Deriving the ADM 3+1 evolution equations from the second variation of arc length

TL;DR

This paper demonstrates that the ADM 3+1 evolution equations with zero shift vector can be derived directly from the second variation of arc-length, providing a geometric foundation for these equations.

Contribution

It introduces a novel derivation of the ADM 3+1 equations from geometric principles, linking them to the second variation of arc-length.

Findings

Derivation of ADM 3+1 equations from geometric considerations

Connection between second variation of arc-length and evolution equations

Provides a geometric interpretation of ADM formalism

Abstract

We will show that the ADM 3+1 evolution equations, for a zero shift vector, arise naturally from the equations for the second variation of arc-length.