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Is There Really a Strictly Conserved Stress-Energy Tensor in GR?

  1. Nov 2, 2012 #1
    This is a fork off the locked thread here: https://www.physicsforums.com/showthread.php?t=648423, and is further a response to a recent blog entry 'Does Gravity Gravitate?' (not sure of the PF rules on blogs re threads so won't post a link to it here).
    The blog presents well what is doubtless a standard argument for why gravitational field is not self-gravitating in GR. One key consequence of that position holds that given ∇aGab=∇aTab=0, the only means whereby the net gravitating mass M of some 'isolated' system can change is via a flux of non-zero Tab stress-energy-momentum in or out of that system. But it seems not hard to readily refute that fundamental GR dogma. Although not widely known, it is well known that a small but finite fraction of the energy pouring out from say a stellar body is owing to HFGW's (high-frequency gravitational waves) as a consequence of thermal jostling between particles

    [Moderator's note: unacceptable reference deleted; acceptable reference needed.]

    This entirely random but overall quite smooth and isotropic outgoing flux of non-Tab energy has an obviously insignificant perturbation on the metric at any given time, yet over time represents a steady conversion from and loss of Tab source. This must be so given argument in that closed thread that all forms of gravitational field - including GW's, are not part of Tab. Thus the continuity eq'n ∇aTab=0 cannot be generally correct - unless one wishes to argue that HFGW's are produced 'for free' - thus a further violation of energy-momentum conservation in order to avoid violation of Tab conservation. In that case one has to ask how it is that the Hulse-Taylor binary-pulsar orbital decay data is cited as evidence in favor of both GR and the GW's it predicts, if energy-momentum accounting is not central to that evidence.

    In that other thread I had cited null Nordtvedt results (involving both Lunar and binary-pulsar observations) as a further line that strongly implied gravity does indeed gravitate, but above single point involving conversion to HFGW's aught to suffice for now. Comments?
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  3. Nov 2, 2012 #2


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    There is no contradiction between the two claims that (1) gravity makes no contribution to the stress-energy tensor, and (2) gravitational radiation causes orbits to decay.

    The difference has to do with general covariance. The generally covariant stress-energy tensor [itex]T^{\alpha \beta}[/itex] is not globally conserved, and has no gravitational contribution. However, in the special case of asymptotically flat spacetime it is possible to choose a specific coordinate system in which one can define a "pseudo-tensor" [itex]t^{\alpha \beta}[/itex] that is globally conserved. It's a "pseudo-tensor" because it is only defined for some coordinate systems, unlike a true tensor, which is defined for any coordinate system. The pseudo-tensor does have a contribution due to gravity.
  4. Nov 2, 2012 #3


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    AFAIK linking to blogs is the same as linking to threads; in any case, I certainly don't mind if anyone links to mine. The entry in question is here:

    https://www.physicsforums.com/blog.php?b=4287 [Broken]

    That's not what I argued. I argued that the question "does gravity gravitate?" can be validly answered *both* ways, "no" *and* "yes".

    No, that's *not* a consequence of the GR position. I will address this in more detail in a follow-up post to the one linked to above (which is in draft now), but the quick answer is that the "net gravitating mass" M of an isolated system *can* change without any flux of non-zero T out of the system; as you correctly note, this is exactly what happens in a system that emits gravitational waves. See next comment.

    Yes, it can, and it is. GW emission does not violate the conservation law. Again, I'll go into this in more detail in the follow-up post (and the questions you've asked here are helping me to draft that post), but the quick answer is that when GWs are emitted, the "amount of source" [itex]T_{ab}[/itex] does not change as viewed from a local inertial frame (which is what the continuity equation requires), but the relationship between a local inertial frame and the global coordinates in which the "net gravitating mass" of the system is evaluated *does* change (so the "net gravitating mass" can change without violating the continuity equation).

    This is really the same general issue as the above: the "net gravitating mass" of an isolated system is something different from the "source" that appears in the EFE.
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  5. Nov 2, 2012 #4


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    Any solid evidence to support this claim?

    The Einstein tensor is divergence free as an identity from Riemannian geometry which is valid for all manifolds, all metrics, any number of dimensions, etc. Since the Einstein Field Equation states that the stress energy tensor is proportional to the Einstein tensor then it is automatically also divergence free. That means any situation which satisfies the EFE guarantees the continuity of the stress energy tensor. Thus, to me it seems that your claim is clearly false.
  6. Nov 2, 2012 #5
    Good - that eases my mind.
    OK but you made it plain there the "yes" part involving quantum gravity reduced to standard "no" GR even for typical BH situation well inside EH, so that was not even a consideration here that sticks to just standard GR.
    Well cannot recall this part ever being presented to me before. Seems highly restrictive - basically only good for an observer in free-fall which has in general implies a very brief use-by date. Anyway, I was about to respond to stevendaryl but you have addressed his points in the meantime, so await further expansion on this matter of just what use the local conservation law is if it is globally flouted - and presumably that could mean for any real-world extended body.
    That you will have to expand on - I cannot see the linkage.
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  7. Nov 2, 2012 #6
    You may have noticed from other posts the issue is now about local vs global - something new to me. Settle down Mr circling buzzard - I'm still kicking!
  8. Nov 2, 2012 #7


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    It's not really restrictive at all, but I agree the key point is not often stressed, even in textbooks. The key point is this: *all* tensor equations are, strictly speaking, written in a local inertial frame. Remember that even an object with nonzero proper acceleration still has a local inertial frame at each event on its worldline; similarly, even inside a strongly gravitating body like a neutron star, there is still a local inertial frame at each event, and all objects "appear" in it, even accelerated ones.

    When you use a global coordinate chart like the Schwarzschild chart to write tensor equations, you're not really writing a single global equation: you're writing an infinite family of local equations, each valid at a particular event with particular values of the tensor components as written in that global chart. So when we write the continuity equation [itex]\nabla^a G_{ab} = \nabla^a T_{ab} = 0[/itex], we are really writing an infinite family of continuity equations, one for each event, and each of them describes how continuity works in a local inertial frame at that event. But *all* physical objects at that event can be described in that local inertial frame.

    In fact, since there are an infinite number of possible local inertial frames at a given event, each corresponding to a particular state of motion being "at rest" instantaneously at that event, a given tensor equation can be written in an infinite number of ways (i.e., with an infinite number of possible sets of values for the tensor components) even at a single event. A global chart like the Schwarzschild chart picks out one particular local inertial frame at each event--in the case of the Schwarzschild chart in the region exterior to the horizon, it's the local inertial frame of a static observer at that event, since that observer is at rest in the global Schwarzschild coordinates.

    Also, it's worth noting that, from the standpoint of tensor equations, a local inertial frame is the only kind of frame there is. There is no such thing as an "accelerated frame"; there is only the local inertial frame of a particular accelerated observer at a given event. A global coordinate chart such as the Schwarzschild chart can make a particular family of accelerated observers appear to be "at rest" for more than an instant, but all that is really doing, as I said above, is picking out the local inertial frames at each event in a particular way.

    A note on technical jargon: when you see people talking about tensor equations being valid in the "tangent space" at a given event, and how every event has its own distinct tangent space, that's a shorthand way of referring to what I said above. So it is in the textbooks, but the aspects of it that I stressed above are not stressed in the textbooks.
  9. Nov 2, 2012 #8
    To say an accelerated (possibly highly non-uniformly accelerated) reference frame is at the same time locally inertial smacks to me of double-talk - not that I'm suggesting that of you personally - just the presumably standard notion in general.
    Instantaneously at rest is one thing, but calling it inertial regardless of proper acceleration is another. Feynman accused philosophers of using words in funny ways - maybe he should have looked closer to home.
    Again, I'm having trouble reconciling an observer static = at rest in a SC (thus experiencing proper acceleration) being at the same time in a locally inertial frame. Always before I have seen locally inertial connected with geodesic motion = free-fall = only tidal forces present, never 'full g'. Wow - this is is a real revelation. The words 'inertial frame' seem to have lost all meaning - after all proper acceleration is an intensive property that affects physics 'at a point' - stress, energy density etc.
    Still gobsmacked - whether in free-fall or violent proper acceleration the situation is always locally inertial? I need to swallow a keep-sane pill right now!
    I get the impression 'tangent space' relates to gradients and higher derivatives of such at a point, which per se doesn't bother me.

    So where does all this lead as to usefulness of the 'conserved' SET? Let me again quote you from that blog:
    Well here's my problem. Above sure seemed to say that, unlike 'ordinary' conservation of energy which *globally* fails in general in GR, here with the SET we have a genuinely conserved quantity. But now I am confronted with that this SET 'conservation law' is valid also only strictly at a point - and therefore fails globally just as 'ordinary' energy-momentum does. Forgive me for concluding that such an at-a-point-only conservation principle is not much of a guide or use. [1]

    Getting back to my scenario in #1, note that with HFGW's gravity can be arbitrarily weak even at local regions of most violent inter-particle accelerations. It is only owing to the vast numbers and huge accelerations that appreciable GW's are generated. In principle one could construct a multilayered heat shield around a HFGW source at sufficient radius that outgoing flux of internal EM radiation matches that of incoming CMBR arbitrarily closely, and essentially the sole outgoing flux is from HFGW's. Given sufficient time, a large proportion of initial mass M within has been converted to GW's, and without a doubt for the remaining gravitating mass M', M'<<M. Yet what seems like to me physics variety of Orwellian Newspeak maintains that SET has been conserved? [1] This doesn't quite add up as a bottom-line accounting procedure imo.

    [1]: Forgot that there is this position that gravitating mass M can diminish while leaving SET unchanged. So my wording may not have been strictly correct there. Whatever the correct wording, it needs to be cleared up just how or whether an arbitrarily reduced system gravitating mass M can leave it's source SET 'conserved' - if that is the official GR position. :confused:
    Last edited: Nov 2, 2012
  10. Nov 2, 2012 #9


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    Warning: somewhat long-winded post, but I think it gives good background on this topic.

    It's not double-talk, but it is insisting on a precision in the use of words that is much greater than usual even in scientific discussions. Here are the precise definitions I am using:

    (1) A "reference frame" is a set of four mutually orthogonal unit vectors, one timelike and three spacelike, *at a given event*. It is only valid at that event. The timelike vector can be physically interpreted as the 4-velocity of an inertial observer that is at rest in the frame; call this observer the "fiducial" observer for the frame. This may also be the 4-velocity, at the given event, of some non-inertial (i.e., accelerated) observer that happens to be (momentarily) at rest relative to the fiducial observer at that event.

    (For practical purposes, we make use of the frame in a small local patch of spacetime surrounding the event; how small depends on how accurate we want our answers to be and how curved the spacetime is in the vicinity of the event.)

    Note that there is no such thing as an "accelerated" vs. "non-accelerated" reference frame by this definition. The frame doesn't care which observers happen to have a 4-velocity at the given event that coincides with the timelike basis vector of the frame, or whether some of them are or are not accelerated. The basis vectors of the frame are just vectors, defined in the tangent space at the given event; there's no such thing as an "inertial" or "accelerated" vector.

    (2) A "coordinate chart" is a mapping of 4-tuples of real numbers to events in a spacetime, or in some patch of a spacetime. If we want to write down actual mathematical expressions for the basis vectors of some frame at some event, we need to define a coordinate chart to write them down in (at least, we do for the most commonly used way of treating such problems). Different coordinate charts covering a patch of spacetime containing a given event will lead to different mathematical expressions for the basis vectors of a frame at that event. But the geometric objects, the basis vectors of the frame, stay the same regardless of which chart we use.

    (3) A "frame field" is a mapping of frames (i.e., sets of 4 mutually orthogonal vectors in a tangent space) to events in a spacetime, or in some patch of a spacetime. The most common way of specifying a frame field is to write down the basis vectors of the frames in the field as functions of spacetime position--i.e., as functions of the coordinates in some coordinate chart. The reason frame fields are useful is that they provide a convenient link between something that has a clear physical interpretation (frames at particular events) and something that has a lot of well-tested mathematical machinery associated with it (coordinate charts). So, for example, if I want to know if a family of observers associated with a particular frame field (such as the frame field of "static" observers in Schwarzschild spacetime--see below) is "accelerated" or not, I can write the frame field in terms of a coordinate chart such as the standard Schwarzschild exterior coordinates, and then compute derivatives of the basis vectors as a function of the coordinates (which in this case means functions of [itex]r[/itex]).

    (4) An "observer" is modeled as a particular timelike worldline in spacetime. However, usually we aren't interested in single observers as much as we are in families of observers that all share some property (such as static observers in Schwarzschild spacetime). Such families of observers are most usefully described by frame fields; the worldlines of particular observers within the family are then given by the integral curves of the frame field (more precisely, of the timelike vector of the frame field, considered as a vector field on spacetime).

    This allows us to make sense of the notion of "inertial" or "accelerated" observers, in terms of the corresponding notions with respect to frame fields (see above): if I take the derivative of the timelike basis vector of the frame field, along the integral curves of that same timelike basis vector, I get the "proper acceleration" of the observers traveling along those integral curves. If it's zero, the observers are inertial; if it's not zero, they are accelerated. However, these terms clearly apply only when we have a full frame *field*; they don't apply if all we have is a single frame (i.e., if we're only looking at a single event). Individual frames can't be "accelerated", because there's no way to compute any derivatives if all you have is vectors at a single event.

    Hopefully that wasn't too long. :smile: But I hope it helps in understanding what's going on. For example:

    The observer experiences proper acceleration in the sense that the frame *field* associated with the family of static observers is accelerated (by the definition given above). But if we are only looking at a single event, then all the observer has at that event is a particular 4-velocity, which is the timelike basis vector of his frame at that event. We can't tell whether the derivative of his timelike basis vector along his worldline is nonzero unless we look at the worldline, i.e., multiple events, not just one event.

    So in this respect, perhaps the term "local inertial frame" is a misnomer as well; it should just be "local frame", with the particular observer whose 4-velocity defines the timelike basis vector specified if necessary. The reason the term "local inertial frame" is often used is that it is often convenient to adopt a coordinate chart in the small local patch of spacetime around the given event in which the metric is (to the desired approximation) the flat Minkowski metric. But there is no requirement that we do this in order to define the basis vectors of the frame. So this is partly my fault for not following my own advice about adopting precise terminology. :redface:

    Strictly speaking, proper acceleration can't be defined "at a point", because strictly speaking, derivatives can't be computed "at a point". Our notation invites the misconception that they can, but they can't. As noted above, when we compute the proper acceleration at an event of a particular observer, we are implictly assuming not just a frame at that particular event, but an entire frame field on the spacetime, with the observer following one integral curve of (the timelike basis vector of) that frame field.

    Kinda sorta. If you want to get more confused, you can try the Wikipedia page: :wink:


    The key point is that, strictly speaking, when we talk about scalars, vectors, tensors, etc. defined "at an event", what we are really talking about is scalars, vectors, tensors, etc. defined *in the tangent space* at that event. Each distinct event has its own distinct tangent space, so in order to compute derivatives of scalars, vectors, tensors, etc., we need to be able to map those objects in the tangent space at one event to the "corresponding" objects in the tangent space at another event. When you see people talking about the "connection", "parallel transport", etc., that's what they're talking about: agreeing on how that correspondence between tangent spaces is to be determined.

    In a curved spacetime, in general there is *no* quantity that is "globally conserved" in the sense you mean here. The SET is only "locally conserved" in the sense you mean here. But if it's "locally conserved" at every event, that amounts to saying that no stress-energy can be created or destroyed anywhere in the spacetime, which is a very useful property for the SET to have, whether or not it satisfies your intuitions. IMO, the cure for that is to change your intuitions; we can't change this aspect of the theory of GR in the general case without breaking it altogether (at least, nobody has figured out a way to yet, and many have tried).

    The SET is conserved as a geometric identity; as DaleSpam pointed out, if the EFE is satisfied, the SET is conserved automatically. That's a general mathematical theorem that applies to any solution of the EFE. We don't have to know the details about "where the stress-energy goes" to know that the theorem holds. Those details may well be useful if you are trying to make the best match you can between your intuitions and what the EFE says, but as I said above, the bottom line IMO is that if the EFE clashes with your intuitions, you need to change your intuitions. Cases like the binary pulsar, which experimentally show energy loss due to GWs, do not call the EFE into question; they *validate* the EFE, because the EFE was used to calculate the predictions that were matched to the experimental data. (In the follow-up blog post I'm working on, I will try to give at least a rough picture of how the calculations work.)
  11. Nov 2, 2012 #10


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    Just to expand on this a bit (I'll save more details for the follow-on blog post), there are two key points to be aware of:

    (1) GR does not use the "accounting procedure" you speak of at all. In other words, when calculating a system like the binary pulsar, we don't calculate the answers by asking "hey, what happens to the gravitating mass M as the GWs are emitted?" and checking to see that the energy carried away by GWs balances with the decrease in M. Nor do we ask, "hey, how can M decrease when the SET is locally conserved?" These questions are *not relevant* to the theory's predictions at all. They are being dictated by your intuitions, *not* by the theory. So *whichever* answer you get to them, it won't make a difference for the validity of GR: GR is already shown to be valid (within its domain of applicability, to the degree of accuracy tested to date) by the fact that we calculate answers using the EFE and they match what we measure. The rest is "interpretation", if you want to call it that, and interpretation is (IMO) always heuristic: it should not be expected to give an exact correspondence with the actual predictions of the theory. (And why should it? Our intuitions did not evolve to handle this kind of stuff.)

    (2) Strictly speaking, the energy carried away by GWs *may not* balance exactly with the decrease in M! That is, this "global conservation" is only *approximately* true anyway, and GR does not predict or require anything more than that. The exact "global conservation" that you are looking for is simply *not required* by the fundamental theory. So if your intuition is telling you that the "global books need to balance", once again, you should change your intuition IMO. (Or you could try to find another theory that matches all of the confirmed predictions of GR but does "balance the global books". Good luck.) I realize that's not going to satisfy you, but we might as well get the truth out on the table: there are definite respects in which GR clashes with your intuitions, and since GR's predictions are confirmed, I think there's a limit to how far we will get with discussion on this issue.
  12. Nov 2, 2012 #11


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    Since Feynman was mentioned at some point in this discussion (it may have been in the previous thread), I thought I would give a quote that seems relevant. It's from his book QED: The Strange Theory of Light and Matter:

    Similar remarks apply here. Some people want to say that if energy isn't "globally conserved", then physics will collapse. But GR says that energy doesn't have to be globally conserved, and yet it gives the right answers: physics has not collapsed.
  13. Nov 2, 2012 #12
    Peter - thanks much for the effort in outputting much info here on these technical definitions. It will take me some time to digest it all, being very non-acquainted with these geometric matters in GR. Meanwhile I just have time to briefly comment on the following, which is still a sticking point:
    (emphasis added)
    Huh?! On the one hand, SET only locally conserved - well, ok. But then, also cannot be created or destroyed anywhere in the spacetime. You are darned right about one thing - that very much bothers my intuition - and my understanding of what consistency and coherency means.
    To labor a point - in my last scenario, sole net, outgoing flux, is strictly non-SET energy in form of GW's. System SET by imo any sane definition has shrunk, maybe not on a one-to-one energy budget basis (I fully allow that 'ordinary' energy conservation can fail), but shrunk nonetheless. Are we being real here in saying the SET is conserved on any rational basis? Like how is it conserved when it has shrunk without any SET flux in or out of system involved?!

    From #10:
    Same comments as above - either SET as a whole is conserved, or not. I'd be fine with either position, as long as we have rational definitions of what 'conserved' means.

    Form #11:
    Again, have become familiar with notion that energy-momentum can globally fail in GR, but cannot follow a line that claims an entity - SET, can be both rigorously conserved, yet arbitrarily shrink via conversion to strictly non-SET GW's that form a non-SET flux of energy-momentum. This is way beyond intuitive problem - this is all about logical consistency - my intuition insists! :yuck: :zzz:
  14. Nov 2, 2012 #13


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    Perhaps the word "locally" is causing confusion. Suppose I have a spacetime with a lot of points (events) in it. Consider a proposition I'll call C(E): "No stress-energy is created or destroyed at event E." Proposition C(E) is just another way of saying that "the SET is locally conserved at event E". The proposition {For all E: C(E)} then corresponds to saying "stress-energy can't be created or destroyed anywhere in the spacetime"; it's just asserting that the "local" version holds at *every* point.

    The SET does not measure the "size" of the system in any useful sense that I can see. You are thinking of it as measuring "the amount of stuff", but the "stuff" is not just what we normally think of as "matter". Not only does it include radiation, it also includes momentum, pressure, and other stresses. Saying that the SET has zero covariant divergence is not saying that none of those components of the SET change at all; it's only saying that the changes in the components have to be related to each other in a particular way. So the conservation law doesn't say "the amount of stuff doesn't change" in any useful way that I can see. It just says that there is a constraint on the "changes in the stuff".

    "Conserved" means "zero covariant divergence". That is a mathematically precise equation which can be computed for any event in any spacetime. I don't see how a definition can be any more "rational" than that. What you really mean by "rational" appears to be "matches my intuition"; that's not a good way to judge whether something is "rational" in science.

    The SET does not "shrink". It changes, but the changes are constrained by the requirement of zero covariant divergence, which, as noted above, is a precise requirement. If the only issue you have is that it doesn't match your intuition, then as I've said several times, IMO you need to change your intuition. (Or at any rate, you need to accept that your intuition is going to clash with a theory, GR, which makes correct experimental predictions, and just deal with it.)
  15. Nov 2, 2012 #14


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    Well, in an asymptotically flat universe, you can, as I said in an earlier post, use a "pseudo-tensor" that is conserved when a system emits gravitational radiation. That's unsatisfying for other reasons (such as being tied to a particular coordinate system) but it does allow for "balancing the books".

    For the universe as a whole, there may be no way to "balance the books", however.
  16. Nov 2, 2012 #15


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    I actually have not seen a lot of literature on the specific requirements for one of the pseudo-tensors to be conserved, so I'm not sure if the conservation holds exactly for *any* asymptotically flat spacetime, regardless of other considerations, or if there are symmetry constraints that have to hold. For example, in a binary pulsar-type system, I'm not sure if the conservation would always hold, or if it would only hold if both objects in the binary system had exactly the same mass, so that the system as a whole had a quadrupole symmetry. Does anyone have any good links?
  17. Nov 2, 2012 #16


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    There's a Usenet Physics FAQ: Is Energy Conserved in General Relativity?

    The SET is a tensor with 16 components, 10 of which are independent. In Minkowski coordinates you can compare two SETs at two different locations just by comparing their components. In curved spacetime you can't do this in a unique, coordinate-independent way. It's the old "parallel transport" problem.
  18. Nov 2, 2012 #17


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    My understanding, from Sam Gralla (co-author of several papers with Wald), who used to post here, is that rigorous conservation of energy, momentum, and angular momentum is possible in GR so long as you have asymptotic flatness (with one of the modern conformal definitions). No other assumption is needed. There is no localization of these quantities - they are only conserved at spatial infinity.

    These approaches account, among other things, for the GW carrying both energy and angular momentum (again, not locally). But you can (according to Sam) separate that portion due to the totality of GW versus other sources via comparison of null infinity integrations versus spatial infinity integrations.
  19. Nov 2, 2012 #18


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    Here is a paper by Wald that references the body of work on conserved quantities in GR given asymptotic conditions. Unfortunately, many of the key results are hard to find on line.

  20. Nov 2, 2012 #19


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    Hm, so maybe I was too pessimistic about exact conservation, at least when evaluated at infinity.

    Yes, this matches my understanding: in spacetimes where the ADM energy and Bondi energy are well-defined, the difference between them is the energy carried away by radiation. In the general case, as I understand it, "radiation" includes all types of radiation, not just GWs, but in the idealized case where there is no other radiation except GWs, the difference between the two energies (ADM energy is evaluated at spatial infinity and Bondi energy at null infinity) gives the energy carried away by GWs.

    The Wald paper you linked to is interesting; I'll have to take some time to digest it, but on a quick skim it looks like it generalizes the kind of scheme I just described to cases where the standard ADM and Bondi energies are *not* well-defined.
  21. Nov 3, 2012 #20
    This surely then amounts to what I wrote earlier; zero divergence of SET is good only for a point and thus has no general validity for a real extended system = global failure, no different then to the better known global failure of energy conservation.
    I never said or implied anywhere that SET only include T00 energy term - always meant the SET inclusive of all terms - and it's the latter that evidently fails in general beyond the point event scale.
    I'm still to here a clear admission that this zero covariant divergence has no generally valid applicability for a real extended system.
    Sorry Peter but you can't use this intuition argument on me here. You say SET doesn't shrink just changes - meaning I take it fully conservative conversions between various components. But the hidden clause here presumably is 'only true for a point event'. In other words, it falls apart for a real world extended system where 'conservation' has real meaning. Are we clear then that strict zero divergence has no general global applicability? Assuming so it gets down to what causes this global failure and under what specific circumstances it may or may not fail. Will expand a bit on that next thread.
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