# $\mathbf{O}(D,D)$ completion of the Einstein Field Equations

## TL;DR

This paper develops an $ ext{O}(D,D)$ covariant formulation of the Einstein equations within Double Field Theory, unifying the equations of motion of the massless NS-NS sector into a single, stringy Einstein equation.

## Contribution

It introduces the off-shell conserved stringy Einstein curvature tensor and on-shell conserved stringy Energy-Momentum tensor, formulating the Einstein Double Field Equations in a covariant manner.

## Key findings

- Unified equations of motion into a single $ ext{O}(D,D)$ covariant form.
- Defined new stringy Einstein curvature and Energy-Momentum tensors.
- Established a framework for 'Stringy Gravity' within Double Field Theory.

## Abstract

Upon treating the whole closed-string massless NS-NS sector as stringy graviton fields, Double Field Theory may evolve into `Stringy Gravity'. In terms of an $\mathbf{O}(D,D)$ covariant differential geometry beyond Riemann, we present the definitions of the off-shell conserved stringy Einstein curvature tensor and the on-shell conserved stringy Energy-Momentum tensor. Equating them, all the equations of motion of the massless sector are unified into a single expression, $G_{AB}{=8\pi G} T_{AB}$, carrying $\mathbf{O}(D,D)$ vector indices, which we dub `the Einstein Double Field Equations'.