## Lorentz transformation based on the first postulate

Some modern derivations of Lorentz transformation are based on the first postulate of Special Relativity.

Its basic point is like this:

Suppose we have a point light source, its wave front will obey this formula:

(1) x^{2}+y^{2}+z^{2}-(ct)^{2}=0

For some other point not on the wave front, it will obey:

(2) x^{2}+y^{2}+z^{2}-(ct)^{2}=A

Here A is a constant.

Based on the first postulate of Special Relativity, any physical law will behave the same in all inertial reference frames,
so the propagation of light in all reference frames should be the same, so the above formulas will be the same in all reference frames.

I don't understand why step (1) can lead to (2), but I can safely claim the basic reasoning is wrong, as it used the first postulate the opposite way.

If we follow the same reasoning, then we can claim that whether you are on the bank or in the river, you will always see the water wave as perfect circles, that is too absurd.

Just as what we have shown in the example of the falling apple, what physical laws decided is acceleration, not speed. In all initial reference frames, the apple alway falls with the same acceleration, but their speed can be quite different.