On conformal Komar currents in LRS spacetimes

TL;DR

This paper explores conformal Komar currents in LRS spacetimes, deriving conditions for their conservation, analyzing their relation to trapped surfaces and horizons, and linking the Noether charge to surface gravity for thermodynamic insights.

Contribution

It introduces two equivalent forms of conformal Komar currents in LRS spacetimes and investigates their conservation conditions and physical implications, including thermodynamic interpretations.

Findings

Conservation of the current relates to trapped surfaces.

In LRS II with constant metric time, the current is always conserved.

Noether charge is proportional to surface gravity near horizons.

Abstract

For locally rotationally symmetric (LRS) spacetimes, we construct two equivalent forms of the Komar current derived from a conformal Killing vector. One is a kinematic construction and the other is in terms of the matter quantities on the spacetime. The required conservation condition for the current is derived and discussed in various instances, and the implications of the conservation of the current, and in the case of a vanishing current, are analyzed. A relationship between the conservation criterion and the presence of trapped surfaces in the spacetime is found and discussed. We also show that for LRS II metrics with constant metric time component, the current is always conserved. In the presence of a conformal Killing horizon, properties of the current are analyzed and restrictions on, and some implications for the physical spacetime variables, in the vicinity of the horizon, are obtained. Finally, with respect to the conformal Killing horizon, the associated Noether charge is shown to be proportional to the surface gravity, establishing the thermodynamic interpretation of the Noether charge.