we notate the 4 – coordinate as x in S frame and y is S’ frame.
let the Lorentz transform be
notice that when writing like this, it is equivalent as a matrix equation on column vector:
For row vector, the Einstein notation read:
be careful the order of the subscript of L.
to derive the Lorentz transform, we only need to know 3 things,
- the wave equation does not change ( i.e. the speed of light does not change )
- the Lorentz transform has inverse
- the relative motion of 2 frame is given by a speed
we need Jacobian when relate the derivative between 2 frames.
(this is a row vector equation )
The wave equation is :
this wave equation can be viewed by using this:
There, we can write the wave equation as :
where the g is a matrix:
Subsitute the Jacobian, we have
the wave equation does not change with frame. there fore, this must be hold:
ntoics that the left Lorentz transform is transpose, and the right one is normal one. For simplicity, for a 1-D spatial dimension. we can let L be:
by solving, we have 3 indenpent equations:
Using the inverse of Lorentz transform, we have another 3 equations:
Solving this 6 equations for non-trivial solution.
Now, using the original of frame S ( ) is moving with speed in the S’-frame.
then we have the Lorentz Transform !