Abstract
The domain of outer communication of an asymptotically flat spactime must be simply connected. In five dimensions, this still allows for the possibility of an arbitrary number of 2-cycles supported by magnetic flux carried by Maxwell fields. As a result, stationary, asymptotically flat, horizonless solutions—“gravitational solitons”—may exist with non-vanishing mass, charge, and angular momenta. These gravitational solutions satisfy a Smarr-like relation, as well as a first law of mechanics. Furthermore, the presence of solitons leads to new terms in the well-known first law of black hole mechanics for spacetimes containing black hole horizons and non-trivial topology in the exterior region. I outline the derivation of these results and consider an explicit example in five-dimensional supergravity.
Highlights
The proof of the black hole uniqueness theorem is a central achievement in general relativity.This result asserts that the any stationary, asymptotically flat analytic solution of the Einstein-Maxwell equations describing a black hole spacetime must belong to the three-parameter Kerr-Newman family of solutions
There are a number of profound consequences of this theorem. It implies that if one starts with a black hole with mass M, angular momentum J and electric charge Q and considers an infinitesimal variation in the phase space of black holes, one must arrive at another member of the Kerr-Newman solution with parameters M + δM, J + δJ and Q + δQ
8π where A H is the area of a spatial cross-section of the event horizon and κ, Ω H and Φ H are the surface gravity, angular velocity and electric potential of the horizon respectively
Summary
The proof of the black hole uniqueness theorem is a central achievement in general relativity This result asserts that the any stationary, asymptotically flat analytic solution of the Einstein-Maxwell equations describing a black hole spacetime must belong to the three-parameter Kerr-Newman family of solutions (see the review [1]). These 2-cycles may carry magnetic flux sourced by Maxwell fields and and contribute towards a non-zero mass, angular momenta, and electric charge of the spacetime. These solitons (variously known as “smooth geometries” or “fuzzballs” in the literature, see e.g., [15] and references therein) appear to play an important role in the lack of black hole uniqueness in D = 5. The consequences of these new terms of black hole thermodynamics is only starting to be explored
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