Relativity Professional

GR requires differential geometry on a smooth manifold M. Key concepts:

Tensors

A tensor T^{μ₁...μₙ}_{ν₁...νₘ} transforms under coordinate changes x^μ → x̃^μ via products of Jacobians. Scalars (rank 0), vectors (rank 1), the metric (rank 2 covariant) are examples.

Covariant derivative

The standard partial derivative ∂_μ does not transform as a tensor. The covariant derivative ∇_μ does:

∇_μ V^ν = ∂_μ V^ν + Γ^ν_{μρ} V^ρ

Christoffel symbols: Γ^σ_{μν} = ½ g^{σρ}(∂_μ g_{νρ} + ∂_ν g_{μρ} − ∂_ρ g_{μν}).

Parallel transport

A vector V^μ is parallel transported along a curve x^μ(λ) if ∇_u V^μ = 0, where u^μ = dx^μ/dλ is the tangent. On a curved manifold, parallel transport around a loop rotates the vector — this is curvature.

Curvature is encoded in the Riemann tensor:

R^ρ_{σμν} = ∂_μ Γ^ρ_{νσ} − ∂_ν Γ^ρ_{μσ} + Γ^ρ_{μλ} Γ^λ_{νσ} − Γ^ρ_{νλ} Γ^λ_{μσ}

Contractions

• Ricci tensor: R_{μν} = R^ρ_{μρν}
• Ricci scalar: R = g^{μν} R_{μν}
• Einstein tensor: G_{μν} = R_{μν} − ½ R g_{μν}

Bianchi identity

∇_λ R_{ρσμν} + ∇_ρ R_{σλμν} + ∇_σ R_{λρμν} = 0. Contracted: ∇^μ G_{μν} = 0 — the Einstein tensor is divergence-free, consistent with ∇^μ T_{μν} = 0 (local energy-momentum conservation).

Sectional curvature

A maximally symmetric space (de Sitter, anti-de Sitter, Minkowski) has R_{μνρσ} = K(g_{μρ}g_{νσ} − g_{μσ}g_{νρ}) for constant K.

The complete field equations with cosmological constant:

G_{μν} + Λg_{μν} = 8πG/c⁴ T_{μν}

The stress-energy tensor

• Perfect fluid: T_{μν} = (ρ + p/c²)u_μ u_ν + p g_{μν}
• Electromagnetic field: T_{μν} = (1/μ₀)[F_{μα}F_ν^α − ¼ g_{μν} F_{αβ}F^{αβ}]
• Cosmological constant: T^Λ_{μν} = −(Λc²/8πG)g_{μν} — acts like a fluid with p = −ρc²

Degrees of freedom

g_{μν} has 10 independent components. The 4 Bianchi identities reduce this to 6 dynamical degrees of freedom. Choosing coordinates (gauge) removes 4 more, leaving 2 physical degrees of freedom — the two polarisations of gravitational waves.

Beyond Schwarzschild, two important exact vacuum solutions extend GR to rotating and charged black holes.

Kerr metric (1963)

For a rotating mass M with angular momentum J = Mac (a = J/(Mc)):

ds² = −(1−r_sr/Σ)c²dt² − 2r_s ar sinθ/Σ c dt dφ + Σ/Δ dr² + Σ dθ² + (r²+a²+r_sa²sin²θ/Σ)sin²θ dφ²

where Σ = r² + a²cos²θ, Δ = r² − r_sr + a².

Frame dragging and ergosphere

Outside the outer horizon (r₊ = r_s/2 + √(r_s²/4 − a²)) is the ergosphere (r_s/2 + √(r_s²/4 − a²cos²θ) > r > r₊), where the Penrose process can extract rotational energy.

Reissner-Nordström

Spherically symmetric charged black hole (mass M, charge Q). Two horizons: r± = GM/c² ± √(G²M²/c⁴ − GQ²/4πε₀c⁴).

Classical GR predicts that generic gravitational collapse inevitably produces singularities — regions where the theory breaks down.

Penrose theorem (1965)

If: (1) the null energy condition T_{μν}k^μk^ν ≥ 0 holds, (2) a trapped surface exists (a closed 2-surface where both families of outgoing null geodesics converge), then (3) the spacetime contains an incomplete null geodesic — a singularity.

Hawking theorem (1970)

If the strong energy condition holds and the universe is globally hyperbolic, a Big Bang singularity was inevitable in our past.

Cosmic censorship conjecture (Penrose)

Singularities are always hidden behind event horizons (no naked singularities). Still unproved in general; counterexamples exist in special cases.

Physical implication

Singularities signal the breakdown of GR itself. A quantum theory of gravity is needed to resolve them.

Bekenstein (1972) and Hawking (1974) discovered a deep connection between black holes, thermodynamics, and quantum mechanics.

Four laws of black hole mechanics

1. Zeroth: surface gravity κ is constant over the horizon (like T in equilibrium).
2. First: dM = κ dA / 8πG + Ω dJ + Φ dQ (like dU = T dS + ...).
3. Second: horizon area A never decreases (like entropy).
4. Third: κ cannot be reduced to zero by a finite process.

Hawking radiation

A black hole radiates as a blackbody with temperature T_H = ℏc³/(8πGMk_B). For a solar-mass black hole T_H ~ 60 nK — undetectably cold. The black hole slowly loses mass and eventually evaporates.

Bekenstein-Hawking entropy

S_BH = k_B A / (4 ℓ_P²), where ℓ_P = √(Gℏ/c³) is the Planck length. The entropy is proportional to area, not volume — the holographic principle.

The detection of GWs opened an entirely new observational window on the universe.

LIGO and Virgo

Two LIGO observatories (Hanford WA, Livingston LA) and Virgo (Pisa) are km-scale Michelson interferometers. Sensitivity: strain h ~ 10⁻²³ / √Hz at 100 Hz. Mirror masses ~40 kg, isolated to 10⁻¹⁸ m.

GW event classes

• Binary black hole (BBH) mergers: most numerous; e.g. GW150914 (36+29 M_☉).
• Binary neutron star (BNS): GW170817 + optical kilonova + gamma-ray burst — confirmed r-process nucleosynthesis site.
• Neutron star–black hole (NSBH): detected since O3.

Tests of GR from GWs

• GW speed: |v_GW − c|/c < 10⁻¹⁵ (GW170817).
• Ringdown spectroscopy: quasi-normal modes of the final Kerr BH confirm the no-hair theorem.
• Bounds on graviton mass: m_g < 1.27 × 10⁻²³ eV/c².

GR and quantum mechanics are incompatible at the Planck scale (ℓ_P ≈ 1.6 × 10⁻³⁵ m, E_P ≈ 10¹⁹ GeV). A theory of quantum gravity remains the central unsolved problem of theoretical physics.

String theory

Replaces point particles with 1D strings. Naturally includes spin-2 gravitons. Predicts extra dimensions. AdS/CFT duality: gravity in d+1-dimensional anti-de Sitter space is dual to a conformal field theory in d dimensions — a concrete holographic framework.

Loop quantum gravity (LQG)

Quantises spacetime geometry directly. Spin networks discretise area and volume at the Planck scale. Resolves the Big Bang singularity into a Big Bounce in some models.

The black hole information paradox

Hawking radiation appears thermal — does a black hole destroy information? Unitarity of quantum mechanics says no. The resolution (Page curve, island formula, ER=EPR) is an active frontier.

Other open problems

• Nature of dark energy (Λ, quintessence, modified gravity?)
• Dark matter particle identity
• The arrow of time and initial low-entropy state

Inflation (Guth 1981, Linde 1982) is a brief period of exponential expansion in the very early universe, driven by a scalar field (inflaton) with negative pressure.

Problems inflation solves

• Flatness problem: why is the universe so precisely flat (Ω ≈ 1)?
• Horizon problem: why is the CMB so uniform across causally disconnected regions?
• Monopole problem: dilutes topological defects from GUT-scale symmetry breaking.

Quantum fluctuations → structure

Quantum vacuum fluctuations during inflation are stretched to cosmic scales, seeding the density perturbations that grow into galaxies. The power spectrum P(k) ∝ k^{n_s−1} with n_s ≈ 0.965 (measured by Planck).

CMB

The cosmic microwave background (T₀ = 2.725 K) is a snapshot of the universe at recombination (z ≈ 1100). Temperature anisotropies ΔT/T ~ 10⁻⁵ encode the primordial power spectrum plus baryon acoustic oscillations.

Primordial gravitational waves

Inflation also produces a stochastic GW background, characterised by the tensor-to-scalar ratio r. Detecting B-mode CMB polarisation from these GWs would be direct evidence for inflation.