Foliation-generating observers under Lorentz transformations

TL;DR

This paper investigates how Lorentz transformations influence the existence of foliations in spacetime, impacting the Hamiltonian formulation of gravity theories beyond General Relativity.

Contribution

It clarifies the relationship between observer kinematics and spacetime foliations, and analyzes the effects of Lorentz transformations on these structures in modified gravity theories.

Findings

Lorentz transformations can alter the existence of orthogonal foliations.

The kinematic properties of observers are crucial for foliation existence.

Implications for the 3+1 formulation of tetrad gravity theories.

Abstract

In this work, we revise the concept of foliation and related aspects that are crucial when formulating the Hamiltonian evolution for various theories beyond General Relativity. In particular, we show the relation between the kinematic characteristics of timelike congruences (observers) and the existence of foliations orthogonal to them. We then explore how local Lorentz transformations acting on observers affect the existence of transversal foliations, provide examples, and discuss the implications of these results for the $3+1$ formulation of tetrad modified theories of gravity.