# Dain's invariant on non-time symmetric initial data sets

## TL;DR

This paper extends Dain's geometric invariant to non-time symmetric initial data sets in vacuum Einstein equations, providing a measure of deviation from stationarity and gravitational radiation content.

## Contribution

It introduces a new invariant applicable to initial data with non-zero extrinsic curvature, generalizing previous stationary-focused constructions.

## Key findings

- Invariant vanishes for stationary data
- Quantifies deviation from stationarity
- Measures gravitational radiation in initial data

## Abstract

We extend Dain's construction of a geometric invariant characterising static initial data sets for the vacuum Einstein field equations to situations with a non-vanishing extrinsic curvature. This invariant gives a measure of how much the initial data sets deviates from stationarity. In particular, it vanishes if and only if the initial data set is stationary. Thus, the invariant provides a quantification of the amount of gravitational radiation contained in the initial data set.