Question

. Consider a frame S

0 moving with a velocity v along its x

0 axis with respect to another frame

S. Also consider the axes of the two frames to be aligned at t = t

0 = 0

(a) First consider Galilean transformations. Write down the matrix which implements this

transformation.

(b) Now let us consider Lorentz transformations. Write down the matrix which implements

the above Lorentz transformation (also called a Lorentz boost).

(c) Suppose you have a Lorentz boost along x by a velocity vx followed by a boost along y

by velocity vy. What is the matrix for the successive transformations? Would one have

got the same matrix if the order of transformations were reversed? What does this say

about Lorentz transformations in general?

Answer 1

As per the given question,

The speed of the frame $S_o$ which is moving with the speed v$_o$ along the x axis with respect to another frame that is S, at t=0 the axis of the two frame is coincides.

i) So the position after the time t is,

$(x,t) \rightarrow (x+vt, t)$

two galiliean transformation $G(R,v,a,S_o)$ and $G(R', v',a', S)$ it compose to form a Galilean transformation,

$G(R' , v' , a' , S' ), G(R, v, a, S_o) = G(R' R, R' v+v' , R' a+a' +v' s, s' +s).$

Hence matrix representation

Where R is the rotational matrix.