Learning #29 – General Relativity & Curved Spacetime: From the Equivalence Principle to Black Holes

General relativity (GR) is Einstein’s 1915 theory of gravitation. It replaces Newton’s force of gravity with the idea that massive objects curve the geometry of spacetime, and freely falling bodies (including light) follow the straightest possible paths (geodesics) through that curved geometry. GR has passed every experimental test to exquisite precision over a century. This post builds from the equivalence principle up to black hole thermodynamics.

1. The Equivalence Principle and Geodesics

Einstein’s Equivalence Principle (EEP) asserts that the effects of gravity are locally indistinguishable from the effects of acceleration. More precisely: in a sufficiently small region of spacetime, the laws of special relativity hold in a freely falling (inertial) frame. This implies that gravity must bend light (if a horizontal light beam is bent in an accelerating rocket, it must also bend in a gravitational field) and cause gravitational time dilation.

Spacetime interval (general metric):
  ds² = g_μν dxμ dxν
  Signature: (−,+,+,+)  [timelike ds²<0, spacelike >0, null =0]

Proper time:  dτ² = −ds²/c²

Geodesic equation (free fall in curved spacetime):
  d²xµ/dτ² + Γµ_(νρ) (dxν/dτ)(dxρ/dτ) = 0

Christoffel symbols (Levi-Civita connection):
  Γµ_(νρ) = (1/2) gµσ (∂_ν g_σρ + ∂_ρ g_σν − ∂_σ g_νρ)
  Symmetric in lower indices: Γµ_(νρ) = Γµ_(ρν)
  Not a tensor (vanishes in locally inertial frame)

Covariant derivative:
  ∇_ν Vµ = ∂_ν Vµ + Γµ_(νρ) Vρ   (contravariant vector)
  ∇_ν ω_µ = ∂_ν ω_µ − Γρ_(νµ) ω_ρ   (covariant vector, 1-form)

Parallel transport:
  DVµ/dλ = (dVµ/dλ) + Γµ_(νρ)(dxν/dλ)Vρ = 0
  Geodesic = self-parallel curve: DUµ/dτ = 0  where Uµ = dxµ/dτ
          

2. Curvature: Riemann, Ricci, and the Einstein Tensor

The Riemann curvature tensor measures the failure of parallel transport around a closed loop — equivalently, the commutator of covariant derivatives. Its contractions, the Ricci tensor and Ricci scalar, appear directly in the Einstein field equations.

Riemann tensor (16&sup4; = 256 components; 20 independent by symmetry):
  Rσ_(μνρ) = ∂_νΓσ_(ρµ) − ∂_ρΓσ_(νµ) + Γσ_(νλ)Γλ_(ρµ) − Γσ_(ρλ)Γλ_(νµ)

Measures curvature via commutator:
  [∇_μ, ∇_ν]Vρ = Rρ_(σμν) Vσ

Symmetries:
  R_μνρσ = −R_νμρσ = −R_μνσρ = R_ρσμν
  Algebraic Bianchi: R_[μνρσ] = 0
  Differential Bianchi: ∇_[λR|μν|ρσ] = 0

Ricci tensor (symmetric, 10 independent):
  R_μν = Rρ_(µρν)  (contraction on 1st and 3rd indices)

Ricci scalar:
  R = gμνR_µν

Einstein tensor (symmetric, divergence-free):
  G_μν = R_μν − (1/2)g_μνR
  ∇ν Gμν = 0  (contracted Bianchi identity → conservation of Tμν)

Weyl tensor C_μνρσ:
  Traceless part of Riemann
  Governs tidal forces in vacuum (R_μν=0 does not imply Rσ_µνρ=0)
          

3. Einstein Field Equations

The Einstein field equations (EFE) describe how the distribution of energy and momentum sources the curvature of spacetime. They are 10 coupled, nonlinear, partial differential equations for the metric tensor g_μν.

Einstein field equations:
  G_μν + Λ g_μν = (8πG/c&sup4;) T_μν

  G_μν = spacetime curvature (geometry)
  T_μν = energy-momentum tensor (matter/energy content)
  Λ ≈ 1.1 × 10−&sup5;² m−² = cosmological constant (dark energy)

Perfect fluid energy-momentum tensor:
  Tµν = (ρ + p/c²) UµUν + p gµν
  Dust: p=0 → Tµν = ρc² UµUν
  ∇_ν Tµν = 0  ↔  Euler equations + energy conservation

Trace-reversed form (useful for linearisation):
  Rµν = (8πG/c&sup4;)(Tµν − (1/2)gµνT)
  Vacuum: Tµν = 0 → Rµν = 0 (Ricci-flat ≠ flat)

Linearised gravity hµν = gµν − ηµν  (|h| ≪ 1):
  Trace-reversed perturbation: h̄µν = hµν − (1/2)ηµνh
  Lorenz gauge condition: ∂νh̄µν = 0

  □h̄µν = −(16πG/c&sup4;) Tµν   (wave equation)
  □ = ηµν∂µ∂ν = −∂&sub2;t/c² + ∇²   (d'Alembertian)

Newtonian limit: g_00 ≈ −(1 + 2Φ/c²), Φ = gravitational potential
  ∇²Φ = 4πGρ  (Poisson equation recovered)
          

4. The Schwarzschild Solution

Karl Schwarzschild found the first exact solution of the EFE in December 1915 — just weeks after Einstein published the final equations — while on the Eastern Front during World War I. The Schwarzschild metric describes the spacetime geometry outside a spherically symmetric, non-rotating, uncharged mass.

Schwarzschild metric (spherical symmetry, vacuum, M, J=0, Q=0):
  ds² = −(1−r_s/r)c²dt² + (1−r_s/r)−¹dr² + r²dΩ²
  r_s = 2GM/c² = Schwarzschild radius

  Examples:
    Sun:   r_s = 2.95 km  (R_sun = 696 000 km → well outside horizon)
    Earth: r_s = 8.9 mm   (R_earth = 6371 km)
    M87*:  r_s = 2 × 10^10 km  (M ≈ 6.5 × 10^9 M_sun)

Effective potential for massive particles:
  V_eff(r) = (c²/2)[(1−r_s/r)(1 + L²/r²c²) − 1]
  where L = specific angular momentum
  Innermost stable circular orbit (ISCO): r_ISCO = 3r_s = 6GM/c²

Orbital precession of Mercury:
  Δφ_prec = 6πGM / [ac²(1−e²)]  per orbit
  Mercury: a = 0.387 AU, e = 0.206 → Δφ = 43.0 arcsec/century  ✓

Gravitational redshift:
  ν_obs/ν_emit = √(g_00(r_emit) / g_00(r_obs))
  z = (1 − r_s/r)^(−1/2) − 1  ≈ GM/(rc²) for r ≫ r_s
  Pound-Rebka (1959): measured z = 2.46 × 10^−15 at h=22.5m  ✓

Photon sphere: r_ps = (3/2)r_s = 3GM/c²
  Unstable circular orbit for photons
  Shadow radius in EHT images: r_shadow = 3√3 GM/c²
  M87* EHT image (2019): shadow diameter 40±3 µas  (predicted 39±1 µas) ✓
          

5. Gravitational Waves

Accelerating mass-energy emits gravitational waves — ripples in spacetime geometry propagating at the speed of light. The transverse-traceless (TT) gauge selects physical degrees of freedom: two polarisations, h₊ and h_×, which alternately squeeze and stretch perpendicular directions as a wave passes through.

TT-gauge solution (far field, r ≫ λ_GW):
  h_ij^TT(t,r) = (2G/rc&sup4;) Ï_ij^TT(t_ret)
  where t_ret = t − r/c  (retarded time)
  I_ij = ∫ ρ x_i x_j d³x  (mass quadrupole moment tensor)

Total radiated power (quadrupole formula):
  P = −G/(5c&sup5;) ⟨Q̈_ij Q̈^ij⟩   where Q_ij = I_ij − (1/3)δ_ij I_kk

Binary system (two equal masses m, separation a, circular orbit f_orb):
  P = (32/5) G^4 m^5 / (c^5 a^5)  (Peters formula)
  a(t) = a_0 (1 − t/t_merge)^(1/4)
  t_merge = (12/19) c_0^4/(B F(e)) ≈ 12/85 (c²0 a^4_0)/(4G^3 m² M)

LIGO GW150914 (2015 Sept 14):
  Two black holes: m_1 ≈ 36 M_sun, m_2 ≈ 29 M_sun
  Peak strain: h ~ 10^−²¹ at 150 Hz
  LIGO arm length: L = 4 km → ΔL = hL/2 ~ 10^−¹&sup8; m ≈ 1/1000 proton radius
  Final BH: M_f ≈ 62 M_sun (3 M_sun → GW energy: ~ 5 × 10^47 J)

Pulsar Timing Arrays (PTAs):
  Nanohertz GW background detected 2023 (NANOGrav, EPTA, PPTA, CPTA)
  f ~1–100 nHz, T_obs ~ 12.5 yr baselines, ~67 millisecond pulsars

LISA planned (2034, ESA/NASA):
  3 spacecraft, 2.5 Mkm arm, 0.1 mHz–1 Hz
  Targets: SMBH mergers z<20, Galactic double white dwarfs, EMRI waveforms
          

6. Black Holes and Hawking Radiation

A black hole is a region of spacetime where the escape velocity exceeds c: no causal signal can escape from within the event horizon. The Kerr solution (1963) extends Schwarzschild to spinning black holes and predicts the ergosphere — a region where spacetime is dragged so violently that nothing can remain stationary. Stephen Hawking showed in 1974 that quantum field theory near the horizon leads to thermal radiation, giving black holes a finite temperature and lifetime.

Kerr metric (Boyer-Lindquist coordinates, M, J=Ma, Q=0):
  ds² = −(1−r_s r/Σ)c²dt² − (2r_s ra sin²θ/Σ)c dt dφ
           + (Σ/Δ)dr² + Σdθ² + (r²+a²+r_s ra²sin²θ/Σ)sin²θ dφ²
  Σ = r² + a²cos²θ,  Δ = r² − r_s r + a²
  a = J/(Mc) = specific angular momentum  (0 ≤ a ≤ GM/c)

Outer event horizon (largest root of Δ=0):
  r_+ = GM/c² + √((GM/c²)² − a²)
  Extremal Kerr: a = GM/c² → r_+ = GM/c² (smallest possible BH radius)

Ergosphere (where g_tt = 0):
  r_ergo = GM/c² + √((GM/c²)² − a²cos²θ)
  Between ergosphere and horizon, angular momentum can be extracted (Penrose process)

Penrose process:
  Particle splits inside ergosphere: one falls in with negative E (w.r.t. infinity)
  Other escapes with E_esc > E_initial → BH loses rotation
  Maximum extractable energy: ~21% of rest mass (for extremal Kerr)
  Blandford-Znajek process: magnetic version, powers AGN jets

Hawking radiation (1974 — quantum effect):
  T_H = ℏc³ / (8πGM k_B)
  Solar-mass BH: T_H ≈ 60 nanokelvins  (negligible vs CMB 2.7 K)
  Asteroid-mass BH (M ~ 10^15 g): T_H ~ 10^11 K → evaporating now

Black hole entropy (Bekenstein-Hawking):
  S_BH = k_B A / (4 l_Pl²)   where l_Pl = √(ℏG/c³) = 1.6 × 10^−35 m
  A = 4πr_+²  (event horizon area; increases in classical processes: 2nd law)

Information paradox:
  Hawking radiation is exactly thermal → no information escapes?
  Violates unitarity of quantum mechanics (Page curve debate)
  Resolutions: fuzzballs (string theory), firewall (AMPS), Island formula (holography)
          

Try These Simulations