We know that in GR it is not possible for arbitrary spacetimes to define a conserved energy by using a 3-integral. There are some obstacles like(adsbygoogle = window.adsbygoogle || []).push({});

- the covariant conservation law DT = 0 (D = covariant derivative; T = energy-momentum-tensor) does not allow for the usual dV integration (like dj = 0)
- only with a timelike killing field k one can define a conserved energy-momentum 4-vector t with t = kT and dt =0
- ...

Now let's forget about this specific case and ask the following more general questions:

Suppose we have a 4-dim. pseudo-Riemannian manifold M with spacelike foliations F, F', F'', ... For each F one can define a family of 3-volumes V_{F}(t) covering M where t indicates a timelike direction (coordinate) perpendicular to V_{F}(t). For each V_{F}(t) one can define 3-integrals

[itex]Q_F[q] = \int_{V_F(T)} \,q[/itex]

using differential forms q?

Under which conditions do these Q represent "reasonable physical obervables" with a well-defined, covariant transformation law? Under which conditions can one find a conservation law

[itex]\frac{dQ_F[\omega]}{dt} = 0[/itex]

Are there some physical relevant examples for q? How can q be constructed from the metric g (or a 3-bein e w.r.t to the 3-volume V_{F})?

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# Covariant global constants of motion in GR?

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