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A classification of near-horizon geometries of extremal vacuum black holes
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55.As written, the above metric is valid for . In fact, the case is also allowed (provided ) and may be obtained by shifting appropriately.
56.These two equations have appeared before, for example, in Ref. 47. We have proven that all components of the space-time Einstein equations are satisfied if and only if (15) and (16) are.
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61.In fact all our results for topology also apply to topology. In particular, since where is a discrete subgroup of the assumed isometry group, one can construct all the metrics on directly from the metrics on by identifying the angles appropriately.
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66.Notice that if then is also coexact.
67.We choose an orientation by setting .
68.Since the functions and are globally defined, the constant must be the same in every coordinate patch and thus this expression is valid globally.
69.A static near-horizon geometry is defined by , see Ref. 22.
70.Note that this does not imply the vector field is nonvanishing everywhere—indeed we will find examples where has isolated zeros [which occurs at the fixed points of (a combination of) the Killing fields].
71.In fact, the solution is the near-horizon limit of Kerr--NUT whose horizon suffers from conical singularities.
72.We could consider the two cases simultaneously, however, for clarity we have chosen not to.
73.In this case, one can solve for from which it follows that . Therefore if there is no root . If then , however, for .
74.The expression for is valid for . For one must use an expression for valid when . This is given in the appendices and simply amounts to a shift in which can be generated by shifting . It thus suffices to check as then the case follows by the appropriate shift in .
75.Note that for the class of -invariant near-horizon geometries, we have been considering that are invariantly defined quantities up to the two constant scaling symmetries (91) and (92). Therefore, the only gauge freedom in our perturbation analysis are these constant scalings. These can be fixed by working with a background solution in which the scaling symmetries have been used to fix the parameterization, as for the boosted Kerr string.
76.Consider a metric of the form with functions of and with having two distinct zeros with in between. The condition for simultaneous removal of the conical singularities at (at which points vanishes) is . Clearly, this is satisfied if are even functions of . Then, in the interval , it is a smooth metric on .
77.In fact, a special case of this with one independent rotation parameter has been constructed.63
78.In General Relativity kinematical arguments such as this are not sufficient to establish symmetry enhancement; one usually uses dynamical input from the Einstein’s equation. In any case this symmetry enhancement occurs in all known examples.
79.As discussed below (32), cannot be constant as otherwise everywhere.
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