# Finding a and b

The goal is to be able to translate between coordinates in K and K'. Thus, we want to find a and b in the following equations (these are the ones we just derived):

x' = ax - ct (*)
ct' = act - bx (**)

Consider the origin of the second coordinate system, x'=0. (*) at x'=0 gives

x= (bc/a)t

This is of the form x = vt; v = bc/a is the velocity of the origin of the second coord system (K') with respect to the first (K). We will use this value in the next derivation.

The principle of relativity dictates that these two lengths must be equal:

• length of rod at rest in K' viewed from K
• length of the same rod at rest in K viewed from K'
To find the first length, take a specific time, say t=0 for simplicity. From (*) this gives x'=ax. So, two points on the x' axis separated by dx'=1 when measured in K is

dx=1/a

This is the first expression. We will now derive the other. Starting with the equations (*) and (**), using the value of v derived above, and eliminating t gives:

x'=a(1-v^2/c^2)x

So if two points on the x axis separated by dx=1, when measured in K' gives

dx'=a(1-v^2/c^2)

This expression for dx' must equal the expression for dx, so

a = (1 - v^2/c^2)^(-1/2)

And so the transform takes the same form as we saw before.