Relativity » Graduate Topics

Graduate Topics

Tensor formalism, exact solutions, and the quantum-gravity frontier

This is the sub-hub for the graduate-level treatment of relativity. The conceptual core lives on Special Relativity and General Relativity; here the material is dense, formal, and split across five focused deep-dive pages. Start with Tensor Formalism for the differential geometry the other four assume, then branch into exact solutions, radiation, cosmology, and frontier topics as needed.

Conventions. Unless stated otherwise these pages use geometric units with $G = c = 1$, so masses, lengths, and times share dimensions and the field equations take their cleanest form (the Schwarzschild factor is $1 - 2M/r$ rather than $1 - 2GM/rc^2$). The metric signature is (−,+,+,+) (“mostly-plus”); it and the (+,−,−,−) convention differ only by an overall sign.

The Five Deep Dives

The graduate material splits into five pages — tensor formalism, black holes, cosmology, gravitational waves, and quantum gravity — each summarized with prerequisites in the table below.

How the Pages Fit Together

The five pages form a dependency chain rooted in the geometry. The tensor formalism supplies the curvature machinery and the field equations; the exact-solution and radiation pages apply that machinery; quantum gravity asks what happens when the classical geometry itself must be quantized.

graph TD
    TF["Tensor Formalism<br/>metric, curvature, field equations"] --> BH["Black Holes<br/>Schwarzschild, Kerr, thermodynamics"]
    TF --> COS["Cosmology<br/>FLRW, Friedmann equations"]
    TF --> GW["Gravitational Waves<br/>linearized gravity, quadrupole formula"]
    BH --> QG["Quantum Gravity<br/>Planck scale, holography"]
    COS --> QG
    GW --> QG
    classDef geom fill:#e0f2f1,stroke:#11998e,stroke-width:2px;
    classDef app fill:#e3f2fd,stroke:#1976d2,stroke-width:2px;
    classDef frontier fill:#fff3e0,stroke:#e65100,stroke-width:2px;
    class TF geom;
    class BH,COS,GW app;
    class QG frontier;
Page What it covers Assumes
Tensor Formalism & the Field Equations Manifolds, the metric, the connection, covariant derivatives, the Riemann/Ricci tensors, Einstein–Hilbert action General Relativity
Black Holes Schwarzschild/Reissner–Nordström/Kerr, horizons, Penrose diagrams, thermodynamics, Hawking radiation, information paradox Tensor Formalism
Relativistic Cosmology FLRW metric, Friedmann equations, $\Lambda$CDM, horizons, (anti–)de Sitter, inflation Tensor Formalism
Gravitational Waves Linearized gravity, TT gauge, quadrupole formula, binary inspiral, LIGO detection Tensor Formalism
Toward Quantum Gravity Planck scale, non-renormalizability, string theory, LQG, asymptotic safety, causal sets, holography Quantum Field Theory

Suggested reading order. Read Tensor Formalism first — it is the foundation the other four assume. After that the pages are independent: Black Holes and Cosmology are the two great families of exact solutions, Gravitational Waves is the weak-field radiative regime, and Toward Quantum Gravity is the open frontier where the classical theory runs out.

See Also

Within relativity:

  • General Relativity — the equivalence principle and the field equations in their conceptual setting.
  • Special Relativity — Minkowski spacetime, four-vectors, and the flat-space limit these pages reduce to.
  • Relativity Hub — overview and navigation.

Elsewhere in physics: