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 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.
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Living Rev. Relativity 15, (2012), 11
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