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Systematic derivation for inertial observers of Relativistic Doppler effect

Let us repeat the derivation more systematically in order to show how the Lorentz equations can be used explicitly to derive a relativistic Doppler shift equation for waves that themselves are not relativistic.

Let there be two inertial frames of reference, $S$ and $S^\prime$ , constructed so that the axes of $S$ and $S^\prime$ coincide at $t=t^\prime=0$ , where $t$ is the time as measured in $S$ and $t^\prime$ is the time as measured in $S^\prime$ . Let $S^\prime$ be in motion relative to $S$ with constant velocity $v$ ; without loss of generality, we will take this motion to be directed only along the x-axis. Thus, the Lorentz transformation equations take the form

$x$	$=$	$\gamma ( x^\prime + \beta ct^\prime )$
$y$	$=$	$y^\prime$
$z$	$=$	$z^\prime$
$ct$	$=$	$\gamma ( ct^\prime + \beta x^\prime )$
$\frac{dx}{dt}$	$=$	$\frac{\frac{dx^\prime}{dt^\prime} + v}{1+\frac{v}{c^2}\frac{dx^\prime}{dt^\prime}}.$

where $\beta = v/c$ and $\gamma = (1-\beta^2)^{-1/2}$ , and $c$ is the speed of light in a vacuum.

The derivation begins with what the observer in $S^\prime$ trivially sees. We imagine a signal source is positioned stationary at the origin, $O^\prime$ , of the $S^\prime$ system. We will take this signal source to produce its first pulse at time $t_1^\prime = 0$ (this isevent 1) and its second pulse at time $t_2^\prime = 1/f^\prime$ (this is event 2), where $f^\prime$ is the frequency of the signal source as the observer in $S^\prime$ sees it. We then simply use the Lorentz transformation equations to see when and where the observer in $S$ sees these two events as occurring:

Observer in $S^\prime$

Observer in $S$

Event 1

$x_1^\prime = 0$

$t_1^\prime = 0$

$x_1 = 0$

$t_1 = 0$

Event 2

$x_2^\prime = 0$

$t_2^\prime = \frac{1}{f^\prime}$

$x_2 = \gamma \frac{v}{f^\prime}$

$t_2 = \gamma \frac{1}{f^\prime}$

The period between the pulses as measured by the $S$ observer is not, however, $t_2 - t_1$ because event 2 occurs at a different point in space to event 1 as observed by the $S$ observer (that is, $x_2 \neq x_1$ ) — we must factor in the time taken for the pulse to travel from $x_2$ to $x_1$ . Note that this complication is not relativistic in nature: this is the ultimate cause of the Doppler effect and is also present in the classical treatment. This transit time is equal to the difference $x_2 - x_1$ divided by the speed of the pulse as the $S$ observer sees it. If the pulse moves at speed $-u^\prime$ in $S^\prime$ (negative because it moves in the negative x-direction, towards the $S$ observer at $O$ ), then the speed of the pulse moving towards the observer at $O$ , as $S$ sees it, is:

$-u = \frac{-u^\prime + v}{1 + (-u^\prime) \frac{v}{c^2}},$

using the Lorentz equation for the velocities, above. Thus, the period between the pulses that the observer in $S$ measures is:

$\tau$	$= t_2 - t_1 + \left( \gamma \frac{v}{f^\prime} \right) \left( \frac{u^\prime - v}{1 - v u^\prime / c^2} \right)^{-1}$
	$= \frac{\gamma}{f^\prime} + \frac{\gamma}{f^\prime} \frac{v}{u^\prime - v} \left(1 - v u^\prime / c^2 \right).$

Replacing $\tau$ with $1/f$ and simplifying, we get the required result that gives the relativistic Doppler shift of any moving wave in terms of the stationary frequency, $f^\prime$ :

$f = \gamma \left( 1 - \frac{v}{u^\prime} \right) f^\prime.$

Ignoring the relativistic effects by taking $v \ll c$ or $c \rightarrow \infty$ (equivalent to $\gamma \rightarrow 1$ ) gives the classical Doppler formula:

$f = \left( 1 - \frac{v}{u^\prime} \right) f^\prime.$

For electromagnetic radiation where $u^\prime = c$ the formula becomes

$f = \gamma \left( 1 - \frac{v}{c} \right) f^\prime = \gamma \left( 1 - \beta \right) f^\prime = f^\prime \sqrt{\frac{1-\beta}{1+\beta}}$

or in terms of wavelength:

$\lambda = \lambda^\prime \sqrt{\frac{1+\beta}{1-\beta}},$

where $\lambda^\prime$ is the wavelength of the source at the origin $O^\prime$ as the observer in $S^\prime$ sees it.

For electromagnetic radiation, the limit to classical mechanics, $c \rightarrow \infty$ , is instructive. The Doppler effect formula simply becomes $f = f^\prime$ . This is the correct result for classical mechanics, although it is clearly in disagreement with experiment. It is correct since classical mechanics regards the maximum speed of interaction — for electrodynamics, the speed of light — to be infinite. The Doppler effect, classical or relativistic, occurs because the wave source has time to move by the time that previous waves encounter the observer. This means that the subsequent waves are emitted further away (or closer) to the observer than they otherwise would be if the source were not in motion. The effect of this is to stretch (or compress) the wavelength of the wave as the observer encounters them. If however the waves travel instantaneously, the fact that the source is further away (or closer) makes no difference because the waves arrive at the observer no later or earlier than they would anyway since they arrive instantaneously. Thus, classical mechanics predicts that there should be no Doppler effect for light waves, whereas the relativistic theory gives the correct answer, as confirmed by experiment.

Black Holes

Sunday, 9 June 2013

Systematic derivation for inertial observers of Relativistic Doppler effect

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