First Graduate Course in General Relativity

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This course is a finite first graduate treatment of general relativity. It develops Lorentzian geometry as a physical language, derives Einstein dynamics, studies Schwarzschild and Kerr spacetimes and their horizons, and then tests the theory across weak fields, radiation, compact objects, and cosmology. The framework can be used independently or adapted for teaching; it implies no institution, instructor, schedule, enrollment, or credential.

Required background. Multivariable calculus and linear algebra are used for tensor-valued differentiation, linear maps, and bilinear forms; ordinary differential equations are used for initial- and boundary-value problems and limiting checks; special relativity is used for invariant intervals, Lorentz transformations, four-vectors, causal structure, and energy–momentum. Check or strengthen the first two skills with Calculus and Linear Algebra for Relativity and Differential Equations for Relativity. For the special-relativity material, review Events, Intervals, and Minkowski Geometry and use the Mechanics and Special Relativity Diagnostic for focused diagnosis and review.

Helpful background. Smooth manifolds and tensors are used to interpret geometric fields and their natural operations; classical field theory is used for actions, boundary terms, and stress–energy. Review Tensor Fields and Natural Operations and Classical Fields and Their Actions for the scientific cores. The Tensor and Geometry Bridge and Fields and Stress-Energy Bridge are required just-in-time steps in the course spine; they also provide bounded review when the diagnostic exposes a gap.

Optional orientation. For a map of the surrounding subjects, browse Spacetime Geometry, Global Structure, and Compact Objects in that order. These overviews are not course steps and do not count toward completion.

General-relativity capabilities

By the end, you should be able to:

use observers, geodesics, curvature, and geodesic deviation in a checked geometric argument;
derive and interpret Einstein’s equation and its action formulation with boundary assumptions explicit;
solve and compare Schwarzschild and Kerr problems involving causal structure, horizons, charges, and black-hole mechanics; and
synthesize classical tests, radiation, compact objects, and cosmology across geometric and observational representations.

Completion requires assessed work for all four outcomes, the transfer task, and the capstone. Time spent or pages visited does not demonstrate understanding.

Workload and pacing

Work through the eight modules in order, treating each module’s reading and assessed work as one progression unit rather than assigning it a fixed number of meetings. Modules 1, 3, 5, and 7 carry the four numbered assignments; Modules 2, 4, and 6 carry shorter checkpoints; Module 8 supplies the final matter and cosmology comparison. Advance after you can reconstruct the decisive reasoning and its check without reading the feedback. Keep the cross-module transfer task and capstone as separate synthesis stages.

The 98 linked pages numbered below are the exact required readings, in their required order. Reading them is necessary but does not by itself establish completion: the assignments and exercises, module checkpoints, cross-module transfer task, and capstone let you demonstrate what you have learned.

Check your preparation

Attempt these first without notes. If that format creates an accessibility barrier, use an equivalent spoken, tactile, enlarged, assistive-technology, or reference-aided form and record the aid; the diagnostic tests reasoning, not recall speed.

For a timelike four-velocity $u^\mu$ , explain the invariant content of $g_{\mu\nu}u^\mu u^\nu=-1$ and how its components change between frames.
For fixed $\omega\gt0$ , solve $y''+\omega^2y=0$ from arbitrary initial data and verify the conserved quadratic energy.
Given a coordinate change $x^\mu\mapsto x^{\mu'}$ , distinguish how a vector, a metric, and a connection coefficient transform.
Starting from a field action, explain what data must be fixed before an integration by parts yields an admissible variational principle.

Response criteria

A secure response identifies $-1$ as the invariant squared norm in the adopted signature, explains that Lorentz transformations change the components but preserve the contraction, and distinguishes the geometric vector from a component column.
A secure response gives $y=A\cos(\omega t)+B\sin(\omega t)$ , fixes $A$ and $B$ from arbitrary initial data, and verifies that $\tfrac12[(y')^2+\omega^2y^2]$ is constant by differentiation.
A secure response gives homogeneous Jacobian transformation laws for the vector and metric but notes the second-derivative, inhomogeneous term in the connection transformation. It therefore does not call the Christoffel symbols a tensor.
A secure response retains the boundary term, names the varied field and boundary data being fixed, and explains why those conditions make the bulk Euler–Lagrange equation admissible.

Mark each item secure, uncertain, or not yet. Any not-yet result calls for its linked review before the dependent module; with no not-yet result, review only the uncertain item; all secure means begin Module 1. The mechanics and special relativity diagnostic is Step 1, followed by the tensor bridge as Step 2. If items 1 or 2 expose a gap, use the special-relativity or differential-equation review linked above; item 3 receives its focused review in the tensor bridge. The fields bridge is the just-in-time core Step 31 immediately before Module 3, not an optional detour. A correct formula without its invariant or boundary explanation is uncertain. You may still enter directly and use Assignment 1 as stronger evidence.

Module 1 — Lorentzian geometry

Study in this order, beginning with the mechanics diagnostic and then the tensor bridge before entering the geometric leaves:

Extract the definition, transformation law, and geometric meaning of every object before calculating components. The module checkpoint is a two-chart calculation in which the scalar invariant agrees although the connection coefficients do not.

Assignment 1. Let $V$ and $W$ be parallel transported along a curve with tangent $u$ . Prove that their inner product is constant for a metric-compatible connection. Then explain why the same statement would fail if $\nabla_\rho g_{\mu\nu}\ne0$ .

Conceptual hint

Differentiate the scalar $g(V,W)$ along the curve. Three quantities can change: the metric itself and the two transported vectors.

Setup hint

Use the covariant Leibniz rule to expand $\nabla_u[g(V,W)]$ into one nonmetricity term and one term for each vector.

Next-step hint

Apply metric compatibility and parallel transport to the three terms separately. Then repeat the last step without setting $\nabla_u g$ to zero.

Solution

Along the curve,

\nabla_u[g(V,W)] =(\nabla_u g)(V,W)+g(\nabla_uV,W)+g(V,\nabla_uW).

Metric compatibility sets the first term to zero and parallel transport sets the other two to zero, so $g(V,W)$ is constant. With nonmetricity, the first term remains even for parallel $V$ and $W$ ; lengths and angles can therefore change under transport. The proof uses all three hypotheses and does not depend on a special coordinate system.

Module 2 — Observers, free fall, and tidal gravity

Continue through:

The central distinction is local versus extended measurement. A freely falling observer can remove connection coefficients along an appropriately chosen worldline, but cannot remove invariant tidal curvature over a finite region. When checking curvature symbolically, retain the metric, coordinate domain, curvature convention, symbolic environment, and an independent invariant check.

Checkpoint. Give an operational protocol, using a tetrad and a small cloud of freely falling probes, that distinguishes acceleration of the reference worldline from geodesic deviation. State what finite-size approximation is being made.

Response criteria

A secure response builds an orthonormal tetrad along the reference worldline and distinguishes two measurements. An accelerometer on that worldline measures its proper acceleration and therefore detects departure from geodesic motion. Relative accelerations among neighboring freely falling probes measure geodesic deviation, $D^2\xi^{\hat i}/d\tau^2=-R^{\hat i}{}_{\hat0\hat j\hat0}\xi^{\hat j}$ , and therefore tidal curvature. The cloud must be small compared with the curvature-variation scale, observed for a short enough interval that a linear separation field remains valid, and light enough that its backreaction can be neglected.

Module 3 — Einstein dynamics and the action

Study the fields bridge just in time, then continue through the dynamics and action leaves:

Connect geometry to dynamics in two directions: the contracted Bianchi identity constrains admissible field equations, while diffeomorphism-invariant matter actions organize stress-energy and conservation. Do not discard a total derivative until the region, boundary, and fixed data have been named.

Assignment 2. Starting from $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}$ , derive covariant stress-energy conservation. Explain what changes if $\Lambda$ is replaced by a prescribed spacetime-dependent scalar $\Lambda(x)$ .

Conceptual hint

Take the covariant divergence of the entire field equation; the geometric left-hand side already contains the relevant identity.

Setup hint

Write $\nabla^\mu G_{\mu\nu}=0$ and $\nabla^\mu g_{\mu\nu}=0$ , but retain $\nabla_\nu\Lambda$ until deciding whether $\Lambda$ is constant.

Next-step hint

Isolate $8\pi G\nabla^\mu T_{\mu\nu}$ and compare the constant and prescribed variable cases. Ask what additional dynamical sector would be needed to make the latter a consistent conservation law.

Solution

Take $\nabla^\mu$ of both sides. The contracted Bianchi identity gives $\nabla^\mu G_{\mu\nu}=0$ , and metric compatibility gives $\nabla^\mu g_{\mu\nu}=0$ . For constant $\Lambda$ , $\nabla^\mu T_{\mu\nu}=0$ . If $\Lambda$ is prescribed as a variable scalar, then

8\pi G\nabla^\mu T_{\mu\nu}=\partial_\nu\Lambda.

Thus ordinary matter stress-energy is not separately conserved unless the varying sector contributes its own dynamics and stress-energy. Merely inserting $\Lambda(x)$ breaks the consistency relation rather than defining a complete theory.

Module 4 — Schwarzschild, Kerr, and coordinate control

Study:

For each metric, separate coordinate singularities, curvature singularities, horizons, stationary-limit surfaces, and asymptotic charges. Across overlapping coordinate charts, verify that at least one scalar invariant and one causal conclusion agree across overlapping charts.

Checkpoint. For Schwarzschild, compare the behavior of the static time coordinate and the Kretschmann scalar at $r=2GM$ and $r=0$ . For Kerr, state the condition for real horizon radii and distinguish the horizon generator from the stationary Killing field.

Response criteria

A secure response notes that Schwarzschild static time degenerates at $r=2GM$ while the Kretschmann scalar $48G^2M^2/r^6$ is finite there and diverges at $r=0$ ; a regular extension is still needed to establish horizon crossing. For Kerr it gives $r_\pm=GM\pm\sqrt{G^2M^2-a^2}$ and the black-hole condition $\lvert a\rvert\le GM$ . It distinguishes the future-horizon generator $\chi=\partial_t+\Omega_H\partial_\phi$ from the stationary Killing field $\partial_t$ , which becomes spacelike inside the ergoregion.

Module 5 — Global causality and black-hole definition

Study:

The assessed object is not a memorized diagram. Reconstruct what conformal rescaling preserves, identify the asymptotic structure being assumed, and state whether a horizon definition is global, quasi-local, observer-dependent, or foliation-dependent.

Assignment 3. Explain why an event horizon cannot in general be located from data on a single finite-time slice, while a marginally trapped surface can be sought on such a slice. Give one reason the latter need not coincide with the former.

Conceptual hint

Contrast a definition using the whole future causal development with one using the null expansion of a two-surface in a chosen slice.

Setup hint

Write the event horizon as a boundary involving the causal past of future infinity. Define a marginally outer trapped surface by the vanishing outward null expansion on one hypersurface.

Next-step hint

Now vary the slicing or consider a dynamical spacetime. Identify which definition can move under that change and which still depends on the entire future.

Solution

An event horizon is the boundary of the region that cannot communicate with the relevant future infinity. Determining that causal past requires the spacetime’s future development, so finite-time local data do not suffice. A marginally trapped surface is defined through null expansions on a chosen slice and can therefore be located quasi-locally. Its position can depend on the slicing, and in dynamical spacetimes trapping horizons can lie away from the globally defined event horizon. The two notions answer different questions; neither is a generic local curvature threshold.

Module 6 — Charges and black-hole mechanics

Study:

Track which symmetry, boundary, normalization, stationarity, and energy condition each charge or law uses. Compare scaling relations with variational first laws rather than treating them as the same derivation.

Checkpoint. Starting from a stated first law, identify the conjugate pairs and test dimensions. Then give a physical-process interpretation and name the assumptions needed before an area increase follows.

Response criteria

A secure response first declares the theory, asymptotics, generator normalization, and neighboring stationary family, then writes the applicable law, for example $\delta M=\kappa\delta A/(8\pi G)+\Omega_H\delta J+\Phi_H\delta Q$ in Einstein–Maxwell theory. It identifies the conjugate pairs $(A,\kappa/8\pi G)$ , $(J,\Omega_H)$ , and $(Q,\Phi_H)$ and checks that every term has energy dimension. The physical-process interpretation additionally assumes a small perturbation of a stationary hole, controlled flux through a regular future horizon, and settling to a neighboring stationary state. An area-increase conclusion also needs the relevant field equations, a suitable energy condition, and the global regularity or censorship assumptions that exclude future horizon endpoints.

Module 7 — Weak fields, empirical tests, and radiation

Study:

Separate a formal perturbation, a gauge choice, an observable, and a data model. A test of GR compares a full theory–source–propagation–instrument chain against alternatives; it is not the observation of a single familiar effect.

Assignment 4. In the slow, weak, stationary limit, write the metric component that contains the Newtonian potential in the $(-,+,+,+)$ convention and state the field equation it must recover. Then explain why the same scalar potential is insufficient to describe propagating gravitational radiation.

Conceptual hint

Separate the quasistatic scalar sector that reproduces Newtonian gravity from the time-dependent tensor sector that transports radiation.

Setup hint

Begin with $g_{00}=-(1+2\Phi)$ and the Poisson equation. List the slow-motion and stationarity assumptions used to obtain them.

Next-step hint

Ask which transverse, propagating degrees of freedom those assumptions remove and which observables—polarization and causal energy transport—therefore cannot be encoded by $\Phi$ alone.

Solution

To leading order, $g_{00}=-(1+2\Phi)$ with $|\Phi|\ll1$ , and Einstein’s equation recovers $\nabla^2\Phi=4\pi G\rho$ for nonrelativistic matter under the usual quasistatic assumptions. Radiation is carried by time-dependent, transverse tensor degrees of freedom in the wave zone; a stationary scalar potential neither captures their polarization nor their causal energy transport. The limit identifies a sector of the theory, not its full propagating content.

Module 8 — Matter and cosmological dynamics

Finish with:

This module tests transfer: the same field equation becomes a stellar boundary-value problem or a homogeneous cosmological evolution problem after different symmetry and matter assumptions. For both, identify the material closure relation and verify a conservation equation independently.

Cross-module transfer task

Choose either a static spherical star or a homogeneous expanding universe. Starting from the same Einstein equation:

state the metric and matter ansatz;
identify the independent reduced equations and conservation law;
specify initial, central, or boundary data;
recover one controlled limit, such as Newtonian hydrostatic balance or ordinary fluid redshifting; and
explain which step fails if the chosen symmetry is relaxed.

Submit a derivation plus a one-page comparison to the other system. A complete answer shows how assumptions—not the field equation alone—create the reduced physics.

Capstone — one phenomenon in four representations

Select one of gravitational redshift, orbital precession, black-hole horizon mechanics, compact-star equilibrium, gravitational-wave tidal response, or cosmic expansion. Write a concise report containing:

the invariant geometric statement;
a coordinate calculation in a declared chart;
an observer or measurement protocol;
the governing field-equation or action argument;
one weak-field, exact-solution, radiation, matter, or cosmological connection; and
two independent checks, one mathematical and one physical.

The capstone is successful when all formulas use consistent conventions, the domain of validity is explicit, coordinate artifacts are separated from invariants, every conclusion follows from shown evidence, and another reader can reproduce the decisive calculation. If one representation disagrees, locate and correct the mismatch rather than averaging the answers.

Continue

Use the flexible Graduate General Relativity Self-Study Path to revisit a weak outcome. Continue to Advanced General Relativity and Black-Hole Physics for formulations and stability, or choose a specialist learning path in black holes, waves, matter, cosmology, computation, or quantum gravity.