A Komar-like Integral For Mass And Angular Momentum Of Asymptotically AdS Black Holes In Einstein Gravity

Komar integral is a definition for the conserved charges of general relativity,proposed by Komar in 1959.It can be adopted to compute the mass and angular momentum for asymptotically flat black holes.However,if the Komar integral is applied to calculate the mass of asymptotically Anti-de Sitter(AdS)black holes,the result is usually divergent.Thus,the usual Komar integral must be modified.In this thesis,in terms of three quadratic differential operators leaving the form degree of an arbitrary differential form unchanged,that is,the d'Alembertian operator and two combined ones from the Hodge coderivative and the exterior derivative,we have proposed various generalizations of the usual Komar current and a Komar-like integral formula for the conserved charges of asymptotically AdS black holes in Einstein gravity.More concretely,First,the usual Komar current for a Killing vector is extended to the ones in the absence of the linearity in the Killing vector field,as well as the ones with various even-derivatives of an arbitrary vector.The applications to some specific vector fields,such as the almost-Killing vectors,the conformal Killing vectors and the divergence-free vectors,are investigated.It is demonstrated that the above generalizations of the Killing vector can be uniformly described by a second-order derivative equation.Next,we obtain a potential that is the linear combination of the usual Komar potential with two third-order derivative terms generated by the action of the d'Alembertian operator and the exterior derivative upon a Killing vector.With help of the higher-order corrected Komar potential,we propose a Komar-like integral for the conserved charges of Einstein gravity with a negative cosmological constant.Finally,we make use of such a formula to compute the mass and the angular momentum of Schwarzschild-AdS black hole,the regular AdS black hole,the asymptotically AdS Kerr-Sen black hole,the Kerr-NUT-AdS black hole,and the asymptotically AdS black hole in five-dimensional minimal gauged supergravity.The results are consistent with the ones given by other standard methods.