Energy Conservation Paradox: Is It True or Not?

[Dale]

My prime interest here is how conservation laws apply to interactions involving charged particles.

No need to worry about GR then. You can assume flat spacetime and all of the standard conservation laws.

 

[Dale]

No, it doesn't, because the theorem itself requires a timelike Killing vector field

This is what led to my mistaken comments about pairs of orbiting black holes. A binary black-hole spacetime doesn't have a timelike Killing vector field, therefore a straight application of Noether's theorem says no conserved energy. But obviously there are definitions of energy in GR that are not limited by that and can be applied to asymptotically flat spacetimes, which I forgot.

 

[wabbit]

On a side note, http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html that Noether's Theorems were originally developped for the purpose of resolving the issue of energy conservation in GR!

 

[bhobba]

My prime interest here is how conservation laws apply to interactions involving charged particles. As an example when an electron approaches a positively charged macroscopic object there is a conversion between PE and KE and when everything is measured and taken into account it seems that energy is conserved.

One would require some VERY strong gravitational fields for it to not be an extremely good approximation to an inertial frame. At that strength the electron would couple to the gravitational field and the whole situation would be far from simple to analyse.

Thanks
Bill

 

[PAllen]

More precisely, the formalism in which the theorem is formulated requires an action and a global foliation. But the conditions for the theorem itself to be true are more stringent than that; see below.
No, it doesn't, because the theorem itself requires a timelike Killing vector field, and FLRW spacetimes don't have one. In terms of your description of the formalism, quoted above, for the theorem to be true, the metric of the spatial hypersurfaces in the foliation would have to be independent of the global time coordinate. In FLRW spacetimes, it isn't, because the scale factor is a function of the time coordinate.

Yes, you are right about the simple form of the theorem that applies, e.g. to the Poincare group. I was mis-remembering the distinctions bertween the simple and the more general versions of the theorem. The more general form can be used to theoretically justify pseudo-tensor conserved energy formulations, as explained here:

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

 

[macrobbair]

one simply defines a new type of energy to make it conserved...
unless one is in gr where one can change frame,
then one needs more information...

 

[phinds]

one simply defines a new type of energy ...

Uh ... "defines a new type of energy" ?