Lorentz Transformation: Wave Equation vs. Interval Invariance

[jk22]

To find the Lorentz transformation, should it start with the invariance of the wave-equation ?

If so, then it gives 5 equations, 2 of them being wave-equations again.

If however the invariance of the space-time interval is demanded only 3 quadratic equations come out.

Which way should be taken to start relativity ?

 

[PAllen]

I don't believe Lorentz started with either of these approaches. Further, can you provide more detail on what you mean by each alternative. The way I would do either approach, I do not get the number and type of equations you claim.

 

[PeterDonis]

To find the Lorentz transformation, should it start with the invariance of the wave-equation ?

If so, then it gives 5 equations, 2 of them being wave-equations again.

If however the invariance of the space-time interval is demanded only 3 quadratic equations come out.

Where are you getting this from? Can you give references?

 

[jk22]

I just start from vague laws claimed by physicists during history : https://en.wikipedia.org/wiki/History_of_Lorentz_transformations

 

[jk22]

I mean in formulas the first way were :

$$
\frac{\partial^2 f}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 f}{\partial t^2}=0\\=\frac{\partial^2 f}{\partial x'^2}\left[\left(\frac{\partial x'}{\partial x}\right)^2-\frac{1}{c^2}\left(\frac{\partial t'}{\partial t}\right)^2 \right]\\-\frac{1}{c^2}\frac{\partial^2 f}{\partial t'^2}\left[-c^2\left(\frac{\partial t'}{\partial x}\right)^2+\left(\frac{\partial t'}{\partial t}\right)^2 \right]\\+2\frac{\partial^2 f}{\partial x'\partial t'}\left[\frac{\partial x'}{\partial x}\frac{\partial t'}{\partial x}-\frac{1}{c^2}\frac{\partial x'}{\partial t}\frac{\partial t'}{\partial t}\right]\\+\frac{\partial f}{\partial x'}\left[\frac{\partial^2 x'}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 x'}{\partial t^2}\right]\\+\frac{\partial f}{\partial t'}\left[\frac{\partial^2 t'}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 t'}{\partial t^2}\right]
$$

as it can be seen the two last equations are wave equations for the change of coordinates.

For example the two last equation would imply :

$$
x'=ax+bt+f(x-ct)+g(x+ct)\\
t'=dx+et+h(x-ct)+k(x+ct)
$$

So is there any hope that the coordinate transformation would allow to know what happens at the speed of light in vacuum $c$, as it was questioned by Einstein at his epoch (I remember having read that but I could not find where again, because Lorentz transformation are diverging at that speed) ?

 

[PeterDonis]

I just start from vague laws claimed by physicists during history

"Vague laws" are not a good starting point. Also Wikipedia is not a good source, you need to be looking at textbooks or peer-reviewed papers that specifically talk about how the Lorentz transformations can be derived and from what axioms. Also that Wikipedia article is very long and I don't see anything in it that corresponds to the claims you made in your OP.

I mean in formulas the first way were

Where are you getting all this from?

 

[jk22]

From the invariance of the wave-equation, by using the chain rule.

 

[PeterDonis]

From the invariance of the wave-equation, by using the chain rule.

In other words, you don't have five equations. You just have one. "Five equations" would mean five independent equations, none of which can be derived from any of the others.

You seem to have a fundamental confusion about how to count "equations" and what it means to "derive" the Lorentz transformations. Searching PF should turn up some good past discussions on this topic. I would strongly recommend checking them out and also looking at the literature on different ways of deriving the Lorentz transformations from particular sets of axioms. That way you will be able to start a new thread with a better basis for discussion.

In the meantime, this thread is closed.