Фізика теорії відносності та космології

⚡ Спеціальна відносність

Lorentz transformations

x' = γ(x - vt), t' = γ(t - vx/c²), γ = 1/√(1-v²/c²). Space-time mixing. Relativity of simultaneity.

Time dilation

Δt' = γ·Δt для двигmoving clocks. Twin paradox. GPS correction: ~7 μs/day.

Length contraction

L' = L/γ для двигmoving rods. FitzGerald-Lorentz contraction. Muon lifetime.

Relativistic energy

E = γmc², p = γmv. E² = p²c² + m²c⁴. Mass-energy equivalence.

🌌 Загальна відносність

Einstein equations

G_μν = 8πGT_μν/c⁴. Ricci tensor, energy-momentum. Spacetime curvature → gravity.

Geodesics

Curved paths: shortest distance. d²x^μ/dτ² + Γ^μ_νλ dx^ν/dτ dx^λ/dτ = 0. Christoffel symbols.

Schwarzschild metric

ds² = -(1-r_s/r)dt² + dr²/(1-r_s/r) + r²dΩ². Static, spherical. Horizon r_s = 2GM/c².

Kerr metric

Rotating black hole: M, J. Ergosphere, frame dragging. Inner, outer horizons.

🌊 Космологія

FLRW metric

ds² = -c²dt² + a(t)²(dr²/(1-kr²) + r²dΩ²). Scale factor a(t). k = ±1,0. Homogeneous, isotropic.

Hubble law

v = H₀·d, H₀ ≈ 70 км/(с·Mpc). Redshift z = Δλ/λ. Expansion history.

Dark energy

~68% energy. Accelerated expansion. Cosmological constant Λ. w ≈ -1.03±0.04.

Big Bang

Hot Big Bang: t₀ ≈ 13.8 Gyr. CMB T ≈ 2.725 K. Nucleosynthesis: ³He, ⁴He, ⁷Li.

⚫ Black holes

Horizon

r_s = 2GM/c². Event horizon: causal boundary. Schwarzschild radius. Penrose diagrams.

Hawking radiation

T_H = ℏc³/(8πGMk_B). Blackbody radiation. Quantum vacuum. Information paradox.

Singularity

r = 0: curvature → ∞. Incomplete geodesics. Quantum gravity needed.

Thermodynamics

S_BH = A/(4Gℏ). Bekenstein-Hawking. Area theorem. Laws of black hole mechanics.

🔬 Tests of relativity

Gravitational waves

LIGO: GW150914 (2015). Black hole mergers. h ~ 10⁻²¹. Einstein's prediction.

Light deflection

Eddington (1919): δ ≈ 1.75″ для Sun. Gravitational lensing. Bending angle.

Perihelion precession

Mercury: 43″/century GR excess. Periastron advance. Binary pulsars.

Redshift

Gravitational: Pound-Rebka. Clocks run slower in gravity. GPS satellites.

🌊 Inflation

Exponential expansion

a(t) ~ e^(H·t) у early Universe. Guts 1980. Scale factor ~e^60. Hot Big Bang.

Problems solved

Flatness, horizon, monopole. Primordial fluctuations. Quantum seeds.

Evidence

CMB isotropy, flat geometry Ω = 1.001±0.005. Gaussian fluctuations.

Inflationary models

Slow-roll inflation. Scalar inflaton field. V(φ) potential. Reheating.

📊 Графіки та діаграми

Lorentz factor

γ vs v/c:

Divergence at v = c. Maximum speed limit. Relativistic effects.

Friedmann equations

Expansion:

Accelerated expansion: dark energy domination. Future: exponential.

γ = 1/√(1-v²/c²) v = 0: γ = 1 v = 0.1c: γ ≈ 1.005 v = 0.9c: γ ≈ 2.3 v → c: γ → ∞

🧪 Практичні приклади

Приклад 1: Time dilation

GPS: v ≈ 4 км/с → γ ≈ 1+9×10⁻¹¹. Clock correction ~7 μs/day. Gravity: ~47 μs/day.

Приклад 2: GW150914

LIGO: black holes 30+20 M☉. M_chirp ≈ 28 M☉. h ~ 10⁻²¹. SNR ~24.

Приклад 3: Light bending

Eddington (1919): δ ≈ 1.75″ @ Sun. GR vs Newton: 2× factor. Gravitational lensing.

Приклад 4: Hubble expansion

H₀ ≈ 70 км/(с·Mpc). Age t₀ ≈ 13.8 Gyr. Distance ladder: Cepheids, SN Ia.

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