Results of Galilean Transformation equations can not be applied for the objects moving with a speed comparative to the speed of the light.
Therefore new transformations equations are derived by Lorentz for these objects and these are known as Lorentz transformation equations for space and time.
Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, they are at the same position that is Observers O and O’ coincides. After that S’ frame starts moving with a uniform velocity v along x axis.
Let an event happen at position P in the frame S’. The coordinate of the P will be x’ according to the observer in S’ and it will be x according to O in S.
The frame S’ has moved a distance “vt” in time t (refer figure).
What should be the relation between x and x’? As we can see from the figure that from frame S’
x’ α x – vt
or x’ = k (x – vt) (1)
where k is constant of proportionality that we will determine.
Similarly from frame S
x = k(x’ + vt’) (2)
Put equation (1) in (2)
x = k[k(x – vt) + vt’]
or x/k = kx – kvt + vt’
or vt’ = x/k – kx + kvt
or t’ = x/kv – kx + kvt
or t’ = kt – kx (1 – 1/k2)/v (3)
Similarly from frame S, time t will be
t = kt’ + kx’ (1 – 1/k2)/v (4)
(This equation can be derived by putting equation 2 in 1 and then solving.)
Calculation of k: Continue reading “Lorentz transformation equations for space and time”