Exploring Ohanian's Approach to GR

[bhobba]

[Moderator's note: spin off from previous thread.]

Confession: I (secretly) think that this "issue" in classical mechanics may provide a deeper insight into what gravity is - in a way that makes the marriage with standard model physics more natural. This idea also implies gravity is an emergent phenomena at "low" energy, that is explained by a balance of universal negotiations which are attractive and inertia which resist this. But after all, GR describes how matter and energy defines curvature, but it does not explain the mechanism in terms of something else.

Have you read Ohanian? He uses an entirely different approach much more in line with the Standard model type of equations and Lagrangian's, and shows how - first you get from an analogy with EM the linearised gravitational equations then how that becomes full GR. It was the first serious book I read on GR but even after reading other books like my favorite Wald, his approach always struck me with its novelty and the light it sheds on questions you mention above.

Thanks
Bill

 

[Fra]

Have you read Ohanian? He uses an entirely different approach much more in line with the Standard model type of equations and Lagrangian's, and shows how - first you get from an analogy with EM the linearised gravitational equations then how that becomes full GR. It was the first serious book I read on GR but even after reading other books like my favorite Wald, his approach always struck me with its novelty and the light it sheds on questions you mention above.

No i haven't read Ohanian, but looking on the table of contents from Amazon the structure looks exactly like a typical book on GR.
1. Introduction to the mathematics of curved spacetime (diff geometry) - this is basically mathematics, no questions here.
2. The part where you argue for the generalization of Newtons gravity to Einsteins field equations.
The part which usually contains a motivation would be chapter 7.1-7.3 in Ohanian, but in what sense does he differ?
3. Then the structure, implications and applications of Einsteins equations are worked out.

#2 is the foundationally critical part. The rest are mathematics and applications (to simply things)
I know well the standard arguments, which i simply to argue from a set of assumptions such as
- Gravity is not a conventional "force", but rather "explained" as a consequence of geodesics in a curved spacetime.
- it must redice to Newtons gravity in weak field and static limit
- laws of physics must be observer invariant (diffeomorphism)

Then there is some farily simply arguments showing that einstens equations is one solution to this, while not unique.

These arguemnts are roughly fine in classical mechanics, but after having thought about unification with quantum mechanics, and choosing the inference perspective seriously the assumptions going into especially the 3rd assumption is a gross simpliciation in my view.

Incidently the inference perspective in mine, is not dissimilar to the geometric perspective, the difference is how you understand the nature of the manifold and its geometry. Geodesiscs are replaced by random walks, so forces are similary "explained" simply from the structure of the space where random walk talks place. The mistake IMHO is to think of the manifolds themselves as observer invariant. The hole topology and all needs to follow IMO from a physical inference in a way that will unify the foundations of GR with those of QM (if you like me, think of them as general inference).

But as this now as i understand was not the focus of Neumaier i will not write mor on this in this thread. I just felt there are commong connection poitns as the foundations of statistical methods and its relevance to physics is indeed, but Neumaier has more the strict mathematical physicists perspective.

/Fredrik

 

[bhobba]

No i haven't read Ohanian, but looking on the table of contents from Amazon the structure looks exactly like a typical book on GR.

Its not.

It analyses EM then via that analysis develops linearised gravity the same way. Here it is in simple terms - in the Lorentz Guage EM is ∂u∂u Av = 4πJv (of course not the correct upper an lower indices - but I am sure you know it).

Then you say based on that what should gravity look like - its pretty obvious ∂u∂u Φuv = kTuv for some constant k and of course Tuv is the stress energy tensor - this is using a gauge where the Hilbert condition holds. Now of course this must be wrong because gravity gravitates ie the gravitational field itself is part of the stress-energy tensor. It is only valid where that can be neglected - of course this does not apply for EM Fields. Of course I have stripped it down to the essentials - he doesn't use those nice equations in those gauges - but argues for the general equations then shows in the appropriate gauges these equations result. Personally its so obvious when done that way I would do the reverse - show the normal linearised equations result from ∂u∂u Φuv = kTuv - I haven't worked out the details personally but it looks obvioius it would not be hard.

You can then show, and this is what chapter 7 does - the full EFE's automatically result. This is pretty amazing actually where the linearised equations implies the full non linear equations.

Thanks
Bill

 

[PeterDonis]

gravity gravitates ie the gravitational field itself is part of the stress-energy tensor

This is not correct, at least not as you state it. There is no "gravitational field" part of the stress-energy tensor. There is a valid sense of "gravity gravitates", but it isn't that the SET includes the gravitational field; it's basically that the EFE is nonlinear, so spacetime curvature can produce more spacetime curvature.

For more details, see this 3-part Insights series (full disclosure, I'm the author):

https://www.physicsforums.com/insights/does-gravity-gravitate/

I have not read Ohanian's textbook, but I suspect that if he talks about the gravitational field being part of a "tensor", he really means one of several possible pseudo-tensors that can be defined that take the standard SET (the one that appears on the RHS of the EFE) and add a term (or terms) that are supposed to model "energy density stored in the gravitational field". But there is no frame-independent way to do this, and doing it, which basically amounts to taking a piece of the Einstein tensor on the LHS of the EFE and moving it to the RHS, breaks a key property of the EFE, that the covariant divergence of both sides is zero. I talk about this in the Insights series.

 

[bhobba]

This is not correct, at least not as you state it.

OK - I will state is precisely as written in the textbook - page 147 (he calls it gravity gravitates - I call it that a well - but it may not mean the same thing to you as to me and Ohanian):

From the textbook:
'In writing the field equation in flat space-time (he gives the linearised gravity equation) we assumed Tuv is the energy momentum tensor of matter. In order to obtain a linear field equation we left out the effect of the gravitational field upon itself. Because of this omission our linear field equation has two related defects. Because of (again the linearised field equation) matter acts on the gravitational fields (changes the fields), but their is no reciprocal action of the gravitational fields on matter, that is the gravitational field can acquire energy-momentum, but nevertheless the energy-momentum of matter is conserved ie ∂uTuv = 0. This is an inconsistency. Secondly, gravitational energy does not act as a source of gravitation in contradiction to the principle of equivalence (although I would express it differently - see later). Thus although (again the linearised equations) may be a fair approximation for weak gravitational fields, it cannot be the exact equation.'

End quote - note when I have said the linearised equation the textbook quotes the actual equation.

My objection to the principle of equivalence here is it's not necessary - the gravitational filed must have its own stress-enrgery tensor - thus it must be part of the stress energy tensor on the RHS.

The equation I gave was ∂u∂u Φuv = kTuv with the Hilbert condition ie ∂u Φuv = 0. To see its more normal form define huv = Φuv - 1/2ηuv*Φ where the indices are raised.

From this, and taking the trace ie contracting h = Φ - 1/2*4*Φ we have h = -Φ. Substituting back you have an equation for Φuv = huv - 1/2ηuv*huv (nice symmetry hey - maybe telling us something important - but what - beats me). In this treatment huv is the gravitational field. Later in the book it is related to the metric by Guv = ηuv + k*huv where k is a constant to be determined in order to get the correct equations of motion when he later considers what Guv means.

So to get the equation of motion you not only have to take into account the stress-energy tensor from matter but the stress-energy tensor form the free field equations themselves:. This is done on page 148. Using both these quantities on the RHS of the gravitational field equation, on page 149-150 where the result is given, the equations of motion are derived. He then writes it in Lagrangian form and low and behold its the same as if space-time was curved and and the motion was geodesic. This is the first argument that space-time is either curved or the gravitational field makes it act like its curved.

He has a second argument. You can write the linear equation out in a general form not simplified by the Hilbert condition. Then you can manifestly demonstrate its gauge invariance. It turn out, from Chapter 7 page 381, the gauge invariance is nothing but invariance wrt infinitesimal changes in coordinates. Heuristically any change in coordinates can be built up from infinitesimal ones - so we end up with the correct equation must be invariant to coordinate transformations,. This equation must reduce of course to the linear equation for weak fields. It turns out the Einstein tensor is the only equation that meets these requirements - and we are done. However while this is very elegant, one can mount a direct attack via a method of successive approximations similar to what was done in deriving the equation of motion via adding the stress energy of the field itself to the stress-energy of the mass. In other words for that approximation you find the new stress energy of the field - and so on - this (from page 387 it was evidently done done by Gupta - Reviews of Modern Physics 29 page 334 (1957).

That's about it. As you can see different from the normal geometrical approach taken. I have written elsewhere the quickest and simplest derivation I know is from Soper - Classical Field Theory. But the above starts with flat space time and curved space-time naturally emerges - and you get the why of the principle of covariance which otherwise is simply aesthetic (I stayed away from the principle of invarience because people here seem to prefer covarience because it's used in MTW)

I have to head off to dinner now - will read your article when I get back - but I hope the above clarifies what I mean by gravity gravitates.

Thanks
Bill

 

[PeterDonis]

I will state is precisely as written in the textbook - page 147

Which part of your post is directly quoted?

 

[bhobba]

Which part of your post is directly quoted?

Will fix it so its clear.

Thanks
Bill

 

[PeterDonis]

the gravitational filed must have its won stress-enrgery tensor

Here's the problem: this isn't possible. There is no tensor that can describe "energy contained in the gravitational field".

The reason is simple: heuristically, the equivalence principle means that you can always make the "gravitational field" vanish at a particular event by an appropriate choice of coordinates. That would mean any tensor describing "energy contained in the gravitational field" would have to also vanish at that event. But a tensor that vanishes in one coordinate chart at a given event must vanish in all coordinate charts at that event. And since this argument applies at any event you choose, it follows that any tensor that purported to describe "energy contained in the gravitational field" would have to vanish everywhere.

This is a well-known result so I'm sure Ohanian is aware of it.

 

[bhobba]

Here's the problem: this isn't possible. There is no tensor that can describe "energy contained in the gravitational field". The reason is simple: heuristically, the equivalence principle means that you can always make the "gravitational field" vanish at a particular event by an appropriate choice of coordinates.

I really have to be off - but I will give a reply when I get back. Just want to mention so far this is flat space-time.

Thanks
Bill

 

[PeterDonis]

Will fix it so its clear.

Ok, that helps. Here is how I would rephrase what is said in the quoted paragraph: the linearized field theory of gravity in flat spacetime is not consistent, because it allows the gravitational field to act on matter but does not allow matter to act back on the gravitational field. This is true, and is also a well-known result (MTW discusses it, for example). In other words, this demonstrates that any valid theory of the gravitational field must have a field equation for the field that is nonlinear.

However, this has nothing to do with whether or not there is a valid stress-energy tensor for the gravitational field. How do we know that? Because in GR, the field equation is indeed nonlinear (the EFE), but there is no valid SET for the gravitational field--the RHS of the EFE does not include any contribution from gravity, only from matter (i.e., non-gravitational fields). So it is obviously possible to have a nonlinear field equation for gravity without having a SET for the gravitational field.

Just want to mention so far this is flat space-time.

Doesn't matter. The end result of the argument is going to be the EFE, and the RHS of the EFE has no contribution from energy stored in the gravitational field. That is a well-known result, and I'm sure Ohanian is aware of it; so whatever Ohanian is claiming, I don't think he's claiming that the RHS of the EFE contains a contribution from the gravitational field. (If he is, he's contradicting every other GR textbook I've seen.)

 

[PeterDonis]

@bhobba, btw, I was able to get the Kindle version of the 3rd Edition of Ohanian--more precisely, Ohanian & Ruffini, Gravitation and Spacetime. Is that the version you're using?

 

[bhobba]

@bhobba, btw, I was able to get the Kindle version of the 3rd Edition of Ohanian--more precisely, Ohanian & Ruffini, Gravitation and Spacetime. Is that the version you're using?

The second edition - been meaning to get the latest one for a while - mine is falling to pieces I have used it so much over the years.

Had a think about the whole thong on the way to dinner and while there.

I believe you have hit on a big issue in his whole approach, logic and consistency - the specific objection you gave is 'sort of' correct, but intimately tied up with the issue I think I have uncovered.

I will explain it in a more detailed post I will pen while watching Kyrios in the Tennis, so please bear with me.

But - yes I now believe his whole approach is, how did Nash but it to his future wife with the answer to the question he set the class - Ingenious, elegant but ultimately wrong. I will detail it soon.

Thanks
Bill

 

[bhobba]

I don't think he's claiming that the RHS of the EFE contains a contribution from the gravitational field. (If he is, he's contradicting every other GR textbook I've seen.)

I am pretty sure he is - but we will see what you think now you have the book.

I am going to write a post detailing what I think the inconsistency in his whole approach is.

Thanks
Bill

 

[Fra]

Its not.

It analyses EM then via that analysis develops linearised gravity the same way. Here it is in simple terms - in the Lorentz Guage EM is ∂u∂u Av = 4πJv (of course not the correct upper an lower indices - but I am sure you know it).

Then you say based on that what should gravity look like - its pretty obvious ∂u∂u Φuv = kTuv for some constant k and of course Tuv is the stress energy tensor - this is using a gauge where the Hilbert condition holds. Now of course this must be wrong because gravity gravitates ie the gravitational field itself is part of the stress-energy tensor. It is only valid where that can be neglected - of course this does not apply for EM Fields. Of course I have stripped it down to the essentials - he doesn't use those nice equations in those gauges - but argues for the general equations then shows in the appropriate gauges these equations result. Personally its so obvious when done that way I would do the reverse - show the normal linearised equations result from ∂u∂u Φuv = kTuv - I haven't worked out the details personally but it looks obvioius it would not be hard.

You can then show, and this is what chapter 7 does - the full EFE's automatically result. This is pretty amazing actually where the linearised equations implies the full non linear equations.

Thanks
Bill

My hunch is that this is not the issues i had in mind, but thanks for the pointer. I agree anyhow that there are most probably many interesting things in the critical part of arriving at EFE that is interesting to analyse. What i am interested in is to clairfy the typically heuristic arguments with a more coherent inference method, centralized around a physical observer. Did Ohanian explain this elsewhere or do you have to buy the book ti get it in context?

About the energy-momentu pseudo-tensor of the gravitational field discussion, that might related to various detours in the constructing logic. Here is one paper that suggests Einstein originally made such a detour as well, but its gone in the end. It seems he started with empty space, and used the "pseudo-tensor" in order to conceptually defined, WHERE in the expression fo all the various derivatives or the metric to add the energy-momentum tensor. This seems a like a natural approach if you start by presuming that the geometric description must be right, its just a question of which "terms" in the abstraction to associate to, and coupled fo the matter terms.

How Einstein Got His Field Equations
" Einstein rewrote this equation in a more physically meaningful form that has an important term in it that represents the energymomentum “tensor” tsm of the gravitational field...
...
Einstein’s next insight: to get the gravitational field equations in the presence of matter and radiation, in the previous equation he replaced the energy momentum tensor tsm of the gravitational field by the total energy momentum tensor tsm +Tsm of gravity and of matter together! Once this is done, he then retraced his steps back the way he arrived to (2.2) to obtain his field equations, which now incorporate matter Tsm . Replacing tsm by tsm +Tsm in the vacuum equation"
-- https://arxiv.org/abs/1608.05752

I haven't spent so much time to analyse that exact inference simply because there are other objections i have that sort of makes it moot for me. But maybe there is an subjective insight-reason for this detour, as there might be several arguements accidently leading to the same thing. Clearly the expression has some similarities to the EM tensors, so i am not sure if this relates to the first point bhobba mentions in the book i don't have.

/Fredrik

 

[bhobba]

Hi Everyone

To get to the bottom of what's going on with Ohanian we have to look at his view of GR

1. He thinks the principle of covariance is basically vacuous - have discussed it before - what people here think of covarience is what Ohanian calls invariance.
2. Ohanian does not agree with the principle of equivalence - take any coordinate system - accelerating, free falling - it doesn't matter - if gravity is space-time curvature then regardless it will be curved - gravity cannot be infinitesimally removed - the curvature is always there. Thus the argument given by Peter about how energy is removed is flawed. But as I will explain the logic related to this is I think the inconsistency in Ohanian's approach.

What he does is start with an inertial frame and looks at a dust of point masses. Pick any mass and go to another inertial frame and we have other terms than mass appear in the stress-energy tensor - so the mass by itself shouldn't really be the source - I suppose you could have the invariant rest mass but that would lead to, as explained later, a rather uninteresting scalar field. So you take the stress-energy as the source.

Now EM is the very well known field theory so let's use it as the basis of our field theory. ∂u∂uA v = 4πJv with the Lorentz gauge condition ∂uAu = 0. Generalizing we have in the case of Tuv as the source rather than Av, ∂u∂uΦvk = kTvk and the gauge condition ∂uΦuv. = 0, also known as the Hilbert condition and some constant to be determined k.. The book goes into more detail in deriving it but IMHO this is short easy and gets to the heart of the matter.

Now remember this is in an inertial frame so Noether can be applied to get the stress energy-tensor without trouble. For reasons I quoted this is added to the stress energy tensor of mass and the equations of motion derived.

Nothing untoward so for - we have and inertial frame so Noether applies.

Now here comes the issue and where he breaks his own principles. He then considers the equation valid in some small region of curved space-time and shows the gauge symmetry is infinitesimal coordinate transformations. But can he do that? Its now not an inertial frame because he has rejected the principle of equivalence. You can't have it both ways. One can argue it's reasonable - and it is - but truth is different than reasonableness. It just happens to also be true - but the reasoning is not correct, it was developed in an inertial frame - but it is now being applied to non inertial frames.

Of course you can fix it up by saying as most do, the curvature should not depend on coordinates and you get the EFE's as per say Soper. You take its linear small field approximation and see they are the same. Fine - extra check on its validity. However he claims the linear equations actually imply the full EFE's. I now however suspect his logic is flawed.

Thanks
Bill

 

[PeterDonis]

The second edition

Then there is an interesting change from your edition to my edition (the third). In the second edition, according to the table of contents I can find online, p. 147, where you got your quote, starts section 3.3, "The Interaction of Gravitation and Matter". This section has been deleted in the third edition; section 3.3 in the third edition is "The Variational Principle and the Equations of Motion" (and Chapter 3 has only 5 sections total instead of 6). So I can't read the text you're quoting from directly because the authors removed it. Nothing in the sections that remain makes the kind of claim you have been quoting.

I can't say why the removal was done, of course, but I wonder if it had something to do with the statements in that section of the 2nd edition being found to be misleading after it was published.

 

[PeterDonis]

Ohanian does not agree with the principle of equivalence - take any coordinate system - accelerating, free falling - it doesn't matter - if gravity is space-time curvature then regardless it will be curved - gravity cannot be infinitesimally removed - the curvature is always there. Thus the argument given by Peter about how energy is removed is flawed.

My argument for how any stress-energy tensor for gravity must be zero does not depend on gravity being "infinitesimally removed". It only depends on being able to choose coordinates that make the metric Minkowski, i.e., ##g_{\mu \nu} = \eta_{\mu \nu}##, at a chosen event. That can always be done, no matter how curved spacetime is.

Now remember this is in an inertial frame so Noether can be applied to get the stress energy-tensor without trouble.

Being able to apply Noether's Theorem does not depend on being in an inertial frame. It depends on the spacetime having time translation symmetry, i.e., on it having a timelike Killing vector field. That is what is needed for Noether's Theorem to give you a conserved energy.

Flat Minkowski spacetime has a timelike KVF. However, the curved spacetime you get when you put a tensor gravitational field ##h_{\mu \nu}## on flat Minkowski spacetime might or might not. If it doesn't, the argument that led to the definition of a stress-energy tensor for gravity breaks down. And if it does, then there is in fact a way to define "energy stored in the gravitational field" in an invariant way--it's just the GR analogue of the Newtonian gravitational potential energy (defined using the norm of the timelike KVF). But this energy is not a tensor; it's a scalar.

 

[PeterDonis]

Flat Minkowski spacetime has a timelike KVF.

Actually it has two families of timelike KVFs, which gives a good further illustration of why "timelike KVF" is not the same as "inertial frame". One of the two families of timelike KVFs corresponds to all the possible inertial frames. But the other family corresponds to all possible Rindler frames--i.e., all possible families of uniformly accelerated observers. These frames are not inertial frames. But you can still use Noether's Theorem to define a conserved energy in them, and it will be a different conserved energy than you would derive for the family of inertial frames. (In fact this is one way to derive the Unruh effect.)

 

[bhobba]

I can't say why the removal was done, of course, but I wonder if it had something to do with the statements in that section of the 2nd edition being found to be misleading after it was published.

I now think so.

As I said my edition is falling to bits.

I prefer books than Kindle for those I refer to frequently like this one - so will get a new book.

Strange isn't how Wald is my favorite for its rigor, but now I am getting older I tend to actually refer to the less rigorous Ohanian - interesting.

Added Later
Ordered it - strangely the book was cheaper than the Kindle edition which most definitely is not common.

Thanks
Bill