Speed of Light is a Property of Massless Particles or Space?

[strangerep]

Gave [Pal's paper] a reading, in that paper he mentions "Isotropy" & "homogeneity" of spacetime as being "given". Is that from the principal of relativity postulate?

No. Spatial isotropy is an extra assumption. Spacetime homogeneity is also an addition assumption to reduce the fractional linear equations to linear equations. But in fact, that common form of the spacetime homogeneity postulate is unnecessary: the more general fractional linear case (not treated by Pal) yields a de Sitter space which has constant curvature. This is a more general version of the idea that space "looks the same" everywhere. I.e., spatial homogeneity (in the sense of constant curvature) need not be assumed.

 

[nitsuj]

No. Spatial isotropy is an extra assumption. Spacetime homogeneity is also an addition assumption to reduce the fractional linear equations to linear equations. But in fact, that common form of the spacetime homogeneity postulate is unnecessary: the more general fractional linear case (not treated by Pal) yields a de Sitter space which has constant curvature. This is a more general version of the idea that space "looks the same" everywhere. I.e., spatial homogeneity (in the sense of constant curvature) need not be assumed.

So there is more than just one postulate in his paper in order to reach the conclusions he did. I'd be able to assume if all physics is the same (inertial) that would imply isotropy and homogeneity of where / when the "physics" plays out. That said I understood it as the invariance of c speaks to isotropy and homogeneity of space.

If as you say isotropy and homogeneity are not from SR relativity principal, did he not include significant assumptions of the invariance of c postulate?

 

[nitsuj]

"Force carriers" implies an interacting theory, but what I described is applicable to the simpler free case, i.e., transformations between inertial (unaccelerated) frames.

Well what do you suppose is traveling between the inertial observers that leads to observing "something happening"? In the simple free case I'd think the observer is relying on Force Carriers (photons) to shed some light (ahaha funny pun).

I'm not sure what you're disagreeing with here. The universal constant happens to be an upper limit on relative speed. It is found simply by looking for fixed points in the velocity boost composition formula that drops out of the analysis.

Whether max speed & it's invariance were determined separately. or is one assumed because of the other?

"The universal constant happens to be an upper limit on relative speed." seems to brush over the difference between invariance and maximum. For me, the "universal constant" is really universal constancy in the order of "co-located" events, or said differently the isotropy / homogeneity of spacetime.

Is his paper not circumnavigating the out right stipulation of the invariance of c postulate, while relying on its key features?

 

[Ibix]

IIRC, Pal uses some sleight of hand to dismiss the K<0 case. Actually, most treatments are a bit weak on that point.

It didn't seem tricksy to me (maybe I've fallen for it). He seems to be saying that if K<0 then it is possible to find frames where the transform does not reduce to the identity operation in the case of zero velocity. He notes that there are similar problems in the Einsteinian K>0 case for velocities greater than c, but observes that those frames are rendered inaccessible by the relativistic velocity addition law. In contrast, the velocity addition law for K<0 doesn't protect you from having to consider the problematic frames.

 

[wabbit]

The case K<0 is weird, you get supraluminal velocities by adding three subluminal velocities, or even infinite ones, e.g. $3×(c/\sqrt{3})=\infty$ and the invariant speed is imaginary. Not very appealing but perhaps it has in interpretation.

 

[strangerep]

So there is more than just one postulate in his paper in order to reach the conclusions he did.

Yes. Apart from the RP, spatial isotropy (and then spatiotemporal homogeneity) there are also assumptions about continuity, including the assumption that the transformations form a Lie group.

I'd be able to assume if all physics is the same (inertial) that would imply isotropy and homogeneity of where / when the "physics" plays out.

Sorry -- I don't know what you mean. The RP does not imply isotropy or homogeneity by itself.

That said I understood it as the invariance of c speaks to isotropy and homogeneity of space.

One requires only the RP, spatial isotropy, and the mathematical features above involving continuity, to deduce the invariant constant c.

 

[strangerep]

Whether max speed & it's invariance were determined separately. or is one assumed because of the other?

The assumed Lie group property of the velocity transformations gives rise to an invariant speed c. Imposition of the group multiplication property gives rise to a velocity addition formula, from which the role of c as a limiting relative speed becomes obvious.

"The universal constant happens to be an upper limit on relative speed." seems to brush over the difference between invariance and maximum. For me, the "universal constant" is really universal constancy in the order of "co-located" events, or said differently the isotropy / homogeneity of spacetime.

The velocity transformation formulas are derived between observers for which the origins of their frames of reference are collocated (and suitably rotated so that the spatial axis direction coincide).

Is his paper not circumnavigating the out right stipulation of the invariance of c postulate, while relying on its key features?

No.

Perhaps you should try another treatment, such as that in Rindler's textbook on special relativity. I get the feeling you have not yet worked through the math, pen-in-hand? There's not much more I can say until you at least attempt this.

 

[strangerep]

It didn't seem tricksy to me (maybe I've fallen for it). [Pal] seems to be saying that if K<0 then it is possible to find frames where the transform does not reduce to the identity operation in the case of zero velocity. He notes that there are similar problems in the Einsteinian K>0 case for velocities greater than c, but observes that those frames are rendered inaccessible by the relativistic velocity addition law. In contrast, the velocity addition law for K<0 doesn't protect you from having to consider the problematic frames.

OK, yes, that's the basic idea. I just felt the way he expressed the argument could be strengthened.

 

[strangerep]

The case K<0 is weird, you get supraluminal velocities by adding three subluminal velocities, or even infinite ones, e.g. $3x(c/\sqrt{3})=\infty$ and the invariant speed is imaginary. Not very appealing but perhaps it has in interpretation.

I think it has no good interpretation, but rather hints at a guiding principle. I'm not sure if it already has a name, but I call it the "Principle of Physical Regularity". The idea is as follows: the relative parameters that we think of a kinematical (or dynamical) variables, such as velocity, spatial orientation, spatial displacement, temporal delay, etc, characterize how one inertial observer differs from another, and hence the particular transformation which must be performed to convert one observer into the other. If (say) observer B has (finite) relative velocity $v_{AB}$ relative to A, and observer C has (finite) relative velocity $v_{BC}$ relative to B, it must be the case that $v_{AC}$ is finite also, else we do not have a good physical theory. Moreover, this must hold for all (relative) velocities in a nontrivial open neighbourhood of $v=0$.

Appealing to this principle is essentially equivalent to what you described: for K<0 it is possible to choose velocities whose composition yields $\infty$. Such embarrassment is only avoided if we restrict the allowed relative velocities to $v=0$, which makes the "theory" trivial.

 

[nitsuj]

I get the feeling you have not yet worked through the math, pen-in-hand? There's not much more I can say until you at least attempt this.

You're absolutely right I have not worked through the math. I can't read math, so is why I was asking about the relations, assumptions ect most of which is likely explicitly shown (or easily deduced from) in the math.

Thanks for explaining it to me, but yea I really should put effort into the math. Concepts & their relations just get to a point where...it's "more than words" are capable of communicating a clear 'n concise way. lol