How much is Special Relativity a needed foundation of General Relativity

In summary: On a background Lorentz spacetime.There are precisely three local geometries we can have on a 4-dimensional pseudo-Riemannian manifold. (the thing we use for space-time in GR)One is that of 4-d Euclidean space.One is that of Minkowski space. (the thing we use for space-time in SR)One corresponds to a signature of (2, 2). (so it's kind of like 2 spatial and 2 temporal...?)
  • #1
lalbatros
1,256
2
If one had to built an invariant theory for gravitation, applicable in any system of coordinate, could it not be possible to create one without knowing about SR (constancy of c, EM, ...).

Could such an off-road journey teach us something, and couldn't SR pop up in some other way?

Thanks for your ideas,

Michel
 
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  • #2
Some approaches to Quantum Gravity are trying to obtain GR in an appropriate continuum limit. One question that comes up... without invoking SR, how could a Lorentzian structure arise?

SR pops up in the tangent space of an event in a GR spacetime manifold.
 
  • #3
lalbatros said:
an invariant theory ... without knowing about SR

What does the word "invariant" mean if you don't know about SR?
 
  • #4
It is always possible to write down a theory in invariant form (though it might be very difficult to do). The most important example is of Newtonian gravity. There is a very elegant formulation of it due mainly to Cartan which is completely covariant.

It turns out that GR actually pops out of this as a very direct generalization. It would actually be unnatural to write down SR as an intermediate step. But of course, it's still in there. The principles of SR are all embedded within GR, so there's no way of completely avoiding it.
 
  • #5
I don't understand the meaning of the word invariant. Invariant with respect to what?
 
  • #7
The 'constancy of c' is not inherent within special relativity, but rather it is a convention used in its standard formulation.
Stingray said:
The principles of SR are all embedded within GR, so there's no way of completely avoiding it.
The 'constancy of c' is not a principle of SR per se, and can be easily avoided if desired.
 
  • #8
actionintegral said:
I don't understand the meaning of the word invariant. Invariant with respect to what?

Invariant with respect to coordinate transformations. The laws of physics can be made to look the same in any coordinate system. Of course doing this requires defining what coordinate transformations mean, etc., which is usually done by giving the theory a geometric structure of some sort. The structure of Newtonian gravity turns out to be more complicated than the structure of general relativity, though it does involve one less parameter (c).
 
  • #9
Aether said:
The 'constancy of c' is not inherent within special relativity, but rather it is a convention used in its standard formulation.
...
The 'constancy of c' is not a principle of SR per se, and can be easily avoided if desired.

...of course, as long as it is replaced by some similar condition, e.g. finite maximum signal speed.
 
  • #10
Aether said:
The 'constancy of c' is not inherent within special relativity, but rather it is a convention used in its standard formulation.The 'constancy of c' is not a principle of SR per se, and can be easily avoided if desired.

I'm not sure what you're trying to say. I meant that SR is just a special case of general relativity, so everything in SR is contained in GR.

Within both special and general relativity, there is an unavoidable constant we call c. Of course it isn't necessary that that parameter has anything to do with electromagnetic phenomena, but experimentally, it does.
 
  • #11
Stingray said:
I'm not sure what you're trying to say. I meant that SR is just a special case of general relativity, so everything in SR is contained in GR.

Within both special and general relativity, there is an unavoidable constant we call c. Of course it isn't necessary that that parameter has anything to do with electromagnetic phenomena, but experimentally, it does.
If you limit this statement to 'round trip average c', then it is a part of the nonconventional content of SR. However, the standard formulation of SR extends this statement to include the one-way istotropy of c and that is not a part of the nonconventional part of SR per se; e.g., that is the conventional part that can be easily avoided if desired.
 
  • #12
actionintegral,

"I don't understand the meaning of the word invariant. Invariant with respect to what?"

I just mean that the laws are the same in any coordinate system.

Michel
 
  • #13
lalbatros said:
I just mean that the laws are the same in any coordinate system.
Michel

The reason I said that was I think "laws" is a synonym for "speed of light".
So SR would be implied in "laws".
 
  • #14
Apparently you can also get GR by looking for a field that describes massless spin-2 particles.
 
  • #15
Thrice said:
Apparently you can also get GR by looking for a field that describes massless spin-2 particles.

On a background Lorentz spacetime.
 
  • #16
There are precisely three local geometries we can have on a 4-dimensional pseudo-Riemannian manifold. (the thing we use for space-time in GR)

One is that of 4-d Euclidean space.
One is that of Minkowski space. (the thing we use for space-time in SR)
One corresponds to a signature of (2, 2). (so it's kind of like 2 spatial and 2 temporal dimensions)
 
  • #17
lalbatros said:
If one had to built an invariant theory for gravitation, applicable in any system of coordinate, could it not be possible to create one without knowing about SR (constancy of c, EM, ...).

Could such an off-road journey teach us something, and couldn't SR pop up in some other way?

Thanks for your ideas,

Michel

Well, to ask whether we needed SR is one thing. But to ask whether we needed an invariant c is another thing altogether. Without invariant c, we are stuck with aether, absolute space, and an independent time. Basically, we would have Newton, where gravity works by magic from afar instantaneously. We'd have no concept of the mechanism which creates the mechanics. Having the correct mechanism then leads to further valid extension of the physics, and unification then becomes more probable.

Maxwell changed it all. His theory had symmetry in it, which required EM to exist at only one rate in vacu. We either ignore this or we don't. If we ignore it, we remain with Newton & Gallileo. But these things cannot be ignored, because it is not in the nature of mankind.

Einstein's Special Theory lead to a number of things, all of which gave the insight to Einstein for his geometric model of space/time & matter/energy. SR lead to Minkowki's notion of a fused spacetime continuum. This allowed Einstein to think in terms of a single spacetime fabric entity. Add the equivalecy principle, providing the insight that the continuum might be warped. Einstein's own E=mc^2 lends support to this notion since it showed that matter is just energy of another form, and the gravity field goes everywhere the mass goes. So gravity wells and rest mass must be mutually coexistent, the rest mass forming at the expense of surrounding medium uniformity. The genius of assuming the medium to support only a speed c change within itself, required gravity to setup, break down, and quake at c ... and so all the limitations of Newton's model are then surpassed as no instantaneous force from afar is required.

Hard to imagine GR in the absence of SR, personally. It's like asking whether SR would have been developed had Maxwell never developed his theory of electromagnetism. Noone would then have believed that Michelson/Morley's null result was anything but a bad test setup. Or if Maxwell could have never developed his theory had Faraday and Gauss never made their contributions first. Or Newton's mechanics in the absence of Gallileo's inertia, gravity, and kinematics.

Had Einstein not existed, we'd have been stuck with Lorentz's aether theory. As close as he was, he fell short. Einstein wasn't stuck on the aether, nor an absolute space. It is possible that Lorentz and Poincare might have eventually got it right, but it may have taken a long time. I doubt anyone would have taken on gravitation though. Einstein was gifted, had keen insight and knew it, was confident as could be, likely spent most his entire life just thinking about these things, and sacroficed his family to do what a group of geniuses were unlikely to even attempt.

That said, my guess is that although many folks would produce many models, none would likely be right had SR not been developed first. If anything close to GR had eventually arisen, I'd bet it would have taken a very very long time with dozens of gifted theoretical physicists to produce a much lesser model, if we were lucky. But then, stranger things have happened :smile:

pess
 
  • #18
pess5 said:
Had Einstein not existed, we'd have been stuck with Lorentz's aether theory.
We are "stuck" with it anyway. To deny this is to deny the nonconventional content of special relativity.
As close as he was, he fell short. Einstein wasn't stuck on the aether, nor an absolute space.
There is no experiment that can distinguish between the view of Lorentz vs. Einstein, and the nonconventional content of special relativity is not only consistent with both, it is inherently dependent on both.
It is possible that Lorentz and Poincare might have eventually got it right, but it may have taken a long time.
Poincare symmetry is the full symmetry of special relativity (according to Wikipedia at least, please cite a better reference if you disagree with this) and not Lorentz symmetry.
 
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  • #19
Aether said:
There is no experiment that can distinguish between the view of Lorentz vs. Einstein, and the nonconventional content of special relativity is not only consistent with both, it is inherently dependent on both.

I'm not familiar with the Lorentz theory. Is there a modern, concise review available?

Is there a Lorentz-based formulation of gravitation which leads to predictions that can be compared to GR?
Is there a Lorentz-based formulation of quantum field theory which leads to predictions that can be compared to standard Quantum Field Theory?
More generally, although (as you claim) no experiment can distinguish between the view of Lorentz vs Einstein, what else can be done with the Lorentz theory [since there's more to the world than special relativity...e.g., gravitation and quantum physics]?
 
  • #20
robphy said:
I'm not familiar with the Lorentz theory. Is there a modern, concise review available?
This is the article that I was implicitly referring to above: In J.A. Winnie, Special Relativity without One-Way Velocity Assumptions: Part I, Philosophy of Science, Vol. 37, No. 1. (Mar., 1970), p. 81 he states: "According to the CS thesis [conventionality of simultaneity], this situation reveals a structural feature of the Special Theory, and thereby of the universe it purports to characterize, which not only makes the one-way speed of light indeterminate, but reveals that its unique determination could only be at the expense of contradicting the nonconventional content of the Special Theory". You can also search for prior posts of mine within this forum for references to Mansouri-Sexl, Zhang, LET, and GGT for extensive discussions and citations on this topic.
Is there a Lorentz-based formulation of gravitation which leads to predictions that can be compared to GR?
LET/GGT and the standard formulation of SR are both special relativity but they are cast in different coordinate systems.
Is there a Lorentz-based formulation of quantum field theory which leads to predictions that can be compared to standard Quantum Field Theory?
I have cited several specific papers on this within this forum, but I think that the most direct evidence may be that the full symmetry of standard Quantum Field Theory is Poincare symmetry rather than Lorentz symmetry. Isn't it?

[add]
Robert M. Wald said:
Major issues of principle with regard to the formulation of the theory arise from the lack of Poincare symmetry and the absence of a preferred vacuum state or preferred notion of ``particles''. -- http://arxiv.org/abs/gr-qc/0608018"
Hmmm...it seems that QFT is lagging behind the times precisely because it lacks Poincare symmetry?

Poincare symmetry allows for both LET/GGT as well as the standard formulation of SR. Lorentz symmetry is a subset of Poincare symmetry that excludes LET/GGT on philosophical grounds, not experimental grounds.[/add]

More generally, although (as you claim) no experiment can distinguish between the view of Lorentz vs Einstein, what else can be done with the Lorentz theory [since there's more to the world than special relativity...e.g., gravitation and quantum physics]?
I am not suggesting that we use Lorentz theory for anything, just pointing out that it is empirically equivalent to the standard formulation of SR; e.g., special relativity is a more general phyiscal theory than just its standard formulation.
 
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  • #21
Hurkyl said:
There are precisely three local geometries we can have on a 4-dimensional pseudo-Riemannian manifold. (the thing we use for space-time in GR)

One is that of 4-d Euclidean space.
One is that of Minkowski space. (the thing we use for space-time in SR)
One corresponds to a signature of (2, 2). (so it's kind of like 2 spatial and 2 temporal dimensions)
This is the one that I am interested in, and I think that I can show at least one physical example that is consistent with two orthogonal dimensions of time (e.g., zitterbewegung). Can you cite any references to a more complete discussion of this?
 
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  • #22
If I am not mistaken, a (2+2)-spacetime [i.e. signature ++--] admits closed timelike curves. In addition, I wonder how Quantum Field Theory would change using SO(2,2).
 
  • #23
Aether said:
This is the one that I am interested in, and I think that I can show at least one physical example that is consistent with two orthogonal dimensions of time (e.g., zitterbewegung). Can you cite any references to a more complete discussion of this?

But we clearly have more than 2 spatial dimensions, so that doesn't describe reality.
 
  • #24
Stingray said:
But we clearly have more than 2 spatial dimensions, so that doesn't describe reality.
No, it is not at all clear that we have more than two spatial dimensions (e.g., see the "holographic principle"). This may seem strange at first, so please be careful to back up what you say by citing a published reference if you wish to dispute this.
 
  • #25
Aether said:
No, it is not at all clear that we have more than two spatial dimensions (e.g., see the "holographic principle"). This may seem strange at first, so please be careful to back up what you say by citing a published reference if you wish to dispute this.

The holographic principle does not say what you think it does. I don't need a published reference to say that we live in (at least) 3 spatial dimensions.
 
  • #26
Stingray said:
The holographic principle does not say what you think it does.
This is what I think that the holographic principle says (in a nutshell), what do you think that it says?
J.D. Bekenstein said:
An astonishing theory called the holographic principle holds that the universe is like a hologram: just as a trick of light allows a fully three dimensional image to be recorded on a flat piece of film, our seemingly three-dimensional universe could be completely equivalent to alternative quantum fields and physical laws "painted" on a distant, vast surface.The physics of black holes--immensely dense concentrations of mass--provides a hint that the principle might be true. -- J.D. Beckenstein, Information in the Holographic Universe, Scientific American:p59, (August 2003).
Stingray said:
I don't need a published reference to say that we live in (at least) 3 spatial dimensions.
Do you disagree with Hurkyl?
Hurkyl said:
There are precisely three local geometries we can have on a 4-dimensional pseudo-Riemannian manifold. (the thing we use for space-time in GR)

One is that of 4-d Euclidean space.
One is that of Minkowski space. (the thing we use for space-time in SR)
One corresponds to a signature of (2, 2). (so it's kind of like 2 spatial and 2 temporal dimensions)
 
  • #27
Aether said:
This is what I think that the holographic principle says (in a nutshell), what do you think that it says?

While I'm not an expert, my understanding is that the holographic principle is a vague idea saying that the quantum mechanical degrees of freedom seem to be less than expected for a 3+1 spacetime. It is largely based on some entropy calculations for black holes, but isn't really well-defined at this point. It suggests that a future formulation of physics may look very different geometrically than our current ones, but who knows what that means.

It does not say that classical physics could be formulated in fewer dimensions with minimal changes. There are clearly 3 spatial dimensions worth of information in classical physics (as we would usually define things), and this cannot all be embedded on a surface without having reconstruction formulae that had extremely odd forms. Try thinking about encoding two dimensions worth of information on a line. It's possible, but highly unnatural. It is possible that it would more natural quantum mechanically, but nobody's sure what that means yet. It would certainly require modifying things by much more than just claiming spacetime had a different signature.

Do you disagree with Hurkyl?

No. Hurkyl was talking about mathematics. If we say the universe is a 4-dimensional pseudo-Riemannian manifold, there are three possible signatures ("local geometries"). Only one has any relation to the observable universe. That statement would be much less strong if we allowed for more dimensions, but that's not what he said.

I think he was also saying that (large parts of) special relativity fall out directly from elegant mathematical structures, so it is interesting and relevant even if you don't care about phyics.
 
  • #28
Stingray said:
It does not say that classical physics could be formulated in fewer dimensions with minimal changes. There are clearly 3 spatial dimensions worth of information in classical physics (as we would usually define things), and this cannot all be embedded on a surface without having reconstruction formulae that had extremely odd forms.
Hey look, is that Stingray doing the backstroke? :biggrin:
Try thinking about encoding two dimensions worth of information on a line. It's possible, but highly unnatural. It is possible that it would more natural quantum mechanically, but nobody's sure what that means yet. It would certainly require modifying things by much more than just claiming spacetime had a different signature.
Ok. I want to seriously study this as a possibility. I do not to claim (yet) that it is a physical certainty, but I won't let others get away with ruling it out without at least citing published references to support their arguments.
No. Hurkyl was talking about mathematics. If we say the universe is a 4-dimensional pseudo-Riemannian manifold, there are three possible signatures ("local geometries"). Only one has any relation to the observable universe.
I will ask you once again, please cite a published reference to support your claim that "only one has any relation to the observable universe".
 
  • #29
Aether said:
I will ask you once again, please cite a published reference to support your claim that "only one has any relation to the observable universe".

For what you quoted this time, I can respond more precisely. But I still can't imagine anyone going to the trouble of publishing anything on this, so I can't quote you anything.

If you take the 4 dimensions to be those of time and space as they're usually understood, we know experimentally that local physics has the geometry of Minkowski space, not Euclidean 4-space. In other words, local physics follows special relativity.

Spacetime does not have 2+2 characteristics either, which is clear from the approximate validity of Euclidean 3-geometry in everyday life. That does not mean that this type of geometry is totally irrelevant in physics. It may find a use at some point, but it will be very different from the way we use 3+1 geometry today.
 
  • #30
Stingray said:
For what you quoted this time, I can respond more precisely. But I still can't imagine anyone going to the trouble of publishing anything on this, so I can't quote you anything.

If you take the 4 dimensions to be those of time and space as they're usually understood, we know experimentally that local physics has the geometry of Minkowski space, not Euclidean 4-space. In other words, local physics follows special relativity.

Spacetime does not have 2+2 characteristics either, which is clear from the approximate validity of Euclidean 3-geometry in everyday life. That does not mean that this type of geometry is totally irrelevant in physics. It may find a use at some point, but it will be very different from the way we use 3+1 geometry today.
What you are claiming is that we can experimentally distinguish between the three options that Hurkyl gave, right? I presume that all three options are allowed under Poincare symmetry, and that no experiment can distinguish between them. Since you can't cite any references to back up your claim, then I will ask you to at least please answer my question directly.
 
  • #31
Aether said:
I presume that all three options are allowed under Poincare symmetry, and that no experiment can distinguish between them.

No. Hurkyl's first choice has (local) Euclidean symmetry. His second choice has local Lorentz symmetry. I don't know of a name for the last possibility, but it is also different.
 
  • #32
Stingray said:
No. Hurkyl's first choice has (local) Euclidean symmetry. His second choice has local Lorentz symmetry. I don't know of a name for the last possibility, but it is also different.
His second choice has Poincaré symmetry, and only takes on Lorentz symmetry if we arbitrarily assume that the one-way speed of light is generally isotropic; this is a convention, and isn't required. I'm not sure about the other two yet.

Wikipedia said:
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime. -- http://en.wikipedia.org/wiki/Poincaré_group
 
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  • #33
Aether said:
His second choice has Poincaré symmetry, and only takes on Lorentz symmetry if we arbitrarily assume that the one-way speed of light is generally isotropic. I'm not sure about the other two yet.

No. His second choice is not necessarily Minkowski space itself, but something with signature -+++. That only has local Lorentz symmetry in general. Poincare symmetry is more general, but only works in Minkowski space.
 
  • #34
Stingray said:
No. His second choice is not necessarily Minkowski space itself, but something with signature -+++. That only has local Lorentz symmetry in general. Poincare symmetry is more general, but only works in Minkowski space.
He said "One is that of Minkowski space. (the thing we use for space-time in SR)", but he did not say that the signature was -+++, you are assuming that.
 
  • #35
Stingray said:
I don't know of a name for the last possibility
You'd probably just name it explicitly, and say it has O(2, 2) symmetry, I suppose. Maybe "generalized orthogonal symmetry" makes sense?


Aether said:
His second choice has Poincare symmetry, and only takes on Lorentz symmetry if we arbitrarily assume that the one-way speed of light is isotropic. I'm not sure about the other two yet.
The Poincaré group includes the Lorentz group -- anything with Poincaré symmetry automatically has Lorentz symmetry.


if we arbitrarily assume that the one-way speed of light is isotropic
And we assume no such thing. The Lorentz group is (isomorphic to) O(1, 3), which is by definition the group of transformations that preserve a +--- metric. (and, of course, also preserve a -+++ metric)

The isotropy of one-way speed of light is a condition on your coordinate charts, not a condition on the geometry.

He said "One is that of Minkowski space. (the thing we use for space-time in SR)", but he did not say that the signature was -+++, you are assuming that.
The three cases I listed are the three possible (equivalence classes of) metric:

Euclidean: ++++ or ----
Minkowski: +--- or +++-
The other: ++--

Every pseudoRiemannian metric falls into one of those 5 cases.
 
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