Relative energy of a black hole.

Q-reeus

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.
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.
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

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.

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.

Agreed.

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

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.

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

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.
Another red herring, apart from the one dealt with above?
Purist response: a massless box cannot withstand the radiant pressure it has to contain. But enough of nitpickery.
So far so good. 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.
And still good. 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.
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.
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

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

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.

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

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:

Q-reeus

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

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

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/...6-3/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.

PeterDonis

Then I'm still confused. But see below.

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.

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.)

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.)

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

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.

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.

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

Sorry Peter, but I agree with yuiop here. How can a 'BH' not be modelled 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.
Again, I agree with yuiop. YouTube provides some nice CGI examples of 'BH merger events':http://www.youtube.com/watch?v=4m-ZV...eature=related http://www.youtube.com/watch?v=L478Z...eature=related (and these from respected numerical GR groups)
Looks impressive, like lava lamp blobs fusing together, with electrostatics thrown in to 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

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.
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.
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.
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:
#46


Yuiop: Me too...A key point for me in reconciling this 'approximate' view with the Einstein formalism was this explanation from Peter:
#51
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: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:
[ok on that piece]

and this example helps...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:
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

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

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.

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.

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

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.

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.

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).

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".

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

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.

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.

Why do you think this?

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

But they also have less energy density. See below.

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

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.

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

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.

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

If GW's are emitted, yes.

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?