# Atom Interferometer Phase Shear and Spacetime Sectional Curvature

## TL;DR

This paper demonstrates that the phase shear in atom interferometry directly measures the integrated sectional curvature of spacetime, linking quantum measurements with geometric properties of gravity through a generalized Gauss-Bonnet theorem.

## Contribution

It introduces a novel relationship between atom interferometer phase shear and spacetime curvature, enabling coordinate-free measurements of gravitational properties.

## Key findings

- Phase shear equals integrated sectional curvature over a spacetime surface.
- Derived correction terms from gravitational redshift effects.
- Provides a computational method for predicting interferometer responses in arbitrary spacetimes.

## Abstract

Atom interferometry is a natural laboratory for precision tests of general relativity, but there is no simple relationship between atom interferometer phase and geometric properties of spacetime. Here we show that a different atom interferometer observable, the phase shear, can be expressed directly as the integrated sectional curvature over a spacetime surface enclosed by the interferometer arms and final beamsplitter. This is a consequence of a generalized Gauss-Bonnet theorem, which also explicitly computes small correction terms arising from gravitational redshift of atom optics pulses. This synthesis of quantum mechanics, relativity, and differential geometry affords a manifestly coordinate-free and representation-free means of measuring spacetime properties. Additionally, it provides a convenient computational tool for predicting atom interferometer properties in arbitrary background spacetimes.