List of Relativistic Equations

Doppler Shift

General doppler shift:

Doppler shift for emitter and observer moving right towards each other (or directly away):

Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them:

Derivation of the relativistic Doppler shift
If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter. If one assumes that the observer is moving with respect to the emitter along the x-axis, then the standard Lorentz transformation of the four-momentum, which includes energy, becomes: Now, if where θ is the angle between p_x and, and plugging in the formulas for frequency's relation to momentum and energy: This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives: $\begin{align} \nu' & = \gamma \nu \left ( 1 - \beta \right )\\ & = \nu \frac{1}{\sqrt{1 - \beta^2}} \left ( 1 - \beta \right ) \\ & = \nu \frac{1}{\sqrt{\left ( 1 - \beta \right ) \left ( 1 + \beta \right ) }} \left ( 1 - \beta \right ) \\ & = \nu \frac{\sqrt{1 - \beta}}{\sqrt{1 + \beta}} \end{align}$ This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the x-axis. The second special case is that where the relative velocity is perpendicular to the x-axis, and thus θ = π/2, and cos θ = 0, which gives: This is actually completely analogous to time dilation, as frequency is the reciprocal of time. So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.

Derivation of the relativistic Doppler shift

If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter. If one assumes that the observer is moving with respect to the emitter along the x-axis, then the standard Lorentz transformation of the four-momentum, which includes energy, becomes:

Now, if

where θ is the angle between p_x and, and plugging in the formulas for frequency's relation to momentum and energy:

This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives:

$\begin{align} \nu' & = \gamma \nu \left ( 1 - \beta \right )\\ & = \nu \frac{1}{\sqrt{1 - \beta^2}} \left ( 1 - \beta \right ) \\ & = \nu \frac{1}{\sqrt{\left ( 1 - \beta \right ) \left ( 1 + \beta \right ) }} \left ( 1 - \beta \right ) \\ & = \nu \frac{\sqrt{1 - \beta}}{\sqrt{1 + \beta}} \end{align}$

This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the x-axis. The second special case is that where the relative velocity is perpendicular to the x-axis, and thus θ = π/2, and cos θ = 0, which gives:

This is actually completely analogous to time dilation, as frequency is the reciprocal of time. So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.

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