How much is Special Relativity a needed foundation of General Relativity

[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.

 

[Aether]

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.

What we seek are the ultimate physical laws that we can validly extrapolate across all space and time. The laws of physics currently assume that the fine-structure constant will be measured to have the same value regardless of time, location, or relative velocity. If the fine-structure constant is confirmed to vary in time (it has already been found to vary in time), then new laws of physics will replace the old laws of physics in order to account for its variation in time. If convenient we may still choose to use the old laws of physics in some circumstances just as we often choose to use Newton's laws today, but we can't validly extrapolate these old laws to extreme conditions because they are known to diverge from reality under extreme conditions.

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

Only in certain frames.

No, this is a coordinate-system independent parameter. The SI base units of time and length are somewhat arbitrary, and these base units themselves may vary over time as a function of the fine-structure constant, but the round-trip speed of light will otherwise be measured to have the same isotropic value in any inertial frame.

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

Ok.

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.

Lorentz covariance (e.g., the basic principle that the laws of physics are invariant under a shift of inertial reference frames) is not valid unless $$\alpha$$ itself is invariant under any shift of inertial reference frames. For example, consider the fine-structure of hydrogen (e.g., Dirac's equation) as an example of an objective law of physics. This law explains the various discrete emission frequencies (or "lines") that are observed in ionized hydrogen plasmas, and one of the ways that it is often used is cosmology is to measure the recession velocities of stars, galaxies, quasars, etc.. The relativistic Doppler-shift is used to describe how any such line frequency transforms between two inertial frames, but this assumes that the line frequency of the emitter is the same as the line frequency of a laboratory reference at the detector (e.g., that the fine-structure constant does not vary with time or space). If $$\alpha$$ varies with time, then we can foliate our pseudo-Riemannian manifold into hypersurfaces of constant $$\alpha$$, and there will be one and only one (locally preferred) inertial frame in which $$\alpha$$ is invariant under spatial translations.

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

How is that observed using a Michelson interferometer?

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

Ok.

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.

Which experiment to probe space-time geometry does not involve particles like electrons and photons? Such experiments can only probe the geometry of spatially extended particles, but not space and time per se:

Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter (particles) in the theory of Newton." (Albert Einstein, 1954)

 

[Hurkyl]

Lorentz covariance (e.g., the basic principle that the laws of physics are invariant under a shift of inertial reference frames) is not valid unless alpha itself is invariant under any shift of inertial reference frames.

Have you considered that alpha may be a scalar field?

If alpha varies with time, then we can foliate our pseudo-Riemannian manifold into hypersurfaces of constant alpha, and there will be one and only one (locally preferred) inertial frame in which alpha is invariant under spatial translations.

Any locally nonconstant scalar field let's you do that. It may be convenient. I don't see why one would consider it preferred.

Incidentally... those who think alpha is varying, do they think it to be a strictly increasing function of time? Or is it permitted to vary back and forth?

How is that observed using a Michelson interferometer?

I have no idea. Why does it matter?

Which experiment to probe space-time geometry does not involve particles like electrons and photons?

When I say "does not involve", I meant the thing we're attempting to measure, not the apparatus doing the measuring.

 

[Aether]

Have you considered that alpha may be a scalar field?

No.

Any locally nonconstant scalar field let's you do that. It may be convenient. I don't see why one would consider it preferred.

$$\alpha$$ is the most fundamental of the physical "constants". Most of the other dimensionful constants, including c are subject to varying in time if $$\alpha$$ varies in time.

You haven't responded to this: "If you don't agree that this would falsify CS, then please give an example of a real experiment that could falsify it." You seem to be arguing that CS, Lorentz covariance, and special relativity aren't falsifiable.

Incidentally... those who think alpha is varying, do they think it to be a strictly increasing function of time? Or is it permitted to vary back and forth?

I think that most people would presume that if alpha is varying, then it is varying with the expansion of the universe. Although this expansion may one day reverse into a contraction, nobody thinks that this has happened yet since the big bang.

I have no idea. Why does it matter?

The criteria used to validate Minkowski space-time (e.g., which includes the results of the Michelson-Morley experiment) over Euclidean space and time should be the same criteria that we continue to use for falsifying Minkowski space-time in view of some other principle like the holographic principle for example. If you make an argument that Minkowski space-time can't be falsified by a certain experiment, then I will deny you the benefit of similar experiments to falsify Euclidean space and time.

When I say "does not involve", I meant the thing we're attempting to measure, not the apparatus doing the measuring.

But what is it that you think we are attempting to measure there? Empty space and time per se aren't physical at all, it is only the geometry of particle fields that we are concerned with:

Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning. The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter (particles) in the theory of Newton." (Albert Einstein, 1954)

 

[Hurkyl]

the round-trip speed of light will otherwise be measured to have the same isotropic value in any inertial frame.

(emphasis mine) In other words, only in certain frames.

You seem to be arguing that CS, Lorentz covariance, and special relativity aren't falsifiable.

You're going to have to state precisely what you mean by CS. But you cannot falsify a convention, by virtue of the fact it's a definition and not a theory.


Lorentz covariance says that the laws of physics remain unchanged under a Lorentz transformation. It does not say that the values of all quantities remain unchanged under a Lorentz transformation. Lorentz covariance doesn't require that alpha = 1/137.03... in all frames; it merely requires that

alpha = e^2 / (hbar * c * 4 * pi * epsilon_0)

in all inertial frames. (Assuming that is, in fact, the correct relation in general)

 

[Hurkyl]

The criteria used to validate Minkowski space-time (e.g., which includes the results of the Michelson-Morley experiment) over Euclidean space and time should be the same criteria that we continue to use for falsifying Minkowski space-time in view of some other principle like the holographic principle for example

The Michelson inferometer mattered for Newton vs SR because they were known to disagree about the results. The MM experiment certainly cannot be used to falsify SR, because SR is consistent with the result. An inferometer is only useful for deciding between SR and something else if SR and something else are known to disagree about what the inferometer says.


Incidentally, as far as general relativity is concerned, the Lorentz group isn't very special. The laws of physics are to remain unchanged under ANY diffeomorphism. That includes anything in the Lorentz group, the Gallilean group, SO(4, 0), SO(2, 2), and any other group of diffeomorphisms you can imagine. The only thing special about the Lorentz group is that, in addition to preserving the laws of physics, it additionally locally preserves lengths and angles for a metric with signature -+++.

 

[Aether]

...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...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.

coalquay404 recently referenced http://arxiv.org/abs/gr-qc/0612118" wherein the authors state:

By studying the 5D geodesic equations we are able to reproduce the usual electrodynamics for a test-particle in a 4D space-time, where the charge-mass ratio is ruled out as follows $$q/m=u_5(1+\frac{u_5^2}{\phi^2})^{-1}$$. In this formula $$u_5$$ is the fifth covariant component of the 5D velocity and can be proved that it is a constant of motion and a scalar under KK transformations...A large scalar field ($$\phi>10^{21}$$ for the electron) allow us to have realistic value for the charge mass ratio avoiding the problem of Planckian mass, and, moreover, allow us to restore the conservation of charge at a satisfactory degree of approximation. Actually, a time-varying charge is very interesting; an isotopic, slow varying $$\phi$$ can explain the time-variation of the fine structure constant over cosmological scale which seems to be inferred by recent analysis.

I can't do calculations in Kaluza-Klein (KK) theory yet, but comparing what these authors say here (implying that $$u_5>5.7\times 10^{30}$$ m/s) to the "extremely large tangential velocity" component that I referred to above for an electron of approximately $$5.1\times 10^{47}$$ m/s (I chose $$R=5.24\times 10^{25}$$ meters here to get the even $$c^2$$ factor below) we can see that they differ by a factor of about $$c^2$$...maybe they are related (or the same). KK theory adds a dimension to unify gravity with electrodynamics, but the holographic principle subtracts a dimension leaving a net four dimensions when combined.

You're going to have to state precisely what you mean by CS. But you cannot falsify a convention, by virtue of the fact it's a definition and not a theory.

I'm reading The Philosophy of Space & Time by Hans Reichenback, and will come back to this discussion later.

 

[Chris Hillman]

Some comments and caveats

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, ...).

From this, it seems you are asking about competitors to gtr which might in some sense not incorporate str. Since str describes the geometry of tangent spaces in Lorentzian manifolds, this would probably require looking at non-metric theories. If so, I find the title puzzling, since gtr is not only a metric theory of gravitation, but a specific such theory.

You mentioned "applicable in any system of coordinates"; in the context of classical gravitation, this is usually interpreted to mean, technically speaking, "diffeomorphism covariance", which gets us back to smooth manifolds. So you probably need to refine what you mean by this in order to consider non-metric theories.

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).

Depends upon what you mean by "complexity", I guess. Interestingly enough, Einstein's notion (sometimes translated as "strength" although a better word would be "richness") attempts to assess the variety of distinct solutions. Then for example Maxwell's theory is actually richer than Newtonian gravitation, as you would expect from the fact that the field equation of Newtonian gravitation (in the classical field theory reformulation) is the same as that of electrostatics, a special case of Maxwell's theory.

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

On a background Lorentz spacetime.

In the case of weak fields, in linearized gtr you write the metric as a linear perturbation from an unobservable Minkowski background metric, so that mathematically speaking we treat "the gravitational field" as a second rank tensor field in Minkowski spacetime, which then suggests a naive quantization. Deser et al. showed that you can systematically introduce higher and higher order corrections, each time obtaining a field theory which is not self-consistent. But in the limit you obtain something self-consistent which is locally equivalent to gtr. However, this is not a true quantum theory of gravitation.

 

[Chris Hillman]

The groups O(4), O(1,3), O(2,2)

Hi, Aether wrote:

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.

The isometry groups of three types of "pseudo-Euclidean" four-manifolds mentioned by Hurkyl, respectively $$E^4, \; E^{1,3}, \; E^{2,2}$$ are semidirect products of the translation group $${\bold R}^4$$ with the isotropy groups $$O(4), \; O(1,3), \, O(2,2)$$.

Let's step back and look at a more familiar example. The isotropy group $$O(3)$$ is the rotation group of $$E^3$$, and the semidirect product of this with the translation group $${\bold R}^3$$ gives the euclidean group $$E(3) = {\bold R}^3 | \! \! \times O(3)$$.

Similarly, the isotropy group $$O(1,3)$$ is the full Lorentz group (the proper orthochronous Lorentz group is an index four subgroup of this), and the isometry group $$E(1,3) = {\bold R}^4 | \! \! \times O(1,3)$$ is the Poincare group.

In each case, the translation group is a normal subgroup, and there is one conjugate of the isotropy group associated with each point in the geometry, corresponding to the freedom to "rotate" about each point.

See for example Jacobson, Basic Algebra I, or Artin, Geometric Algebra.

If I am not mistaken, a (2+2)-spacetime [i.e. signature ++--] admits closed timelike curves.

Yes, in every neighborhood, because $$E^{2,2}$$ admits them.

One place where this geometry naturally arises is the following: suppose we represent M(2,R) (two by two real matrices) as a four dimensional real algebra, and seek the orbits under conjugation. Since conjugation leaves the trace invariant, this suggests rewriting our matrices in new variables on of which is the trace. But conjugation also leaves the determinant invariant, and the determinant in fact gives M(2,R) the structure of $$E^{2,2}$$.

Going up one more dimension, it seems worthwhile to mention that the point symmetry group of the three-dimensional Laplace equation is $$SO(1,4)$$, the (proper) conformal group of $$E^3$$, plus an infinite dimensional group arising from the linearity of the Laplace equation. The point symmetry group of the two-dimensional wave equation (time plus two space variables) is $$SO(2,3)$$, the (proper) conformal group of $$E^{1,2}$$, plus an infinite dimensional group arising from the linearity of the wave equation. And so on. (The conformal groups of the two dimensonal pseudoeuclidean spaces $$E^2, \; E^{1,1}$$ are infinite dimensional, by virtue of helpful "algebraico-analytical accidents".)

Hope this helps.

 

[Aether]

If it's an "explanation" for the uncertainty principle, then it should make the same empirical predictions as the uncertainty principle--if you're saying there's an upper limit to the momentum no matter how much you reduce the uncertainty in the position, that would seem to be a violation of the uncertainty principle.

Let me clarify this (from an old thread). If you measure the velocity of an object as as tends to zero, Heisenberg's uncertainty principle gets in the way. The more accurately you measure x (as 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?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?

In http://arxiv.org/abs/gr-qc/0309134" paper P.S. Wesson describes "Five-Dimensional Relativity and Two Times":

It is possible that null paths in 5D appear as the timelike paths of massive particles in 4D, where there is an oscillation in the fifth dimension around the hypersurface we call spacetime...a cou-
ple of exact solutions of the field equations of 5D relativity have recently
been found which have good physical properties but involve manifolds with
signature [+(− − −)+] that describe two “time” dimensions.

What I'm suggesting is that, in view of the holographic principle, a manifold like this might be equivalent to some other manifold having a signature of [+--+].

See for example Jacobson, Basic Algebra I, or Artin, Geometric Algebra.

Thanks. I am working on getting those books.

 

[Hurkyl]

What I'm suggesting is that, in view of the holographic principle, a manifold like this might be equivalent to some other manifold having a signature of [+--+].

In what sense would they be "equivalent"?!

Here's a quick insanity check:
If a 5-dimensional manifold is "equivalent" to a 4-dimensional manifold via the holographic principle,
and if a 4-dimensional manifold is "equivalent" to a 3-dimensional manifold via the holographic principle,
and if a 3-dimensional manifold is "equivalent" to a 2-dimensional manifold via the holographic principle,
and if a 2-dimensional manifold is "equivalent" to a 1-dimensional manifold via the holographic principle,
and if a 1-dimensional manifold is "equivalent" to a 0-dimensional manifold via the holographic principle,

then why would we ever study anything but 0-dimensional manifolds? They very easy things to understand!