Unsimultaneity and quantum physics

[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.

 

[DevilsAvocado]

@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.

Yep, it looks like the second derivative (acceleration) is a 'multifaceted' property...

 

[DevilsAvocado]

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

Good point and bad reference by me, it should have been:

(page 40 in the 1976/1979 edition)

The confusion surrounding the principle of equivalence led J. L. Synge to remark:
. . . I have never been able to understand this principle. . . .

Anyways, the term "gravitational field" is not well-defined mathematically in GR. The Riemann tensor does not codify the "gravitational field".

The more I read about this, the more complex it seems... I'm definitely not in a position to judge, but it looks like there is a Theory of the Gravitation-Field in Einstein's Foundation of the Generalised Theory of Relativity...

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).

Ahh! Now I see, you mean that the gravitational field (as in electromagnetic field) is not well-defined in its role as mediator of gravitation, as a (direct) force, right? We 'just' get the curved/deformed spacetime, whereas the electromagnetic field is "self-sufficient" in the carrying of electric charges, right? And that gravity in GR is a "fictitious force", or just a 'consequence' of matter/energy following geodetic lines (= no gravitational force), right?

P.S: I like the "fluid part", goes well with my hover!

 

[atyy]

Yep, it looks like the second derivative (acceleration) is a 'multifaceted' property...

Actually, the first derivative is already "local" and "nonlocal". All the derivatives are "local" in the mathematical sense of the limit existing at a point. They are nonlocal in the sense that are first derivative notionally involves the difference between values at two points. In this notional sense, the second derivative is more nonlocal than the first derivative. This is why in physics talk about classical GR, the validity of "free fall = no gravity" provided one ignores "nonlocal measurements" is the same as being able to ignore "second derivatives".

Incidentally, the EP-like idea "acceleration = gravity" is true in the sense that acceleration causes even the first derivatives of the local cartesian coordinates to deviate from flat coordinates. Roughly "acceleration" is "fake gravity" involving the Christoffel symbols (first derivatives of the metric) that WannabeNewton mentioned, and "true gravity" involves the Riemann curvature tensor (second derivatives of the metric).

 

[DevilsAvocado]

(If I'm derailing this thread, please just tell me to stop... )

@WannabeNewton, atyy, bhobba

Maybe it's safest to explain my "curvature outburst" in post #41 ...

First of all – Not Even GNORW!

The naïve 'concept' was this; SR eradicated absolute space/ether and introduced the mathematical abstract flat Minkowski space. GR expanded this 3+1 spacetime with curvature, in the form of the dynamical metric tensor, however still abstract and relative.

Well eh... I just thought the curvature would be something to "hold on to" in finding out "where" you are... in the same way as it's easier to navigate thru Rocky Mountains than the Libyan Desert...

But (of course!) the curved spacetime in GR is still not absolute, and if we imagine two very massive objects (instead of hovercrafts) passing each other in empty space, they will just "drag" their "spacetime bulge" along with them. So when they meet, they will only notice "another bulge" passing by, and no one can say who is moving.

(perfectly clear within the first minute of this video)
https://www.youtube.com/watch?v=MTY1Kje0yLg

What about moving gravitational waves?

Afaik, if we exchange the two objects for two black holes – nothing would change - they will also "drag" their gravitational waves along with them.

What about acceleration?

Well, one obviously will feel the acceleration as a "fictitious force", and others obviously could tell that you are the one moving (in acceleration). End of story.

Almost ...

I found this very fascinating historical paper (that maybe could save my face to some degree): Ether and the Theory of Relativity, Albert Einstein (May 5th, 1920)

My jaw dropped in exponentially for every emphasis (of mine) below.

[13]
More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, i.e. we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it. We shall see later that this point of view, the conceivability of which I shall at once endeavour to make more intelligible by a somewhat halting comparison, is justified by the results of the general theory of relativity.
[14]
Think of waves on the surface of water. Here we can describe two entirely different things. Either we may observe how the undulatory surface forming the boundary between water and air alters in the course of time; or else — with the help of small floats, for instance — we can observe how the position of the separate particles of water alters in the course of time. If the existence of such floats for tracking the motion of the particles of a fluid were a fundamental impossibility in physics — if, in fact, nothing else whatever were observable than the shape of the space occupied by the water as it varies in time, we should have no ground for the assumption that water consists of movable particles. But all the same we could characterise it as a medium.
[24]
Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.

Phew... it looks like I was not completely way out in my "curvy imaginations"... even though it doesn't work...

Thank you Albert!

 

[DevilsAvocado]

Actually, the first derivative is already "local" and "nonlocal". All the derivatives are "local" in the mathematical sense of the limit existing at a point. They are nonlocal in the sense that are first derivative notionally involves the difference between values at two points. In this notional sense, the second derivative is more nonlocal than the first derivative. This is why in physics talk about classical GR, the validity of "free fall = no gravity" provided one ignores "nonlocal measurements" is the same as being able to ignore "second derivatives".

Incidentally, the EP-like idea "acceleration = gravity" is true in the sense that acceleration causes even the first derivatives of the local cartesian coordinates to deviate from flat coordinates. Roughly "acceleration" is "fake gravity" involving the Christoffel symbols (first derivatives of the metric) that WannabeNewton mentioned, and "true gravity" involves the Riemann curvature tensor (second derivatives of the metric).

Thanks for the explanation atyy!

 

[WannabeNewton]

Even the part where the Christoffel symbols are tensors?

*Runs away*

And that gravity in GR is a "fictitious force", or just a 'consequence' of matter/energy following geodetic lines (= no gravitational force), right?

Right. It's good that you put "fictitious force" in quotes because one must use the term in a liberal fashion when speaking of the "gravitational force" in GR. It isn't the exact same thing as the inertial forces from mechanics (in particular the centrifugal and Coriolis forces). See also exercise 16.5 in MTW.

Take note of the following very important quote from MTW (first two pragraphs of section 16.5):

" 'I know how to measure the electromagnetic field using test charges; what is the analogous procedure for measuring the gravitational field?' This question has, at the same time, many answers and none.

It has no answers because nowhere has a precise definition of the term 'gravitational field' been given-nor will one be given. Many different mathematical entities are associated with gravitation: the metric, the Riemann curvature tensor, the Ricci curvature tensor, the curvature scalar, the covariant derivative, the connection coefficients etc. Each of these plays an important role in gravitation theory, and none is so much more central than the others that it deserves the name 'gravitational field'...Another, equivalent term used for them is the 'geometry of spacetime.' "

P.S: I like the "fluid part", goes well with my hover!

What's a hover ?

 

[atyy]

*Runs away*

Actually, that's one of my favourite parts

 

[DevilsAvocado]

Right. It's good that you put "fictitious force" in quotes because one must use the term in a liberal fashion when speaking of the "gravitational force" in GR. It isn't the exact same thing as the inertial forces from mechanics (in particular the centrifugal and Coriolis forces). See also exercise 16.5 in MTW.

Thanks! Finally I got something right in this "curvature-roller-coaster"!

What's a hover ?

Thank god... you've missed the mess of too big thoughts/pictures in post #41... Don't look!