General Relativity & Cosmology

The Equivalence Principle & Curved Spacetime

Newton’s gravity is a force: a falling apple accelerates because of an invisible pull. Einstein’s insight was that a person in a closed elevator cannot distinguish free fall from deep space with no gravity, nor can they tell a stationary elevator on Earth from one accelerating upward at g = 9.8 m s⁻² in empty space. This is the weak equivalence principle: gravitational and inertial mass are identical.

The strong equivalence principle extends this to all physics: locally, in any freely falling reference frame, the laws of physics are those of special relativity. Gravity is not a force but a manifestation of curved spacetime. A freely falling body follows the straightest possible path — a geodesic — through curved four-dimensional spacetime. What we perceive as gravitational attraction is really the geometry of spacetime telling matter where to go, and matter telling spacetime how to curve.

The metric tensor g_μν is the fundamental object in GR. It tells you the distance between nearby events. In flat space g_μν = diag(−1, 1, 1, 1) (the Minkowski metric η_μν). In curved spacetime, g_μν varies from point to point and encodes gravitational effects fully.

Einstein’s Field Equations

The dynamical equations of GR relate the curvature of spacetime to its energy content:

These ten coupled nonlinear PDEs are notoriously difficult to solve exactly. The left side encodes spacetime curvature; the right side encodes the distribution of matter, energy, and momentum. “Spacetime tells matter how to move; matter tells spacetime how to curve” (Wheeler).

The geodesic equation describes the path of a freely falling particle:

Schwarzschild Geometry & Black Holes

In 1916, Karl Schwarzschild found the first exact solution to Einstein’s field equations: the geometry outside a spherically symmetric, non-rotating, uncharged mass. The Schwarzschild metric is:

At r = r_s the metric component g_tt = 0 and g_rr diverges (a coordinate, not physical, singularity). This is the event horizon of a Schwarzschild black hole: the surface beyond which escape velocity exceeds c. Once inside, all geodesics lead to the central singularity at r = 0, where curvature is physically infinite.

Photon Sphere and Innermost Stable Circular Orbit

Massive particles can orbit on circular geodesics only for r > 3r_s/2, with the last stable circular orbit at r = 3r_s (the ISCO). Photons can orbit (unstably) at r = 3r_s/2, the photon sphere. The accretion disk of a black hole extends from the ISCO inward, and radiation emitted there carries the distinctive spectrum seen in VLBI imaging (M87* image by Event Horizon Telescope, 2019; Sgr A* image, 2022).

Kerr Black Holes

Realistic black holes are described by the Kerr metric (Roy Kerr, 1963), which adds rotation parameterised by the spin parameter a = J/Mc (angular momentum per unit mass). The event horizon area shrinks from r_s to r₊ = r_s/2 + √(r_s²/4 − a²). Frame-dragging (the Lense-Thirring effect) drags spacetime around with the rotating body; inside the ergosphere (r > r₊ but outside the static limit) it is impossible to remain stationary. The Penrose process can extract rotational energy from the ergosphere, up to 29% of Mc² for a maximally rotating hole.

Gravitational Lensing

Light follows null geodesics (ds² = 0) in curved spacetime. A massive object between a source and an observer deflects light rays, acting as a gravitational lens. Einstein predicted a deflection angle of:

When source, lens, and observer are perfectly aligned, the image forms a complete ring — the Einstein ring — with angular radius: θ_E = √(4GM d_LS / (c² d_L d_S)). Slight misalignments produce pairs of arc images. The lensing cross-section scales as θ_E², so more massive and closer lenses produce larger effects.

Strong lensing (galaxy clusters, massive ellipticals) produces multiple images, arcs, and Einstein rings visible in HST images. Weak lensing produces subtle coherent shape distortions in background galaxy samples, used to map dark matter. Microlensing monitors individual star brightness for temporary magnification events when a compact foreground object transits; this method discovered exoplanets and probes stellar mass black holes in the Milky Way halo.

Gravitational Redshift & Gravitational Time Dilation

Photons climbing out of a gravitational well lose energy to it, their frequency decreasing: gravitational redshift. From the Schwarzschild metric, the ratio of emitted to received frequency is:

The same formula describes gravitational time dilation: clocks deeper in a gravitational field run slower. A clock at Earth’s surface runs ~6.97 × 10⁻¹⁰ slower per second than a clock infinitely far away. GPS satellites orbit at 20 200 km; their clocks run faster by +45.9 μs/day (gravitational) and slower by −7.2 μs/day (special relativistic time dilation from orbital speed v = 3.87 km/s), giving a net +38.7 μs/day that must be corrected for — otherwise GPS positions would drift by ≈ 11 km/day.

Near a stellar-mass black hole, time dilation becomes extreme. At r = 1.01 r_s, a clock runs 10× slower than at infinity. At r = r_s, a distant observer sees infalling clocks freeze and redshift to zero frequency; the infalling observer crosses the horizon in finite proper time with no local drama, thanks to the coordinate vs. physical singularity distinction.

FLRW Cosmology & the Expanding Universe

On the largest scales, the universe is homogeneous and isotropic (the cosmological principle). The most general metric consistent with this symmetry is the Friedmann–Lemaître–Robertson–Walker (FLRW) metric:

Substituting into Einstein’s equations with a perfect fluid T_μν gives the Friedmann equations:

The Hubble law v = H₀ d emerges in the linear-expansion limit. For ΛCDM (Λ + Cold Dark Matter), the scale factor obeys:

• Radiation domination (z > 3400): a ∝ t^1/2
• Matter domination (3400 > z > 0.3): a ∝ t^2/3
• Λ domination (z < 0.3, today): a ∝ e^H₀t (exponential expansion)

The critical density ρ_c = 3H²/(8πG) defines Ω = ρ/ρ_c. The Planck 2018 results: Ω_m ≈ 0.315 (matter, of which ~84% is dark matter), Ω_r ≈ 9.1 × 10⁻⁵ (radiation), Ω_Λ ≈ 0.685 (dark energy), Ω_total ≈ 1.000 (flat universe).

Cosmic Microwave Background

About 380 000 years after the Big Bang, the universe cooled to ~3000 K: protons and electrons combined into neutral hydrogen (recombination), and the universe became transparent. The photons released at this moment are observed today as the cosmic microwave background (CMB), redshifted to T₀ = 2.725 K by 1089× expansion.

The CMB temperature is isotropic to one part in 10⁵, but tiny anisotropies δT/T ~ 10⁻⁵ encode rich physics. Before recombination, photon-baryon plasma underwent acoustic oscillations: gravity compressed baryons, radiation pressure pushed back. These oscillations left acoustic peaks in the CMB power spectrum C_ℓ at multipoles ℓ ~ 200, 540, 800, … The positions and heights of the peaks constrain cosmological parameters exquisitely.

The CMB also carries polarisation: E-mode from scalar perturbations (measured by Planck, WMAP, ACT) and B-mode from primordial gravitational waves predicted by inflation. Detecting the B-mode will constrain the inflationary energy scale and verify or falsify the inflation hypothesis.

Dark Energy & the Accelerating Universe

In 1998 two supernova surveys (Riess et al., Perlmutter et al.) independently found that Type Ia supernovae at high redshift are dimmer than expected in a matter-dominated universe — the expansion is accelerating. This won the 2011 Nobel Prize. The simplest explanation is Einstein’s cosmological constant Λ, acting as a constant energy density ρ_Λ = Λc²/(8πG) with equation-of-state parameter w = p/ρ = −1.

Observational constraints combine CMB, baryon acoustic oscillations (BAO), and supernovae. Current fits give w = −1.028 ± 0.032 (Planck + DES 2024), consistent with Λ but leaving room for quintessence (scalar field with time-varying w). The fundamental tension: quantum field theory predicts a vacuum energy density ~10¹²⁰ larger than observed — the cosmological constant problem, one of the biggest unsolved problems in physics.

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