Graphical Representation of Invariants and Covariants in General Relativity

TL;DR

This paper introduces a graphical method to describe invariants and covariants in 4-dimensional general relativity, simplifying complex notation and providing new relations and identities relevant for quantum gravity calculations.

Contribution

It presents a novel graphical approach to invariants in general relativity and derives new off-shell relations and identities for quantum gravity applications.

Findings

Two new off-shell relations between mass^6 invariants

Complete list of independent invariants with mass^4 and mass^6 dimensions

Explicit form of 6D Gauss-Bonnet identity

Abstract

We present a grapical way to describe invariants and covariants in the (4 dim) general relativity. This makes us free from the complexity of suffixes . Two new off-shell relations between (mass)$\ast\ast 6$\ invariants are obtained. These are important for 2-loop off-shell calculation in the perturbative quantum gravity. We list up all independent invarians with dimensions of (mass)$\ast\ast 4$\ and (mass)$\ast\ast 6$. Furthermore the explicit form of 6 dim Gauss-Bonnet identity is obtained.