Relativity » Graduate Topics

Graduate Topics

Tensor formalism, exact solutions, gravitational radiation, cosmology and the quantum-gravity frontier

This hub organizes the graduate-level treatment of relativity. The conceptual introduction is on Special Relativity and General Relativity; the five pages below develop the theory in its standard mathematical form and connect it to current observations. Tensor Formalism supplies the differential geometry that the other four assume.

The Five Deep Dives

Page What it covers Assumes
Tensor Formalism & the Field Equations Manifolds, metric, connection and covariant derivative, Riemann/Ricci tensors, geodesic deviation, Einstein–Hilbert action General Relativity
Black Holes Schwarzschild, Reissner–Nordström and Kerr; horizons, ISCO and photon sphere, Penrose diagrams, thermodynamics, Hawking radiation, information paradox, observational status Tensor Formalism
Gravitational Waves Linearized gravity, TT gauge, quadrupole formula, binary inspiral and ringdown, interferometric detection Tensor Formalism
Relativistic Cosmology FLRW metric, Friedmann equations, $\Lambda$CDM, cosmological horizons, de Sitter and anti-de Sitter, inflation Tensor Formalism
Toward Quantum Gravity Planck scale, non-renormalizability, string theory, loop quantum gravity, asymptotic safety, causal sets, holography Quantum Field Theory 
graph TD
    GR["General Relativity<br/>(conceptual)"] --> TF["Tensor Formalism<br/>metric, curvature, field equations"]
    TF --> BH["Black Holes<br/>strong-field exact solutions"]
    TF --> GW["Gravitational Waves<br/>weak-field radiation"]
    TF --> COS["Cosmology<br/>homogeneous exact solutions"]
    BH -. "ringdown" .-> GW
    BH --> QG["Toward Quantum Gravity<br/>where the classical theory ends"]
    COS --> QG
    QFT["Quantum Field Theory"] --> QG

After Tensor Formalism the pages can be read in any order. Black Holes and Cosmology treat the two main families of exact solutions — isolated, strongly curved objects and the homogeneous universe. Gravitational Waves covers the weak-field radiative regime, and its merger and ringdown material links directly to the black-hole page. Toward Quantum Gravity picks up the places where the classical theory predicts its own breakdown: singularities, black-hole entropy, and the initial state of the universe.

Conventions

These pages use geometric units $G = c = 1$, in which mass, length and time share one dimension ($1\,M_\odot \approx 1.48\ \text{km} \approx 4.93\ \mu\text{s}$), so the Schwarzschild factor is $1 - 2M/r$ rather than $1 - 2GM/rc^2$. Signs follow Misner, Thorne & Wheeler (MTW), which is also the convention of Wald, Carroll, Hartle and Schutz.

Choice Used here (MTW) Common alternative
Metric signature $(-,+,+,+)$, “mostly plus” $(+,-,-,-)$, e.g. Landau–Lifshitz and most particle-physics texts
Index ranges Greek $\mu,\nu = 0..3$; Latin $i,j = 1..3$ Some texts put time last ($\mu = 1..4$)
Riemann tensor $R^\rho{}{\sigma\mu\nu} = \partial\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$ Overall sign flipped in some older texts
Ricci tensor $R_{\mu\nu} = R^\rho{}_{\mu\rho\nu}$ Contraction on other index pairs (changes sign)
Field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}$ $-8\pi$ with opposite Ricci sign

Changing signature flips the sign of $g_{\mu\nu}$ and of every expression with an odd number of explicit metrics, but not the physics. When comparing formulas across references, check all three sign choices before suspecting an error.

Key Equations

Topic Equation Details
Field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi T_{\mu\nu}$ Tensor Formalism
Geodesic equation $\ddot x^\mu + \Gamma^\mu_{\alpha\beta}\dot x^\alpha \dot x^\beta = 0$ Tensor Formalism
Schwarzschild horizon $r_s = 2M$; photon sphere $3M$; ISCO $6M$ Black Holes
Kerr horizons $r_\pm = M \pm \sqrt{M^2 - a^2}$, $a = J/M \le M$ Black Holes
Hawking temperature, entropy $T_H = \hbar\kappa/2\pi$, $S = A/4\ell_P^2$ Black Holes
Quadrupole formula $\bar h_{ij} = \dfrac{2}{r}\,\ddot I_{ij}(t - r)$ Gravitational Waves
Friedmann equation $H^2 = \dfrac{8\pi}{3}\rho - \dfrac{k}{a^2} + \dfrac{\Lambda}{3}$ Relativistic Cosmology
Planck length $\ell_P = \sqrt{G\hbar/c^3} \approx 1.6\times10^{-35}\ \text{m}$ Toward Quantum Gravity

Where the Field Stands (2026)

General relativity has passed every precision test to date; the open questions are about the strong-field and cosmological regimes and about its quantum completion.

Area Current picture
Gravitational-wave astronomy LIGO–Virgo–KAGRA completed their fourth observing run (O4) in November 2025, with hundreds of compact-binary candidates; GW250114, the loudest event yet, confirmed Hawking’s area theorem and resolved two ringdown modes. Pulsar-timing arrays (NANOGrav and others, 2023) report evidence for a nanohertz background, most likely from supermassive black-hole binaries. ESA adopted LISA in 2024 for launch in the mid-2030s.
Black-hole imaging The Event Horizon Telescope has imaged M87* and Sagittarius A*, including polarization, and is building multi-epoch observations toward horizon-scale movies.
Cosmology $\Lambda$CDM still fits the cosmic microwave background, but the Hubble tension between early- and late-universe measurements of $H_0$ persists, and baryon-acoustic-oscillation data from DESI (2024–2025), combined with supernovae, show a mild preference for evolving dark energy that is not yet decisive.
Quantum gravity No experimental signature yet. Theoretical progress centres on holography and the black-hole information problem (the island computation of the Page curve), with tabletop proposals to test whether gravity can entangle masses.

See Also

Within relativity:

Elsewhere in physics: