Velocity-addition formula

The velocity-addition formula translates a particle's velocity between two specified inertial frames. Let $S^{\prime }$ move at signed velocity $V$ along the $x$-axis of $S$, with parallel axes and $|V|<c$, where $c$ is the speed of light. Define $u_x=dx/dt$ and $u_x^{\prime }=d{x}^{\prime }/d{t}^{\prime }$ for the same particle's world line, using position and time measured within the corresponding inertial frame. With Lorentz factor ${\gamma }_V=(1-V^2/c^2{)}^{-1/2}$, the forward Lorentz boost gives

For collinear motion, write $u=u_x$ and $u^{\prime }=u_x^{\prime }$. The inverse velocity-addition formula isThis is the composition of the particle's velocity in $S^{\prime }$ and that frame's velocity relative to $S$. The velocities must belong to this same chain of reference frames, with directions specified. Two arbitrary quoted speeds cannot be inserted blindly. Each velocity measurement uses distances and clocks in a single inertial frame, with Einstein synchronisation for separated clocks.

For example, let $V=0.6c$ and let the particle travel at $u^{\prime }=0.6c$ in $S^{\prime }$. Its velocity in $S$ isThe Galilean transformation would instead give $u=u^{\prime }+V=1.2c$, which fails here. For light moving along the positive axis, $u^{\prime }=c$ gives $u=c$; $u^{\prime }=-c$ similarly gives $u=-c$. Negative directions also work for massive particles: at $V=0.6c$, $u^{\prime }=-0.6c$ gives $u=0$, while $u^{\prime }=-0.8c$ gives $u=-5c/13$. Thus “addition” means composition of signed velocities, rather than addition of their magnitudes.

A short derivation differentiates the Lorentz transformation: $d{x}^{\prime }={\gamma }_V(dx-Vdt)$ and $d{t}^{\prime }={\gamma }_V(dt-Vdx/c^2)$. Dividing these differentials gives the longitudinal velocity-addition formula. The transverse formulas use $d{y}^{\prime }=dy$ and $d{z}^{\prime }=dz$, but the same transformed time denominator; omitting it would ignore the relativity of simultaneity. Replacing $V$ by $-V$ gives the inverse. When both relative and particle velocities are small compared with the speed of light, the denominator is close to one and the nonrelativistic limit recovers the Galilean transformation.

The formula lets us compare particle motion in a moving apparatus and a laboratory without losing the invariant speed of light. Einstein's 1905 paper derives the coordinate change in §3 and the composition of velocities in §5, including the collinear formula, its light-speed case, and composition of parallel transformations. ^[ref]. The rapidity parametrization makes the collinear composition particularly simple.

For future-directed unit timelike four-velocities in metric signature $(+---)$, $U_1\cdot U_2/c^2$ is the Lorentz factor of either observer measured in the rest frame of the other. In a common inertial frame it equals ${\gamma }_1{\gamma }_2(1-{\mathbf{\beta }}_1\cdot {\mathbf{\beta }}_2)$. ThereforeExpanding the numerator converts this to the dot-and-cross-product form of relativistic velocity addition. The invariant construction explains why the relative speed is symmetric even when the spatial velocity components belong to different rest frames.

Compare two observers at one event using a common static orthonormal frame outside a Schwarzschild horizon. If one observer moves tangentially at speed $u$ and another radially at speed $w$, their four-velocity contraction is ${\gamma }_u{\gamma }_w{c}^2$. Their relative speed therefore has the displayed form. The radial component measured by the tangential observer is $w/{\gamma }_u$, so Euclidean subtraction of the two original three-velocities fails. For a circular timelike geodesic and radial fall from rest at infinity at radius $R$, the relative speed is $c\sqrt{3\mu /R}$, where $\mu =GM/c^2$ and $R>3\mu$.

Let a receiver be initially at distance $D$ from a stationary responder and approach at speed $\beta c$. A light signal takes $D/c$ to reach the responder, which waits time $\delta$ and replies. Intersecting the reply's null worldline with the receiver's worldline givesThe receiver's proper time is ${\tau }_D$. These expressions describe reception before arrival when the reply is sent before the receiver reaches the responder; otherwise the trajectory after arrival must be specified.

In 1851 Hippolyte Fizeau detected a change in light propagation through moving water using interference fringes. Sunlight entered a telescope through a partially reflecting mirror; two slits selected beams passing through parallel glass tubes, each $1.487\mathrm{m}$ long with $5.3\mathrm{mm}$ internal diameter. A second telescope and focal mirror exchanged the tubes on return, and the beams recombined at a graduated eyepiece. Compressed air drove water between reservoirs, producing opposite flows; reversing both flows reversed the interference fringe displacement. The double transit suppressed static differences in temperature and pressure; filling only one tube with water tested this compensation. Separate supports reduced mechanical disturbance. Hippolyte Fizeau estimated flow speed from volumetric flow rate divided by tube cross section.^[ref]

The Fizeau experiment turns a tiny change in phase velocity into a visible fraction of an interference fringe. The Fizeau experiment path-length convention below explains why both tube passages contribute with the same sign for one beam and the opposite sign for the other. This makes the apparatus sensitive to the part of the phase difference that reverses with the flow, while rejecting disturbances shared by both beams.

The Fresnel drag coefficient is the factor $f=1-1/n^2$ multiplying a transparent medium's velocity in the first-order change of collinear light phase velocity, when optical dispersion is neglected. Here $n$ is the medium's refractive index in its rest frame. The name recalls Augustin-Jean Fresnel's partial-drag law; special relativity obtains the same coefficient from the velocity-addition formula without introducing a luminiferous aether.

Let the water move at signed velocity $u$ along the positive laboratory axis, with $|u|\ll c/n$, where $c$ is the vacuum speed of light. Assume a transparent medium with constant $n>1$, uniform flow and negligible optical dispersion. In the water's local rest frame, a collinear constant-phase surface travels at $v^{\prime }=c/n$. Dividing the Lorentz transformation differentials along that surface gives the same longitudinal velocity-addition formula as for a particle:The correction comes from the transformed time denominator. Simply adding $u$ to $c/n$ would omit it. This argument concerns collinear phase velocity; in a medium with optical dispersion, group velocity is a different quantity.^[ref]

For $n=4/3$, the Fresnel drag coefficient is $f=7/16$. A chosen flow speed of $6\mathrm{m}{\mathrm{s}}^{-1}$ therefore changes the forward phase velocity by approximately $2.625\mathrm{m}{\mathrm{s}}^{-1}$, even though the stationary-medium phase velocity is about $2.25\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These are model values. The invariant vacuum speed of light remains $c$; $c/n$ describes propagation through matter in its own rest frame.

Use the schematic below to count the water traversals explicitly. Each stationary tube has laboratory length $\ell$. In the displayed setting, upper-tube water flows right and lower-tube water flows left, each at speed $u>0$. Beam A goes right through the upper tube, exchanges tubes at the far optics, and returns left through the lower tube: it follows the flow twice. Beam B goes right through the lower tube and returns left through the upper tube: it opposes the flow twice. Each complete beam path therefore contains total water length $L=2\ell$, not $\ell$ or $4\ell$. The connecting air paths and fixed optics are assumed to contribute no flow-dependent phase difference.

Let $v_{+}$ and $v_{-}$ be positive phase velocity magnitudes along and against the water flow:For a laboratory source of frequency $\nu =c/{\lambda }_0$, where ${\lambda }_0$ is its vacuum wavelength, define the signed interference fringe offset $N$ by the propagation phase difference $\Delta \varphi =2\pi \nu \Delta t=2\pi N$. This is an equivalent phase delay between steady beams; no stopwatch resolves their transit-time difference. Subtract the zero-flow baseline and expand to first order:Thus one shifted fringe corresponds to one cycle of phase difference. If the stationary pattern has spatial interference fringe spacing $d$, a simple readout model is $I(x)=I_0[1+\mathcal{V}\cos (2\pi x/d+2\pi N)]$, where $\mathcal{V}$ is the interference visibility. The maxima shift by $-dN$ with this choice of screen coordinate. A sign assigned to “rightward” motion depends on the image orientation; the magnitude in fringe units does not.

Reversing both flows gives $N(-u)=-N(u)$. The displacement between equal and opposite settings therefore has magnitudeThe factor $2$ from comparing opposite flows is additional to the factor $2$ already included in $L=2\ell$. Repeating a round trip in just one flowing tube would instead put a beam with the flow once and against it once; that geometry does not produce the two same-sign contributions drawn here.

Choose $\ell =1.50\mathrm{m}$, $n=4/3$, ${\lambda }_0=550\mathrm{nm}$ and $u=6.00\mathrm{m}{\mathrm{s}}^{-1}$, with $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. These are illustrative choices, not measured values or a reconstruction of Hippolyte Fizeau's data. Each beam traverses $L=3.00\mathrm{m}$ of water. The nondispersive prediction isThe first number is the offset from rest to one flow setting; the second compares equal opposite settings. The very short equivalent delay explains why interference is useful: a measurable fraction of a wavelength replaces an unattainable direct timing measurement.

The Galilean transformation would give phase velocity magnitudes $c/n\pm u$, hence $N_{\mathrm{full}}\simeq 4\ell u{n}^2/(c{\lambda }_0)$. At the chosen values this is $0.38815$ fringe from rest, larger than the special relativity prediction by $n^2/(n^2-1)=16/7$. A model with no flow-induced change of phase velocity instead gives $N=0$. The three alternatives are experimentally distinguishable even when $u/c$ is tiny.

Hippolyte Fizeau's positive displacement favoured Augustin-Jean Fresnel's partial-drag law over zero or full drag. He explicitly distinguished support for that law from establishing its underlying luminiferous aether hypothesis.^[ref] The Fizeau experiment therefore supplies a nonzero relative-motion measurement, not a null test proving an luminiferous aether's absence. The derivation above exhibits the same first-order law within special relativity, showing why agreement with the coefficient alone cannot select an physical interpretation.

Two limitations matter for precision. The velocity profile sampled by the beam need not equal the cross-sectional average inferred from volumetric flow rate. Also, the Doppler effect gives opposite rest-frame frequencies for the counterpropagating beams. With optical dispersion, the coefficient $n^2-1$ in the interference fringe formula becomeswhere the refractive index and its derivative are evaluated for stationary water at the laboratory angular frequency $\omega =2\pi c/{\lambda }_0$. A wavelength-independent $n$ therefore defines an approximation, not a precision fit to real water.^[ref] Flow reversal removes a fixed phase difference, but a mechanical or temperature change that reverses with the flow can still imitate the signal and must be controlled.