Spacetime Geometry

The same spacetime can be described with many coordinate systems. Changing those labels must not change whether two vectors are orthogonal, whether a curve is a geodesic, or whether the geometry is curved. Spacetime geometry provides the objects and operations that make those statements independent of coordinates. This volume builds them from first principles, beginning with smooth spaces and vectors and continuing through curvature, geodesics, frames, hypersurfaces, and global obstructions.

Throughout, we connect three ways of understanding an object: its geometric definition, the coordinate formula used to calculate it, and its components in a chosen frame. You will learn both how to do a calculation and how to recognize which parts of the answer survive a change of description.

The goal is geometric control, not yet a theory of gravitation. Curvature and geodesics are defined here before the Einstein equation assigns physical sources or interprets freely falling bodies. Likewise, causal cones are studied pointwise before global causal relations, horizons, and singularity theorems are introduced. That separation makes it possible to see exactly which conclusions follow from geometry alone.

The structures that make spacetime geometric

A smooth manifold supplies points and compatible coordinate systems without selecting a preferred chart. Tangent and cotangent spaces then support vector fields, covectors, tensors, and differential forms. These constructions are already rich enough to express flows, constraints, integration, orientation, and global flux. They require neither a metric nor a connection. Keeping that fact visible prevents a common mistake: vectors and covectors are already different kinds of objects before a metric is chosen. A metric lets us convert between them by lowering or raising an index; a connection supplies covariant differentiation. Neither operation is part of the smooth structure alone.

A pseudo-Riemannian metric adds a nondegenerate bilinear form to each tangent space. In Lorentzian signature it separates nonzero vectors into timelike, null, and spacelike classes and supplies the pointwise light cone. An affine connection adds differentiation of tensor fields and parallel transport; metric compatibility and vanishing torsion select the Levi–Civita connection. Curvature then measures the failure of two covariant derivatives, or two nearby transports, to commute. Wendl develops the needed sequence in Wendl 2021–2022, §§ 2–13, pp. 11–118, PDF for manifolds, tensors, forms, and integration; Wendl 2021–2022, § 18.5, pp. 154–159, PDF for pseudo-Riemannian metrics and signature; Wendl 2021–2022, §§ 19–21, pp. 161–183, PDF for connections, transport, and torsion; Wendl 2021–2022, § 22.2, Theorem 22.4, pp. 186–187, PDF for Levi–Civita uniqueness; and Wendl 2021–2022, § 26.3, Definition 26.4 and Exercise 26.6, pp. 225–226, PDF for curvature.

The site uses metric signature $(-,+,+,+)$ and the curvature convention

$$R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}.$$

For the Levi–Civita connection—or, more generally, any torsion-free connection—this convention gives $[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma$. A general affine connection instead gives

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

Here the second covariant derivative differentiates the derivative index as well as the vector index. This is different from composing only the two directional operators $\nabla_{\partial_\mu}$ and $\nabla_{\partial_\nu}$, whose commutator on a coordinate basis is the curvature operator even when torsion is present; the distinction is explained on the affine-connection page. The tensor-second-derivative formula is derived in Carroll 1997, § 3, eqs. (3.65)–(3.68), pp. 75–76, PDF. The curvature term on a covector carries the opposite representation sign. A positively oriented spacetime chart has $\epsilon_{0123}=+\sqrt{-g}$, and boundaries use outward-normal-first orientation. Every Hodge-dual formula states its dimension, form degree, signature, and orientation; every hypersurface calculation declares the sign of its extrinsic curvature. These are not decorative choices: changing any one of them changes component formulas while leaving the underlying geometry unchanged.

Preparation and a quick diagnostic

The volume assumes fluency with multivariable calculus and finite-dimensional linear algebra: derivatives of maps, the chain rule, dual vector spaces, bilinear forms, determinants, eigenvalues, and ordinary differential equations. The calculus and linear algebra preparation page is the bounded repair path. Later chapter overviews identify additional required background and point to the exact page that supplies it.

A useful readiness test is to distinguish an invariant object from its components. Let $V=V^\mu\partial_\mu$ and change coordinates from $x^\mu$ to $y^a$. If the basis transforms as $\partial/\partial x^\mu=(\partial y^a/\partial x^\mu)\partial/\partial y^a$, what must happen to the components?

Solution

The vector itself is unchanged, so

$$V=V^\mu\frac{\partial y^a}{\partial x^\mu}\frac{\partial}{\partial y^a} =V'^a\frac{\partial}{\partial y^a}.$$

Therefore $V'^a=(\partial y^a/\partial x^\mu)V^\mu$. The components and the basis both change, but their contraction does not. If this reasoning is unfamiliar, complete the preparation page before beginning the first chapter.

From manifolds to curvature

A running question: what changes with coordinates?

Even a flat plane illustrates the central problem. On a polar-coordinate patch with $r\gt0$, write $x=r\cos\phi$ and $y=r\sin\phi$. A radial vector is

$$\partial_r=\cos\phi\,\partial_x+\sin\phi\,\partial_y.$$

The two sides describe the same vector. The coefficients differ because the bases differ. A metric adds a separate statement about lengths:

$$dx^2+dy^2=dr^2+r^2d\phi^2.$$

The factor $r^2$ says that the coordinate vector $\partial_\phi$ has length $r$; it does not say that the plane is curved. Differentiating a vector field now requires us to account for the changing polar basis, which is the job of the connection. Curvature asks a further question: does parallel transport around a sufficiently small loop return the vector unchanged? It does in this plane, despite its nonzero polar connection coefficients.

The tensor chapter explains the change of components, metrics explain lengths, and connections explain differentiation. The round sphere later supplies a contrasting example with nonzero curvature. Keep this distinction between changing coordinates and changing geometry in mind as the definitions become more abstract.

The opening chapters

The opening six chapters form the geometric core. Smooth Manifolds, Maps, and Tensor Fields constructs charts, tangent vectors, flows, tensors, and submanifolds. It makes the transformation law a consequence of an invariant definition, rather than a rule to memorize. Differential Forms, Integration, and Orientation introduces the exterior derivative, Stokes’ theorem, densities, and de Rham cohomology, thereby connecting local differential identities to global flux.

Metrics, Causal Cones, and Volume Geometry adds signature, musical maps, Lorentzian cones, Hodge duality, divergence, and conformal transformations. Connections, Covariant Differentiation, and Parallel Transport explains why ordinary derivatives of components are not tensorial and how a connection repairs that defect. It also distinguishes arbitrary affine connections from the unique torsion-free, metric-compatible Levi–Civita connection.

Curvature, Holonomy, and Geometric Invariants develops the Riemann tensor, its symmetries and contractions, the Bianchi identities, sectional curvature, holonomy, and the limits of scalar curvature invariants. Geodesics, Exponential Maps, and Jacobi Fields then uses the connection and curvature to study affine geodesics, normal neighborhoods, deviation, conjugate points, completeness, cut loci, and Hamiltonian geodesic flow. Grant treats the exponential map, normal neighborhoods, Jacobi fields, conjugate points, completeness, and the cut locus in Grant 2008, corrected 2009, §§ 4–8, pp. 9–27, PDF, then separates the corresponding Lorentzian causal-extremum statements in §§ 11–13, pp. 33–41, PDF. The cotangent-bundle Hamiltonian description and symmetry-generated constants are developed in Frolov, Krtouš, and Kubizňák 2017, §§ 2.1–2.2, pp. 14–19, PDF. Riemannian minimizing theorems and Lorentzian causal statements are kept separate; signature changes the relevant extrema and global hypotheses.

Symmetry, frames, and families of curves

The next three chapters turn geometric structure into reusable calculational tools. Symmetries, Lie Derivatives, and Killing Fields relates flows to infinitesimal invariance, derives geodesic constants from Killing fields and tensors, and distinguishes local generators from complete group actions. The field-to-constant calculation is given for Killing fields in Carroll 1997, § 5, pp. 135–141, PDF and for hidden-symmetry tensors in Frolov, Krtouš, and Kubizňák 2017, §§ 2.1–2.2, pp. 14–19, PDF. Frames, Tetrads, Spinors, and Clifford Geometry moves between coordinate bases and orthonormal frames, develops Cartan’s structure equations, and separates local Lorentz freedom from the global topological requirement for a spin structure. Carroll gives the local tetrad, spin-connection, and Cartan machinery in Carroll 1997, § 3, pp. 88–94, PDF; Geroch makes the global double-cover problem explicit in Geroch 1968, § I, pp. 1739–1742.

Congruences and Kinematical Decomposition studies entire families of timelike or null curves. Expansion, shear, vorticity, acceleration, screen geometry, the Raychaudhuri identity, and the Sachs equations describe how a small cross-section changes. These equations are geometric identities; the null Jacobi equation, Sachs basis, optical scalars, and caustics are developed in Perlick 2004, updated 2010, § 2.3, pp. 13–17, PDF. Focusing adds congruence and curvature hypotheses, while replacing curvature convergence by an energy condition additionally uses a gravitational field equation; Kar and SenGupta 2007, §§ II.C–II.D, pp. 10–13, PDF makes those steps explicit. The global conclusion of a singularity theorem belongs later in the causal and gravitational development.

Hypersurfaces, phase space, and global obstructions

Hypersurfaces, Extrinsic Curvature, and Embeddings distinguishes intrinsic geometry from bending in an ambient spacetime. It derives projectors, the Weingarten map, Gauss–Codazzi relations, and the lapse–shift description of a foliation, while treating null hypersurfaces as a genuinely degenerate case rather than a formal unit-normal limit. These distinctions and the metric reconstruction are set out in Gourgoulhon 2007, §§ 2.3–2.5, pp. 19–37; §§ 3.2–3.3, pp. 39–46; § 4.2, pp. 54–59, PDF. The Einstein constraint and evolution equations are reserved for Formulations of General Relativity.

Variational and Symplectic Geometry develops the geometry of solution spaces: field variations, Euler–Lagrange and boundary forms, presymplectic currents, Poisson brackets, moment maps, constraints, and reduction. The finite-dimensional Hamiltonian, Poisson, moment-map, and reduction models—and the compact-group, free-action, and regular-value hypotheses of the standard reduction theorem—are developed in Cannas da Silva 2006, Lectures 18–20, pp. 105–126, and Lectures 22–24, pp. 133–154, PDF. The basic field-theory decomposition $\delta L=E_i\delta\phi^i+d\Theta$ follows Iyer and Wald 1994, § III, eqs. (20)–(23), pp. 6–7, PDF, while their eqs. (40)–(43), pp. 8–9, display the relevant exact-form ambiguities. With a spatial boundary, the allowed variations and boundary terms must also make the variational principle well posed; see Harlow and Wu 2020, § 2.2, eqs. (2.15)–(2.21), pp. 11–13, PDF. A transformation is not automatically a gauge degeneracy merely because it is a gauge transformation in the bulk: Harlow and Wu 2020, § 3.3, eqs. (3.24)–(3.25), pp. 26–27, PDF gives a boundary-sensitive Maxwell example.

Topology, Bundles, and Global Geometric Tools closes the volume with homotopy, homology, transition functions, principal and associated bundles, characteristic classes, orientability, spin obstructions, and Morse theory. The point is not a compressed algebraic-topology course. It is to recognize when a locally valid choice—potential, frame, section, orientation, or spin lift—cannot be extended globally. The bounded foundations used here come from Hatcher 2002, Chapter 0, pp. 1–20; § 2.1, pp. 102–133; and § 3.1, pp. 190–205; the bundle, connection, and characteristic-class sequence is supported by Gaiter 2024, §§ 2.2–5.3, pp. 4–36, PDF. Geroch’s spin construction is the separate frame-bundle lifting problem cited above, while Milnor 1963, §§ 1–4, pp. 1–27, PDF supplies the Morse lemma and the topology change across a critical level.

Recurring examples and reading paths

On a first reading, aim to answer three questions on each page: what object is being defined, how its components are calculated, and which assumptions make the result true. Work through a concrete example before trying to memorize a general identity. The solutions are hidden so that you can try the exercises first; a missed exercise is a useful indication of which linked background page to revisit.

Three examples tie the volume together. The round two-sphere begins as a smooth atlas and regular level set, supports oriented integration, acquires a metric and Levi–Civita connection, exhibits curvature and nontrivial holonomy, and supplies explicit geodesics, Jacobi fields, conjugate points, and a cut locus. Minkowski spacetime appears in Cartesian, polar, null, accelerated, and foliated descriptions: connection coefficients may be nonzero even when curvature vanishes, and a null hypersurface can have a degenerate induced metric even though the spacetime metric is nondegenerate. The punctured plane, circle bundles, and tangent bundles over spheres connect closed but nonexact forms and local trivializations to flux, holonomy, and characteristic classes. Spin structures pose a separate lifting problem for the oriented orthonormal frame bundle; they do not follow merely from those examples.

For a first course in relativity, follow Smooth Manifolds, Maps, and Tensor Fields through Geodesics, Exponential Maps, and Jacobi Fields, then study Symmetries, Lie Derivatives, and Killing Fields, Congruences and Kinematical Decomposition, and Hypersurfaces, Extrinsic Curvature, and Embeddings before continuing to General Relativity. Readers interested in tetrads and spinors should add Frames, Tetrads, Spinors, and Clifford Geometry and Topology, Bundles, and Global Geometric Tools. Readers preparing for covariant phase space should emphasize Differential Forms, Integration, and Orientation, Connections, Covariant Differentiation, and Parallel Transport, Curvature, Holonomy, and Geometric Invariants, the hypersurface chapter, and Variational and Symplectic Geometry. For mathematical relativity, complete the geodesic and topology chapters before proceeding to Causality and Global Structure and Mathematical Relativity.

Computational readers should not skip the invariant definitions. Coordinate formulas for Christoffel symbols, curvature, lapse, and shift are reliable only when their tensorial content and sign conventions are understood. This volume develops that foundation; discretization, formulation choice, convergence, and production solvers belong in Computational Relativity.

Boundaries of the geometric account

Several distinctions are essential. A chart singularity is not a curvature singularity. Vanishing scalar polynomial invariants do not classify every Lorentzian metric: vacuum plane waves supply nontrivial examples with vanishing scalar polynomial invariants, as discussed in Coley, Hervik, and Pelavas 2009, § 1, pp. 1–2, and § 8, p. 26, PDF. A caustic marks failure of a congruence description and need not be a singularity of the spacetime. A local tetrad does not guarantee a global tetrad or spin structure. A closed form need not be exact, and a flat connection can have nontrivial global holonomy. Finally, a presymplectic degeneracy becomes a gauge direction only after the allowed variations and boundary conditions are specified.

The Special Relativity and Physical Foundations volume gives the operational meaning of clocks, observers, particle motion, fields, and stress-energy. General Relativity supplies gravitational dynamics and physical interpretation. Exact Solutions of Einstein’s Equations applies the invariant machinery to concrete metrics, while Causality and Global Structure develops chronological relations, domains of dependence, horizons, and global causal theorems. Together, these subjects build on this volume’s account of what geometric statements mean, what they assume, how to calculate them, and where they stop.

For object-first lookup, use the Black Hole Atlas. The Research Atlas holds dated frontier assessments, while Reference provides compact conventions and formulas. The computational chapters above explain how to implement and test numerical methods without replacing the invariant geometry developed here.

References

• Cannas da Silva, Ana. 2006. Lectures on Symplectic Geometry. Revised lecture notes, Lecture Notes in Mathematics 1764. Open PDF.
• Carroll, Sean M. 1997. Lecture Notes on General Relativity. arXiv:gr-qc/9712019. arXiv.
• Coley, Alan, Sigbjørn Hervik, and Nicos Pelavas. 2009. “Spacetimes Characterized by Their Scalar Curvature Invariants.” Classical and Quantum Gravity 26: 025013. DOI; arXiv.
• Frolov, Valeri P., Pavel Krtouš, and David Kubizňák. 2017. “Black Holes, Hidden Symmetries, and Complete Integrability.” Living Reviews in Relativity 20: 6. DOI; Open PDF.
• Gaiter, Jacob. 2024. Characteristic Classes, Principal Bundles, and Curvature. University of Texas at Austin lecture notes. Open PDF.
• Geroch, Robert. 1968. “Spinor Structure of Space-Times in General Relativity. I.” Journal of Mathematical Physics 9: 1739–1744. DOI.
• Gourgoulhon, Éric. 2007. 3+1 Formalism and Bases of Numerical Relativity. arXiv:gr-qc/0703035v1. arXiv.
• Grant, James D. E. 2008, corrected 2009. Global Lorentzian Geometry. University of Vienna lecture notes. Open PDF.
• Harlow, Daniel, and Jie-qiang Wu. 2020. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020: 146. DOI; arXiv.
• Hatcher, Allen. 2002. Algebraic Topology. Cambridge University Press. Official text.
• Iyer, Vivek, and Robert M. Wald. 1994. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50: 846–864. DOI.
• Kar, Sayan, and Soumitra SenGupta. 2007. “The Raychaudhuri Equations: A Brief Review.” Pramana 69: 49–76. DOI; arXiv.
• Milnor, John. 1963. Morse Theory. Annals of Mathematics Studies 51. Princeton University Press. Open PDF.
• Perlick, Volker. 2004, updated 2010. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7: 9. DOI; updated arXiv version.
• Wendl, Chris. 2021–2022. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes. Open PDF.