Special Relativity — formula sheet
Every boxed result from the 17 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Lorentz Transformation from the Two Postulates
A boost mixes space and time linearly. The spatial equation says positions measured in \(S'\) are the Galilean \(x-vt\) rescaled by \(\gamma\ge 1\); the time equation adds a position-dependent term \(-vx/c^2\), so two events simultaneous in \(S\) (equal \(t\), different \(x\)) are not simultaneous in \(S'\). Simultaneity is frame-dependent, and \(\gamma\) governs how strongly clocks and rods disagree. As \(v/c\to 0\), \(\gamma\to 1\) and the map collapses to the Galilean \(x'=x-vt,\ t'=t\).
Invariance of the Spacetime Interval
The interval between two events is a single number that all inertial observers compute to be the same, even though each measures different \( \Delta t \) and different \( \Delta \mathbf{r} \). Its sign is invariant too: \( s^2<0 \) (timelike) means one event lies inside the other's light cone and the two can be causally connected by a sub-light signal; \( s^2=0 \) (null) means they are joined by a light ray; \( s^2>0 \) (spacelike) means no signal can connect them and their time-order is frame-dependent.
Relativity of Simultaneity
Clocks synchronised and separated by proper distance \(L_0\) along the motion do not stay synchronised in a frame through which they move at speed \(v\): the clock that is ahead in the direction of motion (the "leading" clock) reads behind the trailing clock by \(L_0 v/c^2\). More generally, events simultaneous in \(S\) and separated by \(\Delta x\) along the boost acquire a time separation \(-\gamma v\,\Delta x/c^2\) in \(S'\); simultaneity is a property of a frame, not of the events themselves.
Time Dilation
The proper time \(\Delta\tau\) is the interval measured by the single clock present at both events (both ticks happen where it sits). Any frame in which that clock moves assigns a longer coordinate time \(\Delta t = \gamma\,\Delta\tau\) between the same two events, because \(\gamma \ge 1\). Equivalently, watched from \(S\) the moving clock ticks slow, advancing by only \(1/\gamma\) seconds per second of \(S\)-time. Proper time is the smallest of all the coordinate times: the straight (inertial) worldline between two timelike-separated events is the longest in elapsed proper time. The effect is reciprocal — each inertial observer finds the other's clock slow — because "the same two events" differ between the two comparisons.
Length Contraction
An object of proper length \(L_0\) is measured, in any frame moving at speed \(v\) along its length, to be shorter by the factor \(\sqrt{1 - v^2/c^2} = 1/\gamma \le 1\). Proper length is the longest length any observer measures; every moving observer measures less. The contraction is real in the sense of coordinate geometry, not an optical artifact, and it is reciprocal: each observer finds the other's ruler contracted.
Relativistic Velocity Addition
The Galilean numerator \(u_x'+v\) survives, but is suppressed by the denominator \(1+u_x' v/c^{2}\). When both speeds are small compared with \(c\) the denominator is \(\approx 1\) and the ordinary rule returns; when either approaches \(c\) the denominator grows just enough to keep the composite below \(c\). Transverse velocities are not frame-invariant: they inherit the same denominator plus a factor \(1/\gamma\) from time dilation, so a purely transverse motion in \(S'\) is both slowed and tilted in \(S\).
Rapidity and the Additivity of Boosts
The rapidity is the hyperbolic angle of a boost. Successive collinear boosts stack by simple addition of their rapidities, just as successive rotations stack by adding angles. The measurable speed is the hyperbolic tangent of the total rapidity, which automatically enforces \(|\beta|<1\) and regenerates the fractional velocity-addition law — the apparent nonlinearity of velocities is just the nonlinearity of \(\tanh\).
Proper Time and the Four-Velocity
Proper time is the invariant arc length dτ = ds/c along a timelike worldline — the reading of a clock carried by the object. The four-velocity is the unit tangent to that worldline (up to the factor c): its time component γc measures how fast coordinate time flows relative to the clock, and its spatial component γv is the momentum-per-unit-mass direction. The norm is fixed at c² for every observer and every speed — accelerating changes the direction of Uμ in spacetime but never its length.
Relativistic Momentum from Conservation
The relativistic momentum is the Newtonian expression mv multiplied by the Lorentz factor γ ≥ 1. As v → c, γ → ∞, so momentum grows without bound even though speed is capped at c — which is why no finite impulse can push a massive particle to light speed. The factor is not an add-on; conservation in all frames leaves no other choice.
Mass-Energy Equivalence E = mc^2
The total energy of a free body is its Lorentz factor times mc2. Setting v = 0 (so γ = 1) leaves the rest energy mc2 — energy a body carries simply for existing with mass m. The kinetic energy is the excess K = (γ − 1)mc2. The energy-momentum relation ties them together and holds even for massless particles (m = 0 ⇒ E = pc).
The Energy-Momentum Four-Vector
Total energy, momentum, and rest mass sit at the three sides of a right triangle: E is the hypotenuse, pc the "motion" leg, and mc2 the "rest" leg. The rest energy is the momentum-independent floor: at p = 0, E = mc2. Nothing here refers to a frame — m is the invariant length of the four-momentum, so all observers agree on it even while disagreeing on E and p separately.
Relativistic Doppler Effect and Aberration
The bracket (1 − β cos θ) is the ordinary (classical) Doppler factor built from the line-of-sight velocity component; the extra 1/γ in front is pure time dilation of the moving source and is what makes the effect relativistic. At θ = 90° in the observer's frame the classical factor is 1 and only the 1/γ survives — the transverse Doppler redshift. The aberration relation says the same boost tips every ray toward the forward direction: an isotropic emitter beams its light into a forward cone.
The Four-Wavevector and Phase Invariance
The pair (frequency, wavevector) is not two separate quantities but the time and space parts of one four-vector, glued by the invariance of phase. For light in vacuum that four-vector lies on the light cone in k-space: its temporal length exactly cancels its spatial length. Any Lorentz transformation slides kμ around on the cone — changing ω (Doppler) and rotating k (aberration) — but never lifts it off, because a null vector stays null.
Relativistic Force and Four-Acceleration
The four-force is what the four-momentum feels per unit of the particle's own clock; for constant rest mass it is exactly mass times four-acceleration. Projected into the lab, the spatial part is γ times the ordinary force f = dp/dt, and the time part is γ times the power delivered, divided by c. Because momentum carries the factor γ(u), a force along the motion must fight the γ3 stiffness of longitudinal inertia, while a force across the motion sees only γ. A general force is therefore not parallel to the acceleration it produces.
Transformation of Electric and Magnetic Fields
Components along the boost are unchanged; components perpendicular to it are scaled by γ and mixed. Set B = 0 (pure electric field Ey in S): then E′y = γ Ey and B′z = −γ v Ey/c2 ≠ 0. The magnetic field appears purely from motion — there was never a current in S.
Covariant Form of Maxwell's Equations
The inhomogeneous equation ties the field to its sources: its ν = 0 component is Gauss's law and its ν = i components are the Ampère–Maxwell law, with the four-current Jν = (cρ, J) on the right. The homogeneous equation is source-free and follows purely from F being built from a potential: its all-spatial component is ∇·B = 0 and its mixed components are Faraday's law. Eight scalar Maxwell equations collapse into two manifestly Lorentz-covariant tensor statements. Contracting the inhomogeneous equation with ∂ν and using antisymmetry of F gives 0 = μ0∂νJν, so charge conservation is built in, not imposed.
The Four-Current and Charge Conservation
Charge density and current density are two faces of one geometric object, the four-current. Local charge conservation is the frame-independent statement that this four-vector is divergence-free in spacetime. What looks like a scalar-plus-vector conservation law in one frame is a single covariant equation: an observer who measures a pure charge density at rest sees, after a boost, that same charge as a current, and the conservation law is preserved exactly.