Relativistic Doppler Effect - Accelerated Motion

Accelerated Motion

For general accelerated motion, or when the motions of the source and receiver are analyzed in an arbitrary inertial frame, the distinction between source and emitter motion must again be taken into account.

The Doppler shift when observed from an arbitrary inertial frame:

where:

is the velocity of the source at the time of emission

is the velocity of the receiver at the time of reception

is the light velocity vector

is the angle between the source velocity and the light velocity at the time of emission

is the angle between the receiver velocity and the light velocity at the time of reception

If is parallel to, then, which causes the frequency measured by the receiver to increase relative to the frequency emitted at the source . Similarly, if is anti-parallel to, which causes the frequency measured by the receiver to decrease relative to the frequency emitted at the source .

This is the classical Doppler effect multiplied by the ratio of the receiver and source Lorentz factors.

Due to the possibility of refraction, the light's direction at emission is generally not the same as its direction at reception. In refractive media, the light's path generally deviates from the straight distance between the points of emission and reception. The Doppler effect depends on the component of the emitter's velocity parallel to the light's direction at emission, and the component of the receiver's velocity parallel to the light's direction at absorption. This does not contradict Special Relativity.

The transverse Doppler effect can be analyzed from a reference frame where the source and receiver have equal and opposite velocities. In such a frame the ratio of the Lorentz factors is always 1, and all Doppler shifts appear to be classical in origin. In general, the observed frequency shift is an invariant, but the relative contributions of time dilation and the Doppler effect are frame dependent.