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cides with that of the x′-axis; and furthermore, that the origin of the spatial
coordinates of one system at time t = 0 coincides with the origin of the other
system at time t′ = 0, Einstein, on the basis of two postulates, derived the fol-
lowing transformation relating the coordinates of the two systems:

−
x ′ = xvt 2 , y ′ = y, z ′ = z, t ′ = t − c v 2 x (9.26)
2
1 − v 1 − v
c 2 c 2
where c is the velocity of light in vacuum. The derivation of these formulae
are detailed for you in electromagnetic theory or modern physics courses and
are not the subject of discussions here. Our purpose here is to show that
knowing the above transformations, we can deduce many interesting physi-
cal observations as a result thereof.

Preparatory Exercise
Pb. 9.17 Show that, upon a Lorentz transformation, we have the equality:

2
2 2
2
2
x ′ + ′ + ′ −y 2 z 2 c t ′ = x 2 + y 2 + z 2 − c t
This is referred to as the Lorentz invariance of the norm of the space-time
four-vectors. What is the equivalent invariant in 3-D Euclidean geometry?

If we rename our coordinates such that:

x = x, x = y, x = z, x = jct (9.27)
1 2 3 4
the Lorentz transformation takes the following matricial form:

 1 β j 
 2 0 0 2 
 1 − β 1 − β 
L =   0 1 0 0   (9.28)
β
 0 0 1 0 
−  β j 0 0 1 
  1 − β 2 1 − β  
2

v
where β= , and the relations that were given earlier relating the primed
c
and unprimed coordinates can be summarized by:

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