Relativity Advanced

General relativity lives in a curved 4-dimensional Lorentzian manifold. The key tool is the metric tensor g_{μν}(x), which encodes distances:

ds² = g_{μν} dx^μ dx^ν

Geodesics

Free-falling particles follow geodesics — curves that parallel-transport their own tangent vector:

d²x^μ/dτ² + Γ^μ_{νρ} (dx^ν/dτ)(dx^ρ/dτ) = 0

where Γ^μ_{νρ} are Christoffel symbols derived from g_{μν}.

Curvature from the metric

The Riemann tensor R^ρ_{σμν} measures how much parallel transport around a loop rotates a vector. From it: Ricci tensor R_{μν} = R^ρ_{μρν} and Ricci scalar R = g^{μν} R_{μν}.

The exact GR solution for the spacetime outside a spherically symmetric, non-rotating mass M is the Schwarzschild metric (1916):

ds² = −(1 − r_s/r)c²dt² + dr²/(1 − r_s/r) + r²dΩ²

where r_s = 2GM/c² is the Schwarzschild radius and dΩ² = dθ² + sin²θ dφ².

Key features

• At r ≫ r_s: reduces to flat Minkowski space with small corrections.
• At r = r_s (event horizon): g_{tt} = 0 — time appears to stop for a distant observer.
• At r = 0: curvature singularity where the theory breaks down.

Gravitational redshift

A photon climbing out of a potential well loses energy: f_received = f_emitted × √(1 − r_s/r). Confirmed by the Pound–Rebka experiment (1959).

Massive and massless particles follow geodesics in the Schwarzschild metric. The effective potential is:

V_eff(r) = −GM/r + L²/(2r²) − GML²/(c²r³)

Planetary orbits

Unlike Newtonian gravity, closed ellipses are not exact solutions. Mercury's perihelion precesses by an extra 43 arcseconds per century — precisely predicted by GR and confirmed observationally.

Photon orbits

Photons follow null geodesics (ds² = 0). The photon sphere at r = 3GM/c² is an unstable circular orbit. Light deflected by the Sun: δφ = 4GM/(c²b), where b is the impact parameter. Eddington confirmed this in 1919.

ISCO

The innermost stable circular orbit (ISCO) for massive particles is at r = 6GM/c² = 3r_s. Inside this, orbits spiral into the black hole.

A black hole is a region of spacetime from which no causal signal can escape to infinity. Its boundary is the event horizon at r = r_s (for a Schwarzschild black hole).

Crossing the horizon

An infalling observer reaches the horizon in finite proper time and notices nothing special locally (for a large black hole — tidal forces are small). A distant observer sees the infaller slow and redshift asymptotically, never actually crossing.

Penrose diagrams

A maximally extended Schwarzschild solution (Kruskal–Szekeres coordinates) reveals a white hole and a second external universe, connected at the singularity — the Einstein–Rosen bridge (wormhole).

Kerr metric

Rotating black holes are described by the Kerr metric (1963), which introduces frame dragging — space-time itself rotates near the hole — and an ergosphere outside the horizon where energy can be extracted.

From the Schwarzschild metric, a clock at radius r ticks slower than one at infinity:

dτ/dt = √(1 − r_s/r)

Consequences

• A clock on the Sun's surface runs slow by ~2 × 10⁻⁶ relative to one far from any mass.
• GPS satellites at 20,200 km altitude run fast by 45 μs/day due to weaker gravity.
• Atoms near a massive star emit light that is redshifted by the time it reaches us.

Pound–Rebka experiment (1959)

Using the Mössbauer effect to detect the tiny gravitational blueshift (22 m height difference), Pound and Rebka confirmed GR's gravitational time dilation to 1% accuracy.

The Einstein field equations are nonlinear; in the weak-field limit one can linearise around flat space: g_{μν} = η_{μν} + h_{μν}, |h| ≪ 1. In the transverse-traceless gauge:

□h^{TT}_{μν} = −16πG/c⁴ T_{μν}

This is a wave equation. Gravitational waves travel at c.

Polarisations

GWs have two polarisations: + (plus) and × (cross), which stretch and squeeze space perpendicular to propagation.

Sources

Compact binary mergers (black holes, neutron stars) produce the strongest GWs. Power radiated: P = 32G⁴m₁²m₂²(m₁+m₂) / (5c⁵r⁵) (quadrupole formula).

Detection

LIGO detected the first GW event (GW150914) on September 14, 2015 — two ~30 M_☉ black holes merging. The strain was h ~ 10⁻²¹, a distance change of 10⁻¹⁸ m in 4 km arms.

Applying GR to the entire universe (assumed homogeneous and isotropic on large scales — the cosmological principle) yields the FLRW metric:

ds² = −c²dt² + a(t)² [dr²/(1−kr²) + r²dΩ²]

where a(t) is the scale factor and k = −1, 0, +1 (open, flat, closed).

Friedmann equations

(ȧ/a)² = 8πGρ/3 − kc²/a²
ä/a = −4πG/3(ρ + 3p/c²)

The Hubble parameter H = ȧ/a. Today H₀ ≈ 67–73 km/s/Mpc (some tension in measurements).

Components

Matter (p=0), radiation (p=ρc²/3), cosmological constant Λ (p=−ρc²) drive different expansion histories.

GR has passed every experimental test proposed over 110 years.

Classic tests

• Perihelion precession of Mercury: 43 arcsec/century predicted and observed.
• Light deflection by the Sun: 1.75 arcsec at the limb (confirmed 1919, and many times since).
• Gravitational redshift: Pound–Rebka (1959), later with atomic clocks in space.

Modern tests

• Shapiro delay: radar signals passing near the Sun arrive late. Cassini mission: agreement with GR to 0.002%.
• Pulsar timing: Hulse–Taylor binary pulsar loses energy by GW emission exactly as predicted — 1993 Nobel Prize.
• LIGO/Virgo/KAGRA: direct GW detection. GW speed ≈ c to within 10⁻¹⁵. GW170817 (neutron star merger) confirmed multi-messenger astronomy.
• Black hole imaging: Event Horizon Telescope images M87* (2019) and Sgr A* (2022).

The heart of GR is Einstein's field equations (1915):

G_{μν} + Λg_{μν} = (8πG/c⁴) T_{μν}

Left side: geometry (Einstein tensor G_{μν} = R_{μν} − ½Rg_{μν}). Right side: energy–momentum content (stress-energy tensor T_{μν}).

In words

"Matter and energy tell space-time how to curve; curved space-time tells matter how to move."

Cosmological constant Λ

Einstein introduced Λ for a static universe (he later called it his "biggest mistake"). Modern cosmology revived it as dark energy — Λ drives the observed accelerating expansion of the universe.

10 equations, not 16

G_{μν} is symmetric and traceless in a sense, leaving only 10 independent equations. Even so, finding exact solutions requires symmetry assumptions — spherical symmetry gives Schwarzschild; axial symmetry gives Kerr.