Lorentz Transformations Revision Notes

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18.5Lorentz Transformations

In these revision notes for Lorentz Transformations, we cover the following key points:

  • What are the Lorentz spacetime transformations?
  • How do they relate to spacetime Galilean transformations?
  • How to obtain the Lorentz transformations for velocity?
  • In what condition(s) do the Lorentz transformations converge with Galilean transformations?

Lorentz Transformations Revision Notes

The Lorentz Transformation of spacetime coordinates are relativistic equations that consider the fact that length and duration of a given event are not equal in different systems of reference. The equations derived from Lorentz transformations when the event occurs according a single direction (X or X') are:

x' = x - V ∙ t/1 - V2/c2
y' = y
z' = z
t' = t - V/c2 ∙ x/1 - V2/c2

For V << c, the Lorentz transformations point towards Galilean transformations. This is because V is assumed as very close to zero. Giving that the Galilean transformations represent a limit of Lorentz transformations, we can always use the Lorentz transformation but for practical purposes, we often use the Galilean transformations for normal velocities. This approach is similar to situations involving gravitation, in which it is not necessary to use the general formula of gravitation that involves the masses of objects and the distance between them [F = (G · M · m) / r2], but we often use the simplified formula F = m · g instead.

Lorentz transformations of spacetime coordinates allow us find the relativistic formulae of velocity transformations for a particle moving and observed in two inertial systems S and S'. the relationship between velocities in these two systems are:

vx' = vx - V/1 - V ∙ vx/c2
vy' = vy ∙ √1 - V2/c2/1 - V ∙ vx/c2
vz' = vz ∙ √1 - V2/c2/1 - V ∙ vx/c2

If we consider a one dimensional light ray (for example a light ray emitted by a laser) moving only in the X (X') direction, we have vx = c, vy = 0 and vz = 0 (in S). Thus, we find for the velocity in S' (giving that V << c):

vx' = c
vy' = 0
vz' = 0

Therefore, the observer in S' measures the same velocity c in the positive direction of X' while the velocity of light in the other directions is zero (as expected).

The Lorentz transformations for velocity converge with the classical formulae for V << c or for c → ∞. This is obvious given their structure.

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