# Comments on Joint Terms in Gravitational Action

## TL;DR

This paper compares three methods for computing joint terms in gravitational action, revealing their variational invariance and providing conditions for their existence, with applications to general relativity.

## Contribution

It introduces a unified analysis of joint term computation methods in gravity and establishes criteria for their equivalence and existence.

## Key findings

- Differences between methods are variational invariants under fixed boundary conditions.
- Explicit conditions for the existence of joint terms are derived.
- Application to general relativity demonstrates the practical relevance.

## Abstract

This paper compares three different methods about computing joint terms in on-shell action of gravity, which are identifying the joint term by the variational principle in Dirichlet boundary condition, treating the joint term as the limit contribution of smooth boundary and finding the joint term by local SO(1,$d-1$) transformation. In general metric gravitational theory, we show that the differences between these joint terms are some variational invariants under fixed boundary condition. We also give an explicit condition to judge the existence of joint term determined by variational principle and apply it into general relativity as an example.