# Gauge from holography and holographic gravitational observables

## TL;DR

This paper explores how gauge invariance and holographic principles in gravitational theories relate to the calculation of physical observables, emphasizing the role of boundary data and the limitations of purely holographic observables.

## Contribution

It introduces a framework for understanding gauge equivalence and observables in gravitational theories using holography and boundary integrals, extending previous results to include non-holographic observables.

## Key findings

- Holographic observables can be expressed as surface integrals over boundary surfaces.
- Non-holographic observables are necessary to distinguish gauge inequivalent solutions.
- The framework applies to theories including vacuum General Relativity.

## Abstract

In a spacetime divided into two regions $U_1$ and $U_2$ by a hypersurface $\Sigma$, a perturbation of the field in $U_1$ is coupled to perturbations in $U_2$ by means of the holographic imprint that it leaves on $\Sigma$. The linearized gluing field equation constrains perturbations on the two sides of a dividing hypersurface, and this linear operator may have a nontrivial null space. A nontrivial perturbation of the field leaving a holographic imprint on a dividing hypersurface which does not affect perturbations on the other side should be considered physically irrelevant. This consideration, together with a locality requirement, leads to the notion of gauge equivalence in Lagrangian field theory over confined spacetime domains. Physical observables in a spacetime domain $U$ can be calculated integrating (possibly non local) gauge invariant conserved currents on hypersurfaces such that $\partial \Sigma \subset \partial U$. The set of observables of this type is sufficient to distinguish gauge inequivalent solutions. The integral of a conserved current on a hypersurface is sensitive only to its homology class $[\Sigma]$, and if $U$ is homeomorphic to a four ball the homology class is determined by its boundary $S = \partial \Sigma$. We will see that a result of Anderson and Torre implies that for a class of theories including vacuum General Relativity all local observables are holographic in the sense that they can be written as integrals of over the two dimensional surface $S$. However, non holographic observables are needed to distinguish between gauge inequivalent solutions.