Relativistic Transformation of Velocity and the Relativistic Doppler Effect Revision Notes

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18.4Relativistic Transformation of Velocity and the Relativistic Doppler Effect


In these revision notes for Relativistic Transformation of Velocity and the Relativistic Doppler Effect, we cover the following key points:

  • What is the relationship between velocities measured in two different inertial systems of reference?
  • What is the relationship between values of light wavelength measured in two different inertial systems of reference?
  • The same for frequencies of light waves
  • How does Relativistic Doppler Effect helps in understanding how the universe works?

Relativistic Transformation of Velocity and the Relativistic Doppler Effect Revision Notes

In relativistic events we cannot use the formulae of classical physics as they give wrong results. Using the relativistic approach, we find the following relationship between velocities of the same particle when considered from two inertial frames of reference S and S':

v = v' + V/1 + v' ∙ V/c2

where v is the velocity of the moving particle when viewed from the system S at rest, v' the velocity of particle when viewed from the system connected to the source, V is the moving velocity of system S' relative to S and c is the speed of light in vacuum.

For V << c (as usually occurs in daily life), we obtain the classical relationship between velocities v = v' (as expected). If we use light signals instead of material particles in the above experiment (i.e. if we put c instead of v'), we obtain v = c.

The light waves do not require any material medium to propagate. Therefore, any possible Doppler Effect is simply determined by the movement of observer relative to the source (for light vd = vs = 0 because they are too small to be considered and moreover, there is no material medium of propagation in vacuum; the cosmic ether does not exist).

The formula that relates the frequency of original and observed light during red shift (when the star or galaxy is moving away from Earth) is

f = f' ∙ 1-V/c/1 + V/c

while that of blue shift (when the star or galaxy is approaching the Earth) is

f = f' ∙ 1 + V/c/1-V/c

where V is the speed of the moving star, f is the observable frequency and f' is the original frequency of light.

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