Kinematic Consequences

The Lorentz transformations acting on the spacetime interval produce the classic relativistic effects: time dilation, length contraction, velocity addition, and the relativistic Doppler/aberration formulas. Each follows by tracking events between frames. Units: $\beta =v/c$, $\gamma =(1-{\beta }^2{)}^{-1/2}$, $c=1$ where convenient. Derivations cite the boost rules (LT1) (forward) and (LT2) (inverse).

Time dilation

A clock at rest in the primed frame ticks $\Delta \tau$ (proper time); an observer who sees it move at $v$ measures

$$\Delta t=\gamma \Delta \tau \ge \Delta \tau .$$

Derivation. Two ticks of the moving clock occur at the same place in its rest frame, $\Delta x^{\prime }=0$, separated by $\Delta t^{\prime }=\Delta \tau$. By the inverse boost (LT2) $\Delta t=\gamma (\Delta t^{\prime }+\beta \Delta x^{\prime }/c)$, so $\Delta t=\gamma \Delta \tau$ directly.

Moving clocks run slow. The effect is reciprocal — each sees the other slow — without contradiction, because they compare different pairs of events (relativity of simultaneity). Experimentally: muon lifetimes, Hafele–Keating, GPS.

Light clock (geometric derivation)

The same result drops out without the boost, using only the constancy of $c$. A light clock bounces a photon between mirrors a distance $L$ apart; one tick is $\Delta \tau =2L/c$ at rest. Set the clock moving at $v$ perpendicular to the photon's path: the photon must travel a diagonal while the far mirror advances by $v\Delta t/2$.

Pythagoras on the diagonal: $(c\Delta t/2{)}^2=L^2+(v\Delta t/2{)}^2$. With $L=c\Delta \tau /2$, solve for $\Delta t$: $$\Delta t=\frac{\Delta \tau }{\sqrt{1-v^2/c^2}}=\gamma \Delta \tau ,$$ identical to the algebraic result but using only "$c$ is the same for both observers." Reorienting the clock along the motion and demanding it tick the same forces the parallel arm to shrink by $1/\gamma$ — that is length contraction.

Length contraction

An object of rest length $L_0$ moving along its length is measured (ends located simultaneously) as

$$L=\frac{L_0}{\gamma }\le L_0.$$

Derivation. The rod has rest length $\Delta x^{\prime }=L_0$. Measuring both ends simultaneously in the lab means $\Delta t=0$; by the forward boost (LT1) $\Delta x^{\prime }=\gamma (\Delta x-\beta c\Delta t)=\gamma \Delta x$, so $L=\Delta x=L_0/\gamma$.

Light clock (along the motion). Lay the clock parallel to $v$ with lab-frame length $L$. The photon chases the front mirror, then the back mirror chases the photon. Outbound: $c{t}_1=L+v{t}_1$, so $t_1=L/(c-v)$; return: $c{t}_2=L-v{t}_2$, so $t_2=L/(c+v)$. The round trip is

$$\Delta t_{\parallel }=\frac{L}{c-v}+\frac{L}{c+v}=\frac{2L}{c}{\gamma }^2.$$ An identical clock held transverse ticks $\Delta t_{\perp }=\gamma (2L_0/c)$ (time dilation above). The two clocks are identical, so $\Delta t_{\parallel }=\Delta t_{\perp }$, giving $2L{\gamma }^2/c=2L_0\gamma /c$, hence $L=L_0/\gamma$. (This consistency is exactly what the Michelson–Morley null result demands.)

Transverse lengths are unchanged. Contraction is a consequence of relativity of simultaneity, not a physical squeezing.

Velocity addition

For motion along the boost axis, velocities combine as

$$u^{\prime }=\frac{u-v}{1-uv/c^2},$$

Derivation. Differentiate the forward boost (LT1): $u^{\prime }=d{x}^{\prime }/d{t}^{\prime }=\gamma (dx-vdt)/[\gamma (dt-vdx/c^2)]=(u-v)/(1-uv/c^2)$, dividing through by $dt$.

so subluminal speeds never sum to $c$, and light stays at $c$. Transverse components pick up a $\gamma$: $u_{\perp }^{\prime }=u_{\perp }/[\gamma (1-uv/c^2)]$. In rapidity, collinear addition is just ${\varphi }_{\text{tot}}={\varphi }_1+{\varphi }_2$ (see Lorentz transformations § rapidity).

Relativistic Doppler effect

A source frequency $f_0$ moving with $\beta$ is observed (receding) as

$$f=f_0\sqrt{\frac{1-\beta }{1+\beta }}(\text{recession}),f=f_0\sqrt{\frac{1+\beta }{1-\beta }}(\text{approach}).$$

Derivation. Time dilation (LT2) slows the source's emission rate to $f_0/\gamma$ in the lab; classical wavefront stretching from recession adds a $1/(1+\beta )$ factor. Combining, $f=f_0/[\gamma (1+\beta )]=f_0\sqrt{1-{\beta }^2}/(1+\beta )=f_0\sqrt{(1-\beta )/(1+\beta )}$.

Even transverse motion ($\theta =90^{\circ }$) gives a redshift $f=f_0/\gamma$ — the transverse Doppler effect, a pure time-dilation signature absent classically.

Aberration and the headlight effect

Emission angles transform as $\cos {\theta }^{\prime }=(\cos \theta -\beta )/(1-\beta \cos \theta )$, so a fast emitter beams radiation forward — the headlight effect, central to synchrotron sources and relativistic jets.

Summary

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These effects power the paradox resolutions. For energy and momentum see relativistic dynamics.