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Abstract
Although there is little doubt that gravitational waves exist and carry energy as they propagate, it has been notoriously difficult to explain where in spacetime this energy resides. We have summarized a new approach to the localization of gravitational energy-momentum, valid within the linear approximation to general relativity . Built around a local description of the exchange of energy-momentum between matter and linear gravity, the framework defines a unique symmetric gravitational energy-momentum tensor, free of second derivatives, and motivates a natural gauge-fixing programme, which renders the description unambiguous. Once the gauge has been fixed according to this programme, the gravitational energy-momentum tensor obeys the dominant energy condition: gravitational energy-density is never negative, and gravitational energy-flux is never spacelike.
Highlights
Half a century ago, a simple argument established that gravitational waves carry energy and can exchange this energy with matter
For the special case of planewaves, we need not concern ourselves with the gauge freedom that remains after enforcing the harmonic condition: the requirement that the gauge be chosen such that the plane-wave form of the field be manifest is sufficient to unambiguously define the energy-momentum tensor τab from the physical spacetime (M, g)
By constructing a framework to quantify this idea, we have succeeded in localising the energy and momentum of the linear gravitational field, and have shown this energy to be positive and to not flow faster than light
Summary
A simple argument established that gravitational waves carry energy and can exchange this energy with matter. The elusiveness of the “right answer”, and the wrongness of the question, are very often identified as arising from gravity’s gauge freedom, the consequence of which is a one-to-many mapping between physical spacetime and whatever localisation of gravitational energy-momentum might be proposed This issue was cast in terms of coordinate dependence, and the multitude of non-covariant objects that were constructed (first by Einstein [3], and most famously by Landau and Lifshitz [4]) were termed energy-momentum pseudotensors. It is always possible to locate the energy-momentum of matter by measuring the gravity it generates, so one might suggest that gravity’s energymomentum should be localised in a similar fashion, by examining the interaction it has with itself Following this idea to its conclusion, it has been shown [9,10,11,12] that general relativity may be constructed from an initially linear (spin-2) field theory that is systematically coupled to its own (Hilbert) energy-momentum tensor.
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