutions to the Einstein-Maxwell equation, which describe an arbitrary number of charged black holes in a spacetime with positive cosmological constant \(\Lambda\). In the limit \(\Lambda=0\), these solutions reduce to the well known Majumdar-Papapetrou (MP) solutions. Like the MP solutions, each black hole in a \(\Lambda >0\) solution has charge \(Q\) equal to its mass \(M\), up to a possible overall sign. Unlike the \(\Lambda = 0\) limit, however, solutions with \(\Lambda >0\) are highly dynamical. The black holes move with respect to one another, following natural trajectories in the background deSitter spacetime. Black holes moving apart eventually go out of causal contact. Black holes on approaching trajectories ultimately merge. To our knowledge, these solutions give the first analytic description of coalescing black holes. Likewise, the thermodynamics of the \(\Lambda >0\) solutions is quite interesting. Taken individually, a \(|Q|=M\) black hole is in thermal equilibrium with the background deSitter Hawking radiation. With more than one black hole, because the solutions are not static, no global equilibrium temperature can be defined. In appropriate limits, however, when the black holes are either close together or far apart, approximate equilibrium states are established.

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Cosmology vs. Holography

Nemanja Kaloper, Andrei Linde (1999)

The most radical version of the holographic principle asserts that all information about physical processes in the world can be stored on its surface. This formulation is at odds with inflationary cosmology, which implies that physical processes in our part of the universe do not depend on the boundary conditions. Also, there are some indications that the radical version of the holographic theory in the context of cosmology may have problems with unitarity and causality. Another formulation of the holographic principle, due to Fischler and Susskind, implies that the entropy of matter inside the post-inflationary particle horizon must be smaller than the area of the horizon. Their conjecture was very successful for a wide class of open and flat universes, but it did not apply to closed universes. Bak and Rey proposed a different holographic bound on entropy which was valid for closed universes of a certain type. However, as we will show, neither proposal applies to open, flat and closed universes with matter and a small negative cosmological constant. We will argue, in agreement with Easther, Lowe, and Veneziano, that whenever the holographic constraint on the entropy inside the horizon is valid, it follows from the Bekenstein-Hawking bound on the black hole entropy. These constraints do not allow one to rule out closed universes and other universes which may experience gravitational collapse, and do not impose any constraints on inflationary cosmology.

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Publication date Created: 24 March 2010

Publication date Updated: 2010-07-28

Article

DOI: 10.1103/PhysRevD.81.104044

ArXiV ID: 1003.4777

SO-VID: 33411461-8e9c-49eb-8c92-b60f7226063c

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http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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Journal reference Phys.Rev.D81:104044,2010

Comments 23 pages, plain LaTeX, 2 .pdf figures, v3: the finite ingoing time proof improved and generalized, conclusions unchanged

McVittie's Legacy: Black Holes in an Expanding Universe

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Abstract

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Numerical Algebra, Matrix Theory, Differential-Algebraic Equations, and Control Theory

Most cited references 8

Quasi-Local Gravitational Energy

Cosmological Multi-Black Hole Solutions

Cosmology vs. Holography

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