Weiss variation for general boundaries

Abstract

The Weiss variation of the Einstein-Hilbert action with an appropriate boundary term has been studied for general boundary surfaces; the boundary surfaces can be spacelike, timelike, or null. To achieve this we introduce an auxiliary reference connection and find that the resulting Weiss variation yields the Einstein equations as expected, with additional boundary contributions. Among these boundary contributions, we obtain the dynamical variable and the associated conjugate momentum, irrespective of the spacelike, timelike or, null nature of the boundary surface. We also arrive at the generally non-vanishing covariant generalization of the Einstein energy-momentum pseudotensor. We study this tensor in the Schwarzschild geometry and find that the pseudotensorial ambiguities translate into ambiguities in the choice of coordinates on the reference geometry. Moreover, we show that from the Weiss variation, one can formally derive a gravitational Schr{ö}dinger equation, which may, despite ambiguities in the definition of the Hamiltonian, be useful as a tool for studying the problem of time in quantum general relativity. Implications have been discussed.

arXiv commentv2, Published in the GERG memorial volume for Prof. T. Padmanabhan, 25 pages, 1 figure

References in corpus (32)

• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• Gravitational action with null boundaries
• Boundary Terms, Variational Principles and Higher Derivative Modified Gravity
• A Boundary Term for the Gravitational Action with Null Boundaries
• Quasilocal mass in general relativity
• Relativistic conservation laws and integral constraints for large cosmological perturbations
• General Relativity from a Thermodynamic Perspective
• General Teleparallel Quadratic Gravity
• Thermodynamical interpretation of the geometrical variables associated with null surfaces
• The Structure of the Gravitational Action and its relation with Horizon Thermodynamics and Emergent Gravity Paradigm
• Variational Principle for Gravity with Null and Non-null boundaries: A Unified Boundary Counter-term
• Boundary terms of the Einstein-Hilbert action
• Boundary and Corner Terms in the Action for General Relativity
• Relativistic Conservation Laws on Curved Backgrounds and the Theory of Cosmological Perturbations
• Covariant double-null dynamics: $(2+2)$-splitting of the Einstein equations
• Connection independent formulation of general relativity
• A Novel Derivation of the Boundary Term for the Action in Lanczos-Lovelock Gravity
• Volume average regularization for the Wheeler-DeWitt equation
• Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G
• On the energy of gravitational waves
• Noether Current, Black Hole Entropy and Spacetime Torsion
• Null boundary terms for Lanczos-Lovelock gravity
• Gravitational energy is well defined
• The Weiss Variation of the Gravitational Action
• From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimes with a spatial boundary
• On the uniqueness of harmonic coordinates
• The Field-Theoretic Approach in General Relativity and Other Metric Theories. A Review
• Restoration of four-dimensional diffeomorphism covariance in canonical general relativity: An intrinsic Hamilton-Jacobi approach
• Almost-Killing equation: Stability, hyperbolicity, and black hole Gauss law
• Quantum Gravity on a Manifold with boundaries: Schrödinger Evolution and Constraints
• Linearized Constraints in the Connection Representation: Hamilton-Jacobi Solution
• Space, Time, Matter in Quantum Gravity

Cited by in corpus (6)

• Covariant canonical formulations of classical field theories
• Probing the quantum nature of black holes with ultra-light boson environments
• Lorentzian quantum cosmology with torsion
• Einstein gravity from the Einstein action: Counterterms and covariance
• Junction conditions in a general field theory
• Gravity from thermodynamics in vacuum: Lorentz invariance and the role of Bel-Robinson tensor