THE VOCABULARY
Instruments, concepts, and phenomena — the shared vocabulary of the site.
radius of gyration
The distance k = √(I/M) at which a point mass equal to the body's total mass would have the same moment of inertia.
range
Horizontal distance covered by a projectile before it returns to its launch height; maximised at 45° in vacuum.
RC time constant
τ = RC. The characteristic time for an RC circuit to charge to 1−1/e ≈ 63% of its final voltage, or discharge to 1/e ≈ 37% of its initial voltage. In seconds when R is in ohms and C in farads.
Reactance
The imaginary part X of impedance Z = R + jX. Inductive reactance X_L = ωL is positive; capacitive reactance X_C = −1/(ωC) is negative. Reactance stores energy without dissipating it.
Redshift
The stretching of a wave's wavelength when source and observer move apart.
Refractive index
The ratio n = c/v_p of the vacuum speed of light to the phase velocity in the medium.
Relativistic beaming
The forward concentration of radiation from a relativistic source into a cone of half-angle ≈ 1/γ, caused by the Lorentz transformation of solid angles. Responsible for pulsar pulsed emission, blazar flux variability, and the lighthouse behaviour of synchrotron sources.
Relativistic Doppler effect
The frequency shift of light from a moving source observed in any inertial frame, including the time-dilation factor γ alongside the classical motion contribution. For a source receding at radial velocity v: f' = f √((1 − β)/(1 + β)). Reduces to the classical Doppler formula plus a γ correction.
Relativistic velocity addition
The rule for combining velocities in special relativity. For collinear motion: u' = (u − v)/(1 − uv/c²). Replaces the Galilean rule u' = u − v; ensures no combination of subluminal velocities exceeds c.
Relativity of simultaneity
The result that two spatially-separated events judged simultaneous in one inertial frame are not simultaneous in any other frame moving relative to the first. The deepest break with Newtonian intuition; the geometric content of the Lorentz transformation's time-mixing.
Resistance
The ratio R = V/I for a conductor obeying Ohm's law. Measured in ohms (Ω). Determined by the conductor's geometry (R = ρℓ/A) and material resistivity ρ.
Resistor
A two-terminal passive electrical component designed to present a specific resistance to current flow. Values range from milliohms to gigaohms; tolerances from 10% to 0.01%.
resonance
Amplitude peak when driving frequency matches natural frequency.
Rest energy
The energy E₀ = mc² that a massive particle has in its own rest frame, where its three-momentum vanishes and its four-momentum reduces to (mc, 0, 0, 0). The conversion factor between mass and energy; the floor below which a particle's total energy cannot drop.
restoring force
Force proportional to displacement and directed back toward equilibrium: F = −kx.
Retarded time
The earlier time t_r = t − |r − r_s(t_r)|/c at which a signal must have left a moving source in order to arrive at the observer at time t. Built into the retarded potentials and all causal electromagnetic radiation formulae.
Reynolds number
Dimensionless ratio Re = ρvL/η of inertial to viscous forces. Re ≪ 1: creeping flow. Re ≫ 1: turbulent.
Ricci scalar
R = g^{μν} R_{μν} — the contraction of the Ricci tensor with the inverse metric. A scalar field at each point of spacetime. Positive R indicates positively-curved (sphere-like) geometry; negative R indicates negatively-curved (saddle-like). The simplest scalar curvature invariant.
Ricci tensor
R_{μν} = R^λ_{μλν} — the trace of the Riemann tensor over its first and third indices. Symmetric (0,2) tensor with 10 independent components in 4D. Appears directly on the geometric side of Einstein's field equations: R_{μν} − (1/2) R g_{μν} = (8πG/c⁴) T_{μν}.
Riemann curvature tensor
R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}. The (1,3) tensor that fully characterises spacetime curvature; 20 algebraically independent components in 4D. Vanishes if and only if the manifold is flat. Bianchi identity ∇_λ R^ρ_{σμν} + cyclic = 0 underwrites the divergence-free Einstein tensor.
RL time constant
τ = L/R. The characteristic time for current in an RL circuit to rise to 1−1/e ≈ 63% of its steady-state value, or decay to 1/e ≈ 37% of its initial value.
Roche limit
The minimum orbital distance at which tidal forces overcome self-gravity; closer than this, a moon is torn apart.
Rocket equation
Δv = u · ln(m₀ / m_f) — the velocity a rocket gains by expelling propellant, derived from momentum conservation.
Rolling without slipping
The kinematic constraint v = ω·R that locks a wheel's linear velocity to its rotation so the contact point is momentarily at rest.
Runaway solution
A solution of the Abraham-Lorentz equation in which a free charge's acceleration grows exponentially without bound, a ∝ exp(t/τ₀), without any external force or energy source. The most famous pathology of classical radiation-reaction theory.
Running coupling
The energy-dependent value α(E) of the QED fine-structure constant, increasing from α ≈ 1/137 at low energies to α ≈ 1/128 at the Z-pole and toward 1 at the Landau pole. The breakdown of perturbation theory at high energy is one of the doors from QED to deeper theory.
Rutherford scattering
The elastic Coulomb scattering of a charged particle off a fixed point charge, following the hyperbolic trajectory of inverse-square central-force motion. Differential cross-section dσ/dΩ ∝ 1/sin⁴(θ/2). Full treatment in a later branch.
s-polarization
An EM wave incident on an interface with its electric field perpendicular to the plane of incidence (German senkrecht, "perpendicular"). Also called TE (transverse electric) polarisation. Reflection coefficient never vanishes except at grazing.
Self-energy divergence
The infinite electrostatic field energy U = (q²/8πε₀)·∫ dr/r² stored in the field of a point charge, diverging as 1/r at small radii. Root cause of the pathologies of classical radiation-reaction theory and the target of QED's mass-renormalisation procedure.
Self-inductance
The property of a coil that makes it oppose changes in its own current, characterised by L = Φ/I, where Φ is the flux the coil produces through itself. Units of henry (H = V·s/A).