Quasi-Local Energy for Cosmological Models

Abstract

First we briefly review our covariant Hamiltonian approach to quasi-local energy, noting that the Hamiltonian-boundary-term quasi-local energy expressions depend on the chosen boundary conditions and reference configuration. Then we present the quasi-local energy values resulting from the formalism applied to homogeneous Bianchi cosmologies. Finally we consider the quasi-local energies of the FRW cosmologies. Our results do not agree with certain widely accepted quasi-local criteria.

Intrinsic vanishing of energy and momenta in a universe

“…Quoting this conclusion, Vargas [12] used again the pseudotensorial method to calculate the energy of the universe in teleparallel gravity. The conclusion was quoted in [13] too.…”

“…The energy of Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies has been calculated by different authors using divers procedures, like pseudotensorial methods based on specific choices of coordinates [29][30][31][32][33], or Hamiltonian methods imposing boundary conditions [13,21], or by choosing an appropriate background configuration [2,8], or even by other procedures [10]. Quasi-local approaches have also been extensively considered, providing distinct results because of the different used definitions [9,34].…”

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…For class A models (i.e., for I, II, VI 0 , VII 0 , VIII and IX Bianchi types) this is zero, and for class B models (III, IV, V, VI h and VII h Bianchi models) the quasi-local energy is negative , and the energy is proportional to the volume of the domain that is bounded by . (Here a sign error in the previous calculations, reported in [134, 387, 385], is corrected.) The apparent contradiction of the nonpositivity of the energy in the present context and the non-negativity of the energy in general small-sphere calculations indicates that the geometrically distinguished tetrad field in the Bianchi models does not reduce to the ‘natural’ approximate translational Killing fields near a point.…”

Energy of homogeneous cosmologies

“…For those cases, our homogenous results have been compared with those found using a similar approach for the more familiar "isotropic-about-one-point" FLRW formulations [58,59]. The isotropic Bianchi I is isometric to the flat k = 0 model, and both are found to have vanishing energy.…”

“…It had been expected that we would always find non-negative energy. One consequence of this strong belief was that a minor sign error was, quite unfortunately, overlooked in some calculations; this lead to an incorrect claim of positive energy for Bianchi B models being reported in some recent conference proceedings [58,59].…”