Theoretical Analysis On A New Metric Ansatz For Kaluza-Klein Supergravity Theory Black Holes

It has been known for a long time that Einstein’s gravitational field equations are such a very complicated system of non-linearly coupled partial differential equations that finding a rotating exact solution to them is rather difficult. One of the frequently-used approaches to this problem is to assume an appropriate metric form for the unknown line element which is inspired from a known solution in order to simplify the subsequent calculations.This paper presented the study of a new kind of metric ansatz put forward in a previous paper [S.Q. Wu, Phys. Rev. D 83(2011) 121502(R)] to obtain the effective information that helps us to explore new exact black hole solution. That metric ansatz somewhat resembles the famous Kerr-Schild(KS) form, but it is different from the KS one in two distinct aspects. That is, apart from a global conformal factor, the metric ansatz can be written as a vacuum background spacetime plus a “perturbation” modification term, the latter of which is associated with a timelike geodesic vector field rather than a null geodesic congruence in the usual KS ansatz. Replacing the flat vacuum background metric by the(anti–)de Sitter [(A)dS] spacetime, the general rotating charged KK-(A)dS black hole solutions in all higher dimensions have been successfully constructed and put into a unified form. In this paper, we shall study this novel metric ansatz in detail, aiming at achieving some inspiration as to the construction of rotating charged AdS black holes with multiple charges in other gauged supergravity theories. We find that the traditional perturbation expansion method often successfully used in the KS form is no longer useful in our new ansatz, since here no good parameter can be chosen as a suitable perturbation indicator. In order to investigate the metric properties of the general KK-AdS solutions, in this paper we devise a new effective method, dubbed the background metric expansion method, which can be thought of as a generalization of the perturbation expansion method, to deal with the Lagrangian and all equations of motion. In addition to two previously known conditions, namely the timelike and geodesic properties of the vector, we get three additional constraints via contracting the Maxwell and Einstein equations once or twice with this timelike geodesic vector. In particular, we find that these are a simpler set of sufficient conditions to determine the vector and the dilaton scalar around the background metric, which is helpful in obtaining new exact solutions. With these five simpler equations in hand, we rederive the general rotating charged KK-(A)d S black hole solutions with spherical horizon topology and obtain new solutions with planar topology in all dimensions. It turns out that the overall calculations in finding the solution to the KK gauged supergravity can be reduced considerably, compared to the previous process, by directly solving all the field equations. It is then shown that the rotating charged KK-AdS black hole solutions can be further generalized by introducing one or two arbitrary constants, while the black hole solutions with the planar AdS background metric in all higher dimensions are newly obtained.