# Quasi-local energy and ADM mass in pure Lovelock gravity

## TL;DR

This paper extends the concepts of ADM mass and quasi-local energy to pure Lovelock gravity, deriving a new formula for ADM mass and analyzing its behavior in different geometries.

## Contribution

It introduces a new simple formula for ADM mass in pure Lovelock gravity and analyzes the large surface limit of quasi-local energy in this context.

## Key findings

- Large surface limit of quasi-local energy vanishes in asymptotically flat spacetime for Lovelock gravity.
- The variation of quasi-local energy approaches the ADM mass variation, leading to a new formula.
- The new formula is verified in spherically symmetric and asymptotically AdS geometries.

## Abstract

We study how the standard definitions of ADM mass and Brown-York quasi-local energy generalize to pure Lovelock gravity. The quasi-local energy is renormalized using the background subtraction prescription and we consider its limit for large surfaces. We find that the large surface limit vanishes for asymptotically flat fall-off conditions except in Einstein gravity. This problem is avoided by focusing on the variation of the quasi-local energy which correctly approaches the variation of the ADM mass for large surfaces. As a result, we obtain a new simple formula for the ADM mass in pure Lovelock gravity. We apply the formula to spherically symmetric geometries verifying previous calculations in the literature. We also revisit asymptotically AdS geometries.