Lorentz transformation without the speed of light postulate

[lalbatros]

My https://www.physicsforums.com/showthread.php?t=297616" was a pretext to discuss the well-known fact that it is nearly possible to derive the full Lorentz transformation without taking into account the constancy of the speed of light.
I went back to my physics hobby and read that in a few papers recently.
I find this interresting, altough this is probably very well known, if not obvious.

Simply assuming that a transformation must make a one-parameter (v) matrix group leads to the Lorentz-like group. (1)
This transformation depends then on an arbitrary parameter which is an unknown velocity (k).
This parameter could be determined from experience.
But it could also be identified as the propagation speed of a invariant massless vector field.

It is striking that the generic transformation groups must admit a limit velocity.
The Galilean solution appears as a very special case.
Why is it so that a limit velocity is the "most natural" solution?

Clearly it is all more about symmetries than about light.

Any thought, anygood reading about that?
Thanks.

(1) rejecting the rotation group

 

[tiny-tim]

Hi lalbatros!

… But it could also be identified as the propagation speed of a invariant massless vector field.

It is striking that the generic transformation groups must admit a limit velocity.
The Galilean solution appears as a very special case.
Why is it so that a limit velocity is the "most natural" solution?

Clearly it is all more about symmetries than about light.

In your own words, the most natural solution would not be a "very special case".

I'd say it's not that it's the propagation speed of a invariant massless vector field that matters …

it's that it's the propagation speed of the thing that defines time …

clocks are synchronised by light, and if there wasn't a limit speed, they'd be synchronised by something infinitely fast …

also without a limit velocity, there wouldn't be causally unconnected events, whose Hamiltonians commute, and nothing in field theory would converge (see eg Weinberg's Quantum Theory of Fields)

 

[lalbatros]

tiny-tim,

The "generic case" includes a speed limit.
The speed of light appears then to be not more than a conversion factor between space and time coordinates.
The galilean limit is a very special case and it is also very strange given that the conversion factor c->infinity.

 

[tiny-tim]

… The speed of light appears then to be not more than a conversion factor between space and time coordinates. …

No, it has a lot more significance than as just a conversion factor …

as I said before, it provides a distinction between causally connected events and causally unconnected events …

this is very important in quantum field theory.

 

[robphy]

tiny-tim,

The "generic case" includes a speed limit.
The speed of light appears then to be not more than a conversion factor between space and time coordinates.
The galilean limit is a very special case and it is also very strange given that the conversion factor c->infinity.

In my view,
the speed of light has at least two roles:
one as a convenient conversion constant between coordinates,
another as a "maximum signal speed".

Instead of focusing on "numbers",
it might be useful to focus on eigenvectors
of the boost transformations in spacetime.

 

[bernhard.rothenstein]

My https://www.physicsforums.com/showthread.php?t=297616" was a pretext to discuss the well-known fact that it is nearly possible to derive the full Lorentz transformation without taking into account the constancy of the speed of light.
I went back to my physics hobby and read that in a few papers recently.
I find this interresting, altough this is probably very well known, if not obvious.

Simply assuming that a transformation must make a one-parameter (v) matrix group leads to the Lorentz-like group. (1)
This transformation depends then on an arbitrary parameter which is an unknown velocity (k).
This parameter could be determined from experience.
But it could also be identified as the propagation speed of a invariant massless vector field.

It is striking that the generic transformation groups must admit a limit velocity.
The Galilean solution appears as a very special case.
Why is it so that a limit velocity is the "most natural" solution?

Clearly it is all more about symmetries than about light.

Any thought, anygood reading about that?
Thanks.

(1) rejecting the rotation group

I think since long time about a simple proof of the fact that the formulas that account for the different relativistic effects, the Lorentz transformations for the space-time coordinates of the same event and even the postulate of the constancy of the speed of light are a direct consequence of the relativistic postulate.
I state the relativistic principle as
All the laws of physics are the same in all inertial reference frames, mentioning its direct consequences:
1. Distances measured perpendicular to the direction of relative motion have the same magnitude in all inertial reference frames.
2. Considering that observers from I and I' are equipped with identical laser and machine guns at rest relative to them. Measuring the velocity of a light signal and of a bullet fired by their own devices they obtain the same magnitudes c and u respectively.
Inventing scenarios in which the involved observers are not obliged to measure the speed of a light signal emitted by a moving source of light we can consider the obtained results are a direct consequence of the principle of relativity.
Consider as a first example the derivation of the formula that accounts for the time dilation.
It involves a clock synchronization procedure in I,associated with the emission of a light signal by a stationary source and no clock synchronization in I'. So it is a direct consequence of the principle of relativity.
Step by step we could extend the idea for deriving length contraction, Lorentz transformation
and even to show that the principle of the constancy of the speed of light is a direct conseqeunce of the principle of relativity.
Answering my thread please consider only arguments in the limits of an introductory text using kind words and hard arguments.

 

[lalbatros]

bernhard,

The shortest way is probably as follows:

That the transformations are linear results from the assumption of homogeneity.
That the transformations are a group translates the principle of relativity.
In addition, the LT are depending on one parameter: the relative velocity.​

Simple algebra then leads to two generic solutions:

1) the rotation group
2) the Lorentz group​

Only solution (2) represents what we are looking for.
From this solution we learn that -in general- there should be a limit speed.

This way above is very short and reflects precisely the physics that is behind.

But this approach does not discuss the physics and doesn't make it very intuitive.
The original paper by Einstein will never be replaced in this respect, since it really discuss the full physical motivation.
The mathematics are abstract by nature and represent only the pure logical structure of the theory.
Therefore they are a fast way to the LT, but you can't see the landscape very well anymore.

Personally I like this group approach because it tells us why there should be a constant speed.
This is simply because spacetime does really exist.
It is precisely because we assume a transformation involving time and space together could make sense,
that we come to the conclusion of a limit speed.
The fact that this limit speed is also the speed of light is nearly anecdotical at this point (kinematics).

Reshaping the known laws of physics (Newton, Maxwell) to highlight their needed invariant properties leads to further insight about the speed of light.