Abstract
In a spacetime divided into two regions U1 and U2 by a hypersurface Σ, a perturbation of the field in U1 is coupled to perturbations in U2 by means of the holographic imprint that it leaves on Σ. 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 nonlocal) gauge invariant conserved currents on hypersurfaces such that ∂Σ⊂∂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 [Σ], and if U is homeomorphic to a four ball the homology class is determined by its boundary S=∂Σ. 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, nonholographic observables are needed to distinguish between gauge inequivalent solutions.
Highlights
In a spacetime divided into two regions U1 and U2 by a hypersurface Σ, a perturbation of the field in U1 is coupled to perturbations in U2 by means of the holographic imprint that it leaves on Σ
Motivations from string theory lead to the discovery of the gauge/gravity correspondence [5] providing an avenue for defining new quantum gravity theories
This article does not contribute by elongating the promising road towards quantum gravity emerging from the gauge/gravity correspondence, and our work is classical in substance
Summary
Motivated by the development of black hole thermodynamics [1, 2], more than two decades ago pioneers of modern physics put forward a holographic principle that sparked immense interest in the community [3, 4]. From the study of how perturbations propagate through communicating hypersurfaces a condition for gauge equivalence naturally arises We complement it with requirements of locality and relativity of measurement to give rise to a notion of gauge vector fields which is suited to work on spacetime confined domains. The second remark is about the term “local” In this introductory section we already mentioned “spacetime localized” properties of the field; it refers to properties of the field inside a bounded spacetime region U. With this notion of gauge, the version of Lagrangian field theory presented here is completed.
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