# A Covariant Version of Verlinde's Emergent Gravity

## TL;DR

This paper develops a covariant formulation of Verlinde's emergent gravity, introducing a vector field that interacts with matter to replicate dark matter and dark energy effects within a unified framework.

## Contribution

It presents a covariant Lagrangian for emergent gravity, providing a more consistent interpretation and extending the model to include dark energy.

## Key findings

- Derives correction terms to non-covariant expressions in Schwarzschild space.
- Shows the vector field can mimic dark energy effects.
- Provides a covariant framework for emergent gravity theories.

## Abstract

A generally covariant version of Erik Verlinde's emergent gravity model is proposed. The Lagrangian constructed here allows an improved interpretation of the underlying mechanism. It suggests that de-Sitter space is filled with a vector-field that couples to baryonic matter and, by dragging on it, creates an effect similar to dark matter. We solve the covariant equation of motion in the background of a Schwarzschild space-time and obtain correction terms to the non-covariant expression. Furthermore, we demonstrate that the vector field can also mimic dark energy.