Diffeomorphism-invariant Covariant Hamiltonians of a pseudo-Riemannian Metric and a Linear Connection

TL;DR

This paper develops a class of covariant Hamiltonians for pseudo-Riemannian metrics and linear connections that remain invariant under diffeomorphisms, with applications to Palatini and Einstein-Hilbert Lagrangians.

Contribution

It introduces a geometrically defined class of first-order Ehresmann connections ensuring diffeomorphism invariance of covariant Hamiltonians for certain Lagrangians.

Findings

Constructed a class of Ehresmann connections preserving invariance.

Applied results to Palatini and Einstein-Hilbert Lagrangians.

Ensured invariance of covariant Hamiltonians under Diff N.

Abstract

\noindent Let $M\to N$ (resp.\ $C\to N$) be the fibre bundle of pseudo-Riemannian metrics of a given signature (resp.\ the bundle of linear connections) on an orientable connected manifold $N$. A geometrically defined class of first-order Ehresmann connections on the product fibre bundle $M\times_NC$ is determined such that, for every connection $\gamma $ belonging to this class and every $\mathrm{Diff}N$-invariant Lagrangian density $\Lambda $ on $J^1(M\times_NC)$, the corresponding covariant Hamiltonian $\Lambda ^\gamma $ is also $\mathrm{Diff}N$-invariant. The case of $\mathrm{Diff}N$-invariant second-order Lagrangian densities on $J^2M$ is also studied and the results obtained are then applied to Palatini and Einstein-Hilbert Lagrangians.