THE VOCABULARY
Instruments, concepts, and phenomena — the shared vocabulary of the site.
Flux linkage
The total flux threading a multi-turn coil, summed across all turns: λ = N Φ for N turns each enclosing flux Φ. The quantity Faraday's law naturally refers to for real coils rather than single loops.
Focal length
f, the distance from a lens or mirror at which parallel incoming rays converge (or appear to diverge from). Positive for converging optics, negative for diverging. Determines magnification, angle of view, and depth of field.
focus
One of two interior points that define an ellipse; the central body sits at one focus in a Keplerian orbit.
force
A push or a pull; mathematically, the cause of acceleration — F = ma.
Forced oscillation
An oscillator being driven by a periodic external force, settling into a steady state at the drive frequency.
Four-acceleration
The Lorentz four-vector a^μ = du^μ/dτ, the second proper-time derivative of a particle's spacetime trajectory. Always orthogonal to four-velocity in the Minkowski metric (a^μ u_μ = 0); non-zero only on accelerated (non-geodesic) world-lines; reduces to the ordinary three-acceleration in the instantaneous rest frame.
Four-current
The Lorentz four-vector J^μ = (cρ, J_x, J_y, J_z) packaging charge density and current density into a single covariant object. Sources the field tensor F^{μν} via Maxwell's equation ∂_μ F^{μν} = μ₀ J^ν.
Four-force
The Lorentz four-vector F^μ = dp^μ/dτ; the relativistic generalisation of Newton's force. For a particle of constant rest mass, F^μ = m·a^μ. The time component is the rate of energy transfer (power); the spatial components reduce to Newton's second law in the low-β limit.
Four-momentum
The Lorentz four-vector p^μ = (E/c, p_x, p_y, p_z) combining a particle's energy and three-momentum into a single object that transforms covariantly under boosts. Its invariant norm-squared p^μ p_μ = (E/c)² − |p|² = m²c² is the energy-momentum-mass relation; total four-momentum is conserved in any collision.
Four-potential
The Lorentz four-vector A^μ = (φ/c, A_x, A_y, A_z) packaging the scalar potential φ and vector potential A into one covariant object. The fundamental dynamical variable of electromagnetism in the Lagrangian formulation; the EM field tensor F^{μν} = ∂^μA^ν − ∂^νA^μ.
Four-vector
A quantity X^μ = (X⁰, X¹, X², X³) that transforms under Lorentz boosts the same way the spacetime coordinates (ct, x, y, z) do. The natural container for any pair of scalar-plus-three-vector quantities in special relativity.
Four-velocity
The Lorentz four-vector u^μ = dx^μ/dτ = γ(c, v_x, v_y, v_z), the tangent to a particle's timelike world-line parametrized by proper time. Its norm u^μ u_μ = c² is constant on every timelike world-line; differentiating it gives four-acceleration, and m·u^μ is the four-momentum.
Fourier series
The decomposition of an arbitrary periodic function into a sum of sines and cosines.
free fall
Motion under gravity alone, with no air resistance or other forces — every object accelerates at the same rate.
Frequency
Number of oscillation cycles per unit time, symbol f, measured in hertz.
Fresnel equations
The four amplitude coefficients (r_s, r_p, t_s, t_p) giving what fraction of a wave's amplitude reflects from or transmits through a dielectric interface, derived from Maxwell boundary conditions. r_p vanishes at Brewster's angle.
friction
Force that opposes relative motion between two surfaces in contact, converting kinetic energy into heat.
Galilean invariance
The principle that the laws of mechanics take the same form in all inertial frames related by Galilean transformations — uniform translation at constant velocity. The pre-relativistic statement of relativity, valid for low speeds.
Gauge group
The Lie group whose local symmetry transformations leave a gauge theory's Lagrangian invariant. U(1) for electromagnetism (one phase parameter), SU(2) for the weak force (three parameters, three W-bosons), SU(3) for QCD (eight parameters, eight gluons). The Standard Model gauge group is SU(3)×SU(2)×U(1).
Gauge invariance
The principle that the equations of electromagnetism are unchanged under the gauge transformation A_μ → A_μ + ∂_μΛ for any scalar function Λ. Together with Noether's theorem, gauge invariance implies charge conservation. The template for every gauge theory in the Standard Model.
Gauge theory origins
The intellectual lineage from the 1865-1867 observation of gauge freedom in electromagnetism, through Hermann Weyl's 1918 unification attempt and 1929 retooling as a quantum-phase symmetry, to Yang-Mills 1954 and the Standard Model. The gauge principle is the template behind every fundamental force in nature.
Gauge transformation
A change A → A + ∇λ, V → V − ∂λ/∂t in the potentials that leaves all physical fields E and B unchanged. The freedom that defines what 'gauge' means.
Gaussian surface
An imaginary closed surface chosen to exploit symmetry when applying Gauss's law.
Generalised coordinates
Any set of independent variables that fully specifies a system's configuration. Not necessarily Cartesian.
Gradient
A vector that points in the direction of steepest increase of a scalar field, with magnitude equal to the rate of that increase.
gravitational field
The vector field g(r) = −GM/r² r̂ giving the acceleration any test mass would experience at each point in space.
gravity assist
Technique in which a spacecraft gains or loses speed by flying close to a planet, exchanging momentum through the planet's gravitational field.
Group velocity
The speed v_g = dω/dk at which a wave packet's envelope — and therefore its energy and information — propagates.
Group velocity (EM)
v_g = dω/dk. The speed at which a wave packet's envelope — and therefore its energy and information content — propagates. In a dispersive medium v_g differs from the phase velocity v_p = ω/k.
gyroscope
A rapidly spinning rotor on a gimbal or pivot, whose angular momentum resists reorientation — the workhorse of modern inertial navigation.