On the Hamiltonian of gravity theories whose action is linear in spacetime curvature

TL;DR

This paper presents a simple method to compute the Hamiltonian for gravity theories linear in curvature, clarifying the roles of lapse, shift, and metric in the Hamiltonian formulation.

Contribution

It introduces a straightforward approach to derive Hamilton's density for linear curvature gravity theories, emphasizing the constraints and evolution equations.

Findings

Lapse and shift lead to primary constraints.

Induced metric results in nontrivial evolution equations.

Hamilton's density can be formally obtained for these theories.

Abstract

A straightforward method to compute Hamilton's density for theories that are linear in the spacetime curvature is provided. It is shown that the lapse function and shift vector still give rise to primary constraints, while the induced metric gives rise to nontrivial evolution equations. The corresponding Hamilton's density can always be obtained, albeit in a formal sense.