Do Cosmological Assumptions Affect Conservation Laws?

[geoffc]

If the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy, would a cosmology that drops either of those assumptions lead to violations of those conservation laws which in turn could be observed? My first guess would be yes, but from what little I understand it becomes very difficult if not impossible to define and measure those quantities on a global scale. Thoughts?

 

[pervect]

If the assumptions of homogeneity and isotropy lead to the conservation laws of linear momentum, angular momentum, and energy, would a cosmology that drops either of those assumptions lead to violations of those conservation laws which in turn could be observed? My first guess would be yes, but from what little I understand it becomes very difficult if not impossible to define and measure those quantities on a global scale. Thoughts?

Sort of. One still has local conservation laws of linear momentum and energy in a small section of empty space, but if the overall cosmology lacks (for instance) time translation symmetry, there won't necessarily be a globally conserved energy.

Note that in spite of homogeneity and isotropy, the FRW metric does NOT have time translation symmetry, which implies there isn't any global notion of the mass/energy of the universe.

See for instance the sci.physics.faq Is energy conserved in General Relativity?, which I will quote in part. For copyright and other reasons I am only quoting a "fair use" portion of the FAQ, so I encourage the OP and other interested people to read the entire original.

Is Energy Conserved in General Relativity?

In special cases, yes. In general -- it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a finite-sized piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

You might also look at the wikipedia article for a more advanced discussion about the various sorts of mass, energy, and momentum defined in GR.