General Relativity: Geometry, Fields and Applications: /20 Summary

Context and scope

the supplied physics reference, Applications of Classical Physics — Chapters 24–28 From SR to GR · Curved Spacetime · Stars & Black Holes · Gravitational Waves · Cosmology

/20 Summary

• General Relativity = geometric theory of gravity. Gravitation is not a force — it's the curvature of 4-D spacetime sourced by energy-momentum.
• Two-line summary (Wheeler):

  Matter tells spacetime how to curve; spacetime tells matter how to move.

• Einstein's field equations: $$G_{\alpha\beta} \equiv R_{\alpha\beta} - \tfrac{1}{2}g_{\alpha\beta}R = \frac{8\pi G}{c^4} T_{\alpha\beta}$$ 10 nonlinear PDEs for the metric $g_{\alpha\beta}$. The single equation that determines stars, black holes, gravitational waves, and the universe.
• The Equivalence Principle (EP) is the conceptual foundation: locally, free-fall is indistinguishable from inertial motion in SR. Gravity = local consequence of curvature.
• Geodesics are the GR analog of straight lines — paths that parallel-transport their own tangent vector. Free particles follow timelike geodesics; light follows null geodesics.
• Four physical regimes, each a chapter:
  1. Spherical stars — Schwarzschild metric, Tolman-Oppenheimer-Volkoff equation, neutron stars + Chandrasekhar limit.
  2. Black holes — Schwarzschild (static) + Kerr (rotating); event horizons; ergospheres; ISCO; thermodynamics + Hawking radiation.
  3. Gravitational waves — ripples in $g_{\alpha\beta}$ from accelerating masses; LIGO/Virgo/KAGRA/LISA detections from 2015 onward.
  4. Cosmology — FRW metric for homogeneous-isotropic universe; Friedmann equations; ΛCDM ($\Omega_m \approx 0.32$, $\Omega_\Lambda \approx 0.68$).
• Tested to extreme precision: Mercury perihelion, light bending, Shapiro delay, gravitational redshift, frame-dragging (Gravity Probe B), pulsar timing (Hulse-Taylor 0.2% on $\dot P$), GW150914 inspiral-merger-ringdown.

Master Map

mindmap
  root((GR))
    Ch.24 SR→GR
      Equivalence principle
        Weak EP
        Einstein EP
        Strong EP
      Free fall = inertial
      Gravitational redshift
      Light bending
      Spacetime curvature intro
      Newtonian limit
    Ch.25 Fundamentals
      Manifold + tangent space
      Metric g_αβ
      Christoffel Γ
      Covariant deriv ∇
      Parallel transport
      Geodesic
      Riemann R^α_βγδ
      Ricci R_αβ, R
      Einstein G_αβ
      Field equations
      Bianchi identity
      Killing vectors
      Conservation
    Ch.26 Stars & BH
      Schwarzschild metric
      TOV equation
      White dwarfs
      Neutron stars
      Schwarzschild BH
        Event horizon
        Singularity
        Kruskal
      Kerr BH
        Ergosphere
        ISCO
        Penrose process
      BH thermodynamics
        T_Hawking
        S_BH = A/4
        Four laws
    Ch.27 GW & Tests
      Linearized GR
      TT gauge
      Plus / cross polarizations
      Quadrupole formula
      Binary inspirals
      LIGO/Virgo/KAGRA/LISA
      Tests of GR
        Mercury perihelion
        Light bending
        Shapiro delay
        Frame dragging
    Ch.28 Cosmology
      Cosmological principle
      FRW metric
      Friedmann eqs
      Hubble H₀
      Critical density
      ΛCDM
      Distance measures
      Redshift
      CMB
      BBN
      Inflation
      Structure formation
      Dark matter / energy

From Special to General Relativity

The Equivalence Principle — Three Versions

Experimental tests of WEP: Eötvös experiments (10⁻⁹), MicroSCOPE (10⁻¹⁵), Lunar Laser Ranging.

Key consequence (EEP): every local experiment in a freely-falling lab reproduces special-relativistic physics. Gravity is not a force at a point — it's an effect that appears over finite extents through tidal effects (geodesic deviation).

Three Predictions Before the Theory

Using only EEP + SR, Einstein derived three observable consequences. Each has now been verified to high precision.

Gravitational Redshift

Photon climbing out of potential $\Phi$ loses energy. Frequency shift:

$$\frac{\Delta\nu}{\nu} = -\frac{\Delta\Phi}{c^2}$$

(positive $\Delta\Phi$ = higher potential = redshift). Pound-Rebka 1959: 22.6 m tower, $\Delta\nu/\nu \sim 2.5\times 10^{-15}$, agreement to 1%. Modern: atomic clocks at different elevations, 10⁻¹⁸ precision.

Light Deflection

Newtonian (mass with $v = c$): $\Delta\theta = 2GM/(b c^2)$. GR doubles this:

$$\Delta\theta = \frac{4GM}{bc^2}$$

For Sun grazing incidence ($b = R_\odot$): $\Delta\theta = 1.75''$. Eddington 1919 eclipse expedition — confirmed GR, made Einstein world-famous. Modern radio interferometry: $10^{-4}$ precision.

Time Dilation in Gravitational Potential

A clock at potential $\Phi$ ticks slower than one at potential 0:

$$d\tau = dt\sqrt{1 + 2\Phi/c^2}$$

GPS satellites must correct for this: $\sim 45\,\mu$s/day "gain" (gravity) vs $7\,\mu$s/day "loss" (motion, SR). Without correction, position errors $\sim 10$ km/day.

Spacetime as Curved Manifold

Einstein's insight (1907–1915): the EEP demands gravity be encoded in the geometry of spacetime itself, not added as a force.

Tidal effects between nearby free-falling observers reveal local curvature (geodesic deviation, §25.6). A small-enough region looks flat (EEP); but extended structure cannot be transformed away.

Newtonian Limit

For weak fields and slow motion, write $g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}$ with $|h|\ll 1$. The 00-component dominates:

$$g_{00} = -(1 + 2\Phi/c^2), \quad \Phi = -GM/r \text{ (point mass)}$$

The geodesic equation reduces to Newton's: $$\ddot x^i = -\partial^i \Phi$$

Confirming GR contains Newtonian gravity in the appropriate limit.

Fundamentals of General Relativity

Spacetime Manifold

A 4-D manifold is a topological space locally homeomorphic to $\mathbb{R}^4$, with smooth coordinate charts. Tangent space $T_p\mathcal{M}$ at point $p$ is the linear space of vectors at $p$.

Basis vectors: $\mathbf{e}_\alpha = \partial/\partial x^\alpha$ in chart $\{x^\alpha\}$. Dual basis: $\mathbf{e}^\alpha = dx^\alpha$, with $\langle \mathbf{e}^\alpha, \mathbf{e}_\beta\rangle = \delta^\alpha_\beta$.

Metric Tensor

The metric $g_{\alpha\beta}$ defines:

$$ds^2 = g_{\alpha\beta}\, dx^\alpha dx^\beta$$

• Signature $(-,+,+,+)$ (B&T convention).
• Symmetric: $g_{\alpha\beta} = g_{\beta\alpha}$ ⇒ 10 independent components in 4-D.
• Has inverse $g^{\alpha\beta}$ with $g^{\alpha\gamma}g_{\gamma\beta} = \delta^\alpha_\beta$.
• Lowers/raises indices: $V_\alpha = g_{\alpha\beta} V^\beta$.

Examples:

Christoffel Symbols (Connection Coefficients)

The covariant derivative requires a way to compare vectors at different points — the connection. For the Levi-Civita connection (metric-compatible, torsion-free):

$$\boxed{\Gamma^\alpha_{\beta\gamma} = \tfrac{1}{2}g^{\alpha\delta}(\partial_\beta g_{\delta\gamma} + \partial_\gamma g_{\delta\beta} - \partial_\delta g_{\beta\gamma})}$$

Properties:

• Symmetric in lower indices: $\Gamma^\alpha_{\beta\gamma} = \Gamma^\alpha_{\gamma\beta}$.
• Not a tensor — transforms inhomogeneously under coordinate changes (extra $\partial^2 x$ term).
• Vanishes at any point in a freely-falling frame (Riemann normal coordinates).

Covariant Derivative

For a vector field $V^\alpha$:

$$\nabla_\beta V^\alpha = \partial_\beta V^\alpha + \Gamma^\alpha_{\beta\gamma}V^\gamma$$

For a covector $V_\alpha$:

$$\nabla_\beta V_\alpha = \partial_\beta V_\alpha - \Gamma^\gamma_{\beta\alpha}V_\gamma$$

For higher-rank tensors: one $+\Gamma$ for each upper index, $-\Gamma$ for each lower.

Key facts:

• $\nabla_\gamma g_{\alpha\beta} = 0$ (metric-compatibility).
• $\nabla$ is a derivation: Leibniz rule.
• Produces tensor of one higher rank from tensor of given rank.

Notation: sometimes write $\nabla_\beta = ;_\beta$ and $\partial_\beta = ,_\beta$: $V^\alpha{}_{;\beta} = V^\alpha{}_{,\beta} + \Gamma^\alpha_{\beta\gamma}V^\gamma$.

Parallel Transport & Geodesics

Parallel transport of $V^\alpha$ along curve $x^\alpha(\lambda)$:

$$\frac{D V^\alpha}{d\lambda} \equiv \frac{dx^\beta}{d\lambda}\nabla_\beta V^\alpha = \frac{dV^\alpha}{d\lambda} + \Gamma^\alpha_{\beta\gamma}\frac{dx^\beta}{d\lambda}V^\gamma = 0$$

Geodesic equation (curve that parallel-transports its own tangent):

$$\boxed{\frac{d^2 x^\alpha}{d\tau^2} + \Gamma^\alpha_{\beta\gamma}\frac{dx^\beta}{d\tau}\frac{dx^\gamma}{d\tau} = 0}$$

— with $\tau$ proper time (timelike) or affine parameter $\lambda$ (null/spacelike).

Variational form: geodesics extremize $\int d\tau = \int\sqrt{-g_{\alpha\beta}\dot x^\alpha\dot x^\beta}\,d\lambda$.

Riemann Curvature Tensor

Measures failure of parallel transport around a closed loop, or equivalently the failure of $\nabla_\alpha\nabla_\beta = \nabla_\beta\nabla_\alpha$:

$$[\nabla_\gamma, \nabla_\delta] V^\alpha = R^\alpha{}_{\beta\gamma\delta} V^\beta$$

Explicit formula:

$$\boxed{R^\alpha{}_{\beta\gamma\delta} = \partial_\gamma\Gamma^\alpha_{\beta\delta} - \partial_\delta\Gamma^\alpha_{\beta\gamma} + \Gamma^\alpha_{\mu\gamma}\Gamma^\mu_{\beta\delta} - \Gamma^\alpha_{\mu\delta}\Gamma^\mu_{\beta\gamma}}$$

Symmetries (drastically reduce independent components)

1. Antisymmetry first pair: $R_{\alpha\beta\gamma\delta} = -R_{\beta\alpha\gamma\delta}$
2. Antisymmetry last pair: $R_{\alpha\beta\gamma\delta} = -R_{\alpha\beta\delta\gamma}$
3. Symmetry under pair exchange: $R_{\alpha\beta\gamma\delta} = R_{\gamma\delta\alpha\beta}$
4. First Bianchi: $R_{\alpha[\beta\gamma\delta]} = 0$
5. Second Bianchi: $\nabla_{[\epsilon} R_{\alpha\beta]\gamma\delta} = 0$

In 4-D: 20 independent components.

Geodesic Deviation Equation

Tidal force = relative acceleration of neighboring geodesics:

$$\frac{D^2\xi^\alpha}{d\tau^2} = -R^\alpha{}_{\beta\gamma\delta}\, u^\beta u^\gamma \xi^\delta$$

where $\xi^\alpha$ = separation vector, $u^\alpha$ = 4-velocity. This is the precise sense in which curvature = gravity: tidal forces are observable, frame-independent.

Ricci Tensor, Ricci Scalar, Einstein Tensor

Ricci tensor (contraction of Riemann): $$R_{\alpha\beta} = R^\gamma{}_{\alpha\gamma\beta}$$

10 independent components in 4-D; symmetric.

Ricci (scalar) curvature: $$R = g^{\alpha\beta} R_{\alpha\beta}$$

Einstein tensor: $$\boxed{G_{\alpha\beta} \equiv R_{\alpha\beta} - \tfrac{1}{2}g_{\alpha\beta}R}$$

— combination chosen so that $\nabla^\alpha G_{\alpha\beta} = 0$ identically (contracted Bianchi), matching conservation $\nabla^\alpha T_{\alpha\beta} = 0$.

Einstein's Field Equations

$$\boxed{G_{\alpha\beta} + \Lambda g_{\alpha\beta} = \frac{8\pi G}{c^4} T_{\alpha\beta}}$$

with cosmological constant $\Lambda$ (originally added by Einstein for static universe; later discovered to be needed for accelerating expansion).

Properties:

• 10 nonlinear coupled PDEs for $g_{\alpha\beta}$.
• 4 of them are constraints (initial-value): $G^0{}_\beta$ involves only first time derivatives.
• Remaining 6 are dynamical; with diffeomorphism freedom (4 functions) ⇒ 2 physical DOF per spacetime point. Two physical DOF = two polarizations of gravitational waves.
• Reduce to Newton: $g_{00} \approx -(1 + 2\Phi/c^2)$, $G_{00}/c^2 \to \nabla^2\Phi$, $T_{00}/c^2 \to \rho c^2$, giving $\nabla^2\Phi = 4\pi G\rho$.

Lagrangian formulation (Einstein-Hilbert action): $$S = \frac{c^4}{16\pi G}\int R\,\sqrt{-g}\,d^4x + S_{\text{matter}}$$

Variation w.r.t. $g_{\alpha\beta}$ yields the field equations.

Stress-Energy Tensor in Curved Spacetime

Same forms as flat-space (Part I, Ch. 2), with $\eta_{\alpha\beta} \to g_{\alpha\beta}$ and $\partial \to \nabla$:

Conservation: $\nabla_\beta T^{\alpha\beta} = 0$ — guarantees local energy-momentum conservation (with all gravitational effects via $\Gamma$).

Symmetries & Killing Vectors

A Killing vector $\xi^\alpha$ generates a spacetime symmetry; satisfies Killing's equation:

$$\nabla_\alpha\xi_\beta + \nabla_\beta\xi_\alpha = 0$$

Conserved quantity along geodesics:

$$\xi^\alpha p_\alpha = \text{const}$$

where $p^\alpha = m u^\alpha$ (timelike) or photon 4-momentum (null).

Relativistic Stars and Black Holes

The Schwarzschild Solution

Unique spherically symmetric vacuum solution of Einstein's equations (Birkhoff's theorem — even time-dependent spherical solutions reduce to Schwarzschild outside):

$$\boxed{ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2 dt^2 + \frac{dr^2}{1 - r_s/r} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)}$$

with Schwarzschild radius:

$$r_s = \frac{2GM}{c^2}$$

Numerical values:

• Earth: $r_s = 8.87$ mm
• Sun: $r_s = 2.95$ km
• $1 M_\odot$ neutron star: $r_s = 2.95$ km, actual radius ~10 km
• Sgr A* ($4.1\times 10^6 M_\odot$): $r_s = 1.2\times 10^{10}$ m ≈ 0.08 AU
• M87 ($6.5\times 10^9 M_\odot$): $r_s = 1.9\times 10^{13}$ m ≈ 130 AU

Coordinate singularities at $r = r_s$ (removable; just the event horizon) and at $\theta = 0, \pi$ (familiar polar issue). True curvature singularity at $r = 0$ ($R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta} \propto 1/r^6$).

Tests in Schwarzschild Geometry

Perihelion Precession

For bound timelike orbits, periapse advances by

$$\Delta\phi = \frac{6\pi GM}{c^2 a(1-e^2)}\text{ per orbit}$$

Mercury: observed 43''/century unexplained by Newtonian (after extracting other planets' pulls); Einstein's prediction matched immediately — pivotal 1915 confirmation.

Light Bending

Null geodesic with impact parameter $b$ deflects by

$$\Delta\phi = \frac{4GM}{bc^2} + O((GM/bc^2)^2)$$

Shapiro Delay (1964)

Round-trip light travel through gravitational potential is delayed:

$$\Delta t = \frac{4GM}{c^3}\ln\frac{4 r_E r_R}{b^2}$$

Tested: Cassini probe (2002), agreement $\sim 10^{-5}$.

Stable Circular Orbits

• Schwarzschild ISCO at $r = 6GM/c^2 = 3 r_s$.
• Photon sphere at $r = 3GM/c^2 = 1.5 r_s$ (unstable circular photon orbit).
• No stable circular orbits inside ISCO; matter spirals in.

Inside the Star — TOV Equation

For static, spherically symmetric perfect fluid:

$$\boxed{\frac{dP}{dr} = -\frac{G\,(\rho + P/c^2)\,(M(r) + 4\pi r^3 P/c^2)}{r^2(1 - 2GM(r)/rc^2)}}$$

with $dM/dr = 4\pi r^2 \rho$. Tolman-Oppenheimer-Volkoff (TOV) equation is the GR generalization of Newtonian hydrostatic equilibrium $dP/dr = -G M\rho/r^2$.

Three GR corrections (all $> 0$, all destabilizing — make stars less stable):

1. $P/c^2$ adds to mass-energy density.
2. Pressure itself gravitates ($+P$ term in numerator).
3. Spatial curvature in denominator.

Equilibrium Stellar Sequences

Chandrasekhar limit (1931): for fully degenerate, ultra-relativistic electron gas ($P\propto\rho^{4/3}$), no stable hydrostatic solution above

$$M_{\rm Ch} = \frac{0.7732 (\hbar c)^{3/2}}{G^{3/2} m_p^2 \mu_e^2} \approx 1.456 \left(\frac{2}{\mu_e}\right)^2 M_\odot$$

For $\mu_e = 2$ (carbon/oxygen WD): $M_{\rm Ch} \approx 1.44 M_\odot$.

Type Ia supernovae detonate as a WD accretes to near $M_{\rm Ch}$ — standard candles, used to discover dark energy (1998).

TOV limit for neutron stars: depends on EOS; modern limit $\sim 2.3 M_\odot$. Observations (PSR J0740+6620: $2.08\pm 0.07 M_\odot$) constrain dense-matter EOS sharply.

Black Holes — Schwarzschild

Event horizon at $r = r_s$:

• One-way membrane: timelike worldlines inside cannot exit.
• $g_{tt}$ and $g_{rr}$ swap signs ⇒ inside, $r$ becomes timelike and $t$ spacelike. Falling toward $r=0$ is as inevitable as moving forward in time.
• Singularity at $r = 0$ is a future endpoint in spacetime for all interior observers.

Eddington-Finkelstein coordinates remove horizon singularity: $$ds^2 = -(1 - 2M/r)dv^2 + 2 dv\, dr + r^2 d\Omega^2$$ with advanced time $v = t + r + 2M\ln|r/2M - 1|$. Smooth across horizon.

Kruskal extension reveals full maximal extension: includes the BH, an exterior region, a "white hole," and a second asymptotic exterior. Wormhole structure (not physically traversable for Schwarzschild).

Mass formula: the only parameter — Schwarzschild BH is fully characterized by mass $M$ alone. "No-hair" theorem: stationary BH in Einstein-Maxwell theory characterized by $(M, J, Q)$ only — independent of how it formed.

Kerr Black Holes (Rotating)

The unique stationary axisymmetric vacuum BH solution. Two parameters: mass $M$ and angular momentum $J$. Spin parameter $a = J/(Mc)$.

Boyer-Lindquist coordinates: $$ds^2 = -\left(1 - \frac{2GMr}{\Sigma c^2}\right)c^2 dt^2 - \frac{4 GMar\sin^2\theta}{\Sigma c^2}\,c\, dt\,d\phi + \frac{\Sigma}{\Delta}\,dr^2 + \Sigma\, d\theta^2 + \left(r^2 + a^2 + \frac{2GMa^2 r\sin^2\theta}{\Sigma c^2}\right)\sin^2\theta\, d\phi^2$$

with $\Sigma = r^2 + a^2\cos^2\theta$, $\Delta = r^2 - r_s r + a^2$.

Extremal limit: $a \to M$, $r_+ = r_- = M$. For $a > M$: naked singularity (cosmic censorship hypothesis: physically excluded).

Ergosphere & Penrose Process

Within ergosphere, no static observer (frame-dragging). Particle splits into two; one falls in with $E < 0$, other escapes with $E_{\text{out}} > E_{\text{in}}$. Net extracts rotational energy from BH (up to ~29% of mass for extremal Kerr).

Astrophysical relevance: Blandford-Znajek mechanism — electromagnetic Penrose-like extraction powering AGN jets, GRBs.

Innermost Stable Circular Orbit (ISCO)

Depends on spin and direction:

Maximum binding energy at ISCO:

• Schwarzschild: $\sim 5.7\%$ of rest mass.
• Maximally rotating prograde: $\sim 42\%$.
• Compare nuclear burning H→He: $0.7\%$. Accretion onto BHs is the most efficient energy source in nature.

Black Hole Thermodynamics

Bekenstein (1972) + Hawking (1974): BHs are thermodynamic objects.

Four Laws of BH Mechanics

with horizon area $A$, angular velocity $\Omega_H$, electric potential $\Phi_H$. Looks exactly like ordinary thermodynamics if $T \sim \kappa$, $S \sim A$.

Hawking Temperature

Quantum field theory in BH background gives thermal radiation at:

$$\boxed{T_H = \frac{\hbar \kappa}{2\pi c k_B} = \frac{\hbar c^3}{8\pi G M k_B}}$$

For solar-mass BH: $T_H \approx 6\times 10^{-8}$ K — far below CMB; no net evaporation.

Evaporation time: $t_{\text{evap}} \sim M^3 \times 10^{-26}$ yr/kg³. Solar BH: $10^{67}$ yr. Primordial BHs $\sim 10^{15}$ g (small asteroid): evaporating now, end with gamma-ray burst.

Bekenstein-Hawking Entropy

$$\boxed{S_{BH} = \frac{k_B c^3 A}{4 G \hbar} = \frac{k_B A}{4\ell_P^2}}$$

with Planck length $\ell_P = \sqrt{G\hbar/c^3} \approx 1.6\times 10^{-35}$ m.

Holographic principle: entropy of any system bounded by area, not volume. Foundational hint at quantum gravity (AdS/CFT, etc.).

Generalized Second Law

$$d(S_{\text{matter}} + S_{BH}) \ge 0$$

Combined ordinary + BH entropy non-decreasing. Restores 2nd law: ordinary entropy can decrease (e.g., matter falls in) only if BH entropy increases more.

Gravitational Waves & Experimental Tests

Linearized GR

Background flat + small perturbation: $g_{\alpha\beta} = \eta_{\alpha\beta} + h_{\alpha\beta}$, $|h| \ll 1$.

Define trace-reversed: $\bar h_{\alpha\beta} = h_{\alpha\beta} - \tfrac{1}{2}\eta_{\alpha\beta} h$.

In Lorenz gauge $\partial^\alpha \bar h_{\alpha\beta} = 0$:

$$\boxed{\Box \bar h_{\alpha\beta} = -\frac{16\pi G}{c^4} T_{\alpha\beta}}$$

— wave equation with source. In vacuum, plane-wave solutions exist.

Transverse-Traceless (TT) Gauge

Residual gauge freedom fixes: $\bar h^{0\alpha} = 0$, $h^i{}_i = 0$. In TT gauge, $\bar h^{TT}_{\alpha\beta} = h^{TT}_{\alpha\beta}$ has only two independent components corresponding to the two polarizations.

For wave traveling in $z$-direction:

$$h^{TT}_{\alpha\beta} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & h_+ & h_\times & 0 \\ 0 & h_\times & -h_+ & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}\cos(\omega t - kz)$$

Both polarizations propagate at $c$ (in GR; constrained to $|c_{GW}/c - 1| < 10^{-15}$ by GW170817 + GRB).

Quadrupole Formula — Generation

Far-field radiation from a source:

$$\boxed{h^{TT}_{ij} = \frac{2G}{rc^4}\,\ddot Q^{TT}_{ij}(t - r/c)}$$

where $Q_{ij} = \int(x_i x_j - \tfrac{1}{3}\delta_{ij}r^2)\rho\,d^3x$ is the traceless mass quadrupole moment.

No monopole (mass conservation forbids it). No dipole (momentum conservation: 1st time derivative of mass dipole = momentum, conserved → 2nd derivative vanishes). GWs are intrinsically quadrupolar.

Power Radiated

$$P = \frac{G}{5c^5}\,\langle\dddot Q_{ij}\dddot Q^{ij}\rangle$$

The factor $G/c^5 \approx 2.7\times 10^{-53}$ W⁻¹ explains why GWs are so weak. To radiate 1 W requires $\dddot Q^2 \sim 10^{53}$ in SI units — an enormous accelerating quadrupole.

For binary with masses $m_1, m_2$, separation $a$, circular orbit: $$P = \frac{32 G^4}{5 c^5}\,\frac{(m_1 m_2)^2(m_1 + m_2)}{a^5}$$

Order of magnitude for systems:

Binary Inspiral

Orbit decay drives the two bodies together. Chirp signal — increasing frequency and amplitude.

Hulse-Taylor Pulsar PSR B1913+16 (1974)

$$\dot P_{\text{orb}}^{\text{obs}}/\dot P_{\text{orb}}^{\text{GR}} = 1.0013 \pm 0.0021$$

Indirect confirmation, Nobel 1993 (Hulse, Taylor). Best test of GR in strong-field, dynamic regime — until LIGO.

Chirp Mass

For binary observable through GW, the combination

$$\mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}$$

is the chirp mass — directly extracted from GW frequency evolution. Sets amplitude and frequency drift.

Three Stages of Compact Binary Coalescence

flowchart LR
    A[Inspiral<br/>Post-Newtonian] --> B[Merger<br/>Numerical Relativity]
    B --> C[Ringdown<br/>Quasinormal modes]

Direct Detection

LIGO — Laser Interferometer Gravitational-wave Observatory

• 4 km Michelson interferometers (Hanford, Livingston)
• Sensitivity $h \sim 10^{-21}$ — fractional length change $\Delta L / L \sim 10^{-22}$ over 4 km = $4\times 10^{-19}$ m ≈ 1/10000 proton diameter!
• First detection: GW150914 (Sept 14, 2015): merger of $\sim 30 M_\odot$ + $\sim 35 M_\odot$ BHs → $62 M_\odot$ + radiated $3 M_\odot c^2 \approx 5\times 10^{47}$ J in GWs.
• Nobel 2017 (Weiss, Barish, Thorne).

Multi-Messenger: GW170817

Binary neutron-star merger detected coincidently with gamma-ray burst (GRB 170817A) and kilonova (AT 2017gfo) across the EM spectrum. Established:

• Speed of gravity = $c$ to $10^{-15}$ precision.
• Origin of heavy r-process elements (gold, platinum etc.).
• Independent $H_0$ measurement (standard siren).

Current and Future Detectors

Tests of GR (Beyond GW)

No verified deviations from GR to date. Despite decades of attempts at modified gravity (MOND, $f(R)$, scalar-tensor, ...), Einstein wins.

Cosmology

The Cosmological Principle

On scales $\gtrsim 100$ Mpc, the universe is:

1. Homogeneous (same density everywhere)
2. Isotropic (same in every direction)

Verified by CMB temperature uniformity ($\Delta T/T \sim 10^{-5}$) and large-scale galaxy distribution.

The FRW Metric

Most general metric consistent with cosmological principle:

$$\boxed{ds^2 = -c^2 dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2 d\Omega^2\right]}$$

with scale factor $a(t)$ and curvature constant $k \in \{-1, 0, +1\}$ (open, flat, closed).

Comoving coordinates: galaxies (on average) at fixed $(r,\theta,\phi)$; expansion is encoded in $a(t)$.

Hubble parameter:

$$H(t) \equiv \frac{\dot a}{a}, \quad H_0 = H(t_0) \approx 67$–$73$ km/s/Mpc \approx 2.2\times 10^{-18},\text{s}^{-1}$$

(Persistent tension between "early universe" $H_0 \approx 67.4$ from Planck/CMB and "late" $H_0 \approx 73.0$ from local Cepheid+SNIa — known as the Hubble tension.)

Friedmann Equations

Plugging FRW into Einstein equations with perfect-fluid source:

$$\boxed{H^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}}$$

$$\boxed{\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3 P}{c^2}\right) + \frac{\Lambda c^2}{3}}$$

Conservation equation (continuity): $$\dot\rho + 3 H(\rho + P/c^2) = 0$$

(Any two of the three equations are independent.)

The Cosmic Energy Budget

Define density parameters $\Omega_i = \rho_i/\rho_c$ where $\rho_c = 3 H^2/(8\pi G)$. Friedmann becomes:

$$\Omega_m + \Omega_r + \Omega_\Lambda + \Omega_k = 1$$

Best-fit ΛCDM (Planck 2018):

Hubble distance: $D_H = c/H_0 \approx 4.3$ Gpc.

Cosmic History

flowchart LR
    A[t=0<br/>Big Bang] --> B[Inflation<br/>t~10⁻³⁶ s]
    B --> C[Radiation-dom<br/>BBN at t~minutes]
    C --> D[Matter-dom<br/>z~3400]
    D --> E[Recombination<br/>z~1090, t~380 kyr<br/>CMB!]
    E --> F[Dark Ages]
    F --> G[Reionization<br/>z~7]
    G --> H[Structure formation]
    H --> I[Λ-dom<br/>z~0.4]
    I --> J[Today<br/>t=13.8 Gyr]