Is Conservation of Scalar Quantities Possible in General Relativity?

[pivoxa15]

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

 

[lightarrow]

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

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

 

[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" .
For the angular momentum, things are not so clear, I need to check in "Gravitation".

But what means "conserved" finally ?

Michel

 

[lightarrow]

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" .

Ok, but locally only.

But what means "conserved" finally ?

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

 

[lalbatros]

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

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

 

[pervect]

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

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".

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

 

[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?)

 

[pmb_phy]

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

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

 

[notknowing]

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

Please check out this link :
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

 

[pervect]

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?)

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

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as 'the failure of the energy theorem '. In a correspondence with Klein [3], he asserted that this 'failure' is a characteristic feature of the general theory, and that instead of 'proper energy theorems' one had 'improper energy theorems' in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether. In the note to Klein he reports that had requested that Emmy Noether help clarify the matter. In the next section this problem will be described in more detail and an explanation given of how Noether clarified, quantified, and proved Hilbert's assertion. One might say it is a lemma of her Theorem II.

(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).

 

[pervect]

Please check out this link :
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

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.

 

[robphy]

This implies a discreete time translation symmetry, which by Noether's theorem implies a conserved energy.

I think you mean "continuous".

 

[pmb_phy]

Please check out this link :
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

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

 

[pervect]

Good news, bad news.

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.

 

[pervect]

I think you mean "continuous".

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.