What is conserved in GR?

1. Oct 31, 2006

pivoxa15

In SR, relativitic mass, momentum and energy is conserved but what is conserved in GR since the frames in GR are noninertial.

2. Oct 31, 2006

lightarrow

I think you have hit the point! It's the same question I wanted to ask, (but I doubt it has an answer).

3. Oct 31, 2006

lalbatros

The conserved quantities are (more or less) the same as in SR: energy and momentum.
It is a property of the Einstein's equations.
See http://en.wikipedia.org/wiki/Einstein_field_equations" [Broken].
For the angular momentum, things are not so clear, I need to check in "Gravitation".

But what means "conserved" finally ?

Michel

Last edited by a moderator: May 2, 2017
4. Oct 31, 2006

lightarrow

Ok, but locally only.

Do you mean that there isn't any precise definition of it in GR?

Last edited by a moderator: May 2, 2017
5. Oct 31, 2006

lalbatros

No, I just meant that it so simple to derive D.T = 0, that one forgets to think about its meaning. For example: why does it mean conservation locally only? In addition, things are less clear for angular momentum.

Michel

6. Oct 31, 2006

pervect

Staff Emeritus
GR has a differential conservation law for momentum and energy, which works in any small neighborhood (where space-time is flat). In fact, this differential conservation law is built into the theory. However, GR does not have a global energy conservation law that gives a conserved scalar number for the total energy of a space-time EXCEPT for special cases (asymptotically flat space-times and static space-times).

The Schwarzschild metric is both static and asymptotically flat, by the way, so it has a conserved energy in the strong sense and this is commonly used (for instance in calculating geodesics).

See for instance http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

The conservation of energy and momentum in GR is best understood by Noerther's theorem, which was invented for that purpose.

7. Oct 31, 2006

lightarrow

So, thinking globally and in general (not only asymptotically flat space-times or static space-times) is there any conserved quantity, in some appropriate sense? (Maybe with respect proper time?)

8. Oct 31, 2006

pmb_phy

Its important that the divergence of the stress-energy-momentum tensor be given as a covariant statement since it would then hold in all systems of coordinates/frames. Otherwise you could always choose a system of coordinates in whichD.T = 0 but that the sum of the energies/momenta of the particles, even locally, are not conserved.

However, consider a system of coordinates in which none of the components of the metric tensor depends on time. In this case the time component of the momentum 1-form, i.e. the energy of the particle, will be a constant of motion.

Best wishes

Pete

Last edited: Oct 31, 2006
9. Nov 1, 2006

notknowing

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

My (non-expert) view is that the reason why global conservation of energy in GR runs in trouble is that some energy is always hidden in the "gravitational field". Since in GR, one can not speak of a field but only of geometry, there seems no way to do a correct sum of energies.

Rudi

10. Nov 1, 2006

pervect

Staff Emeritus
In the general case, there is no time translation symmetry or space translation symmetry, so there is no known defintion of energy or momentum that is conserved in standard GR. The appropriate symmetries have already been relaxed to asymptotic symmetries for the case of asymptotically flat space-times.

The culprit is the diffeomorphism invariance of the general theory, see for instance http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

(The authors above call 'local' energy conservation what I have been calling global energy conservation. I attribute the difference to the difference between the mathematical and the physical approach).

Some non-standard theories, like SCC, get around this by defining a preferred frame (then the theory no longer has the issue with infinite symmetry groups).

Last edited by a moderator: Apr 22, 2017
11. Nov 1, 2006

pervect

Staff Emeritus
When you have a metric where none of the metric coefficients are a function of time, you have a static space-time. The metric at coordinate time t is the same as the metric at coordinate time t + dt. This implies a discreete time translation symmetry, which by Noether's theorem implies a conserved energy. This is one of the special cases mentioned in the FAQ where energy can be defined in GR, the important case of a static space-time.

Thus orbits in the Schwarzschild metric have a conserved parameter due to the fact that the metric is static. This is discussed in most introductory GR books. The argument will work for any static space-time.

12. Nov 1, 2006

robphy

I think you mean "continuous".

13. Nov 1, 2006

pmb_phy

Yes. I'm overly familiar with that web page thank you.

It was quite clear in the OP's question that he was looking for what is conserved in GR and not what is not conserved in GR. The questioner did not restrict us to strictly global or strictly local considerations or what it was that we were to be considering as conserved quantities. It is rather easy to prove that one such conserved quantity is the energy of a single particle moving in a static/conservative g-field. For proof please see

http://www.geocities.com/physics_world/gr/conserved_quantities.htm

The response regarding the stress-energy-momentum tensor regards the local conservation of a system of particles which are located in a gravitational field. It never appeared from this conversation that he was interested in the energy if the body which generated the g-field itself or the entire energy of the g-field. The term "conserved" quite literally means "does not change in time." My response was to demonstrate when the energy of a particle in free-fall in a g-field was a constant of motion.

Best wishes

Pete

Last edited: Nov 1, 2006
14. Nov 2, 2006

pervect

Staff Emeritus

The good news is that nothing that Pete said earlier contradicts the sci.physics.faq, which is BTW a reasonably reliable source of information in spite of any issues Pete may or may not have with it. (It's hard to interpret his remark about being overly familar with the web page).

The sci.physics.faq is not quite peer reviewed, but it's been reviewed by a large number of people to help stamp out mistakes, typos, poor wording, etc.

The bad news is that you can't necessarily trust anything on Pete's webpages. Many of his web pages ARE quite fine and error free, a few of them are not so fine :-(. Unfortunatley Pete refuses to address concerns raised about his webpages, so they should be taken to represent his personal opinions rather than any sort of "consensus" view. Pete also has publically stated that he has "blocked" reading my posts, because I'm too critical of him. Sorry that you have to get caught in the crossfire here.

15. Nov 2, 2006

pervect

Staff Emeritus
Hmm, my earlier post didn't get through. Discreete is the wrong word as robphy points out, the difference is between one parameter continuous groups (aka finite continuous groups) which give conserved scalar quantites, and infinite continuous groups, which don't.