Generalized Komar currents for vector fields

Abstract

In this paper, on basis 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, the usual Komar current for a Killing vector is formulated into another equivalent form. Then it is extended to more general currents in the absence of the linearity in the Killing vector field. Moreover, motivated by this equivalent of the usual Komar current, we put forward a conserved current corresponding to a generic vector with some constraint. Such a current can be generalized to the one with higher-order derivatives of the 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.

A Komar-like integral for mass and angular momentum of asymptotically AdS black holes in Einstein gravity

“…In the present section, by following the work [47], which provides a way of constructing conserved currents starting from the action of differential operators on an arbitrary vector field, we manage to find out a generalized Komar potential for solutions with asymptotically AdS structure in the context of D-dimensional Einstein gravity. Then the surface integral with respect to the generalized potential will be adopted to define the conserved charges such as the mass and the angular momentum.…”

“…Then the surface integral with respect to the generalized potential will be adopted to define the conserved charges such as the mass and the angular momentum. According to the work [47], the differential operators d and δ, together with = ∇ µ ∇ µ , participate in the construction of the conserved current corresponding to the 1-form vector field. For convenience, here we present their definitions through the action upon an arbitrary p-form…”

“…In what follows, along the lines of the works [46,47], we focus on giving a generalized Komar integral for the asymptotically AdS solutions of the field equation (2.2), with the Riemann curvature tensor at spatial infinity…”

“…Inspired by the expression of the Komar potential in differential forms, being the exterior derivative of a 1-form Killing vector field, a good candidate for the generalization of the Komar potential may be the one encompassing the higher-order derivatives of the Killing vector. Fortunately, we find that the approach to generate conserved currents associated with an arbitrary vector field put forward in the works [46,47] can assist us to accomplish the higher-order corrections to the Komar potential. The relevant idea behind these works goes as follows.…”

“…Within what follows, on basis of the strategy of generating conserved currents in [46,47], we obtain an identically conserved current consisting of the second-and fourth-order derivatives of an arbitrary Killing vector field. In terms of this current, a potential is immediately read off, which is the linear combination of the usual Komar potential with two third-order derivative terms generated by the action of the d'Alembertian and the exterior derivative on the 1-form Killing vector.…”