Relative energy of a black hole.

PeterDonis

Hmm. Do they give any actual examples of SETs? Or worked problems where they explicitly say what the SET is? I'm particularly curious if they give, for example, something like the solution for a static spherically symmetric star, which is one of the paradigmatic cases we've been discussing. In this case, the SET does *not* include any "gravitational field energy" (it's just the standard perfect fluid SET), but nevertheless it's commonly said that "gravitational field energy" needs to be taken into account in determining the externally measured mass M of the star.

(I've explained several times how the standard picture actually deals with this--the mass M is ultimately derived from the standard SET by solving the standard EFE, with no extra "source" terms for "gravitational field energy"--the latter just happens to be one way of describing the relationship between the mass M that appears in the metric and the standard SET that appears on the RHS of the EFE.)

TrickyDicky

Well in the static case highly unrealistic conditions are imposed: staticity, asymptotic flatness, vacuum... and still a very good approximation in solar system scale is reached.
The problem is nobody thinks our universe has those properties listed above, and it is in these cases (basically all GR physics other than the static solution) where the problem with gravitational fields as sources comes up. You cannot negate it because it doesn't appear in static solutions unless you believe our universe is static.
It is something that has been troubling relativists from 1915 when Hilbert referred to it saying that GR generates improper energy theorems. And it hasn't been solved, as I said is at the root of many difficulties with quantum gravity.

Naty1

So glad you guys are homing in on the discrepancy between Tricky's sources and the discucssion here!! .....
I have seen references like Tricky posted but could not locate any again....and yet everything Peter posts is also consistent with what I have seen.

This seems closely related to the issue...I sure don't get it:

http://en.wikipedia.org/wiki/Stress-energy_tensor

Maybe this "classical limit" issue is the one Peter described:

".... it is intuitively appealing (because) we are used to looking at stationary, or nearly stationary, systems, for which two things are true: (1) a meaningful definition of "energy stored in the field" can be given that corresponds, intuitively, to "gravitational potential energy", which is familiar from Newtonian physics; (2) because the system is stationary, there is a very simple relationship between what's there on a spacelike slice and what's there in the past light cone of any particular event. The conceptual issues you are having are basically due to trying to extend the simple viewpoint that works reasonably well for stationary systems to a more general domain, non-stationary systems (systems that collapse, and systems that radiate energy) where items (1) and (2) no longer hold."

George Jones

This is always the case. From 19.8 Gravitational Field Energy of Penrose's Road to Reality
Disembodied, because, from Ryder,
This means that gravitation energy cannot be included in the stress-energy-tensor field, as this is a mapping from spacetime into the space of tensors.

From page 131 of MTW's Gravitation
These quotes do not contradict the other quotes in this thread taken from standard references.

PeterDonis

Agreed, I should have made that clear (thought it ought to be clear from my other posts in this thread).

I don't have my copy handy to check: by "a physical role" for "disembodied" energy in the field, is he referring to gravitational waves carrying energy (for example, the binary pulsar emitting them, as has been discussed in this thread)?

PeterDonis

I wasn't intending to say that my statements about the SET only applied to the static case; they always apply (see George Jones' post and my response). I was only using the static case as a simple example that most textbooks say something about, so it might be a way to get more information about what the authors of this one were thinking.

The "improper energy theorems" bother some relativists because, as I've said in previous posts, they don't fit our intuitions about how "energy" ought to behave. Since standard GR with the standard SET the way it is accounts for all the evidence we currently have, the question of whether the improper energy theorems are a "real problem" or just a sign that our intuitions aren't a good match for this area of physics is, IMO, more a question of philosophy than physics. If we get further evidence that doesn't match the standard GR predictions, then of course that will change, as I've already said.

With regard to quantum gravity, AFAIK the reason this issue creates a problem there is that we don't know how do to quantum theory period with systems that have improper energy theorems. It's quite possible that that is a problem with the way we are doing quantum theory rather than with gravity; we may simply be using the wrong set of tools. Again, unless and until we get further evidence, IMO this is more a question of philosophy than physics.

George Jones

I thought that this is your position. I just wanted to agree, and to give quotes that back this up.
Yes.

PeterDonis

Yes, that's more or less right. Slightly further down the same Wiki page is this comment:

"In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy-momentum vector in a general curved spacetime."

It doesn't say exactly which "spacelike integral" is being talked about, but I assume they mean the continuity equation integral above. In certain special cases, a particular set of spacelike slices is picked out by the symmetry of the spacetime, and the continuity integral using that set of slices defines a "total energy" that behaves the way our "Newtonian" intuitions say energy ought to behave in the presence of gravity--it includes "gravitational energy", *and* energy is "exchanged" between ordinary matter-energy and gravitational energy in such a way that the total is conserved.

But that only holds for spacetimes where the symmetry picks out a particular set of spacelike slices: two examples are a single isolated gravitating body (the "Newtonian" case is a subcase of this), where the time translation symmetry picks out a particular set of slices, and a case like FRW spacetime, where the spherical symmetry defines a set of "comoving" observers that pick out a particular set of slices. (That's why the Usenet Physics FAQ page I linked to earlier includes this case in their discussion.)

Also, note carefully that the way "gravitational energy" enters into the continuity integral is *not* by any change in the SET's definition; it is purely due to the fact that, in curved spacetime, we use covariant derivatives instead of ordinary derivatives. That means extra terms come in due to the connection coefficients, and in certain special cases the extra terms have a simple interpretation in terms of "gravitational energy" being exchanged with ordinary matter-energy.

PeterDonis

George, thanks for the support and clarification!

Naty1

For what little it's worth, I understood George's comment as supportive...

I could not find it again, but Wikipedia has a statement to the effect that the gravitational field CANNOT be associated with any particular component of the Einstein formulation...not the metric, not the Riemann curvature, not Christoffel symbol, etc,etc
and goes to say one entity cannot take precedence over all the others in defining/representing the gravitational field. In addition, Ben Crowell has previously posted in another discussion how the gravitational field representations, and the energy therein, can be subject to varying interpretations....lost that somewhere in my notes, still looking.

These are the kind of tidbits that add clarity:

Again, PeterDonis, thanks for your time and effort....I picked up a lot of good information from your posts....

Naty1

To supplement George's comment from THE ROAD TO REALITY:

Peter explained that quote, I think, in earlier posts here. At least I 'got it'.

Penrose has a bit more detail immediately following George's excerpt [above]which I believe directly complements Peter's previous posts:

[for two massive bodies close together and at rest].....

Extending the above concepts, Penrose closes the chapter:

[I even had some of the above highlighted from a few years ago....too bad I did not remember this source!]

PeterDonis

You're welcome! Glad I was able to help.

Just to expand on this a bit, I believe Penrose is referring here to Noether's theorem: if the Lagrangian of a system has a symmetry, Noether's theorem shows how to construct a conserved current from that symmetry. "Energy" in this interpretation is the conserved current associated with time translation symmetry. Most of the spacetimes discussed in this thread where a useful definition of "total energy" can be made have time translation symmetry; but there are important spacetimes that don't (for example, the FRW spacetimes), which is why this method of defining energy "does not...directly SEEM to give us everything we need in the case of gravitation", as Penrose says.

Q-reeus

But then it makes no sense. You say I should have known from context GW in #77 meant gravitational waves, not gravitational energy. If you take the trouble to trace back that discussion it was referencing to comparing possible pressure vs static field gravitational energy contributions - all in the context of that given in #45 & elaborated in #52. GW's were not involved (there were of course other discussions considering GW's role, but clearly distinct from this matter). So who's to blame for thinking you must logically have meant energy in a static field, not GW's? Maybe you had another entry in mind when writing that.
Last bit is patently untrue, but I guess you forgot to insert 'that I acknowledge'.
Which just amounts to what I say above quoted. Any counterexample, e.g. in #45, cannot be true by definition, so why bother taking it seriously? The way you express that is a little less blatant: 'just apply the standard EFE/SET formula and all must be right. Counterexample X suggesting otherwise must thus be wrong'. This is your procedure to 'defeat' any counterargument, by referring back to the rote formula I complain about! No-win situation gauranteed. i will have another shot at breaking that cyclic dilemma in a later posting.
Is there actually observational evidence here? Would have thought pressure a negligible SET source in stars. Maybe neutron stars, but even there do we have convincing evidence it is needed to account presumably for maximum NS mass (less if pressure is SET source, than if not)? Have come across articles where it is admitted the eqn's of state within NS's are still not fully understood.

Q-reeus

Precisely confirming my suspicions given in #59.

Q-reeus

Hope you can appreciate that from my pov the above is frustratingly empty. On the one hand, a clear statement that gravitational field energy E_g is specifically absent from the SET. But then go on to say it is one way of describing the relationship between measured M and the SET. But nowhere have I seen you attempt to pin down what is then gravitational "energy's" role in a 'way of describing'. What exactly is it that means anything given E_g is utterly absent from the SET? Curvature non-linearity? If so, how about just plainly say so and why, or if something else, say exactly what it is.

Q-reeus

Not energy balance per se - I have consistently acknowledged there is at least nominally a system "energy" balance. Try the 'balance' of total system *gravitating* mass (inclusive of all energy flows including GW's) discussed particularly in #50 and #54. You here in #83 (which in turn references back to #73) have imo clearly set a trap for yourself. Gravitationally collapsed system mass M - the externally observed Keplerian *gravitating* mass, declines by your admission above. Further, by your admission, the decline is owing to GW "energy" emission - which you state clearly is not a part of SET and contributes nothing to M. So please, no appeal to a rote formula here. Admit the inescapable, basic logic - *total* system *observed* mass M thus declines. If your 'answer' is to ignore this request, understand I will feel free to draw obvious conclusions. And recall in past postings you have specifically claimed M cannot decline if all matter+energy is included. Deny that and I will gladly furnish quotes to the contrary. This is relevant to the monopole GW issue btw.

Naty1

Q-reeus posts:


If you READ from posts 88 on....Tricky, my posted quotes, George Jones comments and quotes and Peter's comments explain it to the extent it can be....'non localizable', covarient derivative effects, non localizable,etc,etc ......

these are all complementary, not in conflict.

including these:

from Ryder
and from Penrose:
I could quibble with Peter's comment about problems with energy theorems (in #91) being more 'philosophy' than physics.....but that's waaaaaaaaay too nit picky....

Q-Reeus...While I see why pervect opted out early, I am on the other hand happy to see your persistence:

" It is better to debate a question without settling it than to settle a question without debating it."
.......Joseph Joubert, the 18th century philosopher


I, for one, am 'outta' here....finally!!