[Paper Review] Ultralocal energy density in massive gravity

Vladimir O. Soloviev, Margarita V. Tchichikina|arXiv (Cornell University)|Jun 28, 2011

Black Holes and Theoretical Physics10 references3 citations

TL;DR

This paper presents a spacetime covariant Hamiltonian formulation for massive gravity within the Relativistic Theory of Gravitation (RTG), using Kuchař's approach to derive Poincaré generators as spatial integrals of ultralocal quantities constructed from two metrics. The key result is the identification of a total energy-momentum tensor $ T^{ ueta}_{\text{total}} = -\frac{m^2}{\kappa} f^{\nu\beta} $, which ensures conservation of energy-momentum and Poincaré algebra without central charges, even in the absence of matter, with a positive energy density $ \frac{m^2}{\kappa} $ in Minkowski space.

ABSTRACT

We provide a space-time covariant Hamiltonian treatment for a finite-range gravitational theory. The Kuchar approach is used to demonstrate the bimetric picture of space-time in its most transparent form. This Hamiltonian formalism is applied for the straightforward realization of the Poincaré algebra in Dirac brackets. It uncovers the simplest form of the Poincaré generators expressed as spatial integrals of ultralocal quantities constructed pure algebraically by means of the two space-time metrics.

Motivation & Objective

• To develop a spacetime covariant Hamiltonian formalism for massive gravity that preserves general coordinate invariance and allows flexible spacetime foliations.
• To resolve the causality issue in bimetric gravity by requiring the dynamical metric's null cone to lie within the background metric's null cone.
• To construct Poincaré generators as spatial integrals of ultralocal expressions derived algebraically from the two metrics.
• To demonstrate that the total energy-momentum tensor $ T^{ u\beta}_{\text{total}} = -\frac{m^2}{\kappa} f^{\nu\beta} $ ensures conservation laws and Poincaré algebra without central charges.
• To provide a foundation for canonical quantization and numerical comparison between massive and massless gravity.

Proposed method

• Adopt Kuchař's canonical formalism, treating the dynamical metric and background metric as independent variables with distinct roles.
• Apply $3+1$ decomposition to spacetime tensors using both the dynamical and background metrics, ensuring spacelike hypersurfaces are spacelike in both metrics.
• Derive the Hamiltonian from the Lagrangian, identifying four primary and four secondary second-class constraints from the non-dynamical metric components.
• Solve the second-class constraints algebraically to eliminate four metric components, reducing the phase space and constructing Dirac brackets.
• Express the Poincaré generators (energy, momentum, angular momentum) as spatial integrals of ultralocal quantities built from the two metrics.
• Define the total energy-momentum tensor as $ T^{ u\beta}_{\text{total}} = -\frac{m^2}{\kappa} f^{\nu\beta} $, where $ f^{\nu\beta} $ is the inverse of the induced metric on the constant-time hypersurface.

Experimental results

Research questions

• RQ1 How can a spacetime covariant Hamiltonian formalism be consistently constructed for massive gravity with two independent metrics?
• RQ2 Does the bimetric structure of RTG allow for a consistent realization of the Poincaré algebra without central charges?
• RQ3 What is the nature of the energy-momentum tensor in massive gravity, and does it ensure positive energy and conservation laws?
• RQ4 Can the ultralocal structure of the Poincaré generators be derived purely algebraically from the two metrics?
• RQ5 How does the absence of first-class constraints in massive gravity affect the Cauchy problem compared to massless gravity?

Key findings

• The total energy-momentum tensor is identified as $ T^{ u\beta}_{\text{total}} = -\frac{m^2}{\kappa} f^{\nu\beta} $, which ensures the conservation of energy-momentum and the closure of the Poincaré algebra.
• The energy density in Minkowski space without matter is $ \frac{m^2}{\kappa} $, a positive and finite value, implying a nonzero vacuum energy density.
• The Poincaré generators are expressed as spatial integrals of ultralocal quantities derived purely algebraically from the two metrics, ensuring manifest covariance.
• The Dirac brackets close into the Poincaré algebra without central charges, confirming the consistency of the symmetry algebra.
• The energy-momentum tensor satisfies $ (T^{ u\beta}_{\text{total}})_{;\nu} = 0 $ under the background metric's covariant derivative, ensuring conservation for Killing vectors.
• The theory avoids negative energy flows in spherically symmetric configurations, as scalar wave radiation is absent, supporting stability.