Special and general relativity: two postulates of SR, time dilation, length contraction, mass-energy equivalence, equivalence principle, and Einstein field equations of GR.
relativity
Principle of relativity
The laws of physics take the same form in all inertial reference frames.
Constancy of the speed of light
The speed of light in vacuum is the same in all inertial frames, independent of the motion of source or observer.
Lorentz transformation
Coordinate transformation between inertial frames consistent with the two postulates of special relativity; replaces Galilean…
Time dilation
A clock moving relative to an inertial observer is measured to tick more slowly by a factor γ; gravitational analog exists in GR.
Mass-energy equivalence (E=mc²)
Energy and mass are interchangeable by the factor c²; rest energy E₀ = mc².
Equivalence principle
Locally, gravitational acceleration is indistinguishable from inertial acceleration; inertial and gravitational mass are equal. Heart of GR.
Einstein field equations
Field equations of general relativity: spacetime curvature (Einstein tensor) is sourced by energy-momentum.
Stress–energy tensor T^{μν}
Symmetric 2-tensor source of curvature in general relativity; components encode energy density (T⁰⁰), momentum flux (T⁰ⁱ), and stress…
Geodesic equation (GR)
Equation of motion of a free particle in curved spacetime: ẍ^μ + Γ^μ_{αβ} ẋ^α ẋ^β = 0. Reduces to Newton's 2nd law in the weak-field…
Schwarzschild metric
Unique spherically-symmetric vacuum solution of Einstein's equations. Describes the exterior of non-rotating uncharged spherical masses;…
Kerr metric
Axisymmetric stationary vacuum solution describing rotating black holes. Exhibits ergosphere, inner/outer horizons, and frame-dragging.
Gravitational waves
Propagating ripples in spacetime curvature predicted by GR (Einstein 1916) and first directly detected by LIGO (GW150914, 2015). …
Lorentz boost generators and algebra
K_i = iM_{0i}, J_i = ½ε_ijk M_{jk}; [J,J]=iεJ, [J,K]=iεK, [K,K]=-iεJ; SL(2,ℂ) double cover; spinor reps.
Proper time and time dilation
dτ = √(1-v²/c²) dt = √(-η_μν dx^μ dx^ν)/c; measured Δτ invariant under Lorentz. Muon experiments, GPS.
E² = p²c² + m²c⁴
Invariant dispersion relation; massless particles on light cone; threshold energies for pair production, resonance decays.
Length contraction
Moving rod has length L = L₀/γ in the rest frame of the observer; reciprocal effect; pole-in-barn paradox resolved via relativity of…
Relativity of simultaneity
Events simultaneous in one frame are not in another unless spatially coincident; resolves twin and ladder paradoxes.
Thomas precession
Accelerated frame precesses at Ω_T = (γ²/(γ+1))(a×v)/c²; halves spin-orbit coupling in hydrogen fine structure.
Relativistic Doppler effect
f_obs = f_src √((1-β)/(1+β)) longitudinal; transverse f = f/γ (purely relativistic). Cosmological redshift generalization.
Aberration of light
cos θ' = (cos θ - β)/(1 - β cos θ); headlight effect for relativistic motion; Bradley aberration of stars.
Equivalence principles (WEP/EEP/SEP)
WEP: universality of free fall (Eötvös 10⁻¹³); EEP: local freely-falling frames inertial; SEP includes self-gravity — tested by lunar laser…
Einstein–Hilbert action
S = (c³/16πG) ∫ R √(-g) d⁴x + S_matter; variation gives G_μν = (8πG/c⁴) T_μν; cosmological constant via λg_μν term.
Geodesic equation
d²x^μ/dτ² + Γ^μ_αβ (dx^α/dτ)(dx^β/dτ) = 0; free-fall in curved spacetime; light cones from null geodesics.
Riemann curvature tensor
R^ρ_σμν = ∂_μ Γ^ρ_νσ - ∂_ν Γ^ρ_μσ + ΓΓ - ΓΓ; tidal acceleration via geodesic deviation D²ξ/dτ² = R ξ u u.
Kerr–Newman metric
Rotating charged black hole; parameters M, J, Q; ergosphere; Penrose process extracts rotational energy.
Penrose process
Particle splitting inside ergosphere with negative-energy piece falling into black hole extracts up to 29% of rotational energy.
Hawking radiation
Black holes radiate quasi-thermally at T_H = ℏc³/(8πGMk_B); information paradox; evaporation time ~ M³.
Bekenstein–Hawking entropy
S_BH = k_B A/(4ℓ_P²) with A horizon area; holographic principle; 't Hooft and Susskind motivation for AdS/CFT.
Killing horizons and surface gravity
Hypersurface where Killing vector becomes null; surface gravity κ linked to Hawking T = ℏκ/(2πck_B); first law dM = (κ/8πG)dA + ΩdJ + ΦdQ.
Gravitational redshift and Shapiro delay
Δf/f = gh/c² (Pound–Rebka); Shapiro delay for light grazing massive body; GPS corrections ≈ +45 μs/day.
Gravitational lensing
Bending angle α = 4GM/(bc²); Einstein rings, micro-lensing; weak-lensing cosmic shear probes dark matter distribution.
Gravitational waves — quadrupole formula & LIGO
L_GW = (G/5c⁵)⟨⃛Q_ij ⃛Q^ij⟩; strain h ~ 10⁻²¹; binary BH GW150914 first detection (2015 Nobel 2017).
ADM 3+1 formalism
Foliate spacetime by spacelike slices; dynamical variables h_ij, K_ij; lapse N, shift N^i; basis of numerical relativity.
Cosmological constant problem
Measured Λ ≈ 10⁻¹²² M_P² vs QFT vacuum estimate ~ M_P⁴ — 120-order discrepancy; anthropic/landscape or dynamical dark energy proposals.
Bianchi identity
∇_μ G^μν = 0 automatically from ∇_[λ R_μν]ρσ = 0; ensures conservation of T^μν in GR.
Raychaudhuri equation
dθ/dτ = -θ²/3 - σ² + ω² - R_μν u^μ u^ν; basis of Penrose–Hawking singularity theorems.
Penrose–Hawking singularity theorems
Under energy conditions + trapped surface, timelike/null geodesic incompleteness inevitable. Big Bang and black hole singularities.