Generalized Komar charges and Smarr formulas for black holes and boson stars

TL;DR

This paper generalizes the Komar charge to include matter fields, deriving new Smarr formulas and using these to prove non-existence theorems for gravitational solitons and boson stars.

Contribution

It introduces a generalized Komar charge applicable in matter-filled spacetimes and demonstrates its use in deriving Smarr formulas and non-existence theorems.

Findings

Generalized Komar charges can be defined in presence of matter.

These charges lead to new Smarr formulas for complex systems.

Application to non-existence theorems for boson stars and solitons.

Abstract

The standard Komar charge is a $(d-2)$-form that can be defined in spacetimes admitting a Killing vector and which is closed when the vacuum Einstein equations are satisfied. Its integral at spatial infinity (the Komar integral) gives the conserved charge associated to the Killing vector, and, due to its on-shell closedness, the same value (expressed in terms of other physical variables) is obtained integrating over the event horizon (if any). This equality is the basis of the Smarr formula. This charge can be generalized so that it still is closed on-shell in presence of matter and its integrals give generalizations of the Smarr formula. We show how the Komar charge and other closed $(d-2)$-form charges can be used to prove non-existence theorems for gravitational solitons and boson stars. In particular, we show how one can deal with generalized symmetric fields (invariant under a combination of isometries and other global symmetries) and how the geralized symmetric ansatz permits to evade the non-existence theorems.