# Second Order Perturbation Theory in General Relativity: Taub Charges as Integral Constraints

## TL;DR

This paper investigates second order perturbation theory in General Relativity, focusing on the role of Taub charges as integral constraints on first order solutions in spacetimes with symmetries.

## Contribution

It provides explicit forms of the Taub charge constraints in second order perturbation theory for Einstein gravity with symmetries.

## Key findings

- Derived explicit integral constraints (Taub charges) for first order solutions.
- Identified conditions under which linearized solutions correspond to exact solutions.
- Clarified the role of these constraints in classical and semiclassical contexts.

## Abstract

In a nonlinear theory, such as General Relativity, linearized field equations around an exact solution are necessary but not sufficient conditions for linearized solutions. Therefore, the linearized field equations can have some solutions which do not come from the linearization of possible exact solutions. This fact can make the perturbation theory ill-defined, which would be a problem both at the classical and semiclassical quantization level. Here we study the first and second order perturbation theory in cosmological Einstein gravity and give the explicit form of the integral constraint, which is called the Taub charge, on the first order solutions for spacetimes with a Killing symmetry and a compact hypersurface without a boundary.