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Relative energy of a black hole. 
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#91
Mar112, 02:11 PM

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PF Gold
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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. 


#92
Mar112, 02:24 PM

Mentor
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#93
Mar112, 02:30 PM

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PF Gold
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"In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energymomentum 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 gravityit includes "gravitational energy", *and* energy is "exchanged" between ordinary matterenergy 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 matterenergy. 


#94
Mar112, 02:31 PM

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#95
Mar112, 04:08 PM

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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: 


#96
Mar112, 04:51 PM

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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]..... 


#97
Mar112, 05:56 PM

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#98
Mar212, 04:36 AM

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#99
Mar212, 04:38 AM

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#100
Mar212, 04:41 AM

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#101
Mar212, 04:44 AM

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#102
Mar212, 08:46 AM

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Qreeus posts:
If you READ from posts 88 on....Tricky, my posted quotes, George Jones comments and quotes and Peter's comments explain it to the extent it can be....'non localizable', covarient derivative effects, non localizable,etc,etc ...... these are all complementary, not in conflict. including these: QReeus...While I see why pervect opted out early, I am on the other hand happy to see your persistence: " It is better to debate a question without settling it than to settle a question without debating it." .......Joseph Joubert, the 18th century philosopher I, for one, am 'outta' here....finally!! 


#103
Mar212, 09:05 AM

P: 249

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



#104
Mar212, 09:49 AM

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#105
Mar212, 10:11 AM

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But asking for what "exactly" the concept of "gravitational energy" means is asking too much: the concept is only a heuristic one and it does not have an "exact" meaning. (Or perhaps a better way to say this would be: one could give an exact definition of "gravitational energy", such as the LandauLifgarbagez pseudotensor, but no such definition is unique, and any such definition only "makes sense", only corresponds to our intuition, in a restricted set of cases.) If you want an exact answer, it is this: there is no "gravitational energy" in the SET, so as far as exact calculations of physical predictions are concerned, it doesn't exist. (You'll note, in this connection, that nobody uses any definition of "gravitational energy" to actually make physical predictions: they all use the standard EFE/SET method, and then once they know what the answer is, they overlay their chosen concept of "gravitational energy" on top of it to help them understand intuitively what's going on.) Does this "total system" have a "total energy"? It depends on how you define "energy". The spacetime as a whole does not have a time translation symmetry, so we can't define "energy" that way. The spacetime *may* have a continuous set of spacelike slices that match up well enough with what symmetry does exist (for example, maybe the slices are good approximations to "natural" ones that observers hovering at a large radius R above the binary pulsar system would pick out as "surfaces of constant time") to be useful in defining "energy" by integrating the energy conservation equation (i.e., the covariant divergence of the SET) over each spacelike slice. This could define a "total energy" for the system, and this total energy could turn out to be conserved (i.e., the same on every slice), at least to a good enough approximation (the same level of approximation to which the slices are good "surfaces of constant time" for some set of observers). But will this "conservation of energy" be "exact"? Probably not, since the spacetime does not have any exact symmetry. So if you want an exact answer, it is that there is no "total energy". Now, suppose I decide to draw a boundary at some finite radius R around the binary pulsar system, and say that inside that boundary is "the total system" and outside it is "the rest of the universe". I can pick R large enough that, to a good approximation, the binary pulsar system "looks like" a simple gravitating body with some mass M. More precisely: the metric at R is still not quite in the Schwarzschild form, because the spacetime is not spherically symmetric or static; but it will be close enough that I can "split" it, approximately, into two pieces: a "Schwarzschild" piece and a "gravitational radiation" piece. The Schwarzschild piece, to a good approximation, will look like a gravitating body with a mass M that slowly decreases with time ("time" meaning "proper time according to an observer hovering at radius R). The gravitational radiation piece will be oscillating in quadrupole fashion, and could be measured by, for example, letting the oscillations heat up a detector and measuring the energy taken up. We could then, in principle, do an energy balance: the decrease in M is balanced by the energy carried away by GWs. Will this energy balance be "exact"? Probably not, because the split of the metric into the two pieces probably won't be exact; there will probably be extra terms in the metric that are left outthey aren't included in either the Schwarzschild or the GW piecebecause they are small compared to both of those pieces. So we come back again to what I said above: if you insist on an "exact" answer, then it is this: "gravitational energy" doesn't exist, and the only exact "energy conservation" is what I said earlier: the covariant divergence of the SET (the standard SET) is zero at every event. Anything else is approximate, and breaks down if you try to press it too hard. That includes things I've said previously (like "M cannot decline if all matterenergy is included"); I apologize if I didn't make it clear enough that I was only speaking approximately. 


#106
Mar212, 10:42 AM

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#107
Mar212, 10:58 AM

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I realize that there is an ongoing debate in astronomy as to whether the standard interpretation of observations (like galaxy rotation curves) as signifying the presence of "dark matter" is correct. There are alternate theories that modify the way gravity works (i.e., they are *not* standard GR) in order to account for the observations without postulating dark matter. I am not saying those alternate theories have been proven wrong; they haven't (I consider them all much more unlikely than the standard interpretation, but that's just my opinion). I'm just saying that the observations, by themselves, are not counterexamples to standard GR: standard GR can account for them perfectly well, by just adding the dark matter to the total SET that is being used in the EFE. I realize also that the above is open to another objection: well, sure, you can make any set of observations compatible with standard GR by [Edit: fixed typo, was "but"] just fiddling with the SET. First of all, that's not quite true; mathematically, it can be done, yesyou can postulate any tensor you like as an "SET", put it on the RHS of the EFE, and solve for the metric it will producebut the results may not be very reasonable physically (for example, they may violate energy conditions or other constraints that are widely accepted). Dark matter doesn't do that: the dark matter SET, as I said, is just like that of ordinary matter, so it's perfectly reasonable physically. Second, dark matter fits into the picture in multiple places, not just one; for example, the current "best fit" big bang model requires cold dark matter, in roughly the same proportions ("roughly" because all of these calculations have significant "error bars" at our current level of knowledge) as are required to explain the galaxy rotation curves and other "local" observations. So dark matter is not just being put in ad hoc to fit one piece of data; it has a reasonable place in a comprehensive model, and that comprehensive model uses the standard EFE/SET of GR. (That's one reason, btw, why I think the alternate theories that modify gravity are unlikely to be right; they all monkey with the overall dynamics of the universe in a way that messes up the correspondence with other cosmological observations, so they then have to make other ad hoc assumptions to fix things up. I admit I am not very up to date in this area, so there may be recent developments that I'm not aware of; but that's my understanding of where things stand.) 


#108
Mar212, 01:17 PM

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Peter, appreciate that in #105 you have tackled in your own inimitable style the specifics I raised earlier. There is a sense of deja vu to it all. Collectively we have created a a lengthy record of exchange for any looking on to make their minds up from. Guess you can figure what I'm saying. I will end my participation with a slightly edited cutnpaste from #57 which turned out to be just intermission. End of the show here for me, and I trust no hard feelings between us: From #57:



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