# Prove the Lorentz factor without recourse to the speed of light

• B

## Main Question or Discussion Point

The Lorentz factor shows how fast one frame will judge speeds in another frame to be taking into account the relative motion between the two frames.

The speed of light is a factor in the Lorentz factor but I have heard that this is not because the speed of light is fundamental to it.

So how can this result be attained from first principles and without involving the speed of light as such?

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Nugatory
Mentor
.....but I have heard that this is not because the speed of light is fundamental to it.
You will get better and more helpful answers if you can tell us exactly what you heard and where you heard it.

Thanks. I can't remember.

Do you think I may have misheard?

Even so am fairly confident that I have heard more than once that it is not the speed of light that is responsible for the Lorentz factor (as stated by Einstein).

That being so I wonder why it is such a factor in the transformations -and ,as I say can this result be got without recourse to a scenario involving the transmission of a light signal between the two frames ?

So how can this result be attained from first principles and without involving the speed of light as such?
There are many ways to derive this result, but I don't know of any that work at a B-level, they all require equations of varying degrees of difficulty. If you have the stomach for mathematics, this is my favorite derivation of the Lorentz transform.

There are many ways to derive this result, but I don't know of any that work at a B-level, they all require equations of varying degrees of difficulty. If you have the stomach for mathematics, this is my favorite derivation of the Lorentz transform.
thanks,that looks interesting. At a quick glance over I see c is introduced as some kind of an unspecified scaling factor without reference to light.

It will take me a good while to go through it but it seems very helpful.

Ibix
Pal's paper is also good.
https://arxiv.org/abs/physics/0302045

He starts with the principle of relativity and shows that there are two options: Galilean relativity (with an infinite invariant speed), or Einsteinian relativity (with a finite invariant speed). Checking which transform holds in the real world may be done by experiment (e.g. Bertozzi's experiment measuring the relation between velocity and kinetic energy of electrons). And then we merely note that the implied value of the invariant speed is c.

Pal's paper is also good.
https://arxiv.org/abs/physics/0302045

He starts with the principle of relativity and shows that there are two options: Galilean relativity (with an infinite invariant speed), or Einsteinian relativity (with a finite invariant speed). Checking which transform holds in the real world may be done by experiment (e.g. Bertozzi's experiment measuring the relation between velocity and kinetic energy of electrons). And then we merely note that the implied value of the invariant speed is c.
thanks, I will take a look at that too.

haushofer
Pal's paper is also good.
https://arxiv.org/abs/physics/0302045

He starts with the principle of relativity and shows that there are two options: Galilean relativity (with an infinite invariant speed), or Einsteinian relativity (with a finite invariant speed). Checking which transform holds in the real world may be done by experiment (e.g. Bertozzi's experiment measuring the relation between velocity and kinetic energy of electrons). And then we merely note that the implied value of the invariant speed is c.
It is also nice in that it emphasises that Galilean relativity and Einstein's relativity only differ in one single aspect: the speed of light.

Both relativity theories say that inertial observers are equivalent, and they all measure the same speed of light. In Galilean relativity, this speed is infinite.

It is also nice in that it emphasises that Galilean relativity and Einstein's relativity only differ in one single aspect: the speed of light.
As does the Reflections on Relativity one.

Mister T
This is perhaps related to your original question. If it's discovered that the speed of light depends on the relative motion of the observer or source, then nothing about the theory will change. In a relation like $\gamma=\frac{1}{\sqrt{1-(v/c)^2}}$ we will simply stop referring to $c$ as the speed of light and call it something else. Like the invariant speed, or the ultimate speed. Nothing about the theory will change.
The theory describes an invariant speed $c$ even if light or indeed anything at all, never travels at that speed.