LAUSR

Noether’s 2 theorems [1] were discovered in the context of seeking the answer to the problem of defining the energy of the gravitational field, but the problem remains open today. The main difficulty resides in that general covariance, being a gauge symmetry, does not yield a non-vanishing Noether current that could make a good definition of energy. This problem can be fixed by resorting to some external geometry, and it is widely believed that gravitational energy can be unambiguously defined only with respect to some background structure, such as an asymptotically maximally symmetric solution [2]. The modern view is that gravitational energy can be defined quasi-locally [3], but it is fair to say that there is no physical principle that would distinguish amongst the various quasi-local prescriptions [4, 5]. The canonical resolution was proposed recently [6, 7]. The minimal “universal background structure” that may bestow covariance upon otherwise pseudo-tensorial quantities is, tautologically, the pure-gauge translation connection. The implementation of this connection in Einstein’s formulation of General Relativity [8] results in the reformulation as an integrable gauge theory of translations [9] dubbed Coincident General Relativity [10]. The proposal was to identify the general-relativistic inertial frame, which fixes the additional gauge freedom due to the connection in this theory, and thus uniquely determines gravitational energy, as the frame wherein the energy-momentum tensor of the gravitational field vanishes [6] (in agreement with Cooperstock’s hypothesis [11]). The proposal is consistent with the Noether’s 2 theorem, and with the intuition that gravity is equivalent to inertia. However, the proposal is at odds with the usual considerations of energy in metric-teleparallel gravity [12–15]. Though the gravitational energy-momentum introduced in the metric-teleparallel theory is generally covariant, it is not Lorentz covariant. Therefore calculations of this energy-momentum are as arbitrary as with Einstein’s pseudotensor [8], but as we will argue here, the criterion for the inertial frame [6] should yield uniquely the correct quasi-local energy also when applied in the metric-teleparallel theory. An important lesson of this letter will be that the gravitational energy-momentum and entropy are purely holographic, and this property is independent of the gravity theory and its formulation. Though the gravitational energymomentum current ta vanishes in the inertial frame, we nevertheless find that gravitational waves can, and do carry energy (in contrast to Cooperstock’s hypothesis [11]). The reason is that the conserved charges arise from pure surface integrals. The field equations of Coincident General Relativity written in terms of the gravitational excitation ha and sourced by an energy-momentum ta have the form Dha = ta, and even though in vacuum ta = 0, there may exist non-vanishing conserved charges ∮