On the problem of uniqueness of energy-momentum tensor of gravitational field

For an island-like distribution of matter the gravitational energy-momentum tensor is defined according to Weinberg as a source of metric. If this source is formed by selfinteractions of gravitons, so that nonphysical degrees of freedom are excluded, then this source is a reasonable candidate for the energy-momentum tensor of gravitational field. The disastrous influence of the nonphysical degrees of freedom is demonstrated by comparing the gravitational energy-momentum tensors in the harmonic, isotropic and standard frames for the Schwarzschild solution. The harmonic frame is clearly preferable for defining the gravitational energy-momentum tensor. 1 The gravitational energy-momentum tensor as the source of metric There are several arguments in favor of non-localizability of the energy of the gravitational field, see §20.4 in [1]. They do not seem convincing enough. Following [2], we consider the case when gμν = ημν + hμν , ημν = diag(−1, 1, 1, 1) (1) and hμν → 0 quickly enough when x → ∞, but it is not assumed that hμν ≪ 1 everywhere. The wave equation for hμν is hμν,λ λ − hμ,λν − hν,λμ + h,μν + ημν(hσλ − h,λ) = −16πG(Tμν + tμν), h ≡ hλ; h,σ ≡ ∂ ∂xσ , (2) cf. Ch.3, §17 in [3]. Here tμν is the gravitational energy-momentum tensor. ∗E-mail: [email protected]