Relative energy of a black hole.

[PeterDonis]

Well the thing is in many GR texts the non-linearity of the EFE is attributed precisely to the very thing you are dismissing here as unnecessary or irrelevant for the physics:The gravitational field energy behaviour and the "gravity gravitates" issue. So they must have a different idea , or at least broader of what the physics of GR is.

Without seeing some specific references, I can't say for sure, but I strongly suspect that nothing in the texts you refer to about the nonlinearity of the EFE is in any way inconsistent with what I am saying. But let me try to clarify a bit more how I think "nonlinearity" fits in.

Mathematically, "nonlinearity" simply means that solutions to the EFE can't be superposed: you can't take two solutions, add them together, and get another solution. This is why, for example, the two-body problem can't be solved by simply adding together two one-body Schwarzschild metrics centered on different points: the result is not a solution of the EFE.

Physically, what this means is that fields from different "sources" (where "source" is to be interpreted, strictly speaking, in the precise way I have said: nonzero SET regions in the past light cone) don't just add together: they "reinforce" each other, so to speak. I put "reinforce" in scare-quotes because that word is likely to raise further questions about whether gravity gravitates, etc. So a more precise way of saying it would be: the law that governs how the field "propagates" from multiple sources cannot be derived just by "adding together" multiple copies of the law that governs how the field "propagates" from a single source. The law of field propagation can't be "broken up into pieces" like that. There is nothing physically mysterious about this; it just happens to be the way the law of "field propagation" (the EFE) is structured. The main impact it has is to make it much harder to come up with solutions for spacetimes with multiple sources, because you can't take any shortcuts; you have to look at *all* the sources in the spacetime, all at once, and arrive at a *single* solution to the EFE that takes them all into account. And in doing so, you don't have to add any "extra" sources corresponding to "gravity gravitating"; everything is determined by the standard (nonzero SET) sources.

Dark energy in its most accepted interpretation, that which is compatible with GR, is thought to be precisely a repulsive gravitational field, and as you admit it is a nonzero SET. But you insist that the usual attractive gravitational field doesn't count as SET source, while the standard view is that precisely the fact that gravity gravitates is what makes the EFE non-linear.

The fact that the particular form of the SET that is associated with dark energy happens to create a spacetime which can be viewed as having "repulsive gravity" is a *derived* phenomenon; it is not fundamental. The precise fundamental definition of "dark energy" is just what I said before: the SET is proportional to the metric. That's all. (Btw, dark energy only creates "repulsive gravity" if its SET is a positive number times the metric; if it is a negative number times the metric, such as as negative cosmological constant, the "gravity" it creates is attractive.)

Similar remarks apply to what you are calling "the usual attractive gravitational field"; the fact that it is attractive is a *derived* phenomenon, not a fundamental piece of the physics. The fundamental physics is that the SET of "ordinary" matter or energy (e.g., a perfect fluid or EM radiation) always turns out to obey a number of energy conditions; the strong energy condition is, IIRC, the most important one (since it's the one that, for example, a "dark energy" SET violates). The EFE then ensures that any SET obeying these conditions will produce "attractive" gravity.

(And "gravity gravitates" is an approximate way of looking at a *different* piece of the physics still--the fact that, as I said above, solutions to the EFE can't be superposed. You appear to agree that this is what "gravity gravitates" refers to. But an "attractive gravitational field" can be present even when there is only one "source"--one region of nonzero SET--in the spacetime--which of course is the most commonly analyzed case.)

(It's also worth mentioning that "attractive gravity" vs. "repulsive gravity" is only a portion of the full curvature of the spacetime; there are also tidal effects that can vary in different directions.)

 

[PeterDonis]

Assuming the existence of such a truly causally isolated region, sure and neither I or I assume yuiop are disagreeing with that. But what matters obviously is the entity 'BH' is interacting, as a mass M, with it's surrounds.

But the "BH", in the standard usage of that term, *is* the "causally isolated region", which you have just (apparently) agreed can't interact with anything. So what, exactly, do you mean by saying that the "BH" is "obviously" interacting?

But then you say:

Ergo, the external field is doing this - by logical reduction from your own statements. If the interior isn't interacting, hey, that just leaves the exterior, which is just the field! Call it the SET in the by and by, still boils down to: if not the interior, only one thing left.

No, it doesn't. There is also the nonzero SET region in the past light cone of the exterior vacuum, which is what I've been saying "interacts" all along. Though, as I've also said, the word "interaction" is not a good one; "propagation" would be better, although that also has some undesirable connotations. The point is that the field at any event in the exterior region is entirely determined by solving the EFE using the nonzero SET region in the past light cone of that event as the "source", and then working the solution forward from that region through the intervening vacuum to the event in question. Not once have you shown why this can't work.

Please define precisely the nature of this 'approximation' - and to what numerical extent is it an approximation.

That's up to you, since you're the one using it. Unless you really think it's exact, in which case please show your exact math.

Go back and check the link I gave re Nordtvedt effect in #59. These researchers sure take the concept of gravitational binding energy seriously, and the fact all three masses applied to such are experimentally equal has the consequence I stated there - if there is no active gravitational energy mass, neither is there an inertial mass. Comfortable with that?

I will have to read the link to comment in detail, but basically you appear to be trying to get agreement on this:

And let's get one thing clear. Your repeated claim the standard EFE/SET setup explains everything is not really true. The differences between GR and 'gravity gravitates' theories in general are below the level of detection in all current tests. Baryshev link in another thread sets out some of the details.

In other words, the standard EFE/SET setup explains all the evidence we currently have, but you still think it's wrong because you think there's other evidence waiting out there that's currently below the level of detection. Fair enough; when you have additional evidence that contradicts the standard setup, we can talk further. Until then, I don't see much point in arguing when we don't have Nature's vote yet, since that is the only vote that really counts. I've already agreed that if evidence that clearly contradicts the standard GR setup is found, the standard setup will have to be modified. But that hasn't happened.

Hopefully not needed. From a coordinate observer perspective, looking for this merger event to be over by breakfast, it is infinitely curved - else we say the SC's are lying/useless.

I don't understand what you are trying to say here.

I specified weak gravity regime. Do you deny pressure will there double to all but a tiny and inconsequential corrective factor?

As an approximation, this is probably tolerable. But you are trying to argue that there can't be an *exact* cancellation between the (negligible) pressure term and the (negliglble) GW term. You can't base an argument against *exact* cancellation on that. If both are negligible, then to the given approximation, they cancel (since they're both zero anyway to that approximation). To actually assess whether they cancel for real, you have to either (1) go to a more accurate approximation, where they are *not* negligible (meaning you can't help yourself to convenient assumptions about linearity), or (2) go to a scenario where gravity is stronger, so that pressure (and GWs) become significant (meaning you can't help yourself to convenient assumptions about linearity).

(Also, this all assumes that it would even matter if there *were* a failure of exact cancellation. On the standard viewpoint that I am defending, it doesn't matter in the least. "Gravitational energy" doesn't appear in the standard SET to begin with, so wondering whether it cancels with anything is rather pointless.)

Looks though like it deserves a separate thread - it is spelling death to a key concept in GR and you are saying 'ho hum'!

You are correct that I'm not impressed, but that's only because you are doing vague handwaving, not actual physics. You can't "spell death to a key concept" in a theory with as much experimental confirmation as GR with vague handwaving. And trying to make it less vague and less handwaving definitely deserves a separate thread. (Which would also have to include an argument for why I should even care, since, as I noted above, on the standard viewpoint the question you are asking here is pointless anyway.)

Again, missing the point, which is there is undeniably energy in the GW field. Yet re static field situations you are putting "energy" in inverted commas like that. Why? Do you actually see a fundamental distinction?

Yes--"energy in the GW field" is not associated with a nonzero SET. GWs can propagate in vacuum (zero SET) regions.

No, I am saying with things being as they are we can draw different conclusions. The very fact that pseudo-tensors are needed to get any sort of decent energy definitions in GR should be making that evident.

Which is why I keep sticking to the standard SET, which is *not* a pseudo-tensor but a genuine tensor with a definite physical meaning. It's you who keeps bringing in "energy in the gravitational field" and other concepts that require pseudo-tensors.

You are on record as stating the SET specifically includes only matter contributions, and that means gravitational energy/"energy" is excluded.

Matter, (non-gravitational) energy, pressure, stresses, momentum, cosmological constant, "dark energy"... The SET covers every possible "source" for which an actual, bona fide tensor with a definite physical meaning can be defined. What more do you want? There are no bona fide tensors left; only those pesky pseudo-tensors, which are called "pseudo" for a reason.

Kindly go back then to the scenario in #52 and show me, point by specific point, where you think I an the one getting things wrong.

See my comments above.

 

[PeterDonis]

Yes--"energy in the GW field" is not associated with a nonzero SET. GWs can propagate in vacuum (zero SET) regions.

On re-reading, I think I should elaborate on this some more, since this may be part of the fundamental point at issue here. So let me go over how the standard GR "setup" handles the case of a system like the binary pulsar that is radiating GWs and losing "energy" as it does so.

Here is the precise definition of "energy conservation" in GR:

$${T^{ab}}_{;b} = 0$$

In other words, the covariant divergence of the stress-energy tensor is zero at every event in spacetime. This is a standard tensor equation which transforms covariantly between frames in the standard way. (This equation is enforced as an identity by the EFE, because the Einstein tensor on the LHS of the EFE obeys the Bianchi identities, which guarantee that its covariant divergence is zero. So since the LHS obeys such an equation, the RHS must as well.)

What is this equation trying to say? It says simply that, if we take any infinitesimal 4-volume of spacetime, whatever stress-energy (in the standard "non-gravitational" sense) goes in must come out again. For example, if the SET consists of a small piece of matter at rest, exactly as much matter must "come out" the future surface of the small 4-volume as "went in" the past surface of the 4-volume. "Standard" stress-energy can be localized in the standard way (we can "label" each particle and follow its worldline), so the energy conservation equation can be written as a standard local differential tensor equation.

Why does there have to be a covariant divergence in the standard equation, instead of just an ordinary divergence? Because in a curved spacetime, in order to properly assess a "density" of anything (including stress-energy, which is what the SET is a density of), you have to account for the fact that the coordinates are, in general, non-Euclidean, so a given infinitesimal coordinate 4-volume does not always correspond to the same physical 4-volume. That's what the covariant divergence compensates for; it makes sure that we "count" each infinitesimal piece of stress-energy correctly as we assess whether it remains conserved as it moves through a curved spacetime.

Now consider a system like the binary pulsar. It is radiating GWs. Those GWs "carry energy", in the sense that they can travel across the vacuum, be absorbed by some "detector", and do work--for example, they can vibrate or heat up a piece of matter. But if we look at any particular infinitesimal 4-volume where GWs are being emitted, we find that the above equation holds, even though the GWs themselves have *zero* SET. The matter that emits the GWs changes its "orbital parameters" slightly, so in a sense it has "lost energy"; but the "energy" it has lost is in the form of curvature, so the curvature in the infinitesimal 4-volume changes slightly. The change in curvature exactly compensates for the "energy loss" of the matter, in such a way that the energy "conservation" equation continues to hold. (Mathematically, the individual components of the SET change as the GWs are emitted, but the curvature change changes the way each piece of the SET is "counted", so that the covariant divergence remains constant.)

If you look at all the other cases we've discussed where there is a temptation to say that "gravitational energy" must be added to the standard SET, you'll see that the same sort of effect is involved; changes in individual SET components are exactly compensated for by changes in curvature, so that the covariant divergence of the SET remains constant. This is how the standard picture I have been defending enforces conservation of energy.

One could object, of course, that I have changed the definition of "conservation of energy" to something that doesn't match our common-sense intuitions. In a sense that's true; the binary pulsar system is "conserving energy" in the sense I've given above, yet it is emitting GWs that carry energy and its externally measured mass is decreasing. This is simply a case where our common-sense intuitions are wrong, or at least can't be taken at face value. Things like the Landau-Lifgarbagez pseudotensor are an attempt to create a version of "conservation of energy" that matches up better with our intuitions; but such things are not *necessary* in order to determine the physics according to the best evidence we have today; that can be done entirely within the standard framework.

 

[PeterDonis]

One further note: I just realized that I had forgotten to do what I usually do when I find myself in any lengthy discussion on PF: check the Usenet Physics FAQ to see if it has a page on the subject in question. Turns out it does:

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

I wish I'd thought to link to this earlier; it covers a lot of the same ground we've covered in this thread, but much more compactly.

 

[PeterDonis]

Go back and check the link I gave re Nordtvedt effect in #59. These researchers sure take the concept of gravitational binding energy seriously, and the fact all three masses applied to such are experimentally equal has the consequence I stated there - if there is no active gravitational energy mass, neither is there an inertial mass. Comfortable with that?

Went back and looked at the discussion of the Nordtvedt effect in living reviews (your link just went to the living reviews title page, btw, not to the specific section where the Nordtvedt effect is discussed). As far as I can see, all these experiments do is confirm what I said in the post you quoted from, just before the part you quoted:

"We have *no* evidence of anything in reality with varying "energy/mass ratio" (or different active gravitational mass, passive gravitational mass, and inertial mass, using the terminology you introduce below)."

That's what zero Nordtvedt effect means. The fact that the absence of the effect can be described as showing that "gravitational binding energy" has the same effective "mass" as other types of energy doesn't contradict anything I've said; it just means the people describing the effect are using the same vocabulary as you are (not surprising since it's a common vocabulary). It certainly doesn't invalidate the primary point I've been making all along, that all classical gravitational phenomena can be explained and calculated entirely by looking at the EFE and the standard SET. Try e-mailing Clifford Will and asking him if he thinks the EFE is valid; I bet he'll say yes. Or try asking him if he thinks "gravitational binding energy" appears in the SET; I bet he'll say no.

 

[TrickyDicky]

Without seeing some specific references, I can't say for sure, but I strongly suspect that nothing in the texts you refer to about the nonlinearity of the EFE is in any way inconsistent with what I am saying.

"General Relativity: An Introduction for Physicists" by Michael Paul Hobson,George Efstathiou,Anthony N. Lasenby (a text commonly used and mentioned here at PF): on page 473: "The non-linearity of the Einstein equations is a direct result of the fact that "gravity gravitates". In other words, any form of energy-momentum acts as a source for the gravitational field, including the energy-momentum associated with the gravitational field itself".
Als o on page 189: "The physical reason for this non-linearity is that the gravitational field itself carries energy-momentum and can therefore act as its own source."
Or on page 409 of the GR textbook by Ryder "introduction to General Relativity": "Gravitational waves carry energy (albeit non-localised), and anything carrying energy (or equivalently, mass) acts as the source of a gravitational field. Gravitational waves therefore generate an ‘extra’ gravitational field. This is an aspect of the non-linearity of General Relativity, and is shared by non-abelian gauge theories." In a previous post you directly contradict this assertion about gravitational waves and your reasoning about the gravitational field energy being a different kind of energy that has the property of not being "stress-energy".
Similar statements can be found in other texts and articles but they all can be summarized by stating the fact that gravity couples to energy-momentum, and the gravitational field has energy-momentum (this actually follows from the equivalence principle). But you have written many posts trying to dismiss this apparently basic and accepted fact of GR, or maybe you haven't , at this point I'm not sure. You are more than capable of saying at the same time that the gravitational field is and it isn't a source of the SET, and its energy is and isn't "energy-momentum". Or that gravity doesn't gravitate and it does, but it is irrelevant to the physics, when it is generally acknowledged that many of the problems to come up with a quantum gravity theory come from this conundrum.
It would seem that certainly what you are saying is in some way inconsistent with the quotes above.

 

[Q-reeus]

TrickyDicky (what! Tricky is impersonating my postings?! I'll sue him! Might be another explanation though )
"Ergo, the external field is doing this - by logical reduction from your own statements. If the interior isn't interacting, hey, that just leaves the exterior, which is just the field! Call it the SET in the by and by, still boils down to: if not the interior, only one thing left."

No, it doesn't. There is also the nonzero SET region in the past light cone of the exterior vacuum, which is what I've been saying "interacts" all along. Though, as I've also said, the word "interaction" is not a good one; "propagation" would be better, although that also has some undesirable connotations. The point is that the field at any event in the exterior region is entirely determined by solving the EFE using the nonzero SET region in the past light cone of that event as the "source", and then working the solution forward from that region through the intervening vacuum to the event in question. Not once have you shown why this can't work.

Sorry but all smoke and mirrors imo. Let's drop this BH is/isn't one and move on.

TrickyDicky (no - me!): "I specified weak gravity regime. Do you deny pressure will there double to all but a tiny and inconsequential corrective factor?"
As an approximation, this is probably tolerable. But you are trying to argue that there can't be an *exact* cancellation between the (negligible) pressure term and the (negliglble) GW term. You can't base an argument against *exact* cancellation on that. If both are negligible, then to the given approximation, they cancel (since they're both zero anyway to that approximation). To actually assess whether they cancel for real, you have to either (1) go to a more accurate approximation, where they are *not* negligible (meaning you can't help yourself to convenient assumptions about linearity), or (2) go to a scenario where gravity is stronger, so that pressure (and GWs) become significant (meaning you can't help yourself to convenient assumptions about linearity).

You continue to amaze! I will assume when you write GW above it is not the wave but gravitational energy in a static field. Please clarify. At any rate assuming the latter is true, such reductio ad absurdum argument is absurd. By this standard GW's also are zero to that approximation, so can't use them in any argument, and so on! Further, please take note that the sign of both pressure and gravitational energy E_g are the same in the scenario considered, so no possibility of cancellation in that sense. I thought it self-evident that 'cancellation' was only in the sense that if a pressure contribution could have the same magnitude as that ascribed to E_g, perhaps the latter could be assumed not to exist, because all the deficit worked out in #45 & #52 might then be ascribed to pressure rather than E_g. That was knocked on the head in #65, point 1 there (have more to say on point 2 below). Up to you to argue some flaw in that argument - seems perfectly sound to me. Last time I checked, a parabola and straight line can intersect at no more than two points (only one non-trivial one in the case considered).

(Also, this all assumes that it would even matter if there *were* a failure of exact cancellation. On the standard viewpoint that I am defending, it doesn't matter in the least. "Gravitational energy" doesn't appear in the standard SET to begin with, so wondering whether it cancels with anything is rather pointless.)

And here is what come across as your base position in a nutshell, one that makes it so frustrating for me. That position is 'my version of the EFE/SET in GR is Absolute Truth, if you find differently by any counterexample/counterargument whatsoever, you must be in error - end of story.' Sheesh!

TrickyDicky(no - still me!): "Looks though like it deserves a separate thread - it is spelling death to a key concept in GR and you are saying 'ho hum'!"
You are correct that I'm not impressed, but that's only because you are doing vague handwaving, not actual physics. You can't "spell death to a key concept" in a theory with as much experimental confirmation as GR with vague handwaving. And trying to make it less vague and less handwaving definitely deserves a separate thread. (Which would also have to include an argument for why I should even care, since, as I noted above, on the standard viewpoint the question you are asking here is pointless anyway.)

That last bit echoes my last comment nicely. On your bit implying that the role of pressure in GR has experimental (observational?) confirmation; can you reference any reliable article(s) to that effect?
Now; yet another confession. The example in #65 of back-to-back G-clamps as GW source contained a non-fatal flaw. There needs to be some power source supplying to a pump, motor etc. in each screwed leg - they cannot just self-screw. Looked at in terms of relativistic energy-momentum flow in each G-clamp separately, there is no overall shift in center of mass occurring, hence no mass quadrupole moment generated - assuming however slow motions where inertia is not a significant contributor. Shame, shame on you for not picking me up on that! But note carefully - this finding of null mass quadrupole contribution in no way nullifies that pressure *does* formally contribute there to GW generation. And as pointed out, it cannot be a conservative process owing to complete independence from the arbitrary system elastic constant.

Mechanically vibrating objects, where inertia plays a key role, reintroduce oscillating mass quadrupole contributions. However it is then the formal pressue contributions that are more than a little interesting. One configuration in particular - a spherical mass shell vibrating in monopole 'breathing' mode, spells especially deep trouble for pressure in GR (or to be fair, any other gravity theory similarly incorporating it). But yes it is rightly the topic for another thread.

 

[Q-reeus]

...What is this equation trying to say? It says simply that, if we take any infinitesimal 4-volume of spacetime, whatever stress-energy (in the standard "non-gravitational" sense) goes in must come out again. For example, if the SET consists of a small piece of matter at rest, exactly as much matter must "come out" the future surface of the small 4-volume as "went in" the past surface of the 4-volume. "Standard" stress-energy can be localized in the standard way (we can "label" each particle and follow its worldline), so the energy conservation equation can be written as a standard local differential tensor equation.

And you go on to say GW's are included somehow in the balance despite possessing zero SET contribution themselves. Nice try - looks good on a fast read. But I guess it is a standard position, but from another entry maybe standard position is not exactly the same as unanimity in the GR community.

 

[Q-reeus]

One further note: I just realized that I had forgotten to do what I usually do when I find myself in any lengthy discussion on PF: check the Usenet Physics FAQ to see if it has a page on the subject in question. Turns out it does:

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

I wish I'd thought to link to this earlier; it covers a lot of the same ground we've covered in this thread, but much more compactly.

Another nice try by that writer(s) to explain contradictions. To quote from that piece:

One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are
Gmu,nu = 8pi Tmu,nu
Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below. Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity.

Yep, sure can. Thanks Michael & John for an answerless answer to the connundrum. At least we are made aware of the issue though. Just read TrickDicky's #76 and those authorities quoted might make a nice 'but then there's this contrary pov' entry to that FAQ. What do you say - a fair thing to do in the interests of 'balanced perspective'?

 

[Q-reeus]

Went back and looked at the discussion of the Nordtvedt effect in living reviews (your link just went to the living reviews title page, btw, not to the specific section where the Nordtvedt effect is discussed).

Sorry about that, but will only accept part blame. Go there and click on any part of that multi-page article, and the web address never changes. Weird.

 

[PeterDonis]

It would seem that certainly what you are saying is in some way inconsistent with the quotes above.

It means they are using a different definition of "source" than I'm using. Do they ever define precisely what they mean by "source"?

 

[PeterDonis]

I will assume when you write GW above it is not the wave but gravitational energy in a static field.

Then you assume wrongly; by "GW" I meant specifically "gravitational waves". I thought that was clear from context, but I suppose I should have spelled it out. Please re-read interpreting "GW" specifically as "gravitational waves".

That position is 'my version of the EFE/SET in GR is Absolute Truth, if you find differently by any counterexample/counterargument whatsoever, you must be in error - end of story.'

My position is that the *standard GR* version of the EFE/SET accounts for all the physics. So far you have given no counterexample to that claim. I am not saying that your way of describing certain aspects of the physics is "wrong"; I'm only saying that it's limited to certain aspects of the physics.

That last bit echoes my last comment nicely. On your bit implying that the role of pressure in GR has experimental (observational?) confirmation; can you reference any reliable article(s) to that effect?

It may take a while to find specific references other than textbooks, but two quick general pieces of evidence:

(1) The GR solutions for static or nearly static stars require pressure to contribute to the SET in the standard way--in other words, it's not enough just to put pressure into an equation of hydrostatic equilibrium, you also need to include pressure as a "source" on the RHS of the EFE. These solutions do a good job of predicting the observed masses and other properties of stars.

(2) The FRW cosmologies require pressure to contribute to the SET in the standard way, otherwise the overall dynamics are different. The current hot big bang theory depends on the FRW model and has good experimental confirmation.

But yes it is rightly the topic for another thread.

I think at this point that comment applies to all of your proposed counterexamples to GR.

 

[PeterDonis]

And you go on to say GW's are included somehow in the balance despite possessing zero SET contribution themselves.

What "balance" are you talking about? I said GWs carry away energy in the sense that they can later do work on a detector; and I said that the externally observed mass of the system that emits GWs decreases. But neither of those things affect the "balance" expressed in the energy conservation equation I gave, that the covariant divergence of the SET is zero.

 

[PeterDonis]

Just read TrickDicky's #76 and those authorities quoted might make a nice 'but then there's this contrary pov' entry to that FAQ. What do you say - a fair thing to do in the interests of 'balanced perspective'?

See my response to TrickyDicky above. I will agree that some of the "authorities" (I would prefer the term "pedagogical resources" but I agree it's clumsy--there are no "authorities" in science) are not being as careful and precise as they should be. That's why I've gone to such lengths to precisely explain what I mean by "source", what I mean by "the field", and exactly how the "source" produces the "field" according to the strict standard physical model in GR--i.e., the actual math, not various authors' attempts to express the math in English (which I've said several times is problematic because of the limitations of English). If you look closely at all these "authorities", you will see that they all agree on the precise points I have made: they all agree that the SET on the RHS of the EFE does *not* include "gravitational energy", and that the standard EFE with the SET in that form is sufficient to explain and calculate all the physics. The fact that they then go on to make statements in English that can be construed differently is regrettable, but it doesn't change the physics.

 

[TrickyDicky]

they all agree that the SET on the RHS of the EFE does *not* include "gravitational energy"

No, they don't, or at least they write exactly the opposite if that is of any worth in deciding what they agree about.
They define (page 176 of the first reference) the source as the SET on the RHS of the EFE, and then they explicitly state that the gravitational field (energy) itself acts as a source.

 

[PeterDonis]

They define (page 176 of the first reference) the source as the SET on the RHS of the EFE, and then they explicitly state that the gravitational field (energy) itself acts as a source.

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]

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.

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:

..In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation:

energy is being exchanged with gravitational potential energy, which is not included in the tensor

, and momentum is being transferred through the field to other bodies.

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]

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")

This is always the case. From 19.8 Gravitational Field Energy of Penrose's Road to Reality

Let us return to the question of mass/energy in the gravitational field itself. Although there is no room for such a thing in the energy-momentum tensor T, its is clear that there are situations where a 'disembodied' gravitational energy is actually playing a physical role.

Disembodied, because, from Ryder,

We cannot, then, identify a place or places, where where the gravitational field exists and carries energy, since whether the field carries energy also depends on the frame of reference. Gravitational energy is not localisable.

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

At each point in spacetime, there exists a stress-energy tensor. It is a machine that contains a knowledge of the energy density, momentum density, and stress as measured by any and all observers at that event. Included are energy, momentum, and stress associated with all forms of matter and all nongravitational fields.

These quotes do not contradict the other quotes in this thread taken from standard references.

 

[PeterDonis]

This is always the case.

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

From 19.8 Gravitational Field Energy of Penrose's Road to Reality...

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]

You cannot negate it because it doesn't appear in static solutions unless you believe our universe is static.

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.

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.

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]

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

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

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)?

Yes.

 

[PeterDonis]

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

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]

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

Yes.

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:

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.

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]...

... there will be [negative] gravitational potential energy contribution that makes the total energy and therefore the total mass smaller than it would be if they are far apart. Ignoring much tinier energy effects, such a distortions of each body's shape due to the gravitational field of the other, we see that the total contributions from the actual energy momentum tensor T will be the same whether the two bodies are close together or far apart. Yet the total mass/energy will differ in the two cases and this difference would be attributed to the energy in the gravitational field itself [in fact a negative contribution, that is more sizeable when the bodies are close than when they are far apart.]

...Now let us consider that the bodies are in motion...[he describes the Taylor-Hulse binary thingy]...The energy-momentum tensor in empty space is zero, so the gravitational wave energy has to be measured in some other way that is not locally attributable to an energy 'density'. Gravitational energy is a genuinely non-local entity. This does not imply there is no mathematical description of gravitational energy, however. Although I believe it is fair to say we do yet yet have a complete understanding of gravitational mass/energy, there is an important class of situations in which a very complete answer can be given. These situations are those referred to as asymptotically flat and they refer to gravitating systems that may be regarded as being isolated from the rest of the universe, essentially because of there very large distance from everything else. ...The work of Biondi...generalized by Sachs provided a clear cut mathematical accounting of the mass energy carried away from such a system in the form of gravitational waves and a conservation law for energy-momentum was accordingly achieved. This conservation law does not have a local character of that for non gravitational fields...

Extending the above concepts, Penrose closes the chapter:

...There are general prescriptions for obtaining conservation laws for systems of interacting fields. These come from the Langrangian approach...very powerful,,,,despite the fact that it does not...directly SEEM to give us everything we need in the case of gravitation...

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

 

[PeterDonis]

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

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

Extending the above concepts, Penrose closes the chapter:

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]

Q-reeus: "I will assume when you write GW above it is not the wave but gravitational energy in a static field."
Then you assume wrongly; by "GW" I meant specifically "gravitational waves". I thought that was clear from context, but I suppose I should have spelled it out. Please re-read interpreting "GW" specifically as "gravitational waves".

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.

Q-reeus: "That position is 'my version of the EFE/SET in GR is Absolute Truth, if you find differently by any counterexample/counterargument whatsoever, you must be in error - end of story.'"
My position is that the *standard GR* version of the EFE/SET accounts for all the physics. So far you have given no counterexample to that claim...

Last bit is patently untrue, but I guess you forgot to insert 'that I acknowledge'.

I am not saying that your way of describing certain aspects of the physics is "wrong"; I'm only saying that it's limited to certain aspects of the physics.

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.

(1) The GR solutions for static or nearly static stars require pressure to contribute to the SET in the standard way--in other words, it's not enough just to put pressure into an equation of hydrostatic equilibrium, you also need to include pressure as a "source" on the RHS of the EFE. These solutions do a good job of predicting the observed masses and other properties of stars.

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]

...Disembodied, because, from Ryder,

We cannot, then, identify a place or places, where where the gravitational field exists and carries energy, since whether the field carries energy also depends on the frame of reference. Gravitational energy is not localisable.

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.

Precisely confirming my suspicions given in #59.

 

[Q-reeus]

...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.)

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]

Q-reeus: "And you go on to say GW's are included somehow in the balance despite possessing zero SET contribution themselves."

What "balance" are you talking about? I said GWs carry away energy in the sense that they can later do work on a detector; and I said that the externally observed mass of the system that emits GWs decreases. But neither of those things affect the "balance" expressed in the energy conservation equation I gave, that the covariant divergence of the SET is zero.

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:

...But nowhere have I seen you attempt to pin down what is then gravitational "energy's" role in a 'way of describing'.

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:

There is in fact no way to define a global energy-momentum vector in a general curved spacetime."

from Ryder

We cannot, then, identify a place or places, where where the gravitational field exists and carries energy, since whether the field carries energy also depends on the frame of reference. Gravitational energy is not localisable.

and from Penrose:

... Although I believe it is fair to say we do yet yet have a complete understanding of gravitational mass/energy, there is an important class of situations in which a very complete answer can be given. These situations are those referred to as asymptotically flat...

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

 

[John232]

I couldn't help but wonder if, say for instance a very large star ended up being slung around the suppermassive black hole in the center of the galaxy. Then this star ended up traveling at a very high speed straight for Earth. So then say that the relative speed of the star and its mass creates an event horizon around itself because of the relative mass that was seen from Earth. You could say that it was just the relative mass that made it look like a black hole and that any planets traveling along with the star didn't observe this relative mass so then they could orbit around the star and stay just fine. So then they send a team into the black hole to try and slow it down to prevent the destruction of Earth. They then would travel straight into the black hole at speeds close to the speed of light to prevent becoming spagitified. They then transfer into the frame of reference of the star itself so they no longer observe it being a black hole. And then they land on one of the planets and find life and decide to live there since they failed blowing up the star and live on inside this "black hole" as if they are just fine. So, then do you think something like this scenario would be possible or totally science fiction?

 

[Q-reeus]

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!

Naty1, I was fearing getting only stick from you at first, but sort of ended on a relative high - but I understand your departure. It has got a bit torrid. On your first point, I want to be clear there was no specific attacking the notion of 'non-localizability' in my query. Just can't see the connection on the specifics I raised, and non-localizability seems off the mark in that respect. Just want a clear statement as to whatever connections are implied. May have missed something earlier but can't recall it. Anyway you have inspired me to soldier on, so good!

 

[PeterDonis]

Last bit is patently untrue, but I guess you forgot to insert 'that I acknowledge'.

The insertion would not change the truth value, I suppose. But you apparently don't understand what is actually required for a counterexample. A counterexample would look like this: "Here's an actual physical observable that the standard EFE/SET method doesn't predict or explain." Or: "Here's a prediction made by the standard EFE/SET method that doesn't match this actual physical observable." You have given no such example, because you have never actually tried to figure out what the standard EFE/SET method predicts or explains; you haven't used it. You've insisted on reasoning from your own set of premises (like "gravity gravitates") instead, and then you've tried to claim that if the conclusions you reach don't appear to be consistent with the standard EFE/SET method, the standard method must be wrong. So it's not that I'm saying any counterexample must be wrong by definition: I'm saying you have not actually given counterexamples at all; instead you've given conclusions derived from a different set of premises altogether, and those premises are only approximately true (and even that is only in a limited domain).

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.

Neutron stars are a good example of pressure contributing significantly to the SET, yes. And yes, the maximum NS mass is one area where the pressure contribution is important; we know that even though we don't know the exact equation of state (because we've tested a whole range of possible equations of state numerically).

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.

I appreciate that things look this way from your pov. But now consider how they look from my pov. As I've said several times now, in the standard EFE/SET picture, there is no *need* for the concept of "gravitational energy" at all. All physical predictions can be made without ever using it. So from my pov, the problem is not that I'm not answering your questions, but that you insist on asking them even though I've repeatedly said that they are based on the wrong set of concepts. I have been trying to meet you halfway by at least trying to express how one *might* salvage some kind of correspondence between the concept of "gravitational energy" and the standard EFE/SET method, in a limited domain. But that's only because I understand that the concept of "gravitational energy" has intuitive force, so I'm willing to expend some effort in trying to explore it and its limits.

But asking for what "exactly" the concept of "gravitational energy" means is asking too much: the concept is only a heuristic one and it does not have an "exact" meaning. (Or perhaps a better way to say this would be: one could give an exact definition of "gravitational energy", such as the Landau-Lifgarbagez pseudotensor, but no such definition is unique, and any such definition only "makes sense", only corresponds to our intuition, in a restricted set of cases.) If you want an exact answer, it is this: there is no "gravitational energy" in the SET, so as far as exact calculations of physical predictions are concerned, it doesn't exist. (You'll note, in this connection, that nobody uses any definition of "gravitational energy" to actually make physical predictions: they all use the standard EFE/SET method, and then once they know what the answer is, they overlay their chosen concept of "gravitational energy" on top of it to help them understand intuitively what's going on.)

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.

All right, let's look at this from an *exact* point of view. The exact point of view is this: the "total system" is the entire spacetime, including the region "at infinity". This "total system" does not *have* a "mass M". The exact metric is not in any of the forms where "M" even appears; it's more complicated. (One could try to extract a "piece" of the metric where a coefficient "M" appears, but that's just an approximation-see below.) So from the "exact" point of view, there is *nothing* in the physics corresponding to "total system observed mass". There is a metric at each event, and there is an SET at each event (nonzero in the interiors of the two pulsars themselves, zero everywhere else--if we ignore the EM radiation emitted by the pulsars and assume the only "radiation" in the spacetime is GWs), and the EFE holds at each event. That's it.

Does this "total system" have a "total energy"? It depends on how you define "energy". The spacetime as a whole does not have a time translation symmetry, so we can't define "energy" that way. The spacetime *may* have a continuous set of spacelike slices that match up well enough with what symmetry does exist (for example, maybe the slices are good approximations to "natural" ones that observers hovering at a large radius R above the binary pulsar system would pick out as "surfaces of constant time") to be useful in defining "energy" by integrating the energy conservation equation (i.e., the covariant divergence of the SET) over each spacelike slice. This could define a "total energy" for the system, and this total energy could turn out to be conserved (i.e., the same on every slice), at least to a good enough approximation (the same level of approximation to which the slices are good "surfaces of constant time" for some set of observers). But will this "conservation of energy" be "exact"? Probably not, since the spacetime does not have any exact symmetry. So if you want an exact answer, it is that there is no "total energy".

Now, suppose I decide to draw a boundary at some finite radius R around the binary pulsar system, and say that inside that boundary is "the total system" and outside it is "the rest of the universe". I can pick R large enough that, to a good approximation, the binary pulsar system "looks like" a simple gravitating body with some mass M. More precisely: the metric at R is still not quite in the Schwarzschild form, because the spacetime is not spherically symmetric or static; but it will be close enough that I can "split" it, approximately, into two pieces: a "Schwarzschild" piece and a "gravitational radiation" piece. The Schwarzschild piece, to a good approximation, will look like a gravitating body with a mass M that slowly decreases with time ("time" meaning "proper time according to an observer hovering at radius R). The gravitational radiation piece will be oscillating in quadrupole fashion, and could be measured by, for example, letting the oscillations heat up a detector and measuring the energy taken up. We could then, in principle, do an energy balance: the decrease in M is balanced by the energy carried away by GWs.

Will this energy balance be "exact"? Probably not, because the split of the metric into the two pieces probably won't be exact; there will probably be extra terms in the metric that are left out--they aren't included in either the Schwarzschild or the GW piece--because they are small compared to both of those pieces.

So we come back again to what I said above: if you insist on an "exact" answer, then it is this: "gravitational energy" doesn't exist, and the only exact "energy conservation" is what I said earlier: the covariant divergence of the SET (the standard SET) is zero at every event. Anything else is approximate, and breaks down if you try to press it too hard. That includes things I've said previously (like "M cannot decline if all matter-energy is included"); I apologize if I didn't make it clear enough that I was only speaking approximately.