Relativistic Doppler

The relativistic Doppler effect is different from the true (non-relativistic) Doppler Effect as the equations include the Time Dilation effect of Special Relativity .
They describe the total difference in observed frequencies and possess the required Lorentz Symmetry .

THE MECHANISM (A SIMPLE CASE)

Assume the observer and the source are moving away from each other with a relative velocity

v\,

. Let us consider the problem from the Reference Frame of the source.

Suppose one Wavefront arrives at the observer. The next wavefront is then at a distance

\lambda=c/f_s\,

away from her (where

\lambda\,

is the Wavelength ,

f_s\,

is the Frequency of the source, and

c\,

is the Speed Of Light ). Since the wavefront moves with velocity

c\,

and the observer escapes with velocity

v\,

, they will meet after a time

:

T = rac{\lambda}{c-v} = rac{1}{(1-v/c)f_s}

However, due to the relativistic Time Dilation , the observer will measure this time to be

:

T_o = rac{T}{\gamma} = rac{1}{\gamma(1-v/c)f_s}

where

\gamma = 1/\sqrt{1-v^2/c^2}

, so the corresponding frequency is

:

f_o = rac{1}{T_o} = \gamma (1-v/c) f_s = \sqrt{rac{1-v/c}{1+v/c}}\,f_s

GENERAL RESULTS

For motion along the line of sight

If the observer and the source are moving directly away from each other with velocity

v\,

, the observed Frequency

f_o\,

is different from the frequency of the source

f_s\,

as

:

f_o = \sqrt{rac{1-v/c}{1+v/c}}\,f_s

where

c\,

is the Speed Of Light .

The corresponding Wavelength s are related by

:

\lambda_o = \sqrt{rac{1+v/c}{1-v/c}}\,\lambda_s

and the resulting Redshift

z\,

can be written as

:

z + 1 = rac{\lambda_o}{\lambda_s} = \sqrt{rac{1+v/c}{1-v/c}}

In the non-relativistic limit, i.e. when

v \ll c\,

, the approximate expressions are:

:

rac{\Delta f}{f} \simeq -rac{v}{c} \qquad rac{\Delta \lambda}{\lambda} \simeq rac{v}{c} \qquad z \simeq rac{v}{c}

Note: In all the expressions in this section it is assumed that the observer and the source are moving ''away'' from each other. If they are moving ''towards'' each other,

v\,

should be taken negative.

For motion in an arbitrary direction

If, in the Reference Frame of the observer, the source is moving away with velocity

v\,

at an angle

heta\,

relative to the direction from the observer to the source (at the time when the light is emitted), the frequency changes as

:

f_o = rac{f_s}{\gamma\left(1+rac{v\cos	heta}{c}
ight)}

where

\gamma = rac{1}{\sqrt{1-v^2/c^2}}

However, if the angle

heta\,

is measured in the Reference Frame of the source (at the time when the light is received by the observer), the expression is

:

f_o = \gamma\left(1-rac{v\cos	heta}{c}
ight)f_s

In the non-relativistic limit:

:

rac{\Delta f}{f} \simeq -rac{v\cos	heta}{c}

	CATEGORIES ABOUT RELATIVISTIC DOPPLER EFFECT

	special relativity

	wave mechanics

	doppler effects

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Information About

Relativistic Doppler