Proper mapping of the second postulate into math

[Jeronimus]

The second postulate of SR is telling us that light always travels at C in a vacuum(absent of gravity) measured by any observer independent of the source or inertial frame the observer is measuring the light from.

However, light is made up of photons which do not travel like ping pong balls in a straight line but are "moving" according to the laws of QM with all the weirdness like only being able to the probabilities of finding a photon at a certain volume of space.

Yet, when we derive SR, we mapped the 2nd postulate into mathematics in such a way, which would be equivalent to some "special" ping pong balls being emitted from a source into all directions, always traveling at C into all directions, independent of the observer measuring them.
The formulas we get for SR we derive from the two postulates are exactly the same as if we were to pretend that photons act like those special ping pong balls.

Have there been attempts to map the second postulate of SR in a more precise way, such that it respects the full nature of light being made up of photons with their exact nature, as far as we know it according to QM, rather than treat them as special ping pong balls?

 

[Dale]

In modern formulations the second postulate has nothing to do with light. It is instead expressed simply in terms of an invariant speed.

That this speed happens to be the same as the classical speed of light is not central to the modern formulations of the second postulate. That is instead a derived result which depends on light being massless. The invariant speed can be identified even if light were found to have some small mass.

The formulation in terms of light is outdated, and the use of the term "speed of light" to refer to the "invariant speed" is just a matter of history.

 

[Jeronimus]

In modern formulations the second postulate has nothing to do with light. It is instead expressed simply in terms of an invariant speed.

That this speed happens to be the same as the classical speed of light is not central to the modern formulations of the second postulate. That is instead a derived result which depends on light being massless. The invariant speed can be identified even if light were found to have some small mass.

The formulation in terms of light is outdated, and the use of the term "speed of light" to refer to the "invariant speed" is just a matter of history.

Where did the modern formulations get their empirical data from? Which experiments? The Michelson-Morley experiment established that light (not photons) always travels at C.
Do the modern formulations arrive at more "detailed" formulas than "classical" SR? Where can those modern formulations and more accurate/detailed formulas be found? For example, what kind of Lorentz transformation formulas does one get with the modern formulations compared to treating photons as special ping pong balls always traveling at C and arriving at the "classical" formulas?

edit: Or differently asked. What are the postulates of the modern formulations allowing us to arrive at the Lorentz transformation formulas, be it the same or slightly different ones?

 

[Jeronimus]

I searched for "modern formulation of special relativity" on google but did not find anything relevant.

The only thing i found on the wikipedia article about SR which seems to relate is this

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.[15][16]

From the principle of relativity alone without assuming the constancy of the speed of light (i.e. using the isotropy of space and the symmetry implied by the principle of special relativity) one can show that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum

So the modern formulation is based on just the first postulate? The 2nd postulate isn't needed at all to arrive at the Lorentz transformation formulas?

 

[Nugatory]

However, light is made up of photons which do not travel like ping pong balls in a straight line but are "moving" according to the laws of QM with all the weirdness like only being able to the probabilities of finding a photon at a certain volume of space.

You are right that photons don't move like ping-pong balls, but your description above of light being "made up of" photons is no less misleading. By far the most sensible mathematical description of light ("light" not "all electromagnetic phenomena"!) is electromagnetic waves propagating in accordance with Maxwell's equations; photons appear only when you apply quantum mechanics to the detailed interaction of these electromagnetic waves with matter and hence are irrelevant to the behavior of light in a vacuum.

Yet, when we derive SR, we mapped the 2nd postulate into mathematics in such a way, which would be equivalent to some "special" ping pong balls being emitted from a source into all directions, always traveling at C into all directions, independent of the observer measuring them.
The formulas we get for SR we derive from the two postulates are exactly the same as if we were to pretend that photons act like those special ping pong balls.

They are, but they are also exactly the same as if we were to say that the first postulate applies to Maxwell's laws of electrodynamics which 19th-century physicists used to calculate the speed of light from the observed properties of electrical and magnetic fields. This is much closer to how SR was originally developed than your picture of "special ping-pong balls" - it is not an accident that Einstein's 1905 paper was titled "On the electrodynamics of moving bodies".

Have there been attempts to map the second postulate of SR in a more precise way, such that it respects the full nature of light being made up of photons with their exact nature, as far as we know it according to QM, rather than treat them as special ping pong balls?

After striking out the false alternative, the answer to your question is "yes" - that's what quantum electrodynamics is.

 

[Jeronimus]

By far the most sensible mathematical description of light ("light" not "all electromagnetic phenomena"!) is electromagnetic waves propagating in accordance with Maxwell's equations;

So those electromagnetic waves are "real" and not just a means/part of some mathematical formula to give us the probabilities of finding a photon in a certain volume of space?

 

[Nugatory]

So those electromagnetic waves are "real" and not just a means to give us the probabilities of finding a photon in a certain volume of space?

Electromagnetic waves are as real as the waves on the surface of a body of water.
(And you must not confuse electromagnetic waves with the "wave function" of quantum mechanics - that is something completely different that can be fairly described as "just a means to give us the probabilities...").

 

[Dale]

Where did the modern formulations get their empirical data from? Which experiments?

All of the experiments listed here:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

In particular, the ones based on the strong or weak nuclear forces rather than the EM force would be especially relevant.

What are the postulates of the modern formulations allowing us to arrive at the Lorentz transformation formulas, be it the same or slightly different ones?

Modern formulations would typically postulate the symmetries of spacetime. Based on the symmetries then you would derive the Lorentz transform either directly or through the metric.

 

[pervect]

The Bertozzi experiment is a fairly famous example involving only electrons and an accelerator which demonstrates the modern approach. There's a published paper, and a video. The video, titled "The Ultimate Speed", can be found at



A short summary. We can accelerate electrons to very high speeds. Experiment matches the predictions of relativity, where there would be a limiting speed as to how fast an electron can travel, that limit being c, and does not match Newtonian physics.

 

[Herman Trivilino]

So the modern formulation is based on just the first postulate? The 2nd postulate isn't needed at all to arrive at the Lorentz transformation formulas?

The 2nd Postulate is needed. The original formulation of SR was about the relationship between classical (non-quantum) electric and magnetic fields. That formulation ignores quantum mechanics entirely. Light propagates at speed c in a vacuum. That is a consequence of that relationship.

It is best to think of the speed c as the speed that's the same in all inertial reference frames, which means it's necessarily the fastest possible speed. If it were discovered that light actually travels slower than this speed no changes to the formulation of SR would be necessary. We would simply refer to the speed c as the ultimate (or equivalently the invariant) speed.

I think the answer to your original query is quantum electrodynamics. It's a theory of the interaction between matter and light, and it incorporates both SR and quantum theory.

 

[strangerep]

So the modern formulation is based on just the first postulate? The 2nd postulate isn't needed at all to arrive at the Lorentz transformation formulas?

Yes. Afaict, one needs only the following:

1) The usual 1st postulate, i.e., the relativity principle concerning inertial observers.

2) A technical assumption about continuity and differentiability of spatial and temporal coordinate systems which each observer can establish locally, such that their respective origins coincide.

3) An assumption on the form of transformations between different observers' coordinate systems which makes such transformations intuitively recognizable as corresponding to a relative velocity $v$ between the observers, and that such transformations along any fixed direction form a 1-parameter Lie group, with $v$ as the parameter.

4) Spatial isotropy: there is no intrinsically special direction in space. Mathematically, this is captured by insisting that if all quantities in any equation of the theory are transformed according to an arbitrary spatial rotation, then the resulting equation must be equivalent to the original.

5) Another technical assumption that the domain ${\mathcal V}$ of $v$ includes at least an open neighborhood of 0, and that $v=0$ corresponds to the identity transformation.

6) A (rarely stated) assumption of "physical regularity": if the relative velocities between (inertial) observers A and B, and between B and C are finite, then the relative velocity between A and C must also be finite. This must hold for all velocities in ${\mathcal V}$.

Most textbooks that use only the 1st postulate seem to gloss over one or more of the technical assumptions, else the detailed derivation becomes too long.

 

[vanhees71]

A very clear derivation along the lines mentioned in #11 can be found in

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math. Phys. 10, 1518 (1969)
http://dx.doi.org/10.1063/1.1665000.

 

[pervect]

A very clear derivation along the lines mentioned in #11 can be found in

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math. Phys. 10, 1518 (1969)
http://dx.doi.org/10.1063/1.1665000.

I was looking over an English translation of the Frank and Rothe paper cited in the above, currently at https://ia800304.us.archive.org/13/items/nasa_techdoc_19880069066/19880069066.pdf

They did derive the form of the Lorentz transform, and mention that as a special case, where the constant c (which has not yet been identified with the speed of light) is infinite, that one gets the familiar Galilean transforms. Or more precisely, one takes the limit as c approaches infinity. They made some additional assumptions to get there (reciprocity) which, according to Berzi et all, can be elimiinated.

However, I don't see how one concludes that the constant c must be finite. It seems to me to be equally self consistent to take the limiting case where the constant c is infinite. And that this is really the point under discussion, relativity assumes that this constant c is finite, while the Galilean transform assumes it's infinite.

 

[strangerep]

However, I don't see how one concludes that the constant c must be finite. It seems to me to be equally self consistent to take the limiting case where the constant c is infinite. And that this is really the point under discussion, relativity assumes that this constant c is finite, while the Galilean transform assumes it's infinite.

Indeed. Thus, the Galilean group turns out to be a contraction of the Poincare group.

[But,... oops,... we must remember that this is a "B"-category thread...]

 

[PeterDonis]

this is a "B"-category thread

It was, yes, but on reviewing the OP's question, I'm not sure it's really answerable within the limits of the "B" category. I have recategorized it to "I", which I think is more appropriate.