Relativistic Doppler effect ($\nu =\gamma {\nu }^{\prime }(1+\beta \cos {\theta }^{\prime })$)

The Lorentz transformation changes the photon four-momentum and therefore its frequency, producing the displayed formula for a boost along the first axis. Here the primed ray angle is measured in the emitting frame and $\beta =v/c$. The angle and frequency transforms must be used together: aberration of light changes the direction, and a photon-number angular fraction need not equal the corresponding radiant-energy fraction.

Here $c$ is the vacuum speed of light, $\gamma$ is the Lorentz factor, and the two frequencies are measured in different inertial reference frames. Unlike the classical Doppler effect for sound, there is no material rest reference frame for light in vacuum. For small radial recession speed $v_r\ll c$, the wavelength shift satisfies $\Delta \lambda /\lambda \simeq v_r/c$. This small-speed limit underlies Doppler spectroscopy; it is not an exact sound formula.

If two spectral lines come from the same uniformly moving source along the same line of sight, each wavelength is multiplied by the same longitudinal Doppler factor $D$. Subtracting ${\lambda }_{2,\mathrm{obs}}=D{\lambda }_{2,\mathrm{emit}}$ and ${\lambda }_{1,\mathrm{obs}}=D{\lambda }_{1,\mathrm{emit}}$ proves the displayed identity. Recession has $D>1$ and approach has $D<1$. Defining positive radial velocity for recession gives $\beta =(D^2-1)/(D^2+1)$. A measured increase in wavelength separation therefore cannot be reconciled with strictly approaching longitudinal motion merely by changing the speed's magnitude.

For a forward photon and a positive collinear Lorentz transformation, the Doppler factor is $D=\gamma (1+\beta )$. The identity $\gamma \beta =\sqrt{{\gamma }^2-1}$ gives the exact expression in the title. Expanding the square root by the binomial series gives $D=2\gamma -1/(2\gamma )-1/(8{\gamma }^3)+O({\gamma }^{-5})$. A leading approximation $D\sim 2\gamma$ has relative error of order ${\gamma }^{-2}$; retaining a purported subleading term requires expanding all dependent quantities consistently. For example, $\sqrt{4{\gamma }^2-1}$ has subleading term $-1/(4\gamma )$ and is not the exact longitudinal Doppler factor.

For two inertial frames with equal speed in a laboratory, a photon is measured at equal angular frequency by their comoving observers exactly when its laboratory propagation unit vector $\mathbf{n}$ obeys the displayed condition. Indeed, measured angular frequency is $-k\cdot U=\gamma {\omega }_{\mathrm{lab}}(1-\mathbf{n}\cdot \mathbf{v}/c)$, and equal laboratory Lorentz factors cancel. Thus the permitted directions form a great circle on the unit sphere. In the plane of the two laboratory velocities, only the internal angle bisector line remains. The velocity information alone fixes neither direction along that line nor coplanarity of the ray.

For a freely propagating photon between emitters and receivers rotating with the same angular velocity in flat spacetime, ${\nu }_r/{\nu }_e={\gamma }_r/{\gamma }_e$, provided the photon path is unobstructed. At equal radii the ratio is one. Positions and velocities at the actual emission and reception events must be used.

A source of fixed proper frequency and speed $v$ on a circle has instantaneous frequency extrema ${\nu }_0\sqrt{(1\pm \beta )/(1\mp \beta )}$ for a stationary coplanar receiver outside the circle. The extrema occur when the emission ray is tangent to the source circle, so line-of-sight speed reaches $\pm v$.

Let $\theta$ be the angle between the source velocity and the emitted photon's propagation direction in the observer's frame. The relativistic Doppler effect gives $\nu ={\nu }_0/[\gamma (1-\beta \cos \theta )]$. A ray transverse in that frame has $\lambda =\gamma {\lambda }_0$, a redshift. A ray transverse in the source frame instead has $\cos \theta =\beta$ by aberration of light, and $\lambda ={\lambda }_0/\gamma$, a blueshift.

The time at which a passing source is closest must also be specified. Put the observer at the origin and the source at $(v{t}_e,d)$ at emission, with finite transverse offset $d>0$. Reception occurs atEmission at closest approach means $t_e=0$, a transverse observer-frame ray received at $t_r=d/c$, hence a redshift. Reception at closest approach means $t_r=0$, so $t_e=-\gamma d/c$ and the ray has $\cos \theta =\beta$, hence a blueshift. Sending $d$ to zero does not remove the distinction between these two limiting event prescriptions. ^[ref]

A mirror receding at velocity $u=\beta c$ from normally incident light produces two applications of the relativistic Doppler effect, one into the mirror rest frame and one back to the observer. In that rest frame, reflection preserves frequency. Consequently ${\omega }_r={\gamma }^2(1-\beta {)}^2{\omega }_i={\omega }_i(1-\beta )/(1+\beta )$. Equivalently, continuity of wave phase on the moving surface $z=ut$ gives ${\omega }_i-k_{i}u={\omega }_r+k_{r}u$, with $k_i={\omega }_i/c$ and $k_r={\omega }_r/c$. For $|u|\ll c$, the fractional frequency change is $-2u/c$ to first order.

For a photon emitted from the origin and arriving at receding mirror position $x(t)$, its emission time is $t_e=t-x(t)/c$. Thus $d{t}_e/dt=1-\beta$ and a steady source rate $R$ gives impact rate $R(1-\beta )$ in source-frame coordinate time. This is distinct from the emission rate and from a rate per mirror proper time.

A kinematic redshift is the relativistic Doppler effect from relative motion of emitter and receiver. With positive radial recession speed $v$, its small-velocity limit is $z_{\mathrm{kin}}=v/c+O(v^2/c^2)$. A peculiar velocity gives such a factor in addition to cosmological redshift.