First-year graduateApplied MathematicsLesson available

The geometry of spacetime: special and general relativity

Physics as geometry: Minkowski spacetime and the Lorentz group, proper time and causal structure, Lorentzian manifolds, the equivalence principle, Einstein's field equations, the Schwarzschild solution, geodesics of light and matter, and cosmological models.

math.applied.relativity

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Builds on

Riemannian geometry Classical mechanics

Required prerequisites, all the way down: 46

You will learn to

Use the Minkowski metric to compute proper time, explain time dilation and the twin paradox, and describe the Lorentz group.
Formulate the equivalence principle and Einstein's field equations in the language of Lorentzian geometry.
Derive the motion of light and planets from geodesics of the Schwarzschild metric, including light bending and perihelion precession to leading order.

Definition

Minkowski spacetime is ℝ⁴ with the metric ds² = −c²dt² + dx² + dy² + dz².
A Lorentzian manifold is a smooth manifold with a metric of signature (−, +, +, +).
Einstein's field equations are G_{μν} + Λ g_{μν} = (8πG/c⁴) T_{μν}, relating the curvature of spacetime to its energy–momentum content.

Theorem

Proper time along a timelike curve is maximised, not minimised, by timelike geodesics (locally), which resolves the twin paradox.
The Schwarzschild metric is the unique spherically symmetric vacuum solution (Birkhoff's theorem).
To leading order, general relativity predicts a deflection of starlight grazing the Sun of 4GM/(c²R), twice the value from a naive Newtonian argument.

Common misconceptions

Thinking relativity says 'everything is relative'; the spacetime interval and the speed of light are the same for all inertial observers.
Believing the Schwarzschild radius is a physical singularity; the curvature there is finite, and the singularity of the metric is an artefact of the coordinates.

Optional discoveries

Geometric analysis The positive mass theorem (Schoen–Yau; Witten) is a theorem of geometric analysis with a direct physical meaning: total energy is non-negative.
Gauge theory and the geometry of physics General relativity and gauge theories are both theories of connections and curvature, on the tangent bundle and on principal bundles respectively.
← Differential geometry of curves and surfaces
← Differential forms and Stokes' theorem
← Geometric analysis

Learning pathways

Mathematical physics