# Is Lorentz transformation the only way to preserve speed of light?

Doesn't this transformation also preserve speed of light?:

$$B$$ is moving with speed $$v$$ relative to $$A$$ in the $$+x$$ direction. $$B$$ passes $$A$$ at $$t=0$$, and $$A$$ shines a torch at that moment.

Let $$(x', t')$$, the coordinates of the light beam in $$B$$'s frame, be related to the $$(x,t)$$ coordinates in $$A$$'s frame by these transformations:

$$x'=x-vt =ct-vt =ct\left(1-\frac{v}{c}\right)$$

$$t'=t\left(1-\frac{v}{c}\right)$$

As we can see, this transformation gives $$\frac{x'}{t'}=c$$. So the speed of light is the same in $$B$$'s frame. This transformation tweaks the time but lets distance be the same as given by Galilean transformation.

What issues does a transformation like this cause? And what makes 'Lorentz transformation' the right way to go to account for the constancy of speed of light?

• The Lorentz transformation is global if it's a space-time independent constant matrix. The invariance of physical laws with respect to a global Lorentz transformation is just the principal of the invariance of the speed of light in special relativity. – Cinaed Simson Jul 15 at 9:08
• How would $y$ and $z$ transform? What would be the speed of a light beam that wasn’t traveling in the $x$-direction? – G. Smith Jul 15 at 16:22

Consider observer $$C$$, who is traveling at a speed relative $$-v$$ in the $$x$$-direction relative to $$B$$. By your logic, the coordinates $$x''$$ and $$t''$$ measured by $$C$$ should be $$x'' = x' - (-v) t' = (x - vt) + v (t - v/c) = x - v^2/c \\ t'' = t'( 1 - (-v)/c) = t ( 1 - v/c)(1 + v/c) = t \left( 1 - \frac{v^2}{c^2} \right)$$ But $$C$$'s coordinates should be the same as $$A$$'s coordinates, i.e., $$x'' = x$$ and $$t'' = t$$. This is a contradiction.
The other way to look at it is that the inverse transformation law from $$B$$ to $$A$$, according to your equations, is $$t = \frac{t'}{1 - v/c} \qquad x = x' + \frac{t'}{1 - v/c}$$ But this means that the transformation laws to get between reference frames are different between these two frames. This means that the principle of relativity is broken; the transformation laws should have substantially the same form in all reference frames.