relativity
Michelson–Morley, Lorentz transforms, time dilation & length contraction, relativistic Doppler, dynamics & massless particles, four-vectors & invariants.
frames & transforms · time, length, simultaneity · Doppler · relativistic dynamics · four-vectors & invariance
Frames & transforms
Galilean. \(x'=x-vt\), \(t'=t\), \(u'=u-v\); acceleration invariant. Adequate for \(v\ll c\), but incompatible with a universal light speed.
Michelson–Morley. An interferometer compares round-trip light times along and across the supposed ether wind; rotating it 90° should shift the fringes by \(N=2lv^2/(\lambda c^2)\) (≈ 0.4 fringe for the actual apparatus). Result: null, in every season — there is no ether frame; \(c\) is the same in all inertial frames.
Postulates. Laws of physics the same in all inertial frames; \(c\) the same for all inertial observers. \(\beta=v/c\), \(\gamma=(1-\beta^2)^{-1/2}\).
Lorentz. \(x'=\gamma(x-vt)\), \(t'=\gamma(t-vx/c^2)\), \(y'=y\), \(z'=z\). Inverse: \(x=\gamma(x'+vt')\), \(t=\gamma(t'+vx'/c^2)\). Reduces to Galilean for \(\beta\ll1\); mixes space and time symmetrically.
Velocity transformation. \(u'_x=\dfrac{u_x-v}{1-u_xv/c^2}\), \(u'_y=\dfrac{u_y}{\gamma(1-u_xv/c^2)}\) (and \(z\) alike); inverses swap primes and flip the sign of \(v\). Speeds compose below \(c\): \(0.9c\) "plus" \(0.9c\) gives \(0.99c\), and \(u_x=c\) maps to \(u'_x=c\). First-order consequence: light in a medium moving at \(v\) travels at \(u\approx c/n+v(1-1/n^2)\) — the Fresnel drag Fizeau measured.
Time, length, simultaneity
Time dilation. Proper time \(\Delta t_0\): one clock present at both events; dilated \(\Delta t=\gamma\Delta t_0\). Fast unstable particles reach detectors far beyond \(c\tau\) — decay clocks run slow in the lab frame.
Length contraction. Proper length \(L_0\): object rest frame; contracted \(L=L_0/\gamma\) along the motion only — transverse dimensions are unchanged.
Simultaneity. \(\Delta t'=\gamma(\Delta t-v\Delta x/c^2)\): events simultaneous in one frame are not in another whenever \(\Delta x\ne0\).
Invariant interval. \(s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2\), the same in every inertial frame. Timelike (\(s^2\gt0\)): time order invariant, causal connection possible, proper time \(c^2\Delta\tau^2=s^2\); spacelike (\(s^2\lt0\)): time order frame-dependent, no causal connection — causality survives because signals cannot exceed \(c\).
Twin paradox. The travelling twin logs less proper time and returns younger. No symmetry: the traveller changes inertial frames at turnaround while the stay-at-home never accelerates — the two worldlines between the same events have different \(\int\dd\tau\).
Doppler
Longitudinal. Source and observer approaching at \(v\): \(\nu=\nu_0\sqrt{\dfrac{1+\beta}{1-\beta}}\); receding, invert the ratio. Only the relative velocity enters — unlike sound, where the medium distinguishes source from observer motion; the relativistic result is the geometric mean of the two classical ones.
Observer at angle \(\theta\) (measured in the observer's frame, \(\theta=0\) = approaching): \[\nu=\nu_0\,\frac{\sqrt{1-\beta^2}}{1-\beta\cos\theta}.\] At \(\theta=\pi/2\) the transverse Doppler shift \(\nu=\nu_0\sqrt{1-\beta^2}\) survives — pure time dilation, confirmed by Ives–Stilwell. Classical and relativistic formulas agree to order \(\beta\); differences enter at \(\beta^2\). Beat-frequency measurement of \(\dot r=-c(\nu-\nu_0)/\nu_0\) is the working principle of Doppler tracking/navigation.
Relativistic dynamics
Momentum & energy. \(\vv p=\gamma m\vv v\), \(E=\gamma mc^2\), \(E_0=mc^2\), \(K=(\gamma-1)mc^2\); \(\vv F=\dd\vv p/\dd t\). \(K\) is the work done from rest, and no finite work reaches \(v=c\).
Energy–momentum relation. \[E^2=(pc)^2+(mc^2)^2;\] photon \(m=0\), \(E=pc=hf=hc/\lambda\).
Mass–energy. \(\Delta E=\Delta mc^2\) for any form of energy: an inelastic collision converts kinetic energy into rest mass — the fused body is heavier than its parts, because heat has mass. Binding energy shows up as a mass deficit.
Massless particles. \(E=pc\), moving at exactly \(c\) in every frame. Photon momentum makes light push: radiation pressure \(I/c\) absorbed, \(2I/c\) reflected. Compton scattering off an electron: \(\lambda'-\lambda=\dfrac{h}{mc}(1-\cos\theta)\), with \(h/mc=2.43\,\mathrm{pm}\) — photons collide like particles. A lone photon cannot become an \(e^+e^-\) pair (energy and momentum cannot both balance); pair production needs a nucleus to absorb recoil. The photon mass is zero to strong limits (e.g. no frequency dispersion in pulsar arrival times).
Low-speed limit. \(\gamma\approx1+\tfrac12\beta^2\), \(K\approx\tfrac12mv^2\).
Four-vectors & invariance
Transformation viewpoint. Physical laws must keep their form under coordinate changes; the Lorentz transformation is a rotation-like linear map of spacetime that preserves the norm \(c^2t^2-x^2-y^2-z^2\), as spatial rotations preserve \(x^2+y^2+z^2\). A four-vector is any quadruple \(a_\mu\) transforming like \((ct,\vv r)\); the "dot product" of four-vectors is a scalar invariant, the interval being the norm of \(\Delta x_\mu\).
Four-velocity. \(U_\mu=\dfrac{\dd x_\mu}{\dd\tau}=\gamma(c,\vv u)\) — differentiate with respect to proper time so the result still transforms as a four-vector; norm \(U\cdot U=c^2\). Composing four-velocities reproduces the velocity-addition law.
Momentum–energy four-vector. \[p_\mu=mU_\mu=\Bigl(\frac{E}{c},\,\vv p\Bigr),\qquad p\cdot p=\Bigl(\frac{E}{c}\Bigr)^2-p^2=(mc)^2.\] Energy and momentum conservation merge into conservation of \(p_\mu\): if three components are conserved in every frame, the fourth must be too.
Invariant-mass method. For any system, \(E_{\mathrm{tot}}^2-(P_{\mathrm{tot}}c)^2\) is frame-invariant; the CM frame moves at \(\vv v_{\mathrm{cm}}=\vv P_{\mathrm{tot}}c^2/E_{\mathrm{tot}}\). Evaluate the invariant in the lab and in the CM to solve collision problems without transforming: e.g. \(e^-e^-\to e^-e^-e^-e^+\) needs threshold kinetic energy \(K=6m_ec^2\) on a stationary target (only the CM energy is available for making mass). The photon four-momentum \((h\nu/c)(1,\uvec n)\) reproduces the Doppler formula by transformation.