Relative energy of a black hole.

[PeterDonis]

Peter, you've got my head spinning here again. Thought had this much bedded down: in GR all gravitational fields - static or GW's, contribute nothing to what we would term M, the gravitating mass (inclusive of momentum and pressure) that is the origin of all curvature - Ricci and Weyl etc.

I think you are misunderstanding the meaning of the term "mass"; or, rather, you are conflating two different possible meanings. The "M" that appears in the metric, for example the Schwarzschild metric, is "not" the same as the "mass" (actually "energy density", or "0-0 component") that appears in the stress-energy tensor as a "source" of curvature. It's critical to understand the distinction between these two concepts. See further comments below.

Now, it is common to label a black hole with a certain *gravitating* mass M, right?

It's not just "labeling". The "M" that appears in the metric has a definite physical meaning: it's the mass you would measure if you put a test object in orbit around the black hole, measured its orbital parameters, and applied Kepler's Third Law. The same applies for any gravitating object--the Sun, for example. If we write down an expression for the metric in the vacuum region exterior to the Sun, it will have an "M" in it which is the Sun's mass measured the same way.

But measuring "M" this way tells us nothing about how it relates to the presence of a non-zero SET in the spacetime (except that there must be one *somewhere*). If we only went by the measured mass M, we would not know whether the Sun was a star or a black hole; either would give the same M. See below.

Zero SET, and zero contribution from the field. Oh my. So this gets back to 'past light cone' presumably - there *was* a SET but now... Alright, let's just say a BH's mass M derives from a 'fossil' SET.

Remember how we measure M: we put a test object in orbit. That orbit has to be at some radial coordinate r. To understand "where the M comes from", follow the prescription I gave earlier: pick an event somewhere on the worldline of the test object orbiting at that r; look in the past light cone of that event; and find a region with a nonzero SET. Suppose we have M = M(Sun), and we have used an orbit at r = r(Earth) to measure it. Then if the Sun is actually a star, we will find a region of nonzero SET pretty quickly--only eight light-minutes into the past light cone. But if the Sun is a black hole, we may have to look much further to find the nonzero SET region. It just so happens that, because of the particular symmetry of the situation (remember we are assuming perfect spherical symmetry, since that's a condition of the Schwarzschild solution), the "field" at a particular radial coordinate r in the exterior vacuum region is the same for all times t to the future of the nonzero SET region; so it doesn't matter whether that region is eight light-minutes or a billion light-years into the past light-cone, you get the same field--meaning the same metric, and therefore the same measured mass M--either way.

Is it not still the case, in pre-merger we say start with M₁ + M₂ = M_t, and after merger we have M₃ ~ 60% M_t (the deficit in purely *energy* terms carried off by GW's). All those M's representing gravitating mass. A net reduction, regardless of what we call the source of each M. What am I missing here?

Here we have violated the condition of spherical symmetry during the merger, so the relationship between the measured M and the nonzero SET regions in the past light cone can be more complicated. Before the merger, if we assume each BH was stationary, we can relate M1 and M2 to two nonzero SET regions in the past light cones as above. But after the merger, there is a region of spacetime where there are violent curvature fluctuations because of the violation of spherical symmetry; and those curvature fluctuations carry off energy in the form of gravitational waves. That changes the relationship between the final measured mass M3 and the nonzero SET regions in the past light-cone. It doesn't change anything about the SET regions themselves; no actual stress-energy escapes during the BH merger (it's all trapped behind the horizons of the BH's). But "nonzero SET" and "gravitating mass" in the sense of the value M appearing in the metric are, as I said above, not the same; and the relationship between them depends on the configuration of the spacetime in between the nonzero SET region and the event at which the metric, and thus M, is being measured.

The consequence is that a dispersed system, whether neutron stars or BH's, carries there a maximal total energy/gravitating mass M_t. After collision/merger/ringdown, necesarily a portion of that original M_t has been lost to GW's - the remainder has to be less than before - how can there not be a reduction and maintain conservation of energy?

There is a reduction in M, yes; as you say, there has to be by conservation of energy. There is no "reduction" in the "stress-energy content"--see above. M and "stress-energy content" are not identical.

The problem is you adopted a different meaning to the terms I had originally used in #45. Your M is not the M I used there. If you go back and check carefully I think there will be no conflicting opinion on that issue.

See my comments on the meaning of M, and how it is measured, above.

 

[Q-reeus]

I think you are misunderstanding the meaning of the term "mass"; or, rather, you are conflating two different possible meanings. The "M" that appears in the metric, for example the Schwarzschild metric, is "not" the same as the "mass" (actually "energy density", or "0-0 component") that appears in the stress-energy tensor as a "source" of curvature. It's critical to understand the distinction between these two concepts. See further comments below.

Agreed entirely - and it would aid greatly if the two were given a specific differentiation/identification wrt some simple, static and non-BH system. More on that later.

Remember how we measure M: we put a test object in orbit. That orbit has to be at some radial coordinate r. To understand "where the M comes from", follow the prescription I gave earlier: pick an event somewhere on the worldline of the test object orbiting at that r; look in the past light cone of that event; and find a region with a nonzero SET. Suppose we have M = M(Sun), and we have used an orbit at r = r(Earth) to measure it. Then if the Sun is actually a star, we will find a region of nonzero SET pretty quickly--only eight light-minutes into the past light cone. But if the Sun is a black hole, we may have to look much further to find the nonzero SET region. It just so happens that, because of the particular symmetry of the situation (remember we are assuming perfect spherical symmetry, since that's a condition of the Schwarzschild solution), the "field" at a particular radial coordinate r in the exterior vacuum region is the same for all times t to the future of the nonzero SET region; so it doesn't matter whether that region is eight light-minutes or a billion light-years into the past light-cone, you get the same field--meaning the same metric, and therefore the same measured mass M--either way.

Which is speaking to me that the distinction between M, and source of non-zero SET in the past light cone, is relevant only for a non-static system. A now stable and static planet that formed long ago from a collapsing gas/dust cloud easily qualifies. So I would say there M = volume integral of non-zero SET. The two are here synonymous, agreed? Anyway the following will attempt to clear all confusion about how factors relate and pan out.

Consider please the following scenario: A large bounding box of mass M_b with perfectly reflecting walls. Inside we have diffuse dust of mass M_d >> M_b that over time gravitationally collapses symmetrically to form a stable, static planet of assembled mass M_p < M_d. Heat radiated away during collapse is trapped inside the box. The total energy of this notionally closed system is constant. But the internal state has changed. Without question there has been a partial transfer from non-gravitational to gravitational energy. In GR the latter is 'dead weight' wrt acting as gravitational mass, the former is not. How can it be argued the net gravitating mass, as presented to a region exterior to the box, has not thereby diminished? No need to introduce GW's - whenever gravitational energy of any kind is created, a net reduction in overall system gravitational mass ensues (and note 'system' here means everything including radiation). Or so it seems bleeding obvious to me.
Past light cone is not an issue as this is a stable final system with all contributing sources (and non-sources) accessible in perfectly reasonable time to a region exterior to the box.

See my comments on the meaning of M, and how it is measured, above.

But my 'labelling' convention for M in #45 referred to the dispersed matter prior to collapse/assembly. It was never to be confused with an M for the final gravitating system, which I designated as M' - the assembled mass.

 

[PeterDonis]

Which is speaking to me that the distinction between M, and source of non-zero SET in the past light cone, is relevant only for a non-static system. A now stable and static planet that formed long ago from a collapsing gas/dust cloud easily qualifies. So I would say there M = volume integral of non-zero SET. The two are here synonymous, agreed?

Not quite. The *relationship* between non-zero SET in the past light cone and M, the quantity appearing in the metric, is simplest for the static case; but that still doesn't mean the two are identical.

Consider please the following scenario: A large bounding box of mass M_b with perfectly reflecting walls. Inside we have diffuse dust of mass M_d >> M_b that over time gravitationally collapses symmetrically to form a stable, static planet of assembled mass M_p < M_d. Heat radiated away during collapse is trapped inside the box.

But this changes the scenario from your original one, where the radiated heat can escape to infinity. If the heat is trapped by reflecting walls, then it will "fall" back into the planet, raising its temperature (and hence its energy). So the equilibrium state will be quite different than a "cold" planet with essentially zero temperature and radiation escaping to infinity.

The total energy of this notionally closed system is constant.

Agreed.

But the internal state has changed. Without question there has been a partial transfer from non-gravitational to gravitational energy.

Not necessarily. See my comments above. But in any case, this is a red herring. See below.

In GR the latter is 'dead weight' wrt acting as gravitational mass, the former is not. How can it be argued the net gravitating mass, as presented to a region exterior to the box, has not thereby diminished?

It hasn't. Your specification has ensured that, as you said above and I agreed, the total energy inside the box is constant. That will mean that, if we put a test object in orbit about the box and measured its mass M externally, we would continue to get the same answer regardless of what happens inside the box.

Let's try a simpler system: a box with perfectly reflecting walls but zero mass (so it doesn't affect the curvature of the spacetime) enclosing two objects of equal rest mass m that start at mutual rest at some distance r apart. What will the externally measured mass of this system be? You might think it will simply be 2m, but think again. Suppose we let the system evolve for a while: the two objects fall towards each other, and at the instant right before they hit each other, they both still have rest mass m, but they also (in the center of mass frame, which is just the frame in which they were initially at rest) have each a considerable kinetic energy k. So at this point we would expect the externally measured mass to be 2(m + k).

Now let the two objects collide, and suppose the collision is perfectly inelastic; the two objects plop into each other and come to rest at the point where they collided, which is the center of mass of the combined system. Obviously, by conservation of energy, the final object must have total energy 2(m + k); but the kinetic energy portion is now converted into heat inside the combined object, which will be at some significant temperature (we assume both initial objects started out at zero temperature). If the enclosing box were not there, that heat could eventually be radiated away to infinity, so that we would end up with a final object of mass 2m. But the box prevents that from happening; the heat might be radiated, but it would then be reflected off the walls and converge on the central object again, until finally some thermal equilibrium was established with some portion of the "heat energy" 2k residing in the object and some portion residing in radiation bouncing around inside the box. In any case, the total energy of the system, as measured from outside the box, will continue to be 2(m + k).

What all this tells us is that, by conservation of energy, the *initial* externally observed mass M of the system must have been, not 2m, but 2(m + k). In the initial state, the energy that became the kinetic energy 2k of the objects, and then the heat inside the final combined object, was instead stored as "gravitational potential energy" in the mutual field created by the two objects combined. (At least, this is the usual way of putting it; but as should now be apparent, that way of putting it can lead to considerable conceptual difficulties.) Similar remarks would hold if we replaced the two initial objects by a spherical shell of dust and let it collapse. But notice that, on the "energy stored in the field" interpretation, the "energy stored in the field" is nonzero in the *initial* state, and is *zero* once the objects have collided! In other words, this scenario converts energy *from* "stored field energy" into "tangible" energy, not the other way around!

I'll follow up with more on this in another post when I have more time; but this should at least give some food for thought.

But my 'labelling' convention for M in #45 referred to the dispersed matter prior to collapse/assembly. It was never to be confused with an M for the final gravitating system, which I designated as M' - the assembled mass.

Ah, ok; I was wondering a little about that but didn't read carefully enough. Then my comments were really referring to M', not M.

 

[Q-reeus]

But this changes the scenario from your original one, where the radiated heat can escape to infinity. If the heat is trapped by reflecting walls, then it will "fall" back into the planet, raising its temperature (and hence its energy). So the equilibrium state will be quite different than a "cold" planet with essentially zero temperature and radiation escaping to infinity.

Fair point - I had not specifically addressed that. However it changes nothing in respect of key principle. The easiest counterargument is to allow the enclosing box to grow as large as desired. Equilibrium temperature grows correspondingly small. Secondly, assuming a configuration where concentration of thermal energy in the central mass is high merely slightly increases concentration of the total system energy there, to a generally small degree. It makes no appreciable difference to the existence of an all important partial conversion of non-gravitational mass/energy to gravitational energy via collapse.

Not necessarily. See my comments above. But in any case, this is a red herring. See below.

Another red herring, apart from the one dealt with above?

Let's try a simpler system: a box with perfectly reflecting walls but zero mass (so it doesn't affect the curvature of the spacetime)...

Purist response: a massless box cannot withstand the radiant pressure it has to contain. But enough of nitpickery.

...enclosing two objects of equal rest mass m that start at mutual rest at some distance r apart. What will the externally measured mass of this system be? You might think it will simply be 2m, but think again. Suppose we let the system evolve for a while: the two objects fall towards each other, and at the instant right before they hit each other, they both still have rest mass m, but they also (in the center of mass frame, which is just the frame in which they were initially at rest) have each a considerable kinetic energy k. So at this point we would expect the externally measured mass to be 2(m + k).

[STRIKE]So far so good.[/STRIKE] Oh, too casual reading that. No, if the isolated rest mass, assuming infinite separation, is m each, then the combined mass must be less than 2m when separated by r and stationary. A portion of m+m was lost (as heat, or mechanical work supplied elsewhere) to arrive at the 'initial', partially separated configuration you give. Binding energy is negative, so total mass declines. Subsequent collapse, where all energy is now contained within the enclosure, conserves total energy yes.

Now let the two objects collide, and suppose the collision is perfectly inelastic; the two objects plop into each other and come to rest at the point where they collided, which is the center of mass of the combined system. Obviously, by conservation of energy, the final object must have total energy 2(m + k); but the kinetic energy portion is now converted into heat inside the combined object, which will be at some significant temperature (we assume both initial objects started out at zero temperature). If the enclosing box were not there, that heat could eventually be radiated away to infinity, so that we would end up with a final object of mass 2m. But the box prevents that from happening; the heat might be radiated, but it would then be reflected off the walls and converge on the central object again, until finally some thermal equilibrium was established with some portion of the "heat energy" 2k residing in the object and some portion residing in radiation bouncing around inside the box. In any case, the total energy of the system, as measured from outside the box, will continue to be 2(m + k).

[STRIKE]And still good.[/STRIKE] As per previous edit. If m is substituted with an m' that reflects the reduced system initial mass, the rest of the argument just here I agree with.

What all this tells us is that, by conservation of energy, the *initial* externally observed mass M of the system must have been, not 2m, but 2(m + k). In the initial state, the energy that became the kinetic energy 2k of the objects, and then the heat inside the final combined object, was instead stored as "gravitational potential energy" in the mutual field created by the two objects combined. (At least, this is the usual way of putting it; but as should now be apparent, that way of putting it can lead to considerable conceptual difficulties.)...

Yes those conceptual difficulties come across below. Note here though that the conventional 'modern' Newtonian interpretation of negative gravitational binding energy is to ascribe a corresponding negative energy to the field. What I have shown in #45 is this is incorrect. The fact of redshift demands imo for consistency that field (or curvature, by geometric interpretation) energy is both present and has a positive sign - in keeping with analogous EM, mechanical systems.

Similar remarks would hold if we replaced the two initial objects by a spherical shell of dust and let it collapse. But notice that, on the "energy stored in the field" interpretation, the "energy stored in the field" is nonzero in the *initial* state, and is *zero* once the objects have collided! In other words, this scenario converts energy *from* "stored field energy" into "tangible" energy, not the other way around!

And here imo is the crimson fish indeed. Disagree entirely. If you want to go back and argue a basic flaw in #45 be my guest - I stand by it. It is clear there the difference M'-fM = M(1-f)/2, identified as necessarily gravitational energy (not explicitly stated there but implied), grows monotonically from zero at 'infinite' separation to a positive values at any finite final R. Naturally that expression is just a good approximation for R >> r_s, but in that regime any error amounts to a tiny higher order correction.

The bottom line to all this is stark and simple. To repeat: GR excludes gravitational field energy as source term. At the same time we must as general feature have conversion from non-gravitational to gravitational energy in any collapse scenario. Simple math follows. To avoid this prospect (monopole GW's etc.) while holding to gravity does not gravitate, as stated before one can say there is no energy in static field curvature, while presumably keeping it for GW's. Now it can be postulated that nature truly behaves like that, but I suspect a truly bizarre playground follows. And you well know my suggested cure.

 

[PeterDonis]

GR excludes gravitational field energy as source term.

In the precise sense of "source", meaning nonzero SET, yes.

At the same time we must as general feature have conversion from non-gravitational to gravitational energy in any collapse scenario.

I would phrase this somewhat differently: the externally measured mass M of an isolated system, from one point of view, will, in general, contain a contribution that can be described as "energy stored in the gravitational field". Put another way, if we take a "snapshot" of the system at some instant of time (i.e., on a particular spacelike slice), and try to "count up" the contributions to the total mass M from individual parts of the system, we will find that we have to include a contribution from "gravitational potential energy" to make the final answer come out right.

However, this is only "from one point of view", and there is nothing in the physics that *requires* you to take that point of view, nor do you need to take it to figure out what actually happens. See below.

To avoid this prospect (monopole GW's etc.) while holding to gravity does not gravitate, as stated before one can say there is no energy in static field curvature, while presumably keeping it for GW's. Now it can be postulated that nature truly behaves like that, but I suspect a truly bizarre playground follows.

As I've said before, I believe this is only an issue for you because you are focusing on asking questions that your a priori conceptual scheme wants you to ask, such as "does gravity gravitate?", instead of first looking at the actual physics and then deriving your conceptual scheme from what the actual physics says. If you do the latter, there is no issue; the theory is well-defined and gives definite answers to all the questions you can ask about actual physical observables.

The actual physics, as I've said before, is simple: to figure out what the "observed field" is at a given event (meaning the metric, and therefore all quantities derivable from the metric, which includes the mass M, the "acceleration due to gravity", the "gravitational potential energy", tidal gravity, etc., etc.), it suffices to look in the past light cone of that event, figure out where the ultimate "sources" are (regions of nonzero SET), and then look at the (vacuum--zero SET) spacetime in between the sources and the event of interest to determine how the field generated by the sources "propagates" to the event of interest.

The above can be done without ever having to ask the questions you are asking. You don't need to know whether "gravity gravitates". You don't need to know how to "count up" individual parts of the system on a spacelike slice and add them up to get the externally measured mass M, or whether you need to include "gravitational potential energy" in the total. Those are simply not necessary questions to ask; they aren't needed to figure out what happens (what the observed field is); and "what happens" includes what the externally measured mass M of the system will be at a particular event. For a system whose mass M appears, from one point of view, to contain a contribution from "energy stored in the gravitational field", that same mass M can always be accounted for in the way I have described, without ever having to consider "energy stored in the gravitational field".

Another way of looking at this is to ask why you are so insistent on interpreting the mass M in terms of "adding up sources" on a spacelike slice, instead of doing it the way I have described (looking in the past light cone)--and therefore finding that, to make things "add up" correctly in this way, you need to include "energy stored in the field". I think the reason this way of looking at it is intuitively appealing is that 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.

 

[PeterDonis]

No, if the isolated rest mass, assuming infinite separation, is m each, then the combined mass must be less than 2m when separated by r and stationary. A portion of m+m was lost (as heat, or mechanical work supplied elsewhere) to arrive at the 'initial', partially separated configuration you give. Binding energy is negative, so total mass declines. Subsequent collapse, where all energy is now contained within the enclosure, conserves total energy yes.

This is probably just an issue of definition of terms. For "m" in my post, instead of reading "isolated rest mass at infinity", read "rest mass as-is, in the given initial separation", which, in your terminology, would be (m - e), where e is the portion of the "rest mass at infinity" m that was lost during the process of moving the two objects from infinity to a finite separation.

The key point is that the externally measured total mass M of the system as a whole, in the initial state (objects separated by some distance and, at least momentarily, at rest relative to each other) *cannot* be simply the sum of the "masses" of the two objects individually, if you are trying to compute it the way you are trying to compute it; there *has* to be an additional contribution from "energy stored in the field", because that extra energy will appear as "tangible" energy when the two objects fall towards each other, collide, and form a single object with a positive temperature. You appear to agree with this:

As per previous edit. If m is substituted with an m' that reflects the reduced system initial mass, the rest of the argument just here I agree with.

 

[Q-reeus]

This is probably just an issue of definition of terms. For "m" in my post, instead of reading "isolated rest mass at infinity", read "rest mass as-is, in the given initial separation", which, in your terminology, would be (m - e), where e is the portion of the "rest mass at infinity" m that was lost during the process of moving the two objects from infinity to a finite separation.

The key point is that the externally measured total mass M of the system as a whole, in the initial state (objects separated by some distance and, at least momentarily, at rest relative to each other) *cannot* be simply the sum of the "masses" of the two objects individually, if you are trying to compute it the way you are trying to compute it; there *has* to be an additional contribution from "energy stored in the field", because that extra energy will appear as "tangible" energy when the two objects fall towards each other, collide, and form a single object with a positive temperature. You appear to agree with this:

Peter, thanks for your clarification and with that I agree with the above. On the broader picture, while I respect you are an accomplished master of GR maths and it's application, sad to say there is no final consensus. Bravo though for putting in a lot of effort in trying to evaporate my scepticism. At the least it has given me a clearer understanding on how this issue is seen by the GR community. Have a nice day.

 

[PeterDonis]

On the broader picture, while I respect you are an accomplished master of GR maths and it's application, sad to say there is no final consensus. Bravo though for putting in a lot of effort in trying to evaporate my scepticism. At the least it has given me a clearer understanding on how this issue is seen by the GR community. Have a nice day.

No problem, we can't always reach consensus. I do have one final question, though, about the precise nature of your disagreement. I'm still not entirely clear whether:

(1) You disagree with my contention that the observed field at a given event can always be explained (calculated) entirely in terms of "sources" (regions of nonzero SET) in the past light cone of that event; or

(2) You agree that the observed field at a given event can be explained (calculated) as above, but you don't think this is enough--that something more is needed for a proper physical understanding of what's going on.

 

[Q-reeus]

No problem, we can't always reach consensus. I do have one final question, though, about the precise nature of your disagreement. I'm still not entirely clear whether:

(1) You disagree with my contention that the observed field at a given event can always be explained (calculated) entirely in terms of "sources" (regions of nonzero SET) in the past light cone of that event; or

(2) You agree that the observed field at a given event can be explained (calculated) as above, but you don't think this is enough--that something more is needed for a proper physical understanding of what's going on.

Neither of the above really. (1) is fine in principle, except for the specific contention in GR that SET never includes gravitational energy density W_g - however one wishes to precisely define the latter. And my suspicion is the reason gets down to ambiguities re non-localizability of W_g from the GR geometric perspective (free-fall and it's gone). That I find almost amusing. One could create the same vanishing trick in standard EM. A rather brief 'world' consisting of charged particles all having the same sign and charge-to-mass ratio, allowed to suddenly 'free-fall' apart, will have no charged observer detecting an E field, apart from gradients ('tidal forces'). The particular analogy to gravitation is obvious (of course excludes non-linear features present in GR). Physicists in such a world might justifiably conclude EM field energy was ill-defined and non-localizable, but that would be their mistaken perspective. What I am saying here is it seems natural to make coordinate measure the proper perspective for working out a clear working definition of gravitational energy, with free-fall the improper frame ('inaccessibility' and all that).

Another thing that to me screams 'gravity gravitates' not brought up earlier is the implications I see of a zero Nordtvedt effect: http://relativity.livingreviews.org/Articles/lrr-2006-3/index.html . All three of active mass m_a, passive mass m_p, and inertial mass m_a, are implied exactly equal. Saying gravitational energy does not gravitate (m_a = 0) is one thing, but zero Nordtvedt requires it also have no inertial contribution either. Anyway, why go on - this is just my layman's reasoning. Running late. :zzz:

 

[PeterDonis]

Neither of the above really.

Then I'm still confused. But see below.

(1) is fine in principle, except for the specific contention in GR that SET never includes gravitational energy density W_g - however one wishes to precisely define the latter.

Why is that a problem? The SET has a clear physical meaning, based on being on the RHS of the Einstein Field Equation; and that equation does not work if you try to add "gravitational energy density" into the RHS of the EFE. That was the point of one of the quotes we discussed early on, which AFAIK you agreed with.

Another way of putting this would be to ask: why do you need to even define "gravitational energy" in the first place? See further comments below.

One could create the same vanishing trick in standard EM. A rather brief 'world' consisting of charged particles all having the same sign and charge-to-mass ratio...

But there is a big difference here: we know that this "world" does not match reality; we know that, in reality, there are particles with varying charge/mass ratios (and signs). 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). If we did have such evidence, obviously we would have to change our model of gravity; but we don't. GR predicts that we never will, because all three of those "masses" are really the same thing, so they must all be the same as a matter of physical law.

If we ever found such evidence, GR would be falsified. But if it is actually true that all three "masses" are the same as a matter of physical law, then what GR is basically telling us is that we are asking the wrong question: we are using a conceptual scheme that doesn't quite match reality, because it leads us to ask a question ("why are active m_g, passive m_g, and inertial mass m_a all the same?") that, from a proper conceptual scheme, would never even be asked, because it would be "obvious" that there was only one kind of mass-energy to begin with.

(It's possible, btw, that such a conceptual scheme already exists: I believe there are some versions of quantum gravity in which there is no room for more than one kind of "mass", so to speak. But I'm not very up to date on developments in that area.)

What I am saying here is it seems natural to make coordinate measure the proper perspective for working out a clear working definition of gravitational energy, with free-fall the improper frame ('inaccessibility' and all that).

Once again, why are you trying to find a "proper perspective for working out a clear working definition of gravitational energy"? What physics does it capture that isn't captured in the method I have described (look in the past light cone for nonzero SET regions)? (My answer to this question, of course, is "none".)

Also, trying to use "coordinate measure" as a standard creates a problem: how do I tell *which* state of motion is the "standard" one? For the specific case I just described, the spacetime has a time translation symmetry which picks out the "hovering" observer--but what about, for example, an FRW spacetime, which doesn't have a time translation symmetry--let alone a generic spacetime where there is *no* symmetry? (You will note that these are also cases where it is much harder to come up with a definition of "gravitational energy".)

In other words, the "coordinate measure" criterion, while it is intuitively appealing in the simple cases that we have ordinary, everyday experience of, does not generalize well to more complicated cases. The beauty of the free-fall condition is that it always works: I don't have to assume *anything* about the spacetime. I can always test to see if an object is in free fall by direct physical observation: does the object feel any weight? So I can always use freely falling worldlines as "standard" worldlines to refer things to, no matter what kind of spacetime I am trying to analyze.

(Similar remarks apply, btw, to the prescription to look at the "standard" SET in the past light cone and then work through the vacuum region from there, "propagating" the field to the current event of interest. The definition of the "standard" SET is straightforward and unambiguous, so it can always be applied, regardless of the spacetime, and does not require any symmetry to be present.)

Another thing that to me screams 'gravity gravitates'

And once again, why are you even asking this question to begin with? The actual physical observables, as I pointed out before, can be entirely explained and calculated without ever having to ask this question at all. (Although you still don't appear to entirely accept that this is true--but if it is false, then so is the Einstein Field Equation, since that, as I said above, only includes the "standard" SET, with no "gravitational energy" terms, otherwise it wouldn't work. So if you really want to dispute my #1, you'll need to show that the EFE, as it stands, with no "gravitational energy" terms, gives incorrect predictions.)

 

[yuiop]

I have some sympathy for scepticism expressed by Q-reeus about "fossil gravitational fields". For example let us say a star is just about to collapse to a black hole and it is orbiting a much larger gravitational object. When it finally collapses to a black hole its "frozen" gravitational field continues to orbit. Now if we assume the universe can no longer interact with the mass of the black hole (hidden behind the event horizon) then we have to conclude that the frozen gravitational field of the black hole has all the gravitational and inertial properties of the original mass without requiring the mass to be there. In other words it is just the field that is orbiting. This in turn implies that the momentum of a massive object that we normally associate with its mass, is actually a property of its gravitational field and not of the mass itself.

I also wonder where this leaves Hawking radiation. A very small black hole can evaporate in a matter of minutes, but go along with the idea that the gravitational field does not care if the mass is still there or not, we would be unaware that it had evaporated for a very long time (possibly infinite).

Also consider the merger of 2 or more black holes. The gravitational fields surrounding the merging objects changes in a complex, rapid dynamic way that seems inconsistent with the idea of frozen fossil gravitational fields.

 

[PeterDonis]

Now if we assume the universe can no longer interact with the mass of the black hole (hidden behind the event horizon) then we have to conclude that the frozen gravitational field of the black hole has all the gravitational and inertial properties of the original mass without requiring the mass to be there. In other words it is just the field that is orbiting. This in turn implies that the momentum of a massive object that we normally associate with its mass, is actually a property of its gravitational field and not of the mass itself.

You are making the same conceptual error that Q-reeus is making: you are thinking of the BH as an "object" that has to "interact" with things, instead of thinking of it as spacetime curvature that was produced by some region of nonzero SET somewhere in the past.

For purposes of visualizing what's going on in a scenario like you describe, this is fine; if the BH's externally observed mass is much smaller than that of the object it is orbiting, you can treat the BH like a "test object" orbiting the other object, without having to worry about the BH's internal structure. For practical purposes this can work fine. But it is only an approximation; you are trying to extend the approximation beyond its domain of validity. If you want to think about the fundamentals of the BH, things like "where does its mass come from?", "where does its momentum come from?", etc., you simply can't use this approximation: you have to go back to the fundamentals, the Einstein Field Equation and the specific solution of it that produces the spacetime you are looking at--which is based ultimately on what regions of non-zero SET are present in the spacetime, and where. All of the dynamics of the BH, including how it orbits another body, are ultimately derived from this; there is no need to view the BH as an "object" that has to somehow carry mass and momentum independently of what is propagated to it from the regions of nonzero SET in the past.

I also wonder where this leaves Hawking radiation. A very small black hole can evaporate in a matter of minutes, but go along with the idea that the gravitational field does not care if the mass is still there or not, we would be unaware that it had evaporated for a very long time (possibly infinite).

No, we wouldn't. If we include Hawking radiation, the BH is not stationary; its mass slowly decreases. That means the radius of the horizon also decreases, which means that the time for light rays emitted from "close to the horizon" to get out to far distant observers no longer diverges to infinity.

Also, it is incorrect to say that "the gravitational field does not care if the mass is still there or not". It does. As a given "packet" of Hawking radiation passes a given radius, observers at that radius will see a slightly smaller mass for the BH--as observed, for example, by a decrease in the rocket thrust it takes to hold station at a constant radius.

Also consider the merger of 2 or more black holes. The gravitational fields surrounding the merging objects changes in a complex, rapid dynamic way that seems inconsistent with the idea of frozen fossil gravitational fields.

Once again, you are thinking of the BH's as "objects" instead of as curvature produced and propagated from nonzero SET regions in the past. For a spacetime where 2 BH's merge, there will be two such regions--the two bodies that originally collapsed to form the two BH's. Everything about the merger, including the rapid, dynamic changes in the field as they merge, followed by a "settling down" to a new quasi-stationary state with the final BH, is determined by those initial nonzero SET regions, including their positions relative to each other. There is no need to think about "fossil fields"--again, this is simply a scenario where that approximation breaks down if taken too seriously.

 

[TrickyDicky]

Peter, could you take a look at my #47?

 

[Q-reeus]

You are making the same conceptual error that Q-reeus is making: you are thinking of the BH as an "object" that has to "interact" with things, instead of thinking of it as spacetime curvature that was produced by some region of nonzero SET somewhere in the past.
For purposes of visualizing what's going on in a scenario like you describe, this is fine; if the BH's externally observed mass is much smaller than that of the object it is orbiting, you can treat the BH like a "test object" orbiting the other object, without having to worry about the BH's internal structure. For practical purposes this can work fine. But it is only an approximation; you are trying to extend the approximation beyond its domain of validity. If you want to think about the fundamentals of the BH, things like "where does its mass come from?", "where does its momentum come from?", etc., you simply can't use this approximation: you have to go back to the fundamentals, the Einstein Field Equation and the specific solution of it that produces the spacetime you are looking at--which is based ultimately on what regions of non-zero SET are present in the spacetime, and where. All of the dynamics of the BH, including how it orbits another body, are ultimately derived from this; there is no need to view the BH as an "object" that has to somehow carry mass and momentum independently of what is propagated to it from the regions of nonzero SET in the past.

Sorry Peter, but I agree with yuiop here. How can a 'BH' not be modeled as an object that interacts with things when you have previously described it just that way - has a characteristic mass M according to Keplerian dynamics of an orbiting test mass. Let's not play with words here.

yuiop: "Also consider the merger of 2 or more black holes. The gravitational fields surrounding the merging objects changes in a complex, rapid dynamic way that seems inconsistent with the idea of frozen fossil gravitational fields."
Once again, you are thinking of the BH's as "objects" instead of as curvature produced and propagated from nonzero SET regions in the past.

A matter of definition surely - the two are synonymous by any reasonable score imo. A rose by any other name is still a rose.

For a spacetime where 2 BH's merge, there will be two such regions--the two bodies that originally collapsed to form the two BH's. Everything about the merger, including the rapid, dynamic changes in the field as they merge, followed by a "settling down" to a new quasi-stationary state with the final BH, is determined by those initial nonzero SET regions, including their positions relative to each other. There is no need to think about "fossil fields"--again, this is simply a scenario where that approximation breaks down if taken too seriously.

Again, I agree with yuiop. YouTube provides some nice CGI examples of 'BH merger events':http://www.youtube.com/watch?v=4m-ZVsLf070&feature=related http://www.youtube.com/watch?v=L478ZPy_2Ys&feature=related (and these from respected numerical GR groups)
Looks impressive, like lava lamp blobs fusing together, with electrostatics thrown into achieve that 'necking' effect. But is this real world or just some snippet from a sci-fi flick? Presumably the animations are from a coordinate or near enough to coordinate perspective. But from that perspective we know that Schwarzschild metric predicts infinite time dilation and radial length contraction at the EH of each 'pre-merging' BH. One can argue it's not a physical surface, but point is, logically to deform an infinitely curved region requires infinite coordinate time! So I'm having trouble seeing how an infinitely curved BH boundary is anything other than infinitely rigid in effect. So my idea of 'merger' would be roughly akin to say two basketballs, with a thin foam rubber sheet placed between, approaching each other and deforming slightly but never merging. The squashing bit allows that there is some finite mutual metric distortion of one on the other as seen in coordinate measure, but key word here is *finite*. (just to be clear here; my own interpretation is that it shows the inconsistency of 'BH' in the first place. I am not speaking on anyone else's behalf in saying that.)

 

[Q-reeus]

Q-reeus: "(1) is fine in principle, except for the specific contention in GR that SET never includes gravitational energy density Wg - however one wishes to precisely define the latter."
Why is that a problem? The SET has a clear physical meaning, based on being on the RHS of the Einstein Field Equation; and that equation does not work if you try to add "gravitational energy density" into the RHS of the EFE. That was the point of one of the quotes we discussed early on, which AFAIK you agreed with.

I agreed only that this was the official GR position - never mine as made abundantly clear in many posts. But I have a confession to make. My 'proof' of positive energy in a gravitational field neglected to fully account for one entity. In #45 pressure, which apart from rest mass/energy, forms the only other GR approved contribution to the SET in the scenario considered, was declared negligible, but there was no evidence given that it was negligible wrt gravitational energy as source. I had implicitly lumped pressure together with matter rest mass. My argument there and in #52 was not strictly correct (neither of us picked it up) - conversion, partial or fully, of non-gravitational energy to gravitational energy is not sufficient proof of itself that net system gravitating mass declines. One must account for pressure changes also. On two accounts it turns out my earlier claim holds true overall:

1: Scaling law. Take the thin shell stipulated in #45. Double it's assembled mass M'. Pressure has doubled, and in GR the pressure contribution to curvature is a linear function of that pressure. But the field energy density, in this weak gravity linear region, is quadratic wrt M'. Working out the specifics can be a little messy, but bottom line is, pressure cannot in general act to cancel gravitational contribution to M'. Phew.

2: There is good reason to doubt pressure makes *any* contribution to gravitational mass. Consider the case of two 'G'-clamps welded back-to-back. Tighten both screws evenly. This generates positive stress in the screw sides, and negative stress in the opposite sides. There is also bending and shear stresses present elsewhere, but they are self-cancelling wrt net positive or negative pressure. The pressure distribution by inspection will have a quadrupole character. Hence if the screws are periodically tightened then loosened, we have a harmonic source of quadrupole pressure. It follows this arrangement generates GW's. GW amplitude is linear wrt pressure. But material displacement of the twin clamps under pressure is inversely proportional to the material elastic constant. Plastic clamps will flex far more for a given generated pressure than for say steel clamps. the kicker then is this: any metric back reaction from generating GW's must induce far greater power drain in the plastic clamps case than for the steel ones. There cannot be in general a conservative power balance. Unless that is, pressure is nor a source term in fact! We have not included GW contribution owing to just motion of the clamp material, but that's ok since that contribution will be proportional to material density, which need have no relation to elastic constant. The two contributions are thus decoupled.

Another way of putting this would be to ask: why do you need to even define "gravitational energy" in the first place?...

For all the reasons given before in numerous entries! Only by denying the very existence of gravitational energy can the problem seemingly go away. But then e.g. Hulse-Tayler-binary-pulsar-data-as-proof-of-GW's issue, as before discussed, becomes somewhat problematic.

Q-reeus: "One could create the same vanishing trick in standard EM. A rather brief 'world' consisting of charged particles all having the same sign and charge-to-mass ratio..."
But there is a big difference here: we know that this "world" does not match reality; we know that, in reality, there are particles with varying charge/mass ratios (and signs)...

Which misses the point; this is a valid gedanken experiment. It is possible for such a situation to exist and it implies certain things, which I have stated.

Once again, why are you trying to find a "proper perspective for working out a clear working definition of gravitational energy"? What physics does it capture that isn't captured in the method I have described (look in the past light cone for nonzero SET regions)? (My answer to this question, of course, is "none".)

I'll repeat. With pressure now taken care of, I have shown that there is necessarily such a beast as positive gravitational energy density in a static field. Conversion at least partially from non-gravitational to gravitational energy accompanies any collapse scenario. We now know from the forgoing this logically requires a net reduction in net system gravitating mass. Bingo - monopole GW's etc. Your argument is to just stick with finding the SET in the event past light cone, but what's missing here is crucial. The *recipe* for what constitutes part of the SET. if gravitational energy is missing from that recipe (as GR insists), my last umpteen entries here have been trying to drive home the inconsistencies that then invariably result. Take it or leave it.

 

[Naty1]

Peter: Great explanations, bravo for your patience...I, and perhaps others who have read and not posted, learned a lot from your explanations and while perhaps frustrating to you at times, subsequent explanations with slightly different perspectives added further clarity.

Here are a few summary points I really liked:

#43:

...the planet's mass density and pressure are the only things that contribute to the SET...the SET is only nonzero in the region of spacetime occupied by the planet. In the region exterior to the planet, including the point where the "field" is being measured, the SET is zero--the exterior region is a vacuum.

#46

..In other words, the "energy in the gravitational field" is *not* "stress-energy...

EM waves have zero charge, and gravitational waves have zero stress-energy...

the gravitational waves... *are* curvature, propagated from one region of spacetime to another... without any "source" present.

Yuiop:

I have some sympathy for scepticism expressed by Q-reeus about "fossil gravitational fields". ... When it finally collapses to a black hole its "frozen" gravitational field continues to orbit. ...In other words it is just the field that is orbiting.

Me too...A key point for me in reconciling this 'approximate' view with the Einstein formalism was this explanation from Peter:
#51

...remember we are assuming perfect spherical symmetry...the "field" at a particular radial coordinate r in the exterior vacuum region is the same for all times t to the future of the nonzero SET region; so it doesn't matter whether that region is eight light-minutes or a billion light-years into the past light-cone, you get the same field...

That really clarified for me your earlier explanation of the 'past light cone' perspective.

Further, and it's a minor point, one doesn't even need to 'assume' perfect spherical symmetry for a real black hole somebody proved [Was it Hawking?] spherical symmetry for a Schwarzschild type black hole...any initial irregularities would be smoothed out...


Peter:

...The "M" that appears in the metric, for example the Schwarzschild metric, is "not" the same as the "mass" (actually "energy density", or "0-0 component") that appears in the stress-energy tensor as a "source" of curvature... no actual stress-energy escapes during the BH merger (it's all trapped behind the horizons of the BH's). But "nonzero SET" and "gravitating mass" in the sense of the value M appearing in the metric are, as I said above, not the same...

This business about different M's between the SET and metric is something I need to think about further...it hasn't clicked yet:

edit:
"This is what I am puzzling over...what's the physical background/explanation:
"The *relationship* between non-zero SET in the past light cone and M, the quantity appearing in the metric, is simplest for the static case; but that still doesn't mean the two are identical..."


from #51:

...The "M" that appears in the metric has a definite physical meaning: it's the mass you would measure if you put a test object in orbit around the black hole, measured its orbital parameters, and applied Kepler's Third Law. The same applies for any gravitating object--the Sun, for example. ...But measuring "M" this way tells us nothing about how it relates to the presence of a non-zero SET in the spacetime...

[ok on that piece]

and this example helps...

...But after the merger [of two black holes] there is a region of spacetime where there are violent curvature fluctuations because of the violation of spherical symmetry; and those curvature fluctuations carry off energy in the form of gravitational waves...the relationship between them depends on the configuration of the spacetime in between the nonzero SET region and the event at which the metric, and thus M, is being measured...

I just know I don't have an intuitive grasp yet:
[Is this too naive?: Look dopey (me)!, they are measures at different spacetime points so of course there will be different values.]

In your explanation of SET, a planet's mass density and pressure contribute to the SET...[ok, I get that] In your definition of M [in the metric] they would also be included in that entity, right...How might these differ between the two...what conditions??

#49:

...a real BH is formed from the collapse of a massive object with a nonzero SET, but once the singularity is formed the SET is zero everywhere...

So here is an example of a zero SET and a finite metric M, outside the horizon, right?

I also wondered about trickydicky question. (#47)"
"What about dark energy."

 

[PeterDonis]

Peter, could you take a look at my #47?

Just did; good questions, sorry I hadn't responded before.

Energy is energy, right? Do you mean there are two types of energy, the regular one accounted by the SET and the gravitational one that follows different rules?

The word "energy" can have different meanings. That's why I've tried not to use it in my explanations (and when I have, I've later clarified and qualified what I've said to make clear what specific thing I was using the word "energy" to refer to). Thinking that "energy is energy" is another example of an approximation that works well in the range of our ordinary experience, but breaks down when we try to extend it too far.

Once again, the only way to make sure we're being precise and are properly capturing the physics is to go back to the fundamentals: the Einstein Field Equation and its solutions. The RHS of that equation, the SET, has a precise physical meaning, and the "rules" it follows are also precise (as captured in that equation). That equation is sufficient, as I've said a number of times, to explain and calculate *all* classical gravitational phenomena, including those that are sometimes referred to as "gravitational energy". The latter is *not* a fundamental concept; it is just an approximate way of looking at the physics in a limited domain that works reasonably well in that domain.

Let's consider "Dark energy" for a moment, it is thought to have a gravitational origin (as cosmological constant) and yet everyone agrees it is the source of a SET (with some differences with the usual matter-energy SET). Why one gravitational field energy is "stress-energy" in one case but not in the other?

The precise definition of "dark energy" is a nonzero SET that is proportional to the metric. A nonzero cosmological constant is one possible form of dark energy, but not the only one. (The fact that the SET of dark energy is proportional to the metric is the key difference from an "ordinary" SET derived from matter or EM radiation.) Since dark energy, precisely defined, has a nonzero SET, it appears on the RHS of the EFE, and that is how it affects the physics. Whether or not this kind of energy "counts" as "gravitational field energy" depends on how you define the latter term; but as I've noted already, you don't have to define that term to figure out the physics, so answering the question of whether dark energy counts as gravitational energy is not necessary for the physics.

EM waves have no charge but still carry energy and have nonzero stress-energy so the example is not valid wrt energy.

Just to clarify, I was only using the EM case as a analogy, and I wasn't trying to say that EM fields have zero stress-energy; you are correct that even "source-free" EM fields have a nonzero SET. The analogy I was making was simply that EM radiation can propagate through regions of spacetime that have a zero charge-current 4-vector, which is the "source" in the EM field equations. But the analogy is limited, and I don't insist on it if it doesn't help with understanding the gravity case.

 

[PeterDonis]

Sorry Peter, but I agree with yuiop here. How can a 'BH' not be modeled as an object that interacts with things when you have previously described it just that way - has a characteristic mass M according to Keplerian dynamics of an orbiting test mass. Let's not play with words here.

A matter of definition surely - the two are synonymous by any reasonable score imo. A rose by any other name is still a rose.

I am not playing with words; I am pointing out what the words do and do not refer to, and what can and cannot be derived from them according to the actual, precise physics.

Take your statement about the mass M: it is measured by "Keplerian dynamics of an orbiting test mass". Very true. My point is that that does *not* imply that anything inside the BH horizon is interacting with anything outside. It simply doesn't. That's all. We are used to thinking of gravitating bodies as "interacting" with other bodies (like the Sun and the Earth), without bothering to always remind ourselves that the "interaction" does not occur instantaneously--the Earth is not interacting with the Sun "right now", it is interacting (if that's even the right word) with the Sun eight minutes ago. But the latter is in fact the case. A BH is just a much more extreme case, where we might have to go back billions of years to find the nonzero SET region in the past light cone--but that's still the correct precise description of the physics. Thinking of the BH itself as "interacting" with orbiting bodies is *not*; it's an approximation with a limited domain of validity. You are trying to stretch it beyond its domain of validity, and it is breaking down.

One can argue it's not a physical surface, but point is, logically to deform an infinitely curved region requires infinite coordinate time!

First of all, the event horizon is not "infinitely curved", and the infinite Schwarzschild coordinate time is irrelevant; it's an artifact of the coordinate singularity at r = 2M in Schwarzschild coordinates. Do we need to have a separate discussion on that point, or can I just refer to all the hundreds of previous threads where that subject has been beaten to death?

Next, the horizons do not get "deformed" or "merged"; rather, we have a single spacetime whose horizon (there is only one horizon) happens to be shaped like a pair of trousers, so to speak, rather than a simple cylinder. And this shape of the horizon, once again, is entirely explained by the original configuration of nonzero SET regions that collapsed to form the two BH's that then "merged".

Once again, I'm not saying it's "wrong" to think of BH's as "objects" instead of curvature; I'm just saying that thinking of them as "objects" is an approximation with a limited domain of validity. You are trying to stretch that approximation beyond its domain of validity, and it is breaking down. If you go back to the fundamentals, the actual precise physics based on the EFE, there is no problem.

(just to be clear here; my own interpretation is that it shows the inconsistency of 'BH' in the first place. I am not speaking on anyone else's behalf in saying that.)

I agreed only that this was the official GR position - never mine as made abundantly clear in many posts.

In other words, you basically do not accept that standard GR, based on solutions to the EFE, is valid. If you don't accept that, then we really don't have a good basis for discussion at all, because everything I've said is based on the premise that the EFE is valid. If you don't accept the EFE, then of course you're not going to accept the rest of what I'm saying. But I very much doubt you'll be able to convince me that the EFE is not valid in the domain we have been discussing (though you're welcome to try).

In #45 pressure, which apart from rest mass/energy, forms the only other GR approved contribution to the SET in the scenario considered, was declared negligible...I had implicitly lumped pressure together with matter rest mass.

For the purposes of the discussion we were having, I don't see a problem with this. If we wanted to be precise, we could say that where we were talking about "rest mass", we should instead read "all significant components of the SET".

My argument there and in #52 was not strictly correct (neither of us picked it up) - conversion, partial or fully, of non-gravitational energy to gravitational energy is not sufficient proof of itself that net system gravitating mass declines. One must account for pressure changes also.

Strictly speaking, yes, this is true. But I don't think it affects the general points either of us were making.

1: Scaling law. Take the thin shell stipulated in #45. Double it's assembled mass M'. Pressure has doubled, and in GR the pressure contribution to curvature is a linear function of that pressure.

For this idealized case, yes, the pressure "contribution to curvature" (meaning through the EFE) is linear in the pressure. But the pressure itself is not necessarily linear in the assembled mass (i.e,. doubling the assembled mass does not necessarily double the pressure). You have to actually look at the appropriate solution of the EFE to see how the pressure depends on the assembled mass.

2: There is good reason to doubt pressure makes *any* contribution to gravitational mass.

If by "gravitational mass" you mean the "assembled mass" M of a spherically symmetric gravitating body, you are simply wrong here. Solutions describing, for example, static spherically symmetric stars have been well known for decades, and pressure most certainly does contribute to the "assembled mass" of the star.

GW amplitude is linear wrt pressure.

Why do you think this?

But material displacement of the twin clamps under pressure is inversely proportional to the material elastic constant.

Within a certain range of pressures and displacements (until the material's elastic limit is reached), yes.

Plastic clamps will flex far more for a given generated pressure than for say steel clamps.

But they also have less energy density. See below.

any metric back reaction from generating GW's must induce far greater power drain in the plastic clamps case than for the steel ones...We have not included GW contribution owing to just motion of the clamp material, but that's ok since that contribution will be proportional to material density, which need have no relation to elastic constant.

Really? I agree there is not a straight linear relationship, but there is still some relationship.

Only by denying the very existence of gravitational energy can the problem seemingly go away. But then e.g. Hulse-Tayler-binary-pulsar-data-as-proof-of-GW's issue, as before discussed, becomes somewhat problematic.

Why do you think this? The binary pulsar data is perfectly consistent with standard GR and the EFE, including the fact that the system is emitting GW's and that, consequently, the energy remaining in the system (which would correspond to its externally measured mass, if for example we put a test object in a far orbit about the system and measured its orbital parameters) is decreasing. All of this is perfectly well explained by the configuration of nonzero SET regions in the past light cone.

Which misses the point; this is a valid gedanken experiment. It is possible for such a situation to exist and it implies certain things, which I have stated.

You are basically saying, if the evidence were different than it is, we would draw different conclusions. So what?

I'll repeat. With pressure now taken care of, I have shown that there is necessarily such a beast as positive gravitational energy density in a static field.

You have shown no such thing. You have only shown that you can assign a reasonable meaning to the term "gravitational energy density in a static field" such that that density is positive. Again, so what? This says nothing about the fundamental physics; it only says that you can make a certain approximation work in a certain limited domain. I have never disputed that the approximation works within its limited domain.

Conversion at least partially from non-gravitational to gravitational energy accompanies any collapse scenario.

Again, this is an approximation that works in a limited domain. It is not the fundamental physics.

We now know from the forgoing this logically requires a net reduction in net system gravitating mass.

If GW's are emitted, yes.

Your argument is to just stick with finding the SET in the event past light cone, but what's missing here is crucial. The *recipe* for what constitutes part of the SET. f gravitational energy is missing from that recipe (as GR insists), my last umpteen entries here have been trying to drive home the inconsistencies that then invariably result. Take it or leave it.

You have shown no such inconsistencies. Nothing you have said has rebutted my repeated claim that *all* of the observed physics can be explained and calculated using the standard GR recipe--solve the EFE using the nonzero SET regions (with the standard definition of SET) as the sources on the RHS. Why? Because nothing you have said is actually *derived* from trying to apply the standard recipe. Instead, you keep on applying your own recipe, based on your own approximate version of the physics, and finding that it doesn't work. You're right: it doesn't work.

In other words, all you have illustrated is that other, approximate ways of capturing the physics break down when you try to extend them beyond a limited domain. So what?

 

[TrickyDicky]

Thanks for answering.

The precise definition of "dark energy" is a nonzero SET that is proportional to the metric. A nonzero cosmological constant is one possible form of dark energy, but not the only one. (The fact that the SET of dark energy is proportional to the metric is the key difference from an "ordinary" SET derived from matter or EM radiation.) Since dark energy, precisely defined, has a nonzero SET, it appears on the RHS of the EFE, and that is how it affects the physics. Whether or not this kind of energy "counts" as "gravitational field energy" depends on how you define the latter term; but as I've noted already, you don't have to define that term to figure out the physics, so answering the question of whether dark energy counts as gravitational energy is not necessary for the physics.

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.
Actually your answer to my question amounts to say it is not a relevant question for your idea of the relevant physics.
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.

 

[Q-reeus]

Take your statement about the mass M: it is measured by "Keplerian dynamics of an orbiting test mass". Very true. My point is that that does *not* imply that anything inside the BH horizon is interacting with anything outside. It simply doesn't.

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. 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.

That's all. We are used to thinking of gravitating bodies as "interacting" with other bodies (like the Sun and the Earth), without bothering to always remind ourselves that the "interaction" does not occur instantaneously--the Earth is not interacting with the Sun "right now", it is interacting (if that's even the right word) with the Sun eight minutes ago. But the latter is in fact the case. A BH is just a much more extreme case, where we might have to go back billions of years to find the nonzero SET region in the past light cone--but that's still the correct precise description of the physics. Thinking of the BH itself as "interacting" with orbiting bodies is *not*; it's an approximation with a limited domain of validity. You are trying to stretch it beyond its domain of validity, and it is breaking down.

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

...because it leads us to ask a question ("why are active m_g, passive m_g, and inertial mass m_a all the same?") that, from a proper conceptual scheme, would never even be asked, because it would be "obvious" that there was only one kind of mass-energy to begin with.

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? 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.

First of all, the event horizon is not "infinitely curved", and the infinite Schwarzschild coordinate time is irrelevant; it's an artifact of the coordinate singularity at r = 2M in Schwarzschild coordinates. Do we need to have a separate discussion on that point, or can I just refer to all the hundreds of previous threads where that subject has been beaten to death?

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.

For this idealized case, yes, the pressure "contribution to curvature" (meaning through the EFE) is linear in the pressure. But the pressure itself is not necessarily linear in the assembled mass (i.e,. doubling the assembled mass does not necessarily double the pressure). You have to actually look at the appropriate solution of the EFE to see how the pressure depends on the assembled mass.

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

If by "gravitational mass" you mean the "assembled mass" M of a spherically symmetric gravitating body, you are simply wrong here. Solutions describing, for example, static spherically symmetric stars have been well known for decades, and pressure most certainly does contribute to the "assembled mass" of the star.

You commented before reading the rest!

Q-reeus: "GW amplitude is linear wrt pressure."
Why do you think this?

How could it be otherwise? What do the EFE's say on this any differently (weak gravity regime, remember!)? What do you propose instead?

Q-reeus: "Plastic clamps will flex far more for a given generated pressure than for say steel clamps."
But they also have less energy density...

My turn to say; so what? We are not talking about elastic energy here. One linear in displacement, one parametric. Chalk and cheese.

Q-reeus: "any metric back reaction from generating GW's must induce far greater power drain in the plastic clamps case than for the steel ones...We have not included GW contribution owing to just motion of the clamp material, but that's ok since that contribution will be proportional to material density, which need have no relation to elastic constant."
Really? I agree there is not a straight linear relationship, but there is still some relationship.

Irrelevant. What matters is they are independent parameters, and that is sufficient as part of the proof. Looks though like it deserves a separate thread - it is spelling death to a key concept in GR and you are saying 'ho hum'!

Why do you think this? The binary pulsar data is perfectly consistent with standard GR and the EFE, including the fact that the system is emitting GW's and that, consequently, the energy remaining in the system (which would correspond to its externally measured mass, if for example we put a test object in a far orbit about the system and measured its orbital parameters) is decreasing. All of this is perfectly well explained by the configuration of nonzero SET regions in the past light cone.

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?

Q-reeus: "Which misses the point; this is a valid gedanken experiment. It is possible for such a situation to exist and it implies certain things, which I have stated."
You are basically saying, if the evidence were different than it is, we would draw different conclusions. So what?

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.

...Nothing you have said has rebutted my repeated claim that *all* of the observed physics can be explained and calculated using the standard GR recipe--solve the EFE using the nonzero SET regions (with the standard definition of SET) as the sources on the RHS. Why? Because nothing you have said is actually *derived* from trying to apply the standard recipe. Instead, you keep on applying your own recipe, based on your own approximate version of the physics, and finding that it doesn't work...

This is a repeated theme I deny is accurate. You are on record as stating the SET specifically includes only matter contributions, and that means gravitational energy/"energy" is excluded. I have dealt with pressure and shown it cannot nullify the need of a gravitational field energy ("energy" if you wish). 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. Point by point, not dismissive generalities. That should be real interesting.

 

[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.