Quasilocal conservation laws: why we need them

Abstract

We argue that conservation laws based on the local matter-only stress-energy-momentum tensor (characterized by energy and momentum per unit volume) cannot adequately explain a wide variety of even very simple physical phenomena because they fail to properly account for gravitational effects. We construct a general quasilocal conservation law based on the Brown and York total (matter plus gravity) stress-energy-momentum tensor (characterized by energy and momentum per unit area), and argue that it does properly account for gravitational effects. As a simple example of the explanatory power of this quasilocal approach, consider that, when we accelerate toward a freely-floating massive object, the kinetic energy of that object increases (relative to our frame). But how, exactly, does the object acquire this increasing kinetic energy? Using the energy form of our quasilocal conservation law, we can see precisely the actual mechanism by which the kinetic energy increases: It is due to a bona fide gravitational energy flux that is exactly analogous to the electromagnetic Poynting flux, and involves the general relativistic effect of frame dragging caused by the object's motion relative to us.

Mass in Lovelock unique vacuum gravity theories

“…This is a subtle issue in gravitational physics because of the equivalence principle, which makes localization of gravitational energy (and momentum and angular momentum) fraught with ambiguities [3]. Approaches for addressing this problem date back six decades [4,5] and more, and include a mixture of global [6,7,8,9,10] and quasilocal [11,12,13,14,15] methods. Generalizations to Lovelock gravity have been carried out [16], and a universal form of the boundary term yielding a background-independent definition of conserved quantities for any Lovelock gravity theory with anti-de Sitter (AdS) asymptotics has been constructed [17].…”

Motion of localized sources in general relativity: Gravitational self-force from quasilocal conservation laws

“…holds. In what follows, we introduce the concept of quasilocal frames [46][47][48][49][50][51][52][53] and describe the basic steps for their construction, as well as the energy and momentum conservation laws associated therewith. In Subsection II A we offer an heuristic idea of quasilocal frames before proceeding in Subsection II B to present the full mathematical construction.…”

“…Most of the past work that has been done with quasilocal frames has in fact been done in the rigid case [46][47][48][49][50][51]. We know however that other quasilocal frame choices are also possible, such as geoids-dubbed geoid quasilocal frames [52,53]: these are the general-relativistic generalization of "constant gravitational potential" surfaces in Newtonian gravity.…”

“…The construction of general conservation laws from τ was first achieved in Refs. [46,48], and proceeds along the following lines. Let ψ ∈ T B be an arbitrary vector field in B.…”

Geoids in general relativity: Geoid quasilocal frames

“…In such QFs, the gravitational energy flux is −α · P, which is simply the special relativistic rate of change of energy, −a · p, of an object with fourmomentum p as seen by an observer with four-acceleration a, promoted to a general-relativistic energy flux passing through a rigid boundary containing the object: a becomes α, the acceleration of observers on the boundary, and p becomes P, the quasilocal momentum density measured by those observers (via generalrelativistic frame dragging). An equivalence principle-based argument explaining why this must be so is given in [10,11].…”