List of Footnotes
1 | We work in geometrised units throughout. | |
2 | Of course, a charged extremal black hole can always discharge in the presence of charged matter. | |
3 | The emergence of an AdS_{3} factor for solutions with certain null singularities has been previously observed [68, 18]. We will only consider regular horizons, so the area of spatial cross sections of the horizon is necessarily non-zero. | |
4 | Although spacetimes correspond to Lorentzian metrics, one can often analytically continue these to complete Riemannian metrics. Indeed, the first example of an inhomogeneous Einstein metric on a compact manifold was found by Page, by taking a certain limit of the Kerr–de Sitter metrics [185], giving a metric on . | |
5 | Albeit, under some technical assumptions such as analyticity of the metric. | |
6 | Two oriented manifolds are said to be oriented-cobordant if there exists some other oriented manifold whose boundary (with the induced orientation) is their disjoint union. | |
7 | Similarly, any such black hole in Einstein–Maxwell-dilaton theory with a purely electric field strength must be given by the RN solution [96, 97]. | |
8 | Indeed counterexamples are known in both senses. | |
9 | In fact our constructions only assume the metric is in a neighbourhood of the horizon. This encompasses certain examples of multi–black-hole spacetimes with non-smooth horizons [33]. | |
10 | To avoid proliferation of indices we will denote both coordinate and vielbein indices on by lower case latin letters . | |
11 | A Kundt spacetime is one that admits a null geodesic vector field with vanishing expansion, shear and twist. | |
12 | A Kundt spacetime is said to be degenerate if the Riemann tensor and all its covariant derivatives are type II with respect to the defining null vector field [184]. | |
13 | The remaining components of the Einstein equations for the full spacetime restricted to give equations for extrinsic data (i.e., -derivatives of that do not appear in the near-horizon geometry). On the other hand, the rest of the Einstein equations for the near-horizon geometry restricted to the horizon vanish. | |
14 | The Komar integral associated with the null generator of the horizon vanishes identically. In fact, one can show that for a general non-extremal Killing horizon, this integral is merely proportional to . | |
15 | The borderline case of topology was only excluded later using topological censorship [44]. Furthermore, these results were generalised to the non-stationary case and asymptotically AdS case [90]. | |
16 | A borderline case also arises in this proof, corresponding to the induced metric on being Ricci flat. This was in fact later excluded [88]. | |
17 | An isometry group whose surfaces of transitivity are dimensional is said to be orthogonally transitive if there exists dimensional surfaces orthogonal to the surfaces of transitivity at every point. | |
18 | This can be obtained by analytically continuing Eq. (74) by . | |
19 | A complex manifold is Fano if its first Chern class is positive, i.e., . It follows that any Kähler–Einstein metric on such a manifold must have positive Einstein constant. | |
20 | This is a conserved charge for such asymptotically KK spacetimes. | |
21 | Similarly, higher-dimensional pure-AdS spacetime is unstable to the formation of small black holes [27]. | |
22 | This stems from the fact that such spacetimes admit null geodesic congruences with vanishing expansion, rotation, and shear (i.e., they are Kundt spacetimes and hence algebraically special). | |
23 | We would like to thank Carmen Li for verifying this. | |
24 | There is a vast literature on this problem, which we will not attempt to review here. |