How much is Special Relativity a needed foundation of General Relativity

[lalbatros]

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

 

[robphy]

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.

 

[actionintegral]

an invariant theory ... without knowing about SR

What does the word "invariant" mean if you don't know about SR?

 

[Stingray]

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.

 

[actionintegral]

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

 

[Daverz]

This might be of interest:

http://adsabs.harvard.edu/abs/1974PhRvD..10.2330W

 

[Aether]

The 'constancy of c' is not inherent within special relativity, but rather it is a convention used in its standard formulation.

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.

 

[Stingray]

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).

 

[robphy]

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.

 

[Stingray]

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.

 

[Aether]

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.

 

[lalbatros]

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

 

[actionintegral]

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".

 

[Thrice]

Apparently you can also get GR by looking for a field that describes massless spin-2 particles.

 

[Daverz]

Apparently you can also get GR by looking for a field that describes massless spin-2 particles.

On a background Lorentz spacetime.

 

[Hurkyl]

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)

 

[pess5]

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

pess

 

[Aether]

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.

 

[robphy]

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]?

 

[Aether]

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]

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.

 

[Aether]

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?

 

[robphy]

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).

 

[Stingray]

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.

 

[Aether]

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.

 

[Stingray]

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.

 

[Aether]

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?

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).

I don't need a published reference to say that we live in (at least) 3 spatial dimensions.

Do you disagree with Hurkyl?

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)

 

[Stingray]

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.

 

[Aether]

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?

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".

 

[Stingray]

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.

 

[Aether]

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.

 

[Stingray]

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.

 

[Aether]

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.

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

 

[Stingray]

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.

 

[Aether]

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.

 

[Hurkyl]

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?

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.

 

[Aether]

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.

Were you implying by "Minkowski space-time" that we were restricted to the Lorentz group or the Poincaré group? Poincaré symmetry is the full symmetry of special relativity, and not Lorentz symmetry. I presume that the Poincaré group includes the LET/GGT transformations as well as the Lorentz transformations.

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.

Neither of these is the LET/GGT metric. It has non-zero off-diagonal terms.

 

[Hurkyl]

Neither of these is the LET/GGT metric. It has diagonal terms.

Sorry; I forgot to say that those were the signatures of the metric. (i.e. the number of positive and negative eigenvalues it has)

I presume that the Poincaré group includes the LET/GGT transformations as well as the Lorentz transforations.

I don't know the LET/GGT transformations. (by name)

The Lorentz group is precisely the group of linear transformations that preserve a metric with signature +---. The Poincaré group is that plus translations; i.e. all affine transformations that preserve a metric with signature +---. Does that help?

Were you implying by "Minkowski space-time" that we were restricted to the Lorentz group or the Poincaré group? Poincaré symmetry is the full symmetry of special relativity, and not Lorentz symmetry.

Just the Lorentz; translations don't make sense in this context. Only the "local geometry" is Minkowski; a translation is not local since it moves us from here to there. Even "infinitessimal" translations are problematic, since the curvature of space-time plays a role there, even locally.

 

[Aether]

I don't know the LET/GGT transformations. (by name)

I have a paper on GGT that shows the metric, I could show that tomorrow if necessary; as I recall, it has zeros in the diagonal, and functions of \beta in the off-diagonal terms.

The Lorentz group is precisely the group of linear transformations that preserve a metric with signature +---. The Poincaré group is that plus translations; i.e. all affine transformations that preserve a metric with signature +---. Does that help?

I don't know what the signature of the LET/GGT metric is; it has zeros in the diagonal, so does that give it a signature of 0000?

Just the Lorentz; translations don't make sense in this context. Only the "local geometry" is Minkowski; a translation is not local since it moves us from here to there.

What about round-trips?

Even "infinitessimal" translations are problematic, since the curvature of space-time plays a role there, even locally.

Aren't we assuming flatness above? Are the metric signatures still the same if we don't assume flatness?

 

[Stingray]

I have a paper on GGT that shows the metric, I could show that tomorrow if necessary; as I recall, it has zeros in the diagonal, and functions of \beta in the off-diagonal terms.I don't know what the signature of the LET/GGT metric is; it has zeros in the diagonal, so does that give it a signature of 0000?

I don't know what LET/GGT is either. I've never heard of it. But the signature refers to the signs of the metric's eigenvalues. It is actually possible to have some vanishing eigenvalues, which would expand the possibilities beyond what Hurkyl mentioned.

But that gets you very different structure from SR/GR. And regardless, no meaningful metric could have all of its eigenvalues vanish. Then it would just be zero. So whatever you're talking about does not have signature 0000.

The signature of the metric has nothing to do with flatness. Any metric with signature -+++ can be transformed to Minkowski spacetime at a given point. But away from it, there will of course be differences. That's what's meant by "locally Minkowski."

 

[Aether]

I don't know what LET/GGT is either. I've never heard of it.

It is empirically equivalent to the standard formulation of SR, but maintains absolute simultaneity.

The signature of the metric has nothing to do with flatness. Any metric with signature -+++ can be transformed to Minkowski spacetime at a given point. But away from it, there will of course be differences. That's what's meant by "locally Minkowski."

Does a metric mean anything at a given point? Are we talking about approaching a point without actually reaching it?

 

[Stingray]

It is empirically equivalent to the standard formulation of SR, but maintains absolute simultaneity.

I can't see how that's possible unless it is a whole new theory that looks nothing like SR. But ok.

Does a metric mean anything at a given point? Are we talking about approaching a point without actually reaching it?

Besides the fact that a metric can always be made Minkowski at a point, its first partial derivatives can also be made to vanish at that point. That means that any -+++ spacetime looks nearly Minkowski in sufficiently small regions. Physically, this is basically what is interpreted as the equivalence principle.

 

[Aether]

I can't see how that's possible unless it is a whole new theory that looks nothing like SR. But ok.

It has the same nonconventional content as the standard formulation of SR, but with a different simultaneity convention. See the reference(s) that I gave above. It comes down to this: isotropy of the two-way speed of light is verifiable with a Michelson interferometer, this is a part of the nonconventional content of SR; but, isotropy of the one-way speed of light is not verifiable by any experiment. The standard formulation of SR assumes that the one-way speed of light is generally isotropic and that simultaneity is relative; LET/GGT assumes that the one-way speed of light is only isotropic in one locally preferred inertial reference frame, and that simultaneity is absolute. These two views are empirically equivalent.

Besides the fact that a metric can always be made Minkowski at a point, its first partial derivatives can also be made to vanish at that point. That means that any -+++ spacetime looks nearly Minkowski in sufficiently small regions. Physically, this is basically what is interpreted as the equivalence principle.

Ok, in "sufficiently small regions", but "at a point"? A metric needs a linear "space" to have any meaning, right?

 

[Hurkyl]

What about round-trips?

A round trip induces a Lorentz transformation. Of course, one cannot physically make a round trip. (that would require one to be able to travel stationary or backwards in time) However, we can compare two trajectories that have identical starting and ending points in space-time, and compare those.

translations don't make sense in this context

I'm wrong, I think. I made a mistranslation between the language of tangent spaces and the language of infinitessimal neighborhoods. I'm significantly more comfortable with the former language.

Aren't we assuming flatness above? Are the metric signatures still the same if we don't assume flatness?

The mental problem I was having is that an "infinitessimal translation" in space-time generally involves both a translation and an "infintessimal rotation" of the "local geometry". (By rotation, here, I mean element of the Lorentz group, so that includes boosts)

But, if you're willing to ignore the infinitessimal rotations, then infinitessimal translations do look like translations.

Does a metric mean anything at a given point? Are we talking about approaching a point without actually reaching it?

In differential geometry, yes. The (pseudo)metric allows us to define a differential 2-form. (which, by abuse of notation, we call "the metric") This 2-form acts as an inner product on the tangent spaces.

If we're just working in affine space (such as Euclidean or Minkowski geometry), we typically talk about metrics that can be defined by an inner product. Again, it is common to abuse notation and call the inner product (or the matrix defining it) "the metric".

I don't know what the signature of the LET/GGT metric is; it has zeros in the diagonal, so does that give it a signature of 0000?

If you want to just read the eigenvalues off of a matrix, you have to diagonalize it first. But, since it's supposed to bear resemblance to SR, it will almost certainly be +--- or -+++. I think it's enough to say that your metric is 0 only along lightlike paths; is that the case?

 

[Hurkyl]

but, isotropy of the one-way speed of light is not verifiable by any experiment.

Yes it is. If you can determine the coordinates of an event in a particular coordinate chart, you can determine the one-way speed of light according to that coordinate chart.

If light follows null paths, then light is isotropic in a coordinate chart if and only if its axes are orthonormal.

 

[Stingray]

Ok, in "sufficiently small regions", but "at a point"? A metric needs a linear "space" to have any meaning, right?

That linear space is the tangent space. There is one associated with every point. So yes, metrics are meaningful at a point.

 

[Hurkyl]

If I didn't make it clear earlier, tangent spaces are the way to make rigorous the informal notion of "infinitessimal neighborhood".

 

[Hurkyl]

I don't know what the signature of the LET/GGT metric is

Oh, computing it shouldn't be that hard. You have a matrix right? Just think back to your linear algebra days, and find the eigenvalues.

If you've forgotten, the eigenvalues of A are the solutions to the equation:

Det(A - xI) = 0

The left hand side, once computed, will be a polynomial in x. (I is the identity matrix) A change of basis doesn't change the eigenvalues of a matrix, so it will help to choose a basis in which A has a simpler form.

 

[DrGreg]

I don't know what LET/GGT is either. I've never heard of it.

Aether is referring to co-ordinates T, X, Y, Z defined as

T = t + vx/c^2
X = x
Y = y
Z = z


where t, x, y, z are the standard SR co-ordinates, and v is the velocity (assumed to be in the x direction for simplicity) of the observer relative to some fixed reference frame ("the aether").

These coordinates are not orthogonal. Clocks that are synchronised in this frame are also synchronised in the ether frame, so simultaneity is absolute. In these coordinates the metric is
ds^2 = dX^2 / \gamma^2 + 2 v dXdT + dY^2 + dZ^2 - c^2 dT^2

To determine the signature you need to diagonalise the matrix and once you've done that you are back in orthogonal SR coordinates!

(Why isn't TEX working?)

 

[Aether]

If you want to just read the eigenvalues off of a matrix, you have to diagonalize it first. But, since it's supposed to bear resemblance to SR, it will almost certainly be +--- or -+++.

To determine the signature you need to diagonalise the matrix and once you've done that you are back in orthogonal SR coordinates!

I think this is right. An LET/GGT matrix is just what you get when you work in someone else's (preferred) inertial frame rather than your own so that you are both using their definition of simultaneity; LET/GGT and the standard formulation of SR do transform into one another. By diagonalizing these matrices were able to see the nonconventional geometric content of the theory, and the difference between the matrix before and after it has been diagonalized is the conventional part?

but, isotropy of the one-way speed of light is not verifiable by any experiment.

Yes it is. If you can determine the coordinates of an event in a particular coordinate chart, you can determine the one-way speed of light according to that coordinate chart.

If light follows null paths, then light is isotropic in a coordinate chart if and only if its axes are orthonormal.

But that isn't an experiment, that's a coordinate chart.


Hurkyl and DrGreg, recalling that we had https://www.physicsforums.com/showpost.php?p=731501&postcount=14" conversation over a year ago, it now seems to me that LET/GGT and the "holographic principle" (possibly embodied under a complexified SO(2,2) symmetry) are the key concepts that I was grappling for there.

Stingray, Paul Dirac's assertion there that "the theoretical velocity in the above conclusion is the velocity at one instant of time while observed velocities are always average velocities through appreciable time intervals" indicates that the tangent space that we are most familiar with is an illusion constructed from the "average velocities through appreciable time intervals" while the real tangent space should be constructed from "the velocity at one instant of time". At least in the case of a single particle, our illusion of three observed spatial dimensions is apparently constructed by time-averaging rapidly oscillating "velocites through appreciable time intervals".

 

[Stingray]

So the LET/GGT thing is basically just using a special coordinate system, and claiming that people should base definitions on that system. It seems pretty arbitrary to me, but is in any case identical to special relativity. It's just Minkowski spacetime apparently.

Stingray, Paul Dirac's assertion there that "the theoretical velocity in the above conclusion is the velocity at one instant of time while observed velocities are always average velocities through appreciable time intervals" indicates that the tangent space that we are most familiar with is an illusion constructed from the "average velocities through appreciable time intervals" while the real tangent space should be constructed from "the velocity at one instant of time". At least in the case of a single particle, our illusion of three observed spatial dimensions is apparently constructed by time-averaging rapidly oscillating "velocites through appreciable time intervals".

Of course there will always be a disconnect between theoretical and experimental physics. Real experiments are not infinitely precise, and are always averaging in some sense. Because of that, you might prefer to formulate physical laws from the viewpoint of distribution theory (which was pioneered by Dirac).

But regardless, tangent spaces are mathematical constructs. If you have a manifold, you have a tangent space at each point on it. The only relation to physics is in assuming that spacetime can be modeled as a manifold.

I don't understand your last sentence.