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.

Thus,

(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 !

### Like this:

Like Loading...

## Leave a Reply