How much is Special Relativity a needed foundation of General Relativity

[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!

 

[Aether]

In what sense would they be "equivalent"?!

In the same sense as J.D. Bekenstein intends here:

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).

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!

According to P.S. Wesson the [+(---)+] signature manifold has "good physical properties", and according to J.D. Bekenstein "our seemingly three-dimensional universe could be completely equivalent to alternative quantum fields and physical laws "painted" on a distant, vast surface" so that the (---) part of the [+(---)+] signature manifold might be completely equivalent to the (--) part of a [+(--)+] signature manifold. I am assuming that a [+(--)+] signature manifold having the same "good physical properties" as the [+(---)+] would be something worth studying; what do you think?

 

[lalbatros]

(multiple quotation) ... holographic principle ...

The holographic principle indeed questions the notion of space and dimensions.

I think that the origin of the holographic principles lies in the point of view that physics is about information and that the universe can hold a large but a finite amount of information that are "processed" at a finite rate. (see http://www.phy.duke.edu/~hsg/einstein/seth-lloyd-ultimate-computer.pdf")

Therefore, I think the holographic principle does not suggest us simply to drop one dimension in physics and makes thinks a bit simpler. Actually, it illustrates -for me- the possibility that dimensions themselves might be the simplification while the reality might very well be space and dimension-free.

The ultimate physical description of the universe might very well be a huge amount of qbits and the existence of space and the 4 dimensions might only be a very happy opportunity to make physics simpler.

I am very curious to see if such a "reverse" point of view, from qomputers to the universe, could bring us something useful.

Michel

 

[Aether]

Thanks for the interesting article Michel.

...black holes could in principle be 'programmed': one forms a black hole whose initial conditions encode the information to be processed, let's that information be processed by the Planckian dynamics at the hole's horizon, and extracts the answer of the computation...

At the black-hole limit, computation is fully serial: the time it takes to flip a bit and the time it takes a signal to communicate around the horizon of the hole are the same.

Therefore, I think the holographic principle does not suggest us simply to drop one dimension in physics and makes thinks a bit simpler.

I'm not looking "simply to drop one dimension in physics and make things a bit simpler", but rather to unify physics in terms of "planckian dynamics at the hole's horizon" (or rather at the universe's horizon); either that, or to rule out the possibility of such a thing. The holographic principle doesn't help do that, but rather it may help explain how such a model might appear to us as a projection in four-dimensional space-time.

 

[Sammy k-space]

Special Relativity is just Linear Doppler effect: C is constant by definition; the mass/energy of the photon is conserved for a stationary observer relative to the source, but for a source moving away from the observer there is a decrease in frequency as
v^2/2C^2 ie. vv/2CC where v is the velocity of the source away from the observer. It is a vector quanity; the quantity is added for a source moving toward the observer. This is easily calculated as E=mCC=hn; CC=E/m ie. C squared is the constant of proportionaliy between a mass and it's energy.
For general relativity we can look at two cases: first let a photon move toward the center of mass of an observer, since C is constant the gravitational increase is seen as an increase in frequency; for the other case let the photon be traveling close to a mass but toward an observer stationary to the source; the photon will curve toward the mass, but since C is defined as constant the velocity along the curve is C but the observer will see the curve and the shift in frequency. The change will be as if m=hn/CC.
For a very special case take twin photons with the same frequency and moving in the opposite direction, the observer is stationary relative to the source and "sees" one photon coming and one leaving, but the photons overlap in the view, so E1=hn and E2=hn but sum of the vectors is zero.

 

[jimmy neutron]

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.

By that I assume that you mean that, otherwise, light signals would have to travel at speeds less than c?

 

[Deepak Kapur]

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

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

 

[Fredrik]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

The post you're quoting was written in 2006.

Do you really think that you can refute the most well-understood and most thoroughly tested theory in the history of science with an argument that any kid can come up with, and without making an effort to find out what the theory says?

 

[Deepak Kapur]

The post you're quoting was written in 2006.

Do you really think that you can refute the most well-understood and most thoroughly tested theory in the history of science with an argument that any kid can come up with, and without making an effort to find out what the theory says?

Yes, you are right. It's a childish question. But don't underestimate the importance of such questions as they often turn out to be the germs of excellent theories in case of gifted individuals like Einstein ( he himself was used to such questions).

A few more childish questions.

1. Can we ever understand the mystery of nature. Suppose the String Theory (which in some cases even refutes Einstein's Theories) is able to find the fundamental particle. The immediate question would be 'What is this 'fundamental' composed of?

2. The only saving grace is the adage 'Something is Better than Nothing'. Even scientists are aware of this and never give up their scientific enquiries even if faced with bizarre contradictions like wave-particle duality and all the other paradoxes. Mind you, many scientist dealling with quantum mechanics are still skeptical of it, but they don't want to undermine the superstructure of science and have learned to live with it. Different kinds of politics is also involved in such an attitude. After all only a child can have the audacity to ask God 'Who made you?'

3. How can uniform motion be ever possible, when galaxies are moving away from each other at incredible speeds and time is continuously slowing down. Doesn't science feel it 'convenient' to deal with appoximations and simplifications rather than reality (whatever that is).

4.You would agree that in the laws of mechanics no mention is made of the shape of the body undergoing motion, whereas in real life it makes hell of a difference. Similarly it's extremely difficult to solve three-body problem (what to talk of greater number of bodies), because of the feedback effect. But scientists always tend to avoid such feedbacks and proceed with the simplest of cases. Can it lead us further or enmesh us in a labyrinth of mathematics and incomplete laws? (O! Lord, at least give us a single universal law that is proof against any kind of further enquiry!)

More will follow in case you reply.

 

[Mentz114]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

'Time slows with increasing speed' is not what relativity says - so your argument is based on a complete misunderstanding of relativity. Also, you have no idea what science is about and for.

 

[jtbell]

I doubt the predictions of relativity.

Check out the experimental evidence:

Experimental Basis of Special Relativity

 

[Deepak Kapur]

'Time slows with increasing speed' is not what relativity says - so your argument is based on a complete misunderstanding of relativity. Also, you have no idea what science is about and for.

Plz don't be impatient. Impatience may also signal absence of logic.

There are many interpretations of relativity ( some even contradictory).

To go by your interpretation a very-2 high gravitational field would in theory make a clock stop functioning (or make it extremely-2 slow) for another observer who is far away from the gravity source.

This again amounts to what I have said above. Processes can't come to a stand still just by the presence of super high gravity.

 

[Deepak Kapur]

Check out the experimental evidence:

Experimental Basis of Special Relativity

Thanks. I have already read that.

 

[Mentz114]

Nothing you've said is worth refuting because you don't understand what you are talking about.

For instance

Processes can't come to a stand still just by the presence of super high gravity.

That is not what GR predicts. Again you base your remarks on misunderstandings.

 

[Fredrik]

More will follow in case you reply.

I have answered questions like these many times in the past, but I think I'll pass this time. I don't want to spend 10-15 hours explaining physics (starting with an explanation of what a theory is) to someone who probably would ignore everything I say anyway. This is a forum for people who want to learn stuff, not for people who want to criticize things they don't understand.

Plz don't be impatient. Impatience may also signal absence of logic.

And criticizing the best understood and most thoroughly tested theory in the history of science without making an effort to understand what it says, signifies what exactly? You can't demand that others be patient with you when you show up here with this attitude.

To go by your interpretation a very-2 high gravitational field would in theory make a clock stop functioning (or make it extremely-2 slow) for another observer who is far away from the gravity source.

This again amounts to what I have said above. Processes can't come to a stand still just by the presence of super high gravity.

The problem with your posting this "argument" against relativity (twice!?) isn't that it's extremely naive. It's perfectly OK to ask uneducated questions. The problem is that you clearly know that your argument can't be right, and still talk to us as if you have disproved relativity. If it had been possible to disprove relativity with an argument that any kid can come up with, it would have been done a hundred years ago, and we wouldn't need your help with it. If you continue with this nonsense, you might get banned from the forum.

 

[Deepak Kapur]

I have answered questions like these many times in the past, but I think I'll pass this time. I don't want to spend 10-15 hours explaining physics (starting with an explanation of what a theory is) to someone who probably would ignore everything I say anyway. This is a forum for people who want to learn stuff, not for people who want to criticize things they don't understand.


And criticizing the best understood and most thoroughly tested theory in the history of science without making an effort to understand what it says, signifies what exactly? You can't demand that others be patient with you when you show up here with this attitude.


The problem with your posting this "argument" against relativity (twice!?) isn't that it's extremely naive. It's perfectly OK to ask uneducated questions. The problem is that you clearly know that your argument can't be right, and still talk to us as if you have disproved relativity. If it had been possible to disprove relativity with an argument that any kid can come up with, it would have been done a hundred years ago, and we wouldn't need your help with it. If you continue with this nonsense, you might get banned from the forum.

I am not trying to refute anything but am trying to satisy my curiosity. As far as this forum goes, I haven't got any logical answer till now apart from accusations and non-sensical arguments.

It's a usual strategy of such forums.

1. Goad someone so that he indulges in impatient remarks.

2. If someone is not provoked, dismiss him as being non-sensical.

Plz refrain from 'blind faith' in anything and try to give logical answers to the points (however uneducated they might be) I have raised.

 

[Fredrik]

I am not trying to refute anything but am trying to satisy my curiosity.

I might have believed you if you had said something to show us that you understand that these are the only possible reasons why your understanding of relativity disagrees with your intuition about the real world:

1. You don't actually understand what these theories (SR and GR) say.

2. Your intuition is wrong.

Instead you have been strongly suggesting that the problem is with relativity. That attitude is very inappropriate in this forum.

It's a usual strategy of such forums.

1. Goad someone so that he indulges in impatient remarks.

2. If someone is not provoked, dismiss him as being non-sensical.

Plz refrain from 'blind faith' in anything...

That's definitely not true, but it's a usual strategy of science-haters to make accusations like this one, where they describe their own behavior and claim it's how skeptics and scientists behave. This is also inappropriate here.

...and try to give logical answers to the points (however uneducated they might be) I have raised.

You haven't raised any points. You haven't asked any questions. All you've done is to suggest that your "argument" means that relativity is wrong. (This is also an implicit suggestion that every scientist in the last 100 years was a complete idiot).

 

[atyy]

I doubt the predictions of relativity. To take a simple example, 'Time Slows With Increasing Speed'. If time slows down, so will all the processes like the functioning of our hearts, brains, the solar system, the atomic system and so on. So, the universe will just tend to come to a stand still at extremely high speeds. This is absurd!

Yes, it is absurd. The statements you make are pieced together from popular explanations of relativity. In relativity there are several times - proper time and coordinate time, and there are many coordinate times. If you take a statement about proper time and another statement about coordinate time and draw logical conclusions from them, you will be all mixed up.

So things to distinguish: proper time vs coordinate time, inertial frame versus noninertial frame, local reference frame versus global reference frames, special relativistic time dilation vs general relativistic time dilation. Only predictions about experiments really matter.

 

[Deepak Kapur]

I might have believed you if you had said something to show us that you understand that these are the only possible reasons why your understanding of relativity disagrees with your intuition about the real world:

1. You don't actually understand what these theories (SR and GR) say.

2. Your intuition is wrong.

Instead you have been strongly suggesting that the problem is with relativity. That attitude is very inappropriate in this forum.


That's definitely not true, but it's a usual strategy of science-haters to make accusations like this one, where they describe their own behavior and claim it's how skeptics and scientists behave. This is also inappropriate here.


You haven't raised any points. You haven't asked any questions. All you've done is to suggest that your "argument" means that relativity is wrong. (This is also an implicit suggestion that every scientist in the last 100 years was a complete idiot).

I didn't think 'political correctness' is also required in 'public forums'.

Anyhow, answer my next question (except saying that the question itself is wrong).

When there is no absolute concept of time and distance (as stated by general relativity), how can we talk about an absolute entity like the Speed of light?

 

[Fredrik]

Now that's a real question.

There's just one little problem. The complete answer is long and mathematical. It would take a long time to write it down, and I don't even know if you'd be interested in a mathematical answer. The very short answer is that the speed of light isn't absolute. You can make it whatever you want by choosing an appropriate coordinate system. But there's a class of coordinate systems that are particularly important. They're called inertial frames. The claim that the speed of light is "invariant" actually means that it's the same in all inertial frames, not that it's the same in all coordinate systems.

Why is it the same in all inertial frames? That's just a mathematical property of inertial frames on Minkowski spacetime and null geodesics, the curves that we use to represent the motion of massless particles mathematically.

Why do we use this particular model of space and time? Because the theory based on it makes better predictions about results of experiments than theories based on other models. There is actually only one other model that's consistent with the requirement that inertial observers would describe each other as moving as described by straight lines, and that's the Galilean spacetime, which is used in Newtonian mechanics.

 

[Deepak Kapur]

Now that's a real question.

There's just one little problem. The complete answer is long and mathematical. It would take a long time to write it down, and I don't even know if you'd be interested in a mathematical answer. The very short answer is that the speed of light isn't absolute. You can make it whatever you want by choosing an appropriate coordinate system. But there's a class of coordinate systems that are particularly important. They're called inertial frames. The claim that the speed of light is "invariant" actually means that it's the same in all inertial frames, not that it's the same in all coordinate systems.

Why is it the same in all inertial frames? That's just a mathematical property of inertial frames on Minkowski spacetime and null geodesics, the curves that we use to represent the motion of massless particles mathematically.

Why do we use this particular model of space and time? Because the theory based on it makes better predictions about results of experiments than theories based on other models. There is actually only one other model that's consistent with the requirement that inertial observers would describe each other as moving as described by straight lines, and that's the Galilean spacetime, which is used in Newtonian mechanics.

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

 

[Deepak Kapur]

Nothing you've said is worth refuting because you don't understand what you are talking about.

For instance



That is not what GR predicts. Again you base your remarks on misunderstandings.

Not be able to understand the other person is indeed lack of understanding.

I never said 'it' predicts. It can be one of the implications. A clock so slow (as it apperas to an observer) that all the processes virtually coming to a stand still.

 

[Rasalhague]

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

Speed is a ratio between a distance (how far travelled) and a time (how long it takes). Distances and times change when we switch from measuring them in one inertial reference frame to another, but the ratio between the distance light travels and the time it takes stays the same.

Suppose we measure the speed of light through a vacuum (I'll just call it "the speed of light" from now on) in one inertial reference frame (a non-accelerating spacetime coordinate system, with three coordinates for space and one for time, covering a region small enough and brief enough that the effects of gravity are negligible) and find it to be a certain value c.

I'll use the letter v to stand for the speed of some other inertial reference frame moving parallel to the pulse of light, as measured in our original reference frame. I'll use another variable, defined like this:

\gamma = \frac{1}{\sqrt{1-\left ( \frac{v}{c} \right )^2}}.

The Greek letter \gamma, called "gamma", is just a handy symbol, conventionally used to simplify the equations below. Relativity predicts that intervals of space and time will differ in our new inertial reference frame according to the following equations,

\Delta x' = \gamma \left ( \Delta x - v \; \Delta t \right )

and

\Delta t' = \gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )

where \Delta x is some distance, for example the distance between the place at which a pulse of light is emitted and the place where it's received, measured according to our original reference frame (that is, as measured by rulers at rest in that reference frame), and \Delta x' is the distance between those same points measured according to our new reference frame. Similarly, \Delta t is an interval of time, for example the time between emission and reception of a pulse of light according to our original reference frame (that is, as measured by clocks at rest in that reference frame), and \Delta t' the time between these events according to our new reference frame.

The speed of light in our original reference frame is

c = \frac{\Delta x}{\Delta t},

the distance the light travels divided by the time it takes to travel that distance. The speed of light in our new reference frame is therefore

\frac{\Delta x'}{\Delta t'}=\frac{\gamma \left ( \Delta x - v \; \Delta t \right )}{\gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )}=\frac{\frac{\Delta x}{\Delta t}-v}{1-\frac{v}{c^2}\frac{\Delta x}{\Delta t}}=\frac{c-v}{1-\frac{v}{c}}=\frac{c(c-v)}{c-v}=c

So, without paradox, the speed of light is the same in any inertial reference frame moving at some non-zero speed relative to our original inertial reference frame, even though neither the distance the light traveled nor the time it took are the same as they were in the original inertial reference frame.

 

[Fredrik]

Why is it same in all the inertial frames when there is no absolute definition of time and distance (time dialtion, length contraction) even in the inertial frames. As a matter of fact how can (or should) speed be determined with so much of relativity around (even in inertial frames of reference).

You should think of spacetime as an abstract set of points. Those points are called events. Inertial frames are functions that assign 4-tuples of real numbers (t,x,y,z) to events. Those numbers (the coordinates of events) are used to define velocity the same way as in pre-relativistic physics:

\vec r=(x,y,z)

\vec v=\frac{d\vec r}{dt}

Speed is the magnitude of the velocity:

v=|\vec v|=\sqrt{v_1^2+v_2^2+v_3^3}

It's hard to explain why this formula applied to a null geodesic gives us the result =1 in every inertial frame. To really understand it, you'd have to understand what a geodesic is, which requires that you know some differential geometry. So let me skip most of those technicalities and go directly to what we'd end up with if we went through with all those mathematical details. But before we get started, I should tell you that there's a very natural way to associate an inertial frame with the motion and spatial orientation of an observer that's moving with constant velocity forever. This allows us to identify "observers" with "frames".

We can pick an inertial frame (any inertial frame will do) and use it to identify Minkowski spacetime with \mathbb R^4. Because an inertial frame was used in this identification, we can think of this copy of \mathbb R^4, as representing the point of view of the observer associated with that inertial frame. We clearly need a formula that tells us how to calculate the coordinates that one observer assigns to an event, given the coordinates that another observer assigns to the same event. In 1+1 dimensions (let's keep it as simple as possible), it's a function

x\mapsto \Lambda x+a

where x and a are 2×1 matrices and

\Lambda=\gamma\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}

\gamma=\frac{1}{\sqrt{1-v^2}}

This is in units such that c=1. Such a function is called a Poincaré transformation or a Lorentz transformation. (The term Lorentz transformation is often reserved for the case a=0). We can use this to calculate the speed of light in another inertial frame. The simplest possible scenario we can consider is when both inertial frames have the same origin, and the light we're considering is at x=0 when t=0 (that's in both frames, since they have the same origin). Now it's a really easy excercise to show that the curve that represents the motion of the light looks the same in both coordinate systems (a straight line through the origin with slope 1).

This may not be very satisfying unless you know why we're working with Lorentz transformations. As I said before, the requirement that inertial observers must describe each other's motion as straight lines is much stronger than it looks, and more or less forces us to only consider a spacetime where a coordinate change between inertial frames is done using a Lorentz transformation or a Galilei transformation. But we can actually ignore that completely and just say that using Minkowski spacetime (and therefore the Lorentz transformation) is an axiom of the theory, and is ultimately justified by the fact that theories of matter and interactions in Minkowski spacetime predict the results of experiments so well.

And anyway, you weren't really asking why the speed of light is the same in all inertial frames. You were just trying to find out how it's possible at all. I hope the above shed some light on it, even though I left out a lot. You might also want to have a look at pages 8-9 in Schutz. Light has speed 1 in all inertial frames because a Lorentz transformation tilts the x-axis by the same amount as the t axis. (Schutz's argument on those pages is actually that the invariance of the speed of light implies that tilting of the x axis, but the diagrams would have been the same even if he had been arguing for the converse statement).

 

[Deepak Kapur]

Speed is a ratio between a distance (how far travelled) and a time (how long it takes). Distances and times change when we switch from measuring them in one inertial reference frame to another, but the ratio between the distance light travels and the time it takes stays the same.

Suppose we measure the speed of light through a vacuum (I'll just call it "the speed of light" from now on) in one inertial reference frame (a non-accelerating spacetime coordinate system, with three coordinates for space and one for time, covering a region small enough and brief enough that the effects of gravity are negligible) and find it to be a certain value c.

I'll use the letter v to stand for the speed of some other inertial reference frame moving parallel to the pulse of light, as measured in our original reference frame. I'll use another variable, defined like this:

\gamma = \frac{1}{\sqrt{1-\left ( \frac{v}{c} \right )^2}}.

The Greek letter \gamma, called "gamma", is just a handy symbol, conventionally used to simplify the equations below. Relativity predicts that intervals of space and time will differ in our new inertial reference frame according to the following equations,

\Delta x' = \gamma \left ( \Delta x - v \; \Delta t \right )

and

\Delta t' = \gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )

where \Delta x is some distance, for example the distance between the place at which a pulse of light is emitted and the place where it's received, measured according to our original reference frame (that is, as measured by rulers at rest in that reference frame), and \Delta x' is the distance between those same points measured according to our new reference frame. Similarly, \Delta t is an interval of time, for example the time between emission and reception of a pulse of light according to our original reference frame (that is, as measured by clocks at rest in that reference frame), and \Delta t' the time between these events according to our new reference frame.

The speed of light in our original reference frame is

c = \frac{\Delta x}{\Delta t},

the distance the light travels divided by the time it takes to travel that distance. The speed of light in our new reference frame is therefore

\frac{\Delta x'}{\Delta t'}=\frac{\gamma \left ( \Delta x - v \; \Delta t \right )}{\gamma \left ( \Delta t - \frac{v}{c^2} \; \Delta x \right )}=\frac{\frac{\Delta x}{\Delta t}-v}{1-\frac{v}{c^2}\frac{\Delta x}{\Delta t}}=\frac{c-v}{1-\frac{v}{c}}=\frac{c(c-v)}{c-v}=c

So, without paradox, the speed of light is the same in any inertial reference frame moving at some non-zero speed relative to our original inertial reference frame, even though neither the distance the light traveled nor the time it took are the same as they were in the original inertial reference frame.

Quite helpful! (though equations were not visible)

Since it's a forum, I presume that even if I ask (relply to) extremely large number of questions, I am not going to pester anybody.

1. Now coming to our galaxy (which is accelerating at tremendous speed). Is the speed of light same at every point in the galaxy (between galaxies for that matter). Is this speed equal to the speed of light in an inertial frame. Why?

2. Why does light itself not experience time-dialation. If you say its massless, it's at least energy that has probably resulted from mass annihilation. So, it's "something" after all and don't forget about the 'photons' of light that act like particles.

 

[Fredrik]

(though equations were not visible)

Search the feedback forum. I think others have had the same problem in the past. Maybe you're just using a really old browser and need to upgrade.

1. Now coming to our galaxy (which is accelerating at tremendous speed). Is the speed of light same at every point in the galaxy (between galaxies for that matter). Is this speed equal to the speed of light in an inertial frame. Why?

Our galaxy isn't accelerating significantly. Special relativity holds "locally", in small regions of spacetime. When we're talking about far away galaxies, we need general relativity. At any location in spacetime (in this galaxy or any other), we can consider coordinate systems that I'll call "local inertial frames" (unfortunately there doesn't seem to be a standard name for them). They are coordinate systems that can be associated with the motion of massive particles in a natural way, and the speed of light at the origin of the coordinate system is the same in any of them. However, things are not so simple when you compare things that are far away from each other.

Edit: Apparently it's complicated enough to confuse me too. I had to edit my post to rewrite this part.

The solutions of GR that describe homogeneous and isotropic universes are called FLRW solutions. When we're working with one of them, it's convenient to use a coordinate system in which the galaxies are more or less stationary (they'll have speeds of a few hundred km/s relative to a nearby objects that stay at constant position coordinates). In this coordinate system, it's convenient to define another kind of "speed" to be a measure of how fast distant objects are moving away from each other. Define d(t) to be the proper distance between the two objects in a hypersurface of constant coordinate time t, and define the new kind of speed to be d'(t). The expansion of space ensures that the speed is non-zero for two objects that stay at constant position coordinates, and for distant galaxies, it can even be much higher than c.

This doesn't contradict the invariance of the speed of light, because that's a statement about a different kind of speed in a different kind of coordinate system. If we apply that same definition of speed to the light emitted from a star in a distant galaxy, then it's speed clearly isn't going going to be c. But that light does move at c in a local inertial frame associated with the motion of the star that emits it, no matter how far away it is.

2. Why does light itself not experience time-dialation. If you say its massless, it's at least energy that has probably resulted from mass annihilation. So, it's "something" after all and don't forget about the 'photons' of light that act like particles.

One answer is that particles simply don't have experiences, but that's not the whole story, because we sometimes talk about a massive particle's point of view. When we do, we're specifically referring to the description of events in spacetime using the inertial frame we'd associate with the massive particle's motion. The concept of "the photon's point of view" doesn't quite make sense because there's no natural way to associate an inertial frame with its motion. The problem is that the method we'd like to use to determine which subset of spacetime to call "space, at time t" doesn't work for photons. (This method for massive particles is described in the part of Schutz's book that I linked to in my previous post).

 

[Rasalhague]

A quick fix if you're still having trouble with seeing the equations in earlier posts: you could try copying the LaTeX code (try left clicking on the equation; that should open a window with the code) and pasting it into an online LaTeX editor such as this one:

http://www.codecogs.com/components/equationeditor/equationeditor.php

Or if that doesn't work, there are lots of others.

 

[Deepak Kapur]

Search the feedback forum. I think others have had the same problem in the past. Maybe you're just using a really old browser and need to upgrade.


Our galaxy isn't accelerating significantly. Special relativity holds "locally", in small regions of spacetime. When we're talking about far away galaxies, we need general relativity. At any location in spacetime (in this galaxy or any other), we can consider coordinate systems that I'll call "local inertial frames" (unfortunately there doesn't seem to be a standard name for them). They are coordinate systems that can be associated with the motion of massive particles in a natural way, and the speed of light at the origin of the coordinate system is the same in any of them. However, things are not so simple when you compare things that are far away from each other.

Edit: Apparently it's complicated enough to confuse me too. I had to edit my post to rewrite this part.

The solutions of GR that describe homogeneous and isotropic universes are called FLRW solutions. When we're working with one of them, it's convenient to use a coordinate system in which the galaxies are more or less stationary (they'll have speeds of a few hundred km/s relative to a nearby objects that stay at constant position coordinates). In this coordinate system, it's convenient to define another kind of "speed" to be a measure of how fast distant objects are moving away from each other. Define d(t) to be the proper distance between the two objects in a hypersurface of constant coordinate time t, and define the new kind of speed to be d'(t). The expansion of space ensures that the speed is non-zero for two objects that stay at constant position coordinates, and for distant galaxies, it can even be much higher than c.

This doesn't contradict the invariance of the speed of light, because that's a statement about a different kind of speed in a different kind of coordinate system. If we apply that same definition of speed to the light emitted from a star in a distant galaxy, then it's speed clearly isn't going going to be c. But that light does move at c in a local inertial frame associated with the motion of the star that emits it, no matter how far away it is.


One answer is that particles simply don't have experiences, but that's not the whole story, because we sometimes talk about a massive particle's point of view. When we do, we're specifically referring to the description of events in spacetime using the inertial frame we'd associate with the massive particle's motion. The concept of "the photon's point of view" doesn't quite make sense because there's no natural way to associate an inertial frame with its motion. The problem is that the method we'd like to use to determine which subset of spacetime to call "space, at time t" doesn't work for photons. (This method for massive particles is described in the part of Schutz's book that I linked to in my previous post).

What would be the scenario if we consider that the galaxies are not moving at all but the space between them is expanding.

Mind you, this is not the opposite of what we have discussed before.

 

[Fredrik]

I'm not sure I understand the question (because it seems to me that the answer is in the text you quoted).