General Relativity

General relativity is the theory in which spacetime geometry is dynamical. Its central achievement is not merely a new force law: it joins clocks, freely falling bodies, light rays, curvature, matter, and conservation in one framework whose predictions can be tested by local experiments and astronomical observations. The aim here is to build that framework from operational questions—what an observer measures, what an accelerometer reads, and how neighboring trajectories separate—before deriving Einstein’s field equation and its consequences.

Helpful background. Special-relativistic spacetime and invariant intervals provide useful preparation, but the operational distinctions needed here are recalled. The geometric arguments are easier to follow after studying tensor fields and their natural operations, although the needed constructions are recalled when their physical meaning matters.

For a first course spanning the Library, use the First Graduate Course in General Relativity for modules, assignments, and a capstone, or the Graduate General Relativity Self-Study Path for readiness checks and a guided reading sequence. Both connect the local field theory developed here to exact black holes, global structure, radiation, matter, and cosmology. You can also enter this volume at the physical question you need and use each page’s background notes to prepare.

From universal free fall to spacetime geometry

You do not need to master every tensor identity before starting. Begin with a physical question, identify the observer and the quantity measured, and then ask which part of the geometry determines the answer. The same four questions will organize much of the volume:

Physical question	Mathematical tool	First place to work it out
What does one clock or detector read?	Proper time and tetrad projections	Measured energy and frequency
How does a small freely falling body move?	A geodesic with specified initial data	Ideal timelike free fall
How do neighboring bodies move relative to one another?	Curvature and the Jacobi equation	Geodesic deviation
How does matter determine the geometry?	Einstein’s equation and matter dynamics	The Einstein equation

For each derivation, first identify what is given, what is being solved for, and which approximation is in use. Then check a simple limit and attempt the exercise before opening its solution. A sign or index that initially feels abstract often becomes clear when translated back into a clock reading or a relative acceleration.

The experimental starting point is the universality of free fall for suitable test bodies. In Newtonian language, this is the observed equality of passive gravitational-to-inertial mass ratios; in relativistic language, it is the statement that neutral, nonspinning, negligibly self-gravitating bodies released with the same initial data follow the same unparametrized trajectory. The equivalence-principles chapter separates that weak equivalence principle from the stronger Einstein equivalence principle. The latter also demands local Lorentz invariance and local position invariance for nongravitational experiments.

This distinction matters. Universal trajectories suggest that gravity can be encoded by a common geometric structure rather than by a composition-dependent force, but the equivalence principle does not uniquely imply Einstein’s field equation. At any event one can choose a freely falling frame in which the metric is Minkowskian and the Levi-Civita connection vanishes. Curvature survives in the quadratic variation of the metric across an extended laboratory. Gravity can therefore be removed locally to first order, while tidal gravity cannot.

The precise logical domain and modern experimental language are developed alongside durable, nonnumerical tests of universality; see Reall 2013, Chapter 1, pp. 1–10, PDF and Will 2014, § 2.1, pp. 6–15, PDF.

Observers, clocks, free fall, and light

A tensor becomes a measurement only after the measuring observer and apparatus have been specified. The observers-and-measurement chapter begins with a timelike worldline, a unit four-velocity, and an orthonormal tetrad. It then constructs proper time, radar time and distance, Fermi coordinates, measured particle energy, photon frequency, redshift, accelerometer readings, and gyroscope transport. The recurrent rule is simple: coordinate components describe a chart, whereas tetrad components describe readings in a calibrated local frame.

Radar coordinates require suitable two-way null connections and can lose global uniqueness when light cones develop caustics or self-intersections; Fermi normal coordinates instead organize a tubular neighborhood of a reference geodesic, with curvature entering at quadratic order. See Perlick 2008, §§ 1–2 and 4, pp. 1–12, PDF and Manasse and Misner 1963, pp. 735–745.

The free-fall chapter derives ideal timelike and null motion from reparametrization-invariant actions. It distinguishes the geometric curve from a chosen affine parameter, proves local existence for specified initial data, and explains why a freely falling laboratory is locally inertial even when separated probes have relative acceleration. The geodesic conclusion is explicitly limited to a structureless test body with negligible backreaction; external forces or failure of that approximation require a separate equation of motion.

The geometric-optics chapter derives rather than assumes the connection between electromagnetic waves and null geodesics. A short-wavelength expansion gives an eikonal equation, amplitude and polarization transport, and beam-area evolution. The single-phase leading-order field representation becomes singular at a caustic even though the underlying null geodesics and Jacobi fields can be continued, and the expansion ceases to control the field when the wavelength is comparable to the background scale. A dispersive material requires a constitutive model beyond vacuum metric optics. Thus a metric null geodesic is a leading vacuum ray, not an exact description of every electromagnetic signal. The curved-spacetime hierarchy is developed in Dolan 2018, §§ 2–4, pp. 3–11, PDF.

Curvature and tidal gravity

Connection coefficients can be nonzero in flat spacetime and can vanish at one event in curved spacetime. Curvature is the invariant discriminator. The curvature-as-gravitation chapter develops this statement through normal-coordinate expansions, infinitesimal holonomy, commutators of covariant derivatives, and the Ricci–Weyl decomposition. Relative to an observer, the electric and magnetic parts of the Weyl tensor organize locally measured tidal and differential-precession information; the connection-to-curvature chain in the same signature and Riemann convention is developed in Carroll 1997, § 3, pp. 61–96, PDF.

The tidal-gravity chapter turns curvature into a laboratory prediction. The Jacobi equation relates the Riemann tensor to the relative acceleration of neighboring geodesics. Its Newtonian limit recovers the Hessian of the gravitational potential, while Fermi-normal expansions estimate the accuracy of an extended local laboratory. Generalized deviation along nongeodesic reference curves introduces the reference and probe accelerations explicitly; a sufficiently specified ideal gravitational compass can then solve for curvature components from relative-acceleration data. The construction and its measurement assumptions are given in Puetzfeld and Obukhov 2016, §§ II–IV, pp. 1–13, PDF.

These chapters also establish a theme used later: Ricci curvature and Weyl curvature play different roles. Einstein’s equation algebraically relates Ricci curvature to local stress-energy, while the Weyl tensor contains the remaining tidal degrees of freedom. Neither part should be identified too quickly with “matter here” or “radiation here”; those interpretations require additional hypotheses.

Field equations, matter, and the action principle

The Einstein-equations chapter introduces

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu},$$

with Newton’s constant $G$ retained explicitly. The contracted Bianchi identity makes the covariant divergence of the left-hand side vanish, and matter dynamics supplies the corresponding on-shell conservation law. Trace reversal, the cosmological term, local equation counting, constraints, and the sharply delimited four-dimensional Lovelock uniqueness result are treated separately so that no one slogan carries more logical weight than it should.

The uniqueness statement assumes a smooth, symmetric, divergence-free, second-order natural tensor constructed locally from the metric in four dimensions; changing the dimension, derivative order, fields, torsion, locality, or naturality leaves its domain. See Navarro and Navarro 2011, Definitions 1.1–1.10, Theorem 1.11, Corollary 2.3, and Theorem 2.6, pp. 2–9, PDF.

Stress-energy is not an arbitrary source inserted after the geometry is chosen. The matter-coupling chapter defines it by metric variation of a matter action and tests the definition on particles, dust, perfect fluids, scalar fields, and electromagnetism. Conservation then splits into recognizable equations of motion: continuity plus geodesic flow for dust, and energy plus relativistic Euler equations for a perfect fluid. The metric-variation definition, on-shell conservation, and representative matter equations are developed in Reall 2013, Chapters 5–6, pp. 49–66, PDF.

The Einstein–Hilbert-action chapter derives the field equation from

$$S_{\mathrm{grav}}=\frac{1}{16\pi G}\int_{\mathcal M} d^4x\,\sqrt{-g}\,(R-2\Lambda),$$

while retaining the divergence produced by varying the Ricci scalar. Non-null Dirichlet boundaries require the Gibbons–Hawking–York term; null segments and corners require additional choices of generators, orientations, and normalization. These segment and joint terms are derived in Lehner, Myers, Poisson, and Sorkin 2016, §§ II.B–II.C and II.H, pp. 6–27, PDF. The Palatini comparison makes explicit when independent metric and connection variations do—and do not—return the Levi–Civita theory; Dadhich and Pons 2012, §§ 1–2 and 4.1, pp. 2–7, PDF supplies the arbitrary-connection and torsion-free cases.

Gauge, focusing, and observable predictions

General covariance is a property of how equations are written; diffeomorphism gauge redundancy is a statement about which field configurations represent the same physical situation under specified boundary conditions. The diffeomorphisms chapter distinguishes active pullbacks from passive coordinate changes, derives the Noether–Bianchi identity, and constructs Killing currents. There is no unique generally covariant pointwise tensor built only from the metric that has the universal source role of matter stress-energy. This does not erase gravitational energy: pseudotensors, superenergy tensors, Hamiltonian boundary charges, quasi-local masses, and ADM or Bondi quantities answer different questions with additional structure or restricted domains. See Jaramillo and Gourgoulhon 2010, § 1.2, pp. 4–6, PDF. Boundary and asymptotic symmetries are therefore left visible rather than discarded as “just gauge.”

The focusing-and-energy-conditions chapter uses the Raychaudhuri equation to control the expansion of timelike and null congruences. An initially converging, geodesic, twist-free bundle obeying curvature convergence cannot remain regular beyond a finite parameter bound; if its generators extend that far, a focal point must occur by then. The conclusion is local: a focal point is not by itself a curvature singularity or proof of geodesic incompleteness. Pointwise energy conditions are stated with their quantifiers, translated carefully through Einstein’s equation, and compared with classical and quantum violations. The distinction between the geometric identity, focusing hypotheses, and field-equation input follows Kar and SenGupta 2007, §§ II.C–II.D, pp. 10–13, PDF.

Finally, the weak-field-predictions chapter returns the theory to observation. Linearization about Minkowski spacetime recovers Newtonian gravity and organizes gauge freedom. Controlled stationary and Schwarzschild expansions yield gravitational clock shifts, perihelion advance, light deflection, Shapiro delay, geodetic precession, and frame dragging. Each formula is paired with its approximation parameters, so the classic result is not mistaken for a modern high-order model of a binary or an extended source. See Reall 2013, §§ 12.2–12.4, pp. 134–140, PDF and Will 2014, §§ 4.1–4.4, pp. 40–51, PDF.

Conventions and ways through the material

The metric signature is $(-,+,+,+)$, timelike observer velocities satisfy $u^\mu u_\mu=-1$, and measured energy and frequency are $E=-p_\mu u^\mu$ and $\omega=-k_\mu u^\mu$. The Riemann tensor is fixed by

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma,$$

with $R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}$. Unless a controlled geometric-unit specialization is explicitly announced, $G$ remains visible. Boundary signs are never portable without the declared normal orientation and extrinsic-curvature convention.

For a first course, follow the chapters in order, postponing geometric optics if necessary. Readers interested mainly in operational measurement can move from equivalence principles through observers, light, curvature, and tides. Readers approaching classical field theory may begin with the field equations, then matter coupling, the action, and diffeomorphism identities. Every ordinary page states the exact background it uses, so a targeted route is also possible without reading the volume linearly.

For object-first lookup, continue to the Black Hole Atlas. The Research Atlas keeps dated frontier assessments separate from this stable exposition, while Reference provides compact formulas and conventions. Formulation choices and continuum data problems continue in Formulations of General Relativity, and implementation and verification workflows belong to Computational Relativity.

Check your preparation

Coordinates and curvature.

Explain why the ability to set $\Gamma^\rho{}_{\mu\nu}=0$ at one event does not make the Riemann tensor vanish there.

Solution

The connection is not a tensor: its coordinate transformation contains an inhomogeneous term involving second derivatives of the coordinate map, which can cancel its value at one chosen event. Curvature is a tensor built from derivatives and quadratic combinations of the connection. In normal coordinates the connection vanishes at the central event, but its first derivatives generally do not; their tensorial combination is $R^\rho{}_{\sigma\mu\nu}$. Thus freely falling coordinates remove uniform gravitational acceleration locally, not tidal gravity.

A weightless laboratory.

Two small laboratories have zero central accelerometer reading. One is inertial in Minkowski spacetime; the other falls freely in the weak exterior field of a spherical mass. Can the central reading distinguish them? Describe a measurement using several nearby freely falling probes that can.

Solution

The central reading cannot distinguish them: both reference worldlines have zero proper acceleration. Release neighboring probes with zero initial relative velocity and measure how their separations change. In Minkowski spacetime they remain fixed in an inertial frame. In the weak point-mass field, the local tidal equation predicts radial stretching and transverse compression, even though each probe has zero proper acceleration. The distinction is between acceleration of one worldline and relative acceleration across a finite separation, not between two different readings of the same accelerometer. The observer tidal tensor quantifies the effect. The probes must be close enough for geodesic deviation and the stated weak-field approximation to apply.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Dadhich, Naresh, and Josep M. Pons. “On the Equivalence of the Einstein–Hilbert and the Einstein–Palatini Formulations of General Relativity for an Arbitrary Connection.” General Relativity and Gravitation 44 (2012): 2337–2352. DOI; arXiv.
• Dolan, Sam R. “Geometrical Optics for Scalar, Electromagnetic and Gravitational Waves on Curved Spacetime.” International Journal of Modern Physics D 27 (2018): 1843010. DOI; arXiv.
• Jaramillo, José Luis, and Éric Gourgoulhon. “Mass and Angular Momentum in General Relativity.” In Mass and Motion in General Relativity, edited by Luc Blanchet, Alessandro Spallicci, and Bernard Whiting, 87–124. Springer, 2011. DOI; arXiv.
• Kar, Sayan, and Soumitra SenGupta. “The Raychaudhuri Equations: A Brief Review.” Pramana 69 (2007): 49–76. DOI; arXiv.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94 (2016): 084046. DOI; arXiv.
• Manasse, F. K., and C. W. Misner. “Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry.” Journal of Mathematical Physics 4 (1963): 735–745. DOI.
• Navarro, Alberto, and José Navarro. “Lovelock’s Theorem Revisited.” Journal of Geometry and Physics 61 (2011): 1950–1956. DOI; arXiv.
• Perlick, Volker. “On the Radar Method in General-Relativistic Spacetimes.” In Lasers, Clocks and Drag-Free Control, edited by Hansjörg Dittus, Claus Lämmerzahl, and Slava G. Turyshev, 131–152. Springer, 2008. DOI; arXiv.
• Puetzfeld, Dirk, and Yuri N. Obukhov. “Generalized Deviation Equation and Determination of the Curvature in General Relativity.” Physical Review D 93 (2016): 044073. DOI; arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17, 4 (2014). DOI; arXiv.