How much is Special Relativity a needed foundation of General Relativity

[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.

 

[Aether]

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.

Nobody is claiming that people should base definitions on the LET/GGT system. As I said https://www.physicsforums.com/showpost.php?p=1154714&postcount=20":

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.

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).

He isn't talking about the precision of measurements there, what he is saying (point blank) is that the classical velocities of particles that we think that we are observing aren't real at all, they are synthesized from something that is always moving at exactly c but having a direction that is changing so rapidly that the average velocity over a long period seems to be a lower number.

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".

I don't understand your last sentence.

The velocity of a classical particle is the average of something that is always moving at c, but rapidly changing direction; it (classical velocity) does not really exist at any given instant. I am distiguishing between the instantaneous coordinates of this "thing", and the illusory coordinates obtained from averaging the instantaneous coordinates over a long time period. The holographic principle teaches that:

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 think that these two concepts may be related, and am looking for the right way to model this.

 

[Hurkyl]

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

But we can measure the coordinates of an event in a coordinate chart. And if we can measure coordinates, we can experimentally determine whether or not the one-way speed of light is isotropic in that chart.

 

[joruz1]

So... You mean time is speeding up too?

 

[Aether]

But we can measure the coordinates of an event in a coordinate chart. And if we can measure coordinates, we can experimentally determine whether or not the one-way speed of light is isotropic in that chart.

https://www.physicsforums.com/showpost.php?p=1154714&postcount=20" is what I am talking about:

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".

If what you are talking about isn't a "unique determination" of the one-way speed of light, then it isn't an experimentally determined quantity; e.g., the quantity is pre-determined by your choice of coordinates. For example, isotropy of the two-way speed of light is actually measurable using a Michelson interferometer, but isotropy of the one-way speed of light isn't actually measurable by anything (unless that thing is "at the expense of contradicting the nonconventional content of the Special Theory") because to actually do that you have to synchronize two clocks at two different locations. Any such synchronization that is not "at the expense of contradicting the nonconventional content of the Special Theory" is arbitrary; this is what is implied by the "conventionality of simultaneity".

So... You mean time is speeding up too?

Time is what a clock measures, and in http://physics.nist.gov/cuu/Units/current.html" the "second" is defined as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom." It doesn't mean anything to say that "time is speeding up" unless you propose a different kind of clock with which to compare the tick-rate of an atomic clock. You could then ask "is the tick-rate of a cesium 133 atomic clock speeding up with respect to my different kind of clock?", and actually get an answer by comparing the atomic clock to your different kind of clock.

"www.phys.unsw.edu.au/astro/research/PWAPR03webb.pdf"[/URL] is a discussion of some experiments to compare the tick-rate of different atomic clocks.

[QUOTE=John Webb]The European Space Agency has plans to fly an atomic-clock experiment – called the Atomic Clock Ensemble in Space (ACES) – on the International Space Station. In addition to various tests of general relativity, ACES will be 100 times more sensitive to changes in α than terrestrial experiments. ACES will comprise two atomic clocks: a cesium clock called PHARAO (see photograph) built by a team led by Christophe Salomon of the ENS and Andre Clairon of the Observatoire de Paris, and a hydrogen maser built by Alain Jornod of the Observatoire Cantonal de Neuchâtel in Switzerland.[/QUOTE]

 

[DrGreg]

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".

Let me clarify this (from an old thread). If you measure the velocity of an object as \delta x / \delta t as \delta t tends to zero, Heisenberg's uncertainty principle gets in the way. The more accurately you measure x (as \delta x tends to zero), the less accurately can you measure momentum. In the limit, the momentum tends to infinity (implying a velocity of c).

Aether seems to think this means the object really is traveling at c, in a rapidly changing direction that averages out to the measured velocity. I would say that is a misinterpretation of quantum theory.

 

[Aether]

Let me clarify this (from an old thread). If you measure the velocity of an object as \delta x / \delta t as \delta t tends to zero, Heisenberg's uncertainty principle gets in the way. The more accurately you measure x (as \delta x tends to zero), the less accurately can you measure momentum. In the limit, the momentum tends to infinity (implying a velocity of c).

Some of the concepts that you have referred to above (e.g., to "measure the velocity of an object", "the less accurately can you measure momentum", and "a velocity of c") are coordinate-system dependent. Isn't how one is obliged to interpret Heisenberg's uncertainty principle determined by their choice of coordinate system?

Aether seems to think this means the object really is traveling at c, in a rapidly changing direction that averages out to the measured velocity. I would say that is a misinterpretation of quantum theory.

That is how I interpreted what Paul Dirac said, but I suspect that interpretations may differ according to one's choice of coordinate system. Do you think that the difference between your interpretation of quantum theory and mine might ultimately be reduced to a difference in our choice of coordinate-systems?

If http://www.phys.unsw.edu.au/astro/research/PWAPR03webb.pdf" experiments confirm that the fine-stucture constant really does vary in time, then we will be able to foliate our pseudo-Riemannian manifold into hypersurfaces of constant \alpha. This immediately establishes a locally preferred definition of simultaneity, and falsifies the principle of the conventionality of simultaneity. How would we then be obliged to interpret Heisenberg's uncertainty principle and quantum theory?

1. SO(4,0) symmetry is not consistent with the Michelson-Morley experiment;
2. SO(3,1) symmetry does not admit a locally preferred frame, so it isn't viable if \alpha varies with time;
3. I think that it may be possible to show that SO(2,2) symmetry is consistent with the Michelson-Morley experiment, admits a locally preferred frame, and admits a geometric interpretation of Heisenberg's uncertainty principle that is consistent with the holographic principle;

 

[DrGreg]

Aether,

It's true that values of velocity and momentum depend the coordinate system you use. But the concepts of distance tending to zero or momentum tending to infinity are the same in both S.R. coordinates and ether coordinates. So I don't think the choice of coordinate system affects the argument much.

Bear in mind that in ether coordinates there is no unique speed of light, as light is no longer isotropic except in the ether frame. So what exactly do you mean by c? When I say c, I mean the speed of light (in vacuum) measured in the ether frame or in any S.R. inertial frame.

I know enough about quantum theory to realize that there's a lot I don't know. You might like to try posting a suitable question to the Quantum Physics forum on this site.

I find it hard to imagine what 2-dimensional time might be (which seems to be what you imply by SO(2,2) symmetry).

 

[Hurkyl]

This immediately establishes a locally preferred definition of simultaneity

In what sense is it preferred? Picking a direction based on the direction of changing alpha seems analogous to picking a direction on the surface of the Earth based upon which way the land is sloping, or which way the magnetic field lines point. In other words, it's a convenient definition, rather than being somehow "preferred".

SO(3,1) symmetry does not admit a locally preferred frame

SO(3, 1) symmetry means that local consideration of space-time itself cannot pick out a preferred frame. It says nothing about whether or not you can prefer something based on other considerations.

 

[Aether]

It's true that values of velocity and momentum depend the coordinate system you use. But the concepts of distance tending to zero or momentum tending to infinity are the same in both S.R. coordinates and ether coordinates. So I don't think the choice of coordinate system affects the argument much.

Exercising the freedom to choose one's own coordinate system doesn't affect the argument as long as simultaneity is merely conventional, but this freedom no longer exists if there is a locally preferred definition of simultaneity. In that case all velocities are demonstrably absolute, and then the question arises: "...all velocities are demonstrably absolute with respect to what?".

Bear in mind that in ether coordinates there is no unique speed of light, as light is no longer isotropic except in the ether frame. So what exactly do you mean by c? When I say c, I mean the speed of light (in vacuum) measured in the ether frame or in any S.R. inertial frame.

c is the isotropic round-trip speed of light in both SR and LET/GGT, and it is the isotropic one-way speed of light in the locally preferred frame of LET/GGT. c is also the rate at which the radius of our causally connected space expands, and this helps define the time-coordinate in SO(3,1)...I suppose that this defines a substantially similar time-coordinate in SO(2,2), and I am using the existence of a locally preferred frame to give this time coordinate an explicit geometric meaning; e.g., this coordinate defines a distant vast spherical surface. The two spatial coordinates are latitudes and longitudes on that surface.

I know enough about quantum theory to realize that there's a lot I don't know.

Me too.

You might like to try posting a suitable question to the Quantum Physics forum on this site.

I may go there eventually.

I find it hard to imagine what 2-dimensional time might be (which seems to be what you imply by SO(2,2) symmetry).

I will speculate here a little to help you imagine what (I think) 2-dimensional time might be like: Consider the raster-scanned image on a CRT monitor or TV screen for example. A 3D image is re-constructed as pixels having coordinates (x,y,t) that are encoded within a serial data stream; the (x,y) coordinates on the screen are mapped to synchronized cyclic time coordinates within the serial data stream.

If the instantaneous value of a first time-coordinate represents the radial velocity c of a hypothetical object located on a distant vast spherical surface pointing in a direction (theta, phi), then the instantaneous value of a second time-coordinate might represent an extremely large tangential velocity (4*pi*R*mc^2/hbar) for this object. Integrate these two velocities over absolute time to get instantaneous positions. Yes, the holographic principle does imply that there is some wildness going on under the hood of our manifold.

This immediately establishes a locally preferred definition of simultaneity

In what sense is it preferred?

It is locally preferred in the sense that all observers can agree on this definition of simultaneity, and actually realize it within a laboratory, using only local physical properties; e.g., no exchange of photons, etc.. It is also convenient in cosmology because we get information about the local value of alpha from distant quasars.

Picking a direction based on the direction of changing alpha seems analogous to picking a direction on the surface of the Earth based upon which way the land is sloping, or which way the magnetic field lines point. In other words, it's a convenient definition, rather than being somehow "preferred".

Alpha is the most fundamental of the physical "constants", it is dimensionless and can be measured in a coordinate-system independent way without reference to any particular system of units. If it turns out that alpha does not vary in time, then the principle of the conventionality of simultaneity (CS) is safe. However, if it does turn out to vary in time then CS is falsified. If you don't agree that this would falsify CS, then please give an example of a real experiment that could falsify it.

SO(3, 1) symmetry means that local consideration of space-time itself cannot pick out a preferred frame. It says nothing about whether or not you can prefer something based on other considerations.

Would you agree that SO(3,1) symmetry beats SO(4,0) symmetry in view of the Michelson-Morley experiment? Is that experiment an example of what you mean by a "local consideration of space-time itself"? Doesn't SO(3,1) symmetry stand or fall with the CS principle in the same way that SO(4,0) symmetry stands or falls with the principles of time dilation and length contraction? I suppose that a complexified SO(3,1) symmetry might admit a locally preferred definition of simultaneity without also implying the holographic principle.

 

[Hurkyl]

but this freedom no longer exists if there is a locally preferred definition of simultaneity.

Why not? The ability to talk about absolute simultaneity does not force you to abandon the notion of relative simultaneity.

c is the isotropic round-trip speed of light in ... SR

Only in certain frames.

and this helps define the time-coordinate in SO(3,1)

SO(3, 1) is a symmetry group; it doesn't have a time-coordinate. As for Minkowski spacetime (a.k.a. 3+1-dimensional spacetime), all the light-cones tell you is that (for an orthonormal basis) the time axis must lie inside the cones.

I suppose that this define a substantially similar time-coordinate in SO(2,2)

Again, SO(2, 2) doesn't have a time-coordinate. You mean 2+2 spacetime.

It is locally preferred in the sense that all observers can agree on this definition of simultaneity, and actually realize it within a laboratory, using only local physical properties

That's not very special. For example, any bit of matter in the universe allows you to give a local definition of simultaneity, and all observers will agree upon that definition.

Would you agree that SO(3,1) symmetry beats SO(4,0) symmetry in view of the Michelson-Morley experiment?

No. The hypothesis of 4+0 space fails because there is an observable geometric difference between "forward in time" and, say, "North".

(the symmetry group of pre-relativistic mechanics is not SO(4, 0))

Is that experiment an example of what you mean by a "local consideration of space-time itself"?

When I say that, I mean experiments that (attempt to) involve only geometric things, like lengths and angles. In particular, they do not involve non-geometric things like the observed matter distribution, CMB temperature, or the local values of (non-geometric) constants like alpha.