Learning #23 — General Relativity: Spacetime, Gravity and Black Holes

Newton's gravity works — for planets, projectiles and most engineering problems. But it breaks down near dense objects, at high velocities and over cosmological distances. Einstein replaced the concept of a gravitational force with something deeper: mass and energy curve spacetime, and freely falling objects follow the straightest possible paths (geodesics) in that curved geometry. The result is a theory that predicted phenomena no one had seen yet — gravitational time dilation, light deflection by the Sun, frame dragging, gravitational waves — all later confirmed to extraordinary precision.

This post builds from the bottom up: start with special relativity (the flat-spacetime limit), then add gravity through the equivalence principle, derive the Schwarzschild metric around a point mass, and use it to explain black hole orbits, gravitational redshift, Einstein rings and the chirp signal from merging neutron stars.

1. Special Relativity — Flat Spacetime First

Before curving spacetime we need to understand flat spacetime — the arena of special relativity. The two postulates are simple: the laws of physics are the same in all inertial frames, and the speed of light in vacuum is the same for all inertial observers. Every relativistic effect follows from these two statements.

Lorentz factor:  γ = 1 / √(1 − v²/c²)
  v → 0: γ → 1 (Newtonian limit)
  v → c: γ → ∞ (impossible to reach c for massive objects)

Time dilation (moving clock ticks slowly):
  Δt = γ · Δτ
  Δτ = proper time (time on the moving clock)
  At v = 0.87c: γ ≈ 2 (clock runs at half speed)
  At v = 0.99c: γ ≈ 7.1

Length contraction (moving rod appears shorter):
  L = L₀ / γ
  L₀ = proper length (rest-frame length)
  Only in the direction of motion; transverse dimensions unchanged.

Lorentz transformation (boost along x):
  t' = γ(t − vx/c²)
  x' = γ(x − vt)
  y' = y,  z' = z

Invariant spacetime interval (same in all frames):
  ds² = −c²dt² + dx² + dy² + dz²
  ds² < 0: timelike separation (same object at different times)
  ds² = 0: lightlike (null) — on the light cone
  ds² > 0: spacelike (no causal connection possible)
  |ds²| = invariant "spacetime distance" = proper time² for timelike paths

Relativistic momentum and energy:
  p = γmv        E = γmc²
  Rest energy:   E₀ = mc²
  Full relation: E² = (pc)² + (mc²)²

The twin paradox is the cleanest demonstration: one twin travels to a star at 0.87c and returns; the other stays home. Each twin sees the other's clock running slow — but the travelling twin is the one who changes inertial frame (decelerates and accelerates), accumulating less proper time. After a 10-light-year round trip at 0.87c the stay-at-home twin ages 23 years; the traveller ages only 11.5.

Minkowski diagrams make the geometry transparent. Time runs vertically; space horizontally. Light cones are fixed at 45° (in c = 1 units). A moving observer's time axis tilts toward the light cone; their simultaneous surfaces tilt the other way. The relativity of simultaneity — two events that are simultaneous in one frame are not simultaneous in another — is the most counter-intuitive consequence and it follows directly from the tilt of those simultaneity lines.

2. The Equivalence Principle — Gravity as Geometry

Einstein's crucial insight: a person in a sealed box cannot distinguish between resting on Earth's surface (gravitational field g downward) and accelerating upward at g in empty space. Gravity and acceleration are locally equivalent. This extends the relativity principle from inertial to accelerating frames — hence General Relativity.

The equivalence principle has an immediate consequence: light must be deflected by gravity. In an accelerating rocket, a photon shot horizontally curves toward the floor. Therefore gravity must bend light too — confirmed by Eddington's 1919 measurement of star positions during a solar eclipse. It also implies gravitational time dilation: a photon climbing out of a gravitational well loses energy and redshifts. Clocks deeper in a gravitational field run slower.

Freely falling frame = locally inertial frame (no gravity felt).
  Globally, gravity cannot be transformed away → spacetime curvature.

Geodesic equation (free-fall trajectory):
  d²xᵘ/dτ² + Γᵘ_αβ · (dxᵅ/dτ)(dx^β/dτ) = 0
  Γᵘ_αβ = Christoffel symbols — encode how coordinates curve.
  In flat spacetime: Γ = 0, geodesic = straight line.
  In curved spacetime: geodesic = "straightest possible path" (extremal proper time).

Metric tensor g_μν:
  ds² = g_μν dx^μ dx^ν   (Einstein summation convention)
  In flat spacetime (Minkowski): g_μν = diag(−1, +1, +1, +1)
  Near a mass: g_μν departs from diag — coordinates "stretch" around the mass.

Einstein field equations:
  G_μν + Λg_μν = (8πG/c⁴) T_μν
  G_μν = R_μν − ½Rg_μν  (Einstein tensor — curvature of spacetime)
  T_μν = stress-energy tensor (source: mass, energy, pressure, momentum)
  Λ = cosmological constant (dark energy)
  G = Newton's gravitational constant
  10 coupled nonlinear PDEs — immensely hard to solve in general.

The full Einstein field equations are ten coupled nonlinear partial differential equations. Exact solutions are rare and precious. The Schwarzschild solution — the spacetime geometry around a single, non-rotating, uncharged point mass in vacuum — was found within weeks of Einstein publishing his theory in 1915. It remains the most important exact solution in general relativity.

3. The Schwarzschild Metric — Spacetime Around a Mass

Karl Schwarzschild solved the vacuum Einstein equations for a spherically symmetric static mass while serving on the Russian front in World War I. His solution, sent to Einstein shortly before Schwarzschild died of illness in 1916, describes the spacetime geometry outside any spherical mass — Earth, Sun, neutron star, or black hole.

Schwarzschild metric (c = 1 units):
  ds² = −(1 − Rₛ/r)dt² + (1 − Rₛ/r)⁻¹dr² + r²dΩ²
  dΩ² = dθ² + sin²θ dφ²   (angular part, unchanged from flat)

Schwarzschild radius:
  Rₛ = 2GM/c²
  Sun:   Rₛ ≈ 2.95 km   (actual radius 696,000 km — not a black hole)
  Earth: Rₛ ≈ 8.9 mm
  A black hole forms when the entire mass is within Rₛ.
  At r = Rₛ: g_tt = 0, g_rr → ∞ (coordinate singularity, not physical)
  At r = 0: true curvature singularity (R_μνρσ R^μνρσ → ∞)

Circular photon orbit (photon sphere):
  r_photon = (3/2)Rₛ = 3GM/c²
  Photons can orbit the black hole but the orbit is unstable.
  Observed as the bright ring in EHT images of M87* and Sgr A*.

Innermost Stable Circular Orbit (ISCO):
  r_ISCO = 3Rₛ = 6GM/c²
  For r_ISCO: orbital speed v_ISCO = c/√3 ≈ 0.577c
  Accretion disc terminates here; infalling material plunges inward.
  Energy released per unit mass in ISCO orbit: ε = 1 − √(8/9) ≈ 5.7%
  (vs. nuclear fusion: ~0.7% — black holes are far more efficient engines)

Newtonian limit (r ≫ Rₛ):
  g_tt ≈ −(1 − 2GM/rc²) → recovers Newtonian potential Φ = −GM/r
  Orbital period: T² = 4π²r³/(GM)  (Kepler's third law)

Geodesics in Schwarzschild spacetime generalise Keplerian orbits. For massive particles there are three qualitatively different trajectories depending on energy and angular momentum: stable circular orbits (r > 3Rₛ), unstable circular orbits (Rₛ < r < 3Rₛ), and plunging orbits that cross the event horizon. Precession of the perihelion — famously observed in Mercury — is a natural consequence: the geodesic does not close, the orbit slowly rotates. Einstein's prediction of 43 arcseconds per century matched the anomalous precession that had puzzled astronomers for decades.

4. Gravitational Time Dilation and Redshift

The g_tt component of the Schwarzschild metric tells us directly how time runs at different radii. A clock deep in a gravitational well runs slower than one far away. This is not a coordinate artefact — it is physically measurable. The Pound-Rebka experiment (1959) confirmed gravitational redshift with a 22.6-metre fall in a Harvard laboratory, matching the prediction to 1% precision.

Proper time ratio (static clocks at radii r₁ and r₂, r₁ < r₂):
  dτ₁/dτ₂ = √(1 − Rₛ/r₁) / √(1 − Rₛ/r₂)
  Clock closer to the mass runs slower.

Gravitational redshift of a photon:
  λ_obs / λ_emit = √(g_tt,emit / g_tt,obs) = √((1 − Rₛ/r_emit)/(1 − Rₛ/r_obs))
  For weak field (r ≫ Rₛ):  λ_obs/λ_emit ≈ 1 + GM(1/r_emit − 1/r_obs)/c²
  Photon climbing out of well: redshifted (loses energy, longer wavelength).
  Infinite redshift at r = Rₛ: photons emitted at event horizon never escape.

GPS satellite corrections (two competing relativistic effects):
  Gravitational time dilation (satellite at r = 26,560 km):
    Clock runs FAST by +45.9 µs/day  (weaker gravity at altitude)
  Special-relativistic time dilation (orbital v ≈ 3.87 km/s, γ ≈ 1 + 8.3×10⁻¹¹):
    Clock runs SLOW by −7.2 µs/day
  Net effect:  +38.7 µs/day
  Uncorrected: GPS position drifts ~10 km/day — general relativity is not optional.

Shapiro delay (light travel time increase near a mass):
  Δt_Shapiro = (2GM/c³) ln(4r₁r₂/b²)   for closest approach b ≪ r₁,r₂
  Measured in Viking lander radar delays (1976): matches GR to 0.05%.

The GPS example is the most practical demonstration that general relativity is engineering: the system must correct for both gravitational time dilation (+45.9 µs/day) and special-relativistic time dilation (−7.2 µs/day). The net +38.7 µs/day, if uncorrected, would accumulate a 10 km position error per day. Every navigation device in the world runs a general-relativistic correction.

5. Gravitational Lensing — Light Follows Curved Spacetime

Light has no rest mass, but it follows geodesics in curved spacetime. Near a massive object those geodesics bend — and from a distant observer's perspective, the mass acts like an imperfect lens. Einstein predicted the precise deflection angle for the Sun in 1916; Eddington measured it during the 1919 solar eclipse and confirmed it within experimental error. The confirmation made global headlines and turned Einstein into a celebrity overnight.

Schwarzschild deflection angle (light passing at impact parameter b):
  α̂ = 4GM / (c²b) = 2Rₛ/b
  (Newton's prediction gives half this: α̂_Newton = 2GM/c²b)
  For the Sun (b = R_⊙ ≈ 696,000 km):
    α̂ = 1.75 arcseconds  (confirmed by Eddington 1919)

Einstein ring (perfect alignment: source, lens, observer collinear):
  θ_E = √(4GM D_LS / (c² D_L D_S))
  D_L = distance from observer to lens
  D_S = distance from observer to source
  D_LS = distance from lens to source
  When mis-alignment angle β < θ_E: two arcs appear on either side.
  When β → 0: full Einstein ring (observed for galaxy clusters).

Lensing magnification:
  μ = |θ/β · dθ/dβ| (ratio of solid angles with/without lens)
  For point lens:  μ = (u² + 2) / [u√(u² + 4)]
    where u = β/θ_E
  Microlensing (u → 0):  μ → ∞ (star temporarily brightens — OGLE/Kepler surveys)

Types of gravitational lensing:
  Strong lensing:  multiple images, arcs, rings (galaxy/cluster scale)
  Weak lensing:    small coherent distortions of background galaxies
  Microlensing:    temporary magnification of a point source (planet detection)
  All three are now key probes of dark matter distribution.

Gravitational lensing is one of the most powerful tools in modern cosmology. Clusters of galaxies act as giant cosmic telescopes, magnifying galaxies behind them by factors of 10 to 100. The distortion pattern of background galaxy shapes (weak lensing) maps the dark matter distribution without assuming anything about its nature. The Hubble Space Telescope, JWST and Euclid satellite all use gravitational lensing as a primary science tool.

6. Gravitational Waves — Ripples in Spacetime

The Einstein field equations — like Maxwell's equations in electrodynamics — have wave solutions. Accelerating masses radiate gravitational waves: ripples in the metric that carry energy away at the speed of light, stretching and squeezing spacetime transversely as they pass. Einstein predicted them in 1916 and doubted they were physically real at various points. LIGO detected them directly for the first time on 14 September 2015, a century later.

Linearised GR: g_μν = η_μν + h_μν,  |h_μν| ≪ 1
  h_μν satisfies the wave equation: □h̄_μν = −16πG/c⁴ T_μν

Quadrupole radiation formula (leading order, TT gauge):
  h_ij^TT(t, r) = (2G/c⁴r) · d²Q_ij/dt²  |_{t_ret}
  Q_ij = ∫ ρ(x_i x_j − ⅓δ_ij r²) d³x  (reduced quadrupole moment)
  Monopole (changing total mass) and dipole (centre of mass motion)
  do not radiate in GR — minimum multipole is quadrupole.

Strain amplitude from a compact binary inspiral (chirp):
  h ≈ (4G/c²r) · (GM_c/c²) · (πGM_c f/c³)^(2/3)
  M_c = chirp mass = (m₁m₂)^(3/5) / (m₁+m₂)^(1/5)
  f = gravitational wave frequency = 2 × orbital frequency
  r = luminosity distance to the source

LIGO GW150914 (first detection — two ~30M⊙ black holes):
  Strain at Earth: h ≈ 10⁻²¹  (detector arms changed by ~10⁻¹⁸ m, 1/1000 proton diameter)
  Peak frequency: ~150 Hz at merger
  Distance: ~410 Mpc (1.3 billion light years)
  Chirp mass M_c ≈ 28.3 M⊙

Gravitational wave luminosity (quadrupole formula):
  P = −(32/5)(G⁴/c⁵)(m₁m₂)²(m₁+m₂) / r⁵
  Negative sign: energy radiated away → orbit shrinks → inspiral accelerates
  Taylor-Hulse binary pulsar (1974): orbital decay matches prediction to 0.1%
  (Nobel Prize 1993 — first indirect detection of gravitational waves)

The gravitational wave signal from a compact binary has a characteristic shape: a slowly chirping sinusoid that sweeps upward in frequency as the pair spirals together, ending in a brief ringdown as the merged object settles into a Kerr black hole. The chirp mass can be read directly from the frequency sweep rate, and the distance from the amplitude. By 2028, LIGO-Virgo-KAGRA have catalogued hundreds of binary mergers — a new astronomy of the violent universe invisible to electromagnetic telescopes.

The Shape of Curved Spacetime

General relativity is fundamentally geometric. The Einstein field equations are not a force law — they say: mass-energy tells spacetime how to curve; curved spacetime tells matter how to move. John Wheeler's two-sentence summary remains the clearest statement of the theory. What makes GR so difficult mathematically — and so profound physically — is that the source (mass-energy) is also affected by the field it produces. The equations are self-referential, non-linear, and almost impossible to solve in general.

Yet the few exact solutions we have — Schwarzschild, Kerr (rotating black hole), Friedmann-Lemaître-Robertson-Walker (expanding universe) — each describe observed phenomena in detail. And weak-field approximations (post-Newtonian, linearised GR) give accurate predictions for binary pulsars, GPS corrections and gravitational wave templates used by LIGO. The frontier now is numerical GR: simulating black hole mergers on supercomputers to generate the theoretical waveforms matched against detector data.

The cosmological side of GR — Hubble expansion, the cosmic microwave background, dark energy and the large-scale structure of the universe — is explored in Spotlight #25 — Cosmology & the Universe. The quantum gravity frontier, where GR and quantum field theory conflict, is one of the deepest open problems in physics and is touched on in Learning #18 — Quantum Mechanics.