# Energy in GR - side question

1. Jun 21, 2012

### Antiphon

I don't want to derail it. But if energy is "poorly defined" globally in GR as explained in that thread then let's exploit it. Let's devise a "machine" that creates energy.

I pose this in such a provocative way to underscore the point- its not enough in todays physics to say that global energy conservation isn't well defined. We should either lead this notion into a contradiction or start figuring out how to manufacture energy with it.

2. Jun 22, 2012

### spoirier

Energy in GR can no more be described as a precise quantity simply obtained by integrating some explicit field along the space-like 3D surface we consider. This does not mean that there is no conservation of energy. But it takes a more subtle, complex form.
In particular we can roughly define the mass inside a sphere with size r, as r times the integral of the intrinsic Riemannian curvature of space through this sphere. As this is computed from the surface only and not from the inside, it is not possible to modify it by purely local processes : it is necessary to do something that can affect the surface. Thus if you take a large sphere away from the system you consider, you can only increase your local energy by bringing it from far away.

3. Jun 22, 2012

### tom.stoer

The problem is that energy as a volume integral cannot be defined in GR. You cannot create something that isn't defined ;-)

Energy is conserved locally i.e. there is a covariantly conserved energy-momentum tensor i.e. conserved energy-momentum density.

4. Jun 22, 2012

### Antiphon

But it sure seems counterintuitive that a locally conserved quantity isn't globally conserved.

If I understand the replies, it may in fact be globally conserved but we haven't figured out a proper integration method for it yet.

5. Jun 24, 2012

### tom.stoer

Definitly not.

The conserved current density related to Noether's theorem would have to be integrated over a volume - which is not possible mathematically and covariantly with a tensor density!

On the other hand the surface integrals which have been constructed for non-local definitions of a "mass" or "energy" cannot be related to Noether's theorem.

So what would help is the derivation of the surface integrals from a first principle like Noether's theorem, not the search for a volume integral.

http://relativity.livingreviews.org/Articles/lrr-2009-4/ [Broken]

Last edited by a moderator: May 6, 2017