0

I'm studying the deduction of Lorentz transformations through spacetime diagrams and I have encountered the following:

The mathematical formulas describing the transformations for both of the coordinates are as follows (notice the symmetry between them): $$ x_B=x-\frac{v}{c}x^0 $$ $$ x_B^0=x^0-\frac{v}{c}x $$ But this is not actually quite correct yet. We don’t know if these equations are actually true for every transformation. Thus, more generally we multiply the equations by a scaling factor γ.

Why do we suspect that these equations are not always true? Isn't the constancy of c already considered when we enforce the equality of angles between t and t' and x and x'? (I know that time dilation and lenght contraction must be taken into account, but I'd like to understand how we come to that conclusion through the space-time diagram deduction)

Qmechanic
  • 201,751
JH RP
  • 37

1 Answers1

0

Isn't the constancy of c already considered when we enforce the equality of angles between t and t' and x and x'?

Remember, the invariance of $c$ is only one of the two postulates. The other is the principle of relativity. This requires that the inverse transform must also be the same as the forward transform (just with an opposite $v$), and that any coefficients, like $\gamma$ must be even functions of $v$.

If you assume $\gamma=1$, then the inverse transformation you get will not be the same as the forward transform. Specifically, in your notation we can simply invert your transform to obtain $$x=\frac{1}{1-v^2/c^2}\left( x_B + \frac{v}{c} x^0_B\right)$$$$x^0=\frac{1}{1-v^2/c^2} \left(x^0_B+\frac{v}{c}x_B \right)$$ This violates the first postulate, the principle of relativity. The forward transform differs from the inverse transform in more than just the sign of $v$. Furthermore, it gives us a clear indication of how to fix it. We have found a function which is an even function of $v$, and if we simply multiply the forward transform by this factor then the forward and inverse transforms will be the same.

Dale
  • 99,825