Constructing a well-posed variational principle is a non-trivial issue in general relativity. For spacelike and timelike boundaries, one knows that the addition of the Gibbons–Hawking–York (GHY) counter-term will make the variational principle well-defined. This result, however, does not directly generalize to null boundaries on which the 3-metric becomes degenerate. In this work, we address the following question: What is the counter-term that may be added on a null boundary to make the variational principle well-defined? We propose the boundary integral of $$2 \sqrt{-g} \left( \Theta +\kappa \right) $$
 as an appropriate counter-term for a null boundary. We also conduct a preliminary analysis of the variations of the metric on the null boundary and conclude that isolating the degrees of freedom that may be fixed for a well-posed variational principle requires a deeper investigation.

A boundary term for the gravitational action with null boundaries

The Palatini formalism, which assumes the metric and the affine connection as independent variables, is developed for gravitational theories in flat geometries. We focus on two particularly interesting scenarios. First, we fix the connection to be metric compatible, as done in the usual teleparallel theories, but we follow a completely covariant approach by imposing the constraints with suitable Lagrange multipliers. For a general quadratic theory we show how torsion naturally propagates and we reproduce the Teleparallel Equivalent of General Relativity as a particular quadratic action that features an additional Lorentz symmetry. We then study the much less explored theories formulated in a geometry with neither curvature nor torsion, so that all the geometrical information is encoded in the non-metricity. We discuss how this geometrical framework leads to a purely inertial connection that can thus be completely removed by a coordinate gauge choice, the coincident gauge. From the quadratic theory we recover a simpler formulation of General Relativity in the form of the Einstein action, which enjoys an enhanced symmetry that reduces to a second linearised diffeomorphism at linear order. More general theories in both geometries can be formulated consistently by taking into account the inertial connection and the associated additional degrees of freedom. As immediate applications, the new cosmological equations and their Newtonian limit are considered, where the role of the lapse in the consistency of the equations is clarified, and the Schwarzschild black hole entropy is computed by evaluating the corresponding Euclidean action. We discuss how the boundary terms in the usual formulation of General Relativity are related to different choices of coordinates in its coincident version and show that in isotropic coordinates the Euclidean action is finite without the need to introduce boundary or normalisation terms. Finally, we discuss the double-copy structure of the gravity amplitudes and the bootstrapping of gravity within the framework of coincident General Relativity.

Teleparallel Palatini theories

Noether's general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance. Bulk charges are replaced by fluxes at a suitable singularity (in general, at infinity) of so-called superpotentials, namely local functions of the gauge fields (or more generally of the gauge forms). Some gauge-invariant bulk charges and current densities may subsist when distinguished one-dimensional subgroups are present. We shall study mostly local consequences of gauge invariance. Quite generally there exist local superpotentials analogous to those of Freud or Bergmann for general relativity. They are parametrized by infinitesimal gauge transformations, but are afflicted by topological ambiguities which one must handle on a case-by-case basis. The choice of variational principle: variables, surface terms and boundary conditions is crucial. As a first illustration we propose a new affine action that reduces to general relativity upon gauge fixing the dilatation (Weyl-1918-like) part of the connection and elimination of auxiliary fields. We can also reduce it by similar considerations either to the Palatini action or to the Cartan-Weyl moving frame action and compare the associated superpotentials. This illustrates the concept of Noether identities. We formulate a vanishing theorem for the superpotential and the current when there is a (Killing) global isometry or its generalization. We distinguish between asymptotic symmetries and symmetries defined in the bulk. A second and independent application is a geometrical reinterpretation of the convection of vorticity in barotropic non-viscous fluids first established by Helmholtz-Kelvin, Eckart and Ertel. In the homentropic case it can be seen to follow by a general theorem from the vanishing of the superpotential corresponding to the time-independent relabelling symmetry. The special diffeomorphism symmetry is, in the absence of dynamical gauge field and spin, associated with a vanishing internal transverse momentum flux density. We also consider the non-homentropic case. We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an impulsive forcing for creating or destroying selectively helicity (respectively enstrophies) in odd (respectively even) dimensions. This is an example of a new and general forcing rule.

http://cds.cern.ch/record/351871/files/9804029.pdf

Currents and superpotentials in classical gauge-invariant theories: I. Local results with applications to perfect fluids and general relativity

Constructing a well-posed variational principle is a non-trivial issue in general relativity. For spacelike and timelike boundaries, one knows that the addition of the Gibbons-Hawking-York (GHY) counter-term will make the variational principle well-defined. This result, however, does not directly generalize to null boundaries on which the 3-metric becomes degenerate. In this work, we address the following question: What is the counter-term that may be added on a null boundary to make the variational principle well-defined? We propose the boundary integral of $2 \sqrt{-g} \left( \Theta+\kappa \right)$ as an appropriate counter-term for a null boundary. We also conduct a preliminary analysis of the variations of the metric on the null boundary and conclude that isolating the degrees of freedom that may be fixed for a well-posed variational principle requires a deeper investigation.

A Boundary Term for the Gravitational Action with Null Boundaries

We present a detailed examination of the variational principle for metric general relativity as applied to a ``quasilocal'' spacetime region $\M$ (that is, a region that is both spatially and temporally bounded). Our analysis relies on the Hamiltonian formulation of general relativity, and thereby assumes a foliation of $\M$ into spacelike hypersurfaces $\Sigma$. We allow for near complete generality in the choice of foliation. Using a field--theoretic generalization of Hamilton--Jacobi theory, we define the quasilocal stress-energy-momentum of the gravitational field by varying the action with respect to the metric on the boundary $\partial\M$. The gravitational stress-energy-momentum is defined for a two--surface $B$ spanned by a spacelike hypersurface in spacetime. We examine the behavior of the gravitational stress-energy-momentum under boosts of the spanning hypersurface. The boost relations are derived from the geometrical and invariance properties of the gravitational action and Hamiltonian. Finally, we present several new examples of quasilocal energy--momentum, including a novel discussion of quasilocal energy--momentum in the large-sphere limit towards spatial infinity.

Action and Energy of the Gravitational Field

The authors discuss the modifications needed to free the Einstein-Hilbert action of gravitation from all second derivatives of fields, and give explicitly the resulting action applicable to either metric or vierbein variables. Variation of this action leads to Einstein's equations without boundary conditions. It vanishes for flat space-time and contains one arbitrary real parameter.

https://iopscience.iop.org/article/10.1088/0305-4470/16/8/013/pdf

Surface Integrals and the Gravitational Action

The canonical Hamiltonian for vierbein general relativity is derived, starting from an expression for the action which differs from the Hilbert-Einstein action by a surface term. Results are given both in the proper time formalism and in time gauge.

The canonical Hamiltonian for vierbein general relativity

Discusses the canonical treatment of general relativity in a vierbein formulation. The authors derive the primary constraints, and find that they satisfy an algebra the same as that of the generators of local Poincare transformations, namely the Poincare algebra.

https://iopscience.iop.org/article/10.1088/0305-4470/16/14/026/pdf

J. M. Charap

Papers

Surface Integrals and the Gravitational Action

The canonical Hamiltonian for vierbein general relativity

Canonical general relativity: the primary constraint algebra