# On the space of initial values strictly satisfying the dominant energy condition

## TL;DR

This paper investigates the topological structure of the space of initial data satisfying the strict dominant energy condition in Lorentzian geometry, introducing an index difference and establishing its non-trivial homotopy groups.

## Contribution

It introduces an index difference for initial values satisfying the dominant energy condition and demonstrates the non-trivial topology of this space.

## Key findings

- The space of initial values has non-trivial homotopy groups.
- An index difference is defined and compared to classical Riemannian cases.
- The results imply rich topological structure of initial data satisfying the energy condition.

## Abstract

The dominant energy condition imposes a restriction on initial value pairs found on a spacelike hypersurface of a Lorentzian manifold. In this article, we study the space of initial values that satisfy this condition strictly. To this aim, we introduce an index difference for initial value pairs and compare it to its classical counterpart for Riemannian metrics. Recent non-triviality results for the latter will then imply that this space has non-trivial homotopy groups.