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Thermodynamics

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Lorentz Transformations

Consider two frames of reference A and B, A is fixed and B is moving with a constant speed v along the direction of the x-axis. Initially, both the frame of reference B has moved a distance OO’ = vt. For a point P in space, the co-ordinates are (x, y, z) with reference to the frame A and (x’, y’, z’) with reference to the frame B.

According to Galilean’ frame of reference

x’ = x – vt

y’ = y

z’ = z

t’ = t                              (i)

Differentiating the first equation in (i)

dx'/dt = dx/dt – v                     (ii)

c’ = c – v                       (iii)

It means that if a person is moving in a spaceship, the speed of the passing light will be (c – v). But all attempts to show that the velocity of light changes with the motion of the observer have failed. The velocity of light remains constant in free space.

When the transformation of equations (i) are made in Maxwell’s equations, their form does not remain the same. To overcome this, Lorentz suggested the following transformations.

y’ = y                          (v)

z’ = z                          (vi)

These equations are known as Lorentz transformations. When substituted in Maxwell’s equations, the latter remain unchanged.

Therefore, using Lorentz transformation,

where m₀ is the rest mass.

With this transformation for mass, there will be harmony between the laws of electrodynamics and Newton’s laws. Thus all equations will appear similar in the two frames of reference A and B.