Unsimultaneity and quantum physics

[exponent137]

Distinction between general relativity and quantum mechanics is also, that QM time is absolute, but GR time is relative.

But, special relativity also contains unsimultaneity, but SR is quantized.
How it is solved the problem of unsimultaneity, or it cause some problems?

I know Dirac equation, but I do not find how it treats and solves the problem of unsimultaneity.

 

[my2cts]

Dirac's theory fully complies with special relativity so there is no issue.

 

[exponent137]

Dirac's theory fully complies with special relativity
Yes, i know, but how it can be visualized how relativistic quantum theory deals with unsimultaneity?

I suppose that Klein-Gordon equation is easier for visualization of this than Dirac equation.

 

[bhobba]

Yes, i know, but how it can be visualized how relativistic quantum theory deals with unsimultaneity?

It doesn't - or rather it sidesteps the issue by treating everything as a field. That's why we need QFT where both time and position are just parameters.

There is a fundamental issue with QM and relativity. Time is a parameter, and position an observable - but relativity treats them the same. To get around it you either need to promote time to an operator or demote position to a parameter. Time as operator leads to severe mathematical difficulties, but position as a parameter gives QFT.

Thanks
Bill

 

[my2cts]

It doesn't - or rather it sidesteps the issue by treating everything as a field. That's why we need QFT where both time and position are just parameters.

There is a fundamental issue with QM and relativity. Time is a parameter, and position an observable - but relativity treats them the same. To get around it you either need to promote time to an operator or demote position to a parameter. Time as operator leads to severe mathematical difficulties, but position as a parameter gives QFT.

Thanks
Bill

I am not sure, this may be a separate issue.
The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

 

[bhobba]

I am not sure, this may be a separate issue.
The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

The Klein Gordon Equation has problems with negative probabilities:
http://www.phy.ohiou.edu/~elster/lectures/advqm_3.pdf

The Dirac Equation had problems with negative energies - the hole theory only partly solved the issues (it created others worse eg what about the charge of these negative energy electrons in the infinite sea):
http://www.phy.ohiou.edu/~elster/lectures/advqm_4.pdf

These are all fixed in QFT.

Thanks
Bill

 

[my2cts]

The Klein Gordon Equation has problems with negative probabilities:
http://www.phy.ohiou.edu/~elster/lectures/advqm_3.pdf

The Dirac Equation with negative energies - the hole theory only partly solved the issues:
http://www.phy.ohiou.edu/~elster/lectures/advqm_4.pdf

These are all fixed in QFT.

Thanks
Bill

I know but how are these issues connected with simultaneity?
By the way, in KG theory there is no probability density.
psi^2 is just charge density which is not of definite sign.

 

[bhobba]

I know but how are these issues connected with simultaneity?
By the way, in KG theory there is no probability density.
psi^2 is just charge density which is not of definite sign.

Lack of simultaneity is a symptom that we need space-time not space and an absolute time. Space and time must be treated on the same footing - no way out. I don't know what more I can really say. If you don't do it you get symptoms like I mentioned - one must second quantisize the field to remove them ie go to QFT.

Cant follow your comment about KG - what I said is well known eg from the paper I linked to:
This means that the Klein-Gordon equation allows negative energies as solution. Formally,
one can see that from the square of Eq. (2.60) the information about the sign is lost.
However, when starting from Eq. (2.71) all solutions have to be considered, and there is
the problem of the physical interpretation of negative energies.

Thanks
Bill

 

[WannabeNewton]

By the way, in KG theory there is no probability density.

Eh? In Relativistic QM, solutions $\varphi(x)$ to the KG equation are particle states and the probability 4-current is given by $j^{\mu} = i[(\partial^{\mu}\varphi )\varphi^{\dagger}- (\partial^{\mu}\varphi^{\dagger})\varphi]$ where $\int i[(\partial^{0}\varphi)\varphi^{\dagger} - (\partial^{0}\varphi^{\dagger})\varphi]d^{3}x = 1$ (in contrast to $\int \varphi^{\dagger}\varphi d^{3}x = 1$ in non-relativistic QM). The current conservation $\partial^{\mu}j_{\mu} = 0$ ensures that $\int j^0 d^3 x = 1$ is a Lorenz scalar.

Reinterpreting $j^0$ as charge density arises in QFT where $\varphi(x)$ are fields instead of particle states.

But I also do not see how relativity of simultaneity is an issue here...

 

[my2cts]

Multiply that with e and you have the charge current. It is the Noether current associated with invariance under global phase transition. Note that charge current is an observable while probability is not.

 

[Bill_K]

The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

The problem arises in the context of first quantization (where you try to use ψ as a single particle wavefunction) and include an interaction term. Solutions to the inhomogeneous Dirac and Klein-Gordon equations are nonzero outside the light cone. It's only in QFT, with reinterpretation of the negative energy states as positive energy antiparticles, that this causality violation cancels.

 

[exponent137]

One of the biggest obstacles has been that general relativity and quantum mechanics treat time very differently. In the former theory, time is another dimension alongside space and can bend and stretch, speed up and slow down, in different circumstances. Quantum theories, however, usually assume that time is set apart from space and ticks at a set rate.

one of the key paradigms of quantum and classical mechanics: the paradigm of a state evolving in time,

It is clear that this different time is a problem in merging general relativity and quantum theory. But, why it is not a problem at merging of special relativity and quantum theory? One speciality of special relativity is unsimultaneity? Why it is not a problem at quantum theory? How it is with evolution of wave function is special relativity, where different locations means different time, if a rocket is moving?

 

[WannabeNewton]

Again, in relativistic QM as well as QFT the meta-theory of space-time structure is that of special relativity and not Galilean relativity.

 

[Bill_K]

It is clear that this different time is a problem in merging general relativity and quantum theory. But, why it is not a problem at merging of special relativity and quantum theory? One speciality of special relativity is unsimultaneity? Why it is not a problem at quantum theory?

No, I don't think it is. Instead of regarding the state ψ as a function of hyperplanes t = const, you regard it as a functional on a spacelike hypersurface. In place of ∂/∂t, you use the functional derivative δ/δτ that varies the surface.

There is no essential problem, as you seem to think there is, in doing quantum mechanics on a classical curved background. This is the way Hawking radiation is derived. The problem in merging QM with GR comes when you allow the background to be a further dynamical degree of freedom.

 

[exponent137]

Again, in relativistic QM as well as QFT the meta-theory of space-time structure is that of special relativity and not Galilean relativity.
Yes, but different time in general relativity is problem for quantum theory, but time in special relativity is not a problem for quantum theory. Where is this key difference between special and general relativity? Yes, curved space-time is this difference, but why such impact on quantum theory of this difference?

 

[WannabeNewton]

I think you're sort of confusing two aspects here. We can do QFT on a flat background just fine; this background is Minkowski space-time and the meta-theory is SR. We can also do QFT on a curved background just fine wherein the meta-theory is GR. The "problem" is with quantization of GR itself (very loosely speaking) and this is a whole different ball game.

 

[bhobba]

Note that charge current is an observable while probability is not.

Sure - one can interpret the conserved quantity as a charge density - but then out goes the probability interpretation of QM. Either way indicated a sickness QFT is required to remove.

Thanks
Bill

 

[bhobba]

It is clear that this different time is a problem in merging general relativity and quantum theory.

Not really. Feynman and others have shown spin 2 particles leads to linearized gravity which mathematically is the same as an infinitesimal space-time curvature and leads directly to GR. You will find this approach in Ohanian's textbook on GR:
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

The issue with GR and Quantum theory is its not renormalizeable - but still is a perfectly valid theory as an effective theory up to a cutoff about the Plank scale:
http://arxiv.org/abs/1209.3511

But even for theories such as QED that is renormalizeable to all orders we don't expect it to be valid at the Plank scale anyway. Indeed well before that scale the Electroweak takes over and we expect another theory to take over from that until we, hopefully, fingers crossed, have a TOE.

Thanks
Bill

 

[my2cts]

Sure - one can interpret the conserved quantity as a charge density - but then out goes the probability interpretation of QM. Either way indicated a sickness QFT is required to remove.

Thanks
Bill

The probability interpretation is basically non-relativistic. The charge-current interpretation is covariant.

 

[exponent137]

Not really. Feynman and others have shown spin 2 particles leads to linearized gravity which mathematically is the same as an infinitesimal space-time curvature and leads directly to GR. You will find this approach in Ohanian's textbook on GR:
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

The issue with GR and Quantum theory is its not renormalizeable - but still is a perfectly valid theory as an effective theory up to a cutoff about the Plank scale:
http://arxiv.org/abs/1209.3511

But even for theories such as QED that is renormalizeable to all orders we don't expect it to be valid at the Plank scale anyway. Indeed well before that scale the Electroweak takes over and we expect another theory to take over from that until we, hopefully, fingers crossed, have a TOE.

Thanks
Bill

I read this fine book (Feynman Lectures on Gravitation), but I did not remember all details. I read some articles of Isham, where he found the problems in all theories of quantum gravity. Do you maybe know where the are the problems in Feynman quantum gravity, except of renormalizability.

Otherwise, I also think that the space-time is grained, so this case there is no problems with renormalizability.

 

[atyy]

Yes, the unsimultaneity is a potential problem in special relativity, not just general relativity. How the problem is solved is understood for many cases, but there are still interesting things like

http://arxiv.org/abs/hep-th/0110205
"We show that the nondemolition measurement of a spacelike Wilson loop operator W(C) is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of W(C) would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality."

The authors say the problem does not occur for abelian theories (like QED):
"The causality problem arose for the nondemolition measurement of non-Abelian Wilson loops because multiplication of conjugacy classes is ill-defined. Since this problem does not arise if G is Abelian, one might expect that a spacelike Wilson loop operator should be measurable in an Abelian gauge theory (or more generally, if the Wilson loop is evaluated in a one-dimensional irreducible representation of the gauge group). We will see that this is the case."

 

[DevilsAvocado]

Yes, but different time in general relativity is problem for quantum theory, but time in special relativity is not a problem for quantum theory. Where is this key difference between special and general relativity? Yes, curved space-time is this difference, but why such impact on quantum theory of this difference?

The key difference between SR/QM and GR is that in SR/QM we have a fixed spacetime background (Minkowski space/Hilbert space), while in GR the spacetime geometry is dynamic and do not refer to a specific coordinate system.

It might not sound like a big deal, but if you regard the uncertainty principle and the probabilistic nature of QM, involving fluctuations, you have a well-defined background against which to define these fluctuations, while in GR any "fluctuations" will "loopback" on the dynamic background (including space and time) and define itself!

Hence, one could say that GR gives the picture of some sort of "relational loopback" – matter is being located with respect to the gravitational field, and vice versa – and you could maybe imagine what happens when incorporating QM fluctuations into this "loopback brew"...

Tricky huh?

General covariance (or diffeomorphism of smooth manifolds) is not the same as (global) Lorentz covariance, because it also applies to accelerated relative motions.

The image of a rectangular grid on a square under a
diffeomorphism from the square onto itself.

They say QM is weird, but some aspect of GR is at least as bizarre; "Beyond my wildest expectations" to quote Einstein...

Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

Here is more on the so-called "problem of time":
http://plato.stanford.edu/entries/quantum-gravity/#5.1

(Of course, QG only applies to extreme situations of energy and/or gravity, like Big Bang or Black Holes)

 

[bhobba]

Do you maybe know where the are the problems in Feynman quantum gravity, except of renormalizability.

To the best of my knowledge there is none.

Without the problems of renormalisability GR is a perfectly good quantum theory as evidenced by the fact we have a EFT valid up to about the Plank scale.

But we think about renoprmalisability differently these days using the EFT paradigm of Wilson. We are pretty sure all our theories break down at the Plank scale so GR isn't really special. Its just that the interesting phenomena with gravity lies beyond that.

Thanks
Bill

 

[bhobba]

The key difference between SR/QM and GR is that in SR/QM we have a fixed spacetime background (Minkowski space/Hilbert space), while in GR the spacetime geometry is dynamic and do not refer to a specific coordinate system.

Actually its not a problem.

From the properties of spin 2 particles one can derive linearized gravity in a flat space-time. Then you show linearized gravity is exactly the same as if spacetime had an infinitesimal curvature. Then from requiring the principle of invarience (a correct varient of the principle of covarience) - lo and behold - you get GR.

You will find a discussion of this in the Ohanian text mentioned previously in this thread, as well as a key error of Einstein with the principle of covarience - it's basically a vacuous statement and he was chided by Kretschmann for it:
http://en.wikipedia.org/wiki/Erich_Kretschmann

But see the book for the detail. And your quote is correct - no-prior geometry is the correct basis of GR - but not the only one.

The key point here is there is really no difference between a field that makes space-time behave as if it as curved and if it actually is curved.

Thanks
Bill

 

[DevilsAvocado]

But we think about renoprmalisability differently these days using the EFT paradigm of Wilson. We are pretty sure all our theories break down at the Plank scale so GR isn't really special. Its just that the interesting phenomena with gravity lies beyond that.

Oh yes, very differently, this guy obviously didn't have a clue:

"The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate." -- Richard Feynman (1985)​

Seriously bhobba, I love your – "business as usual" – positivism, but maybe sometimes you push it just a little bit too far? Afaik, an effective field theory should be interpreted as an approximation (reflecting human ignorance), right? It does not (generally) claim to be fundamental or self-consistent, right?

And to make it even more complicated, renormalization fails for GR, and the (only?) way to overcome this problem is to use Steven Weinberg's "Asymptotic Safety", which is promising, but still only a concept, without any rigorous proof of existence.

(Gravitation & Asymptotic Safety starts @53:50)
https://www.youtube.com/watch?v=1mQ0Yu2x9vE
http://www.youtube.com/watch?v=1mQ0Yu2x9vE&t=53m50s

So even if promising, it is still "work in progress", and while Weinberg himself is betting his money on strings, I prefer to keep mine in the wallet until the first physical experiment is about to run...

 

[DevilsAvocado]

Actually its not a problem.

From the properties of spin 2 particles one can derive linearized gravity in a flat space-time. Then you show linearized gravity is exactly the same as if spacetime had an infinitesimal curvature. Then from requiring the principle of invarience (a correct varient of the principle of covarience) - lo and behold - you get GR.

Okay, but then again; linearized gravity is only an approximation, not the whole picture, right?

You will find a discussion of this in the Ohanian text mentioned previously in this thread, as well as a key error of Einstein with the principle of covarience - it's basically a vacuous statement and he was chided by Kretschmann for it:
http://en.wikipedia.org/wiki/Erich_Kretschmann

I don’t get this at all... the discussion between Kretschmann and Einstein was of 'philosophical' nature, but GR is constructed using tensors and covariant derivative, so it would be quite inane to claim there's something wrong about GR without any (experimental) proof...??

But see the book for the detail. And your quote is correct - no-prior geometry is the correct basis of GR - but not the only one.

The key point here is there is really no difference between a field that makes space-time behave as if it as curved and if it actually is curved.

Okay, so what's the name of this successful competing gravitational theory, and will it produce something like this?

Gravitational Lensing in LRG 3-757

(I'm probably wrong, but to me it looks like the main problem is not if spacetime is actually curved or not, but the fundamental GR "feedback loop" between spacetime & matter, which causes severe difficulties when applying QM uncertainty/randomness/fluctuation, which unconditionally will propagate to the dynamic background, creating a "bubbling-random-geometry-brew", in which we're supposed to define everything else, including space & time... gravity "falls down onto itself", so to speak. Einstein also took for granted definite properties of the world, whereas Bell has proven that we cannot have both definite properties and locality, which to me seems very difficult for GR gravity to overcome, in any case... but this of course is just guessing from my side, safest not to pay any serious attention to...)

 

[WannabeNewton]

Okay, but then again; linearized gravity is only an approximation, not the whole picture, right?

Yes. It simply gives you perturbed solutions to Einstein's equation starting from some flat or curved background solution. The perturbed solutions have curvature scales varying to small degrees from the curvature scales of the background metric. They are not exact solutions.

I don’t get this at all... the discussion between Kretschmann and Einstein was of 'philosophical' nature, but GR is constructed using tensors and covariant derivative, so it would be quite inane to claim there's something wrong about GR without any (experimental) proof...??

Actually Kretschmann's comments led to confusions regarding "general covariance" for several decades down the road. Even today laymen tend to fall into the same confusing trap. If you want a beautiful discussion of what covariance vs. invariance really entails I would highly recommend section 3.5 of Straumann "General Relativity". If you don't have access to this text (which is a brilliant text on general relativity by the way) then read this: http://www.pitt.edu/~jdnorton/papers/decades.pdf

 

[DevilsAvocado]

Thanks² WannabeNewton! :thumbs:

 

[atyy]

Actually Kretschmann's comments led to confusions regarding "general covariance" for several decades down the road. Even today laymen tend to fall into the same confusing trap. If you want a beautiful discussion of what covariance vs. invariance really entails I would highly recommend section 3.5 of Straumann "General Relativity". If you don't have access to this text (which is a brilliant text on general relativity by the way) then read this: http://www.pitt.edu/~jdnorton/papers/decades.pdf

Doesn't Norton's paper say that Kretschmann was basically right, and that it was Einstein that was initially confused?

 

[WannabeNewton]

Doesn't Norton's paper say that Kretschmann was basically right, and that it was Einstein that was initially confused?

Yes I wasn't trying to say that Kretschmann himself was confused but rather this his comments led to confusions regarding what "general covariance" was, mainly regarding the difference between invariance and covariance.

 

[atyy]

Yes I wasn't trying to say that Kretschmann himself was confused but rather this his comments led to confusions regarding what "general covariance" was.

Ah, I see. I thought maybe Straumann had a different take (I've never read Kretschmann for my self, so have just bought the standard line that he was basically right.)

 

[bhobba]

Yes. It simply gives you perturbed solutions to Einstein's equation starting from some flat or curved background solution. The perturbed solutions have curvature scales varying to small degrees from the curvature scales of the background metric. They are not exact solutions.

Yes - but the strange but true thing about GR is, while most linear approximations do not imply the full non linearised equations, GR is the odd man out - the linearised equations imply the full EFE's.

You will find the detail in Ohanion (see section 7.2 - page 380):
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

Ohanion explains very carefully and clearly exactly what Einsteins error was as well as Krectmann's comments on General Covarience being vacuous.

In modern times most textbooks on GR, correctly IMHO, lean heavily on the geometrical approach to GR. But it is not the only approach, and IMHO is not the best approach in understanding GR's relation to QFT which is a field theory. Ohanian takes this approach and I think anyone into GR needs to be aware of it.

Thanks
Bill

 

[bhobba]

Doesn't Norton's paper say that Kretschmann was basically right, and that it was Einstein that was initially confused?

Yes I wasn't trying to say that Kretschmann himself was confused but rather this his comments led to confusions regarding what "general covariance" was, mainly regarding the difference between invariance and covariance.

Wannabe is correct.

However its a subtle issue that can lead to confusion, and most textbooks these days present the geometrical view without fully discussing the issue.

Ohanian is the odd man out. His approach is radically different and you get an understanding of what's really going on by seeing both approaches. He also carefully explains the difference between covarience and invarience. Covarience is vacuous - invarience in physically prescriptive.

In understanding GR's relationship to QFT Ohanions approach (ie the field theory approach analogous to Maxwell's equations) is much more illuminating eg the lineraised GR equations are derived as a generalization of Maxwell's equations and it is well known how to quantize that.

Thanks
Bill

 

[WannabeNewton]

You will find the detail in Ohanion (see section 7.2 - page 380)

I think we may have different editions of Ohanian because for me it's section 7.3-page 262.

Ohanion explains very carefully and clearly exactly what Einsteins error was as well as Krectmann's comments on General Covarience being vacuous.

Ohanian's book is actually where I first learned about the difference between general covariance and general invariance but Straumann has a much more mathematically rigorous discussion so if you ever come across the book check out the relevant section I referenced earlier.

 

[bhobba]

Oh yes, very differently, this guy obviously didn't have a clue

Yea - Feynman knew nothing

Seriously though there has been a lot of work done on renormaliztion by guys like Wilson (who got a Nobel Prize for it) showing that dippy process is not really that dippy after all, but it took a while to sink in, and it's not surprising at the time Feynman wrote that he was of that view:
http://www.preposterousuniverse.com/blog/2013/06/20/how-quantum-field-theory-becomes-effective/
'Wilson’s viewpoint, although it took some time to sink in, led to a deep shift in the way people thought about quantum field theory. Pre-Wilson, it was all about finding theories that are renormalizable, which are very few in number. (The old-school idea that a theory is “renormalizable” maps onto the new-fangled idea that all the operators are either relevant or marginal — every single operator is dimension 4 or less.) Nowadays we know you can start with just about anything, and at low energies the effective theory will look renormalizable. Which is useful, if you want to calculate processes in low-energy physics; disappointing, if you’d like to use low-energy data to learn what is happening at higher energies. Chances are, if you go to energies that are high enough, spacetime itself becomes ill-defined, and you don’t have a quantum field theory at all. But on labs here on Earth, we have no better way to describe how the world works.'

Seriously bhobba, I love your – "business as usual" – positivism, but maybe sometimes you push it just a little bit too far? Afaik, an effective field theory should be interpreted as an approximation (reflecting human ignorance), right? It does not (generally) claim to be fundamental or self-consistent, right?

I think itself consistent, but its an approximation and not fundamental.

The point of the EFT approach is even QED is like that. In what follows I will reference the following paper:
http://arxiv.org/pdf/hep-th/0212049.pdf

First there is a basic sickness inherent in QFT - see page 5:
'Because F(x) has the same dimension as g0, it also is dimensionless and so are the Fi,(x). The only possibility for a dimensionless quantity like F to be a function of a dimensional variable like x is that there exists another dimensional variable such that F depends on x only through the ratio of these two variables. Apart from x, the only other quantity on which F depends is the cutoff, which must therefore have the same dimension as x. This is indeed the case in our example, Eq. (5).'

The cause of the infinity is seen by basic dimensional analysis - one must introduce another parameter, the most obvious choice being a cutoff, to prevent this. The theory is wrong, just like GR is wrong, it is not valid to all energies.

Its right at the foundation of QFT which shows the vacuum has infinite energy. That's wrong - simple as that. A cutoff must be imposed.

The thing that makes QED special over gravity is its property of renormaizability. This means it doesn't matter what cutoff we use - finite values can always be extracted regardless of the energy levels. Its a special and very nice property - yes you need to have a cutoff - but it doesn't really matter what it is. We do know however that beyond a certain energy level QED is replaced by the Electroweak theory so its fundamentally wrong ie merely an approximation. It too is renormalizeable, but of course is equally as sick, the infinities require a cutoff.

The difference with gravity is its not renormalizable - a specific cutoff is required about the Plank scale. But theories that are renormalizeable are equally as sick - they all require a cutoff as the dimensional analysis argument shows.

The difference with gravity is the specific cutoff we need to have at the Plank level - its not up in the air like renormalizable theries. But the main, the real problem, the key issue is the interesting physics occurs at and below the Plank scale with gravity. The interesting physics with QED and the electroweak theory occur well before that.

Thanks
Bill

 

[bhobba]

Ohanian's book is actually where I first learned about the difference between general covariance and general invariance but Straumann has a much more mathematically rigorous discussion so if you ever come across the book check out the relevant section I referenced earlier.

For me it was a little different.

I had finished my math degree - actually it was a double degree in math and computer science, there were tons of computer science jobs at the time so I thought what the heck - let's go out and get a job, and chase a bit of that evil luker, rather than do post grad work like my professors wanted.

I had read some GR lay books and thought let's see some true technical books. Ohanian had this picture of the CMBR on the cover like the lay book I had read - its as good as any reason to choose a book - so got it.

It was only later I learned his approach was atypical - most are highly geometrical. I got Wald, which I really liked, and MTW, which I thought was not really a patch on Wald - good - but Wald was way ahead. However I always felt there was distinct advantages in knowing Ohanians approach especially in regard to quantizing gravity.

Thanks
Bill

 

[DevilsAvocado]

You will find a discussion of this in the Ohanian text mentioned previously in this thread,

Thank you very very much for this hint bhobba, I managed to get my hands on the "digital express" () version. It’s a wonderful book, and I’m completely flabbergasted – Einstein was wrong??

(all bolding mine)

The physical interpretation of the principle of special relativity is that velocity is relative; no experiment can detect any intrinsic difference between reference frames in uniform motion with different velocities. It is tempting to give the principle of general invariance the interpretation that acceleration is also relative. Einstein found it hard to resist this temptation; he named his theory of gravitation the theory of general relativity because he thought that (locally) the phenomena observed in a gravitational field are indistinguishable from those observed in an accelerated system of reference and that according to this conception one cannot speak of the absolute acceleration of a system of reference, just as in the ordinary theory of relativity one cannot speak of the absolute velocity of a system.

However, we have seen in Section 1.7 that the tidal effects allow us to make an absolute distinction between the gravitational forces and the pseudo-forces found in accelerated reference frames. It is therefore false to speak of a general relativity of motion.

This and Wannabe’s link to Norton’s paper was definitely a total revelation... Einstein’s own curved spacetime in General Relativity makes it not that general at all... amazing, absolutely amazing... this must mean that the Equivalence Principle is slightly wrong (in some way), or...?

When it comes to linearized gravity, it still looks like the Linear Field Approximation is an approximation that works well for "other approximations" (but nothing more, hint hint ).

We will now seek out the relativistic field equations for gravitation. We begin with what is simple by assuming a linear approximation and neglecting the effects of the gravitational field on itself. Of course, if Newton's principle of equivalence (mI = mG) is to hold as an exact statement, gravitational energy must gravitate and the exact field equations must be nonlinear. Although it is true that the most spectacular results of gravitational theory depend in a crucial way on the nonlinearity of the field equations, almost all of the results that have been the subject of experimental investigation can be described by the linear approximation.

A black hole forms. During the collapse process, gravitational radiation can be emitted. Since it is useless to pretend that the linear approximation is applicable to this case, we only quote an estimate given by Misner, Thome, and Wheeler.

I.e. linear approximation/linearized gravity do not work in cases where QG becomes interesting, and could therefore not be the solution to QG.

It’s probably some 'defect' in my very tiny brain, but I don’t understand this:

That the linear equations imply the full nonlinear equations is a very remarkable feature of Einstein's theory. Given some complicated set of nonlinear equations, it is easy to derive the corresponding linear approximation, but in general if we only know the linear approximation, we cannot reconstruct the nonlinear equations.

That the exact nonlinear equations are implied by the linear equations (the converse is of course trivial) is a very remarkable feature of Einstein's theory. This very tight connection between the exact equations and the linear approximation would not exist were it not for the principle of general invariance.

What does Ohanian mean?


(I'll be back asap on post #35)

 

[atyy]

Thank you very very much for this hint bhobba, I managed to get my hands on the "digital express" () version. It’s a wonderful book, and I’m completely flabbergasted – Einstein was wrong??

(all bolding mine)


This and Wannabe’s link to Norton’s paper was definitely a total revelation... Einstein’s own curved spacetime in General Relativity makes it not that general at all... amazing, absolutely amazing... this must mean that the Equivalence Principle is slightly wrong (in some way), or...?

When it comes to linearized gravity, it still looks like the Linear Field Approximation is an approximation that works well for "other approximations" (but nothing more, hint hint ).




I.e. linear approximation/linearized gravity do not work in cases where QG becomes interesting, and could therefore not be the solution to QG.

It’s probably some 'defect' in my very tiny brain, but I don’t understand this:




What does Ohanian mean?


(I'll be back asap on post #35)

The linear equations are inconsistent (which is ok for an approximation), but if one asks if we can add terms to make the equations consistent, then one does end up with general relativity. A famous derivation is that of Deser's http://arxiv.org/abs/gr-qc/0411023 .

The equivalence principle is fine if one thinks of general relativity (without the cosmological constant) as a field on flat spacetime. If one further assumes the field is a quantum field, Weinberg has given an argument that the equivalence principle can be derived. http://arxiv.org/abs/1007.0435 (section 2.2.2 and appendix A).

Another review which has useful stuff is Hinterbichler's http://arxiv.org/abs/1105.3735:

"The path Einstein followed, on the other hand, is a leap of insight and has logical gaps; the equivalence principle and general coordinate invariance, though they suggest GR, do not lead uniquely to GR. (p5)"

"The real underlying principle of GR has nothing to do with coordinate invariance or equivalence principles or geometry, rather it is the statement: general relativity is the theory of a non-trivially interacting massless helicity 2 particle. The other properties are consequences of this statement, and the implication cannot be reversed. (p6)"

"As a quantum theory, GR is not UV complete. It must be treated as an effective field theory valid at energies up to a cutoff at the Planck mass, M_P, beyond which unknown high energy effects will correct the Einstein-Hilbert action. (p6)"

"An important fact about GR is that there exists this parametrically large middle regime in which the theory becomes non-linear and yet quantum eects are still small. This is the region inside the horizon r = r_S but farther than a Planck length from the singularity. In this region, we can re-sum the linear expansion by solving the full classical Einstein equations, ignoring the higher derivative quantum corrections, and trust the results. This is the reason why we know what will happen inside a black hole, but we do not know what will happen near the singularity. (p53)"

 

[bhobba]

What does Ohanian mean?

Atty explained it pretty well.

But the gist is this; from page 380 of my text:

To be precise the exact nonlinear field equation satisfied by the metric tensor Guv can be derived from the following assumptions:

1.The equation is invariant under general coordinate transformations
2. The equation reduces to the linear ear equation in the linear approximation
3. The equation is of second differential order, and is linear in second derivatives.

Just a few comments about the other stuff.

Yes Einstein was wrong - but the way he was wrong in fact shows just how penetrating and insightful his thinking was. You probably have heard of a great mathematician called John Von Neumann. This guy was so bright geniuses, even magicians like Feynman who are beyond genius, would sit in awe and describe him as the only guy on the planet that was fully awake. His technical mathematical brilliance was simply breathtaking - way beyond Einstein, whose mathematical ability was good (but not great), and even mathematical virtuosos like Feynman (his mathematical ability was genuinely great). His ability to penetrate a problem simply left mere mortals shaking their heads. But that ability to penetrate, as great as it was, and it was great, paled into insignificance compared to Einstein. Einstein simply saw things no one else could.

That comment about the inverse square character of gravity being a dead give away you are dealing with gravity and not acceleration is not quite correct - special distributions of matter will give a constant gravitational field - but they are generally not what is out there. A physicist that finds a constant gravitational field can be pretty sure its the result of acceleration - but not 100% positive.

Glad however you found it interesting and thought provoking.

Thanks
Bill

 

[DevilsAvocado]

The equivalence principle is fine if one thinks of general relativity (without the cosmological constant) as a field on flat spacetime. If one further assumes the field is a quantum field, Weinberg has given an argument that the equivalence principle can be derived. http://arxiv.org/abs/1007.0435 (section 2.2.2 and appendix A).

Thank you for the links atty! I was too hasty about the equivalence principle, there are obviously three of them (weak, Einstein and strong) and Einstein’s has the following constraint:

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.​

"Local" means that the experiment must be small compared to any variations in the gravitational field (tidal forces), which saves the situation. (Einstein was way too smart to make my suspected avocado-blunder! )

 

[DevilsAvocado]

Atty explained it pretty well.

But the gist is this; from page 380 of my text:

To be precise the exact nonlinear field equation satisfied by the metric tensor Guv can be derived from the following assumptions:

1.The equation is invariant under general coordinate transformations
2. The equation reduces to the linear ear equation in the linear approximation
3. The equation is of second differential order, and is linear in second derivatives.

OMG , I’m sorry bhobba... English is not my mother tongue, and when turbo-skimming (in an "excited state" with neurons in the outer shell wobbling freely ;), I mixed up "implied" with "derived", which would have made the two statements clashing... gosh...

Just a few comments about the other stuff.

Yes Einstein was wrong - but the way he was wrong in fact shows just how penetrating and insightful his thinking was. You probably have heard of a great mathematician called John Von Neumann. This guy was so bright geniuses, even magicians like Feynman who are beyond genius, would sit in awe and describe him as the only guy on the planet that was fully awake. His technical mathematical brilliance was simply breathtaking - way beyond Einstein, whose mathematical ability was good (but not great), and even mathematical virtuosos like Feynman (his mathematical ability was genuinely great). His ability to penetrate a problem simply left mere mortals shaking their heads. But that ability to penetrate, as great as it was, and it was great, paled into insignificance compared to Einstein. Einstein simply saw things no one else could.

That comment about the inverse square character of gravity being a dead give away you are dealing with gravity and not acceleration is not quite correct - special distributions of matter will give a constant gravitational field - but they are generally not what is out there. A physicist that finds a constant gravitational field can be pretty sure its the result of acceleration - but not 100% positive.

Glad however you found it interesting and thought provoking.

Yes, it's a very nice book that provides the means for a (hopefully) deeper understanding of this complex subject (even for a foreign avocado like me :shy:). It’s almost 'scary' the way Einstein had an intuitive feeling for what is going on in nature, I mean... the mathematics wasn't really there, and no one had made any attempts like this before, and no experiments to guide – and yet he 'knew' what was right. It's almost as if some guy in 1850, on his own, had come up with QED. Amazing.

I must read the book thoroughly, but until then, I must ask if one could set these 'aliases' for SR and GR:

Special Relativity –– Geometric Lorentz Invariance
General Relativity –– Geometric Lorentz Invariance + General Dynamic Symmetry​

Would that be acceptable?

If this is okay, there is one thing that puzzles me – How can SR & GR be correct at the same time? (Please don’t laugh, I'm probably wrong again... ;)

Let’s make a "conceptual image" (i.e. a heavily simplified version of 3+1 spacetime):

Special Relativity
You're driving a "relativistic hovercraft" on borderless and perfect ice, in complete fog, at fixed speed. The hover is constructed to make no sound or vibrations, and inside the cabin there is no way to tell if you are standing still or moving at high speed. Suddenly another hover passes by, in opposite direction, just outside the window. There is no way for you to decide if the other hover is standing still, or if it is you, or if both are moving. No way. The ice would be the imagined Minkowski space.

(Illegal virtual deer ;)

General Relativity
Same vehicle, same fog; but now the ice has melted and the hover, according to the laws, is now forced to drive thru the water and interact with it, according to the new dynamical metric tensor G, which will make ripples and waves. Suddenly another hover passes by, in opposite direction, just outside the window...

Question: Doesn't the GR setup give us the ability to detect who is moving, if we had sensitive enough instruments? And if this is the case, where is the Geometric Lorentz Invariance then??


NOTE: This is NOT "personal speculations", just a (very dumb) question.


P.S. I wouldn’t have dared to ask this question if it wasn't for J. L. Synge:

. . . I have never been able to understand this principle. . . . Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer's acceleration? If so, it is false. In Einstein's theory, either there is a gravitational field or there is none, according as the Riemann tensor* does or does not vanish. This is an absolute property; it has nothing to do with any observer's worldline. . . . The Principle of Equivalence performed the essential office of midwife at the birth of general relativity. . . . I suggest that the midwife be now buried with appropriate honours and the facts of absolute spacetime be faced.

... am I really getting this right ... ?

 

[bhobba]

Question: Doesn't the GR setup give us the ability to detect who is moving, if we had sensitive enough instruments? And if this is the case, where is the Geometric Lorentz Invariance then??

Well your melted ice would correspond in a rough sort of way to the Higgs field that gives things its mass. It doesn't violate SR. Massive objects accelerating however would create gravitational waves.

... am I really getting this right ... ?

You are on the right track.

Regarding the Synge thing I am in two minds about it. If a physicist doesn't find tidal forces he is correct in concluding it's acceleration and not Gravitation with a bit of a caveat. If you are in the middle of two large vertical slabs of matter you will get a constant gravitational field. But such is not what we generally see out there, so you can say I am pretty sure its not a gravitational field - but pretty sure is not 100% sure.

Food for thought - its a bit of a controversial issue.

Thanks
Bill

 

[atyy]

@DevilsAvocado, regarding Synge.

Yes, of course Synge is right - for one definition of the equivalence principle - the definition which says gravity = acceleration, and that free falling gets rid of the gravitational force. In formal terms, this can be seen from the local cartesian coordinates of a free-falling observer. The spacetime metric in local cartesian coordinates is the same as in flat spacetime for first derivatives, but for second derivatives deviations appear, even at the origin, if spacetime is curved. This is why the equivalence principle is said to only hold locally, for some definition of "locally". Because the second derivative is mathematically "local", for other definitions of "local", the equivalence principle does not hold, not even "locally".

Edit: See WannabeNewton's post below. His $\Gamma^{\mu}_{\nu\delta}$ are what I was thinking about when I loosely said "first derivatives".

But in another sense the equivalence principle holds exactly (as a postulate) in general relativity, if the equivalence principle is stated as the principle of minimal coupling.

 

[WannabeNewton]

P.S. I wouldn’t have dared to ask this question if it wasn't for J. L. Synge:

Is that your question or a quote from Synge's text? Anyways, the term "gravitational field" is not well-defined mathematically in GR. The Riemann tensor does not codify the "gravitational field". It codifies gravitational tidal forces (and tidal torques); it tells us information about shear, vorticity, and expansion of swarms of test particles in free fall (using the exact same mathematical apparatus as fluid mechanics except now in curved space-time). This was actually something Synge first elucidated in his text. The closest thing to the "gravitational field" is the derivative operator $\nabla_{\mu}$ (formally called the Levi-Civita connection); more precisely, the coefficients $\Gamma^{\mu}_{\nu\delta}$ of $\nabla_{\mu}$ are the closest thing to the "gravitational field" and these vanish at the origin of local inertial frames (more precisely: local inertial coordinate systems).

 

[bhobba]

Is that your question or a quote from Synge's text?

Its a quote attributed to Synge from Ohanian.

Ohanian, as you know, is highly critical of basing GR on the principle of equivalence.

Personally I am not that concerned about it, but I am glad I learned GR both ways - from Wald and Ohanian (plus a few others).

It codifies gravitational tidal forces (and tidal torques); it tells us information about shear, vorticity, and expansion of swarms of test particles in free fall (using the exact same mathematical apparatus as fluid mechanics except now in curved space-time). This was actually something Synge first elucidated in his text.

That is the sense I most emphatically agree with Synge.

But people sometimes take it to be an attack on the equivalence principle itself which I think is pushing it a bit far. It seems to me Ohanian's textbook and its whole approach was to skirt this issue by giving it a different foundation. The positive is he does give a good discussion of it.

Landau in his Classical Theory Of Fields is a bit more sanguine about the issue, calling them 'special' gravitational fields.

I had a long chat about this on sci.physics.relativity with genuine GR experts like Steve Carlip many many moons ago now. The upshot was its probably best to learn GR from modern books like Wald rather than ancient texts like Landau

Thanks
Bill

 

[WannabeNewton]

Three cheers for Wald

 

[bhobba]

Three cheers for Wald

Yea - its my favorite GR text.

Thanks
Bill

 

[atyy]

Three cheers for Wald

Even the part where the Christoffel symbols are tensors?

 

[DevilsAvocado]

Yea - Feynman knew nothing

Surely You're Joking, Mr. Bhobba!

Seriously though there has been a lot of work done on renormaliztion by guys like Wilson (who got a Nobel Prize for it) showing that dippy process is not really that dippy after all, but it took a while to sink in, and it's not surprising at the time Feynman wrote that he was of that view:

Well yes, but Wilson did his work in the early seventies and got his NP in 1982 ... and Feynman dropped his "dippy verdict" in 1985 so the "sinking in", later on, must have been 'substantial'...

I think itself consistent, but its an approximation and not fundamental.

I think we are more or less on the same footing. A low-energy effective field theory is an "approximation tool" that gives us the ability to get work done, without worrying about high-energy infinites.

Then maybe it's a matter of "taste" if one regard cutoffs as compatible with self-consistency... but to me it looks like the fine-structure "constant" and the Landau pole put some 'constraint' on this path...

Maybe time will tell.

 

[DevilsAvocado]

Well your melted ice would correspond in a rough sort of way to the Higgs field that gives things its mass. It doesn't violate SR. Massive objects accelerating however would create gravitational waves.

Huum, the Higgs field, that's cool. I wish Einstein was alive today, so many interesting things going on...

You are on the right track.

Thanks! Thinking some more, it looks like the basis for my question was just wrong/naïve.