In an effective field theory approach, one concentrates on long-range interactions, which are described by the physical Einstein–Maxwell theory. However, it is instructive in testing ideas about quantum gravity models of black holes to embed our familiar Einstein–Maxwell theory into the larger framework of supergravity and study the generic properties of rotating black holes as toy models for a physical string embedding of the Kerr–Newman black hole.

Another independent motivation comes from the AdS/CFT correspondence [205, 265]. Black holes in anti-de Sitter (AdS) spacetime in dimensions can be mapped to thermal states in a dual CFT or CFT in dimensions. Studying AdS black holes then amounts to describing the dynamics of the dual strongly-coupled CFT in the thermal regime. Since this is an important topic, we will discuss in this review the AdS generalizations of the Kerr/CFT correspondence as well. How the Kerr/CFT correspondence fits precisely in the AdS/CFT correspondence in an important open question that will be discussed briefly in Section 7.2.

In this review we will consider the following class of four-dimensional theories,

possibly supplemented with Planck-suppressed higher-derivative corrections. We focus on the case where and are positive definite and the scalar potential is non-positive in (1). This ensures that matter obeys the usual energy conditions and it covers the case of zero and negative cosmological constant. Some theories of interest contained in the general class (1) are the Einstein–Maxwell gravity with negative or zero cosmological constant and the bosonic sector of = 8 supergravity. Note that the phenomenology described by the action (1) is limited by the absence of charged scalars, massive vectors, non-abelian gauge fields and fermions.The explicit form of the most general single-center spinning–black-hole solution of the theory (1) is not known;however, see [270, 221] for general ansätze. For Einstein and Einstein–Maxwell theory, the solutions are, of course, the Kerr and Kerr–Newman geometries that were derived about 45 years after the birth of general relativity. For many theories of theoretical interest, e.g., supergravity, the explicit form of the spinning–black-hole solution is not known, even in a specific U-duality frame (see, e.g., [49] and references therein). However, as we will discuss in Section 2.3, the solution at extremality greatly simplifies in the near-horizon limit due to additional symmetries and takes a universal form for any theory in the class (1). It is for this reason mainly that we find convenient to discuss theory (1) in one swoop.

Living Rev. Relativity 15, (2012), 11
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