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Approximate symmetries and the energy content of spacetimes

There is a problem of defining energy in time-varying spacetimes as energy is not conserved in them. This problem is particularly severe for gravitational waves as there is no conventional stress-energy tensor. The gravitational field is given by the Weyl tensor but it does not provide a measure of the energy. Pseudo tensors are introduced but are not invariant quantities. Another approach uses the breaking of time translational invariance for a measure of the energy. No attempt has proved unambiguously better than the others in that they provide calculational convenience but no physical implications. We here use approximate Lie symmetries of the geodesic equations and the Noether symmetries for the arc length Lagrangian to obtain a definition of energy that does provide physical implications.

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