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Two distinct spacetime events have timelike separation when their displacement has negative Minkowski interval in metric signature $(-,+,+,+)$, equivalently $c|\Delta t|>|\Delta \mathbf{x}|$. A massive observer can connect them along a straight timelike curve. If the later spacetime event has $\Delta t>0$, every Lorentz transformation in the orthochronous Lorentz group keeps it later. The elapsed proper time on the straight inertial segment is $\sqrt{-s^2}/c$.

In the light-second units of the Minkowski spacetime example, $O$ to $A$ has $s^2=(-4^2+3^2){\mathrm{ls}}^2=-7{\mathrm{ls}}^2$. Its straight moving clock therefore records $\sqrt{7}\mathrm{s}$, approximately $2.646\mathrm{s}$, while the coordinate time difference is $4\mathrm{s}$. Coordinate time and proper time answer different questions.

A runner enters a stationary tunnel of length $L$ at the tunnel-frame instant a photon is emitted towards them from the far end. The interception fraction is $f=\beta /(1+\beta )$, where $\beta =v/c$. In the runner frame, with entry at the origin, emission is at $(x^{\prime },t^{\prime })=(\gamma L,-\gamma vL/c^2)$ and the entrance clock simultaneously reads $-vL/c^2$. The entrance is then ahead by $\gamma {\beta }^{2}L$, so the photon travels $\gamma L$, not the contracted tunnel length $L/\gamma$. Subtracting that initial entrance gap from the entrance's motion during the photon flight gives the same interception fraction. This is a concrete application of relativity of simultaneity, length contraction and the Lorentz transformation.

During either inertial leg, the traveller can correctly say that the moving home clock runs slowly. The missing step is deciding which distant home-clock reading is simultaneous with turnaround. By the Lorentz transformation, the outward inertial frame's constant-time line through $P$ obeys $t-vx/c^2=\text{constant}$. At $x=0$ it meets the home world line atFor the return inertial frame, use $t+vx/c^2=\text{constant}$ through the same spacetime event $P$; its home-clock assignment isThe dashed lines in the spacetime diagram illustrate this relativity of simultaneity. For $v=0.6c$ and $T=10\mathrm{yr}$, they assign $3.2\mathrm{yr}$ and $6.8\mathrm{yr}$ to the home clock. The difference of $3.6\mathrm{yr}$ is a change in the traveller's distant-time assignment, not a sudden physical advance of the home clock.

On each leg, the traveller's $4\mathrm{yr}$ of proper time corresponds to $4/\gamma =3.2\mathrm{yr}$ of home-clock advance according to that leg's inertial frame. Including the twin paradox simultaneity change gives $3.2+3.6+3.2=10\mathrm{yr}$, exactly the reading at reunion. Actual received light signals remain continuous across a physical turnaround; their arrival times are distinct from the distant spacetime events assigned as simultaneous by Einstein synchronisation.

Decay of a massive parent into exactly two photons. In its rest frame, momentum conservation requires opposite photon momenta of equal magnitude, so each photon has energy $M{c}^2/2$. A boost along their common axis gives laboratory photon energies $\gamma M{c}^2(1\pm \beta )/2$. Their product stays $(M{c}^2/2{)}^2$ in that collinear configuration; other emission angles require the corresponding angular Lorentz transformation.

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.

Suppose standard collinear boosts have the Lorentz transformation form, but initially assign speed scales $a$ and $b$ to two successive transformations with relative velocities $v$ and $u$. Their composition has spatial and temporal diagonal factors proportional to $1+uv/a^2$ and $1+uv/b^2$, respectively. Closure within the same calibrated transformation family requires these factors to agree, forcing $a^2=b^2$ for generic nonzero boosts.

This argument fixes a universal reciprocal squared speed; it does not establish that it is nonzero. The common limit $1/a^2=0$ gives the consistent Galilean transformation family. A finite invariant speed of light is an additional empirical property of spacetime and electromagnetism, expressed by the second of the postulates of special relativity. It cannot be deduced by excluding the Galilean limit through boost composition alone.

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.

Two noncollinear Lorentz boosts produce a Lorentz transformation with both a boost and a rotation. In the SL(2,C) lift, the product of two positive Hermitian boost matrices generally ceases to be Hermitian. Its polar decomposition of an invertible complex matrix separates a positive Hermitian boost factor and a unitary rotation factor. The rotation depends on both boosts and their order, and is called the Wigner rotation.