Gravitational energy for GR and Poincaré gauge theories: A covariant Hamiltonian approach

Abstract

Our topic concerns a long standing puzzle: the energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy-momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein's general relativity (GR), noting especially Noether's contribution. In the second part we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincaré gauge theories of gravity (PG), with GR as a special case. The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and, furthermore it determines the boundary conditions for the dynamical variables. Energy-momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincaré transformations. They are best described in a local Poincaré gauge theory. The type of spacetime that naturally has this symmetry is Riemann-Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy-momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to the PG. d One reason was his famous "hole" argument. 61,98 e Einstein and Hilbert had quite different agendas; 112,115,136,141 Hilbert in his Foundation of Physics papers, based on the work of Einstein and Mie, was using his axiomatic method with the objective of finding a unified field theory of gravity and electromagnetism. 15,28,58,108,109,114 July 28, 2015 0:26 World Scientific Review Volume -9.75in x 6.5in 100GR˙Ch4˙150726v1 page 5Gravitational energy for GR and Poincaré gauge theories:a covariant Hamiltonian approach 5 l The sign in this expression is dictated by the condition for positive energy determined by the Hamiltonian using our local Minkowski signature convention: Pµ = (−E/c, p).

Freud’s superpotential in general relativity and in Einstein-Cartan theory

“…In our deduction, the relation to the Einstein 3-form is apparent and quite natural. A similar observation can be found in [5] where it was pointed out that this derivation of the superpotential is in "remarkable constrast" to the tensor calculus approach.…”

“…General relativity does not lend itself naturally to the definition of a gravitational energy. This led to the formulation of energy via pseudo-tensors or an energy-momentum density complex, see Chen, Nester, and Tung [5] for a brief historical account; earlier contributions, chronologically ordered, include [31,19,6,8,30,13,14,20,27]. These objects appear more or less naturally when studying gravitational field equations, depending on one's approach.…”

A brief history of gravitational wave research

“…From the work of Hilbert, Klein and Noether it was established that there is no proper energy density for general relativity (or indeed for any generally covariant theory of gravity); for an extensive discussion of this point see Ref. [54]. For the density of gravitational energy-momentum Einstein had found a pseudotensor expression, i.e., not a proper tensor but rather an expression that was inherently reference frame dependent.…”

“…Regarding gravitational energy, there are two main ambiguities: (1) there is no unique expression, (2) each expression depends on the choice of reference frame and there is no preferred reference frame (both of these issues are addressed in our Hamiltonian boundary approach [54]).…”

“…In our understanding from Hilbert, Noether and our Hamiltonian approach this idea is in fact quite correct-but a density is not the whole story. There is more to energy-momentum-associated with any region there is also a (generally non-vanishing) surface term: gravitational energy is quasi-local [54].…”