Pseudo Riemannian ManifoldEdit
Pseudo Riemannian manifolds sit at the crossroads of differential geometry and mathematical physics. They generalize the familiar positive-definite metrics of Riemannian geometry by allowing indefinite signatures, textures that encode both spatial and temporal directions in a single geometric object. The resulting framework is the natural habitat for theories that require a notion of causality and lightlike separation, most famously general relativity, but it also serves as a rich arena for pure mathematics. At each point of a smooth manifold, a pseudo Riemannian metric assigns a symmetric, non-degenerate bilinear form on the tangent space, varying smoothly from point to point. This data gives rise to most of the standard geometric constructions—connections, curvature, geodesics—while also shaping the causal and global features of the space.
In a nutshell, a pseudo Riemannian manifold is a pair (M, g) where M is a smooth manifold and g is a smooth section of the bundle of symmetric bilinear forms that is non-degenerate at every point. The requirement of non-degeneracy implies that no nonzero tangent vector can be everywhere orthogonal to all vectors, which in turn yields a well-defined notion of length and angle in a way that depends on the point. The key difference from a Riemannian manifold is that g need not be positive definite; it may have a prescribed number of negative signs—its signature (p, q) with p + q = dim M—so that there are directions in which the metric evaluates to negative values, zero-value directions in the sense of null vectors, and others where it is positive. The case where q = 1 and p = n − 1 is the Lorentzian setting that underpins much of modern physics.
Definition and basic concepts - Manifold and metric: Let M be a smooth n-manifold. A pseudo Riemannian metric g assigns to each p ∈ M a symmetric bilinear form g_p on the tangent space T_pM, varying smoothly in p. The collection {g_p} is non-degenerate: if g_p(v, w) = 0 for all w in T_pM, then v = 0. - Signature and index: The signature of g_p is a pair (p, q) with p + q = n, indicating p positive directions and q negative directions in an appropriate basis. This index p or q is constant on connected components of M for a smooth metric. - Exemplar cases: If q = 0, one recovers the familiar Riemannian case with a positive-definite metric. If q = 1, one obtains a Lorentzian metric, the mathematical backbone of spacetime models in General relativity and related theories. - Local vs global: Locally, pseudo Riemannian geometry agrees with linear algebra on tangent spaces, but global properties—such as causality, completeness, and the existence of global coordinates—depend on the global arrangement of the metric.
Signature and examples - Lorentzian manifolds: The signature (n−1, 1) (or equivalently (1, n−1)) is the standard Lorentzian form. The classic flat example is Minkowski space, whose metric encodes the light cone structure that divides vectors into time-like, null, and space-like classes. - de Sitter and anti-de Sitter spaces: These are constant-curvature Lorentzian manifolds with high symmetry, frequently used in cosmology and theoretical physics to model expanding universes and conformal boundaries. - Other semi-Riemannian cases: Many geometries of interest have signatures with more than one negative direction, such as (p, q) with q > 1, relevant in certain mathematical models and in higher-dimensional theories. - Relationship to Riemannian geometry: A pseudo Riemannian metric reduces to a Riemannian metric when its signature is purely positive. The broader umbrella, sometimes called semi-Riemannian geometry, covers all signatures.
Levi-Civita connection and curvature - Levi-Civita connection: Each pseudo Riemannian manifold (M, g) carries a unique torsion-free connection ∇ that is compatible with the metric, i.e., ∇g = 0. This Levi-Civita connection defines parallel transport and covariant differentiation in a way that respects the metric structure. - Curvature: The Riemann curvature tensor R measures the failure of second covariant derivatives to commute and encodes how the space bends. From R one forms the Ricci tensor and the scalar curvature, which summarize curvature content in shorter expressions. In the pseudo Riemannian setting, the curvature objects behave similarly to the Riemannian case, but their interpretation interacts with the indefinite metric in distinctive ways. - Coordinate-free viewpoint: Much of pseudo Riemannian geometry emphasizes invariant, coordinate-free descriptions. This attitude is particularly fruitful in physics, where equations must be valid independently of coordinate choices, and in the modern approaches to global analysis.
Geodesics and causal structure - Geodesics: Curves whose tangent vectors are parallel transported along themselves are the natural generalization of “straight lines” to curved spaces. In the Lorentzian context, geodesics describe the free-fall trajectories of particles and light rays in spacetime. - Causal classification: In Lorentzian geometry, tangent vectors can be time-like, null (light-like), or space-like, depending on whether g_p(v, v) is negative, zero, or positive. This division yields a light cone at each point, organizing the causal structure of the manifold. - Global properties: Concepts such as global hyperbolicity, the existence of Cauchy surfaces, and various forms of causal completeness place strong constraints on the geometry and have consequences for predictability and determinism in physical theories.
Applications and connections - In physics: The most prominent application is in General relativity, where the spacetime manifold is modeled as a four-dimensional Lorentzian manifold with the metric solving the Einstein field equations. Solutions like the Schwarzschild, Friedmann–Lemaître–Robertson–Walker, or Kerr metrics illustrate how geometry encodes gravitational fields, black holes, cosmology, and gravitational waves. The metric also determines the propagation of light and matter, linking geometry to observable phenomena. - In mathematics: Pseudo Riemannian geometry supports a wide range of mathematical topics, from the study of geodesic flows and spectral theory on noncompact manifolds to the analysis of global topological properties constrained by curvature. While certain powerful theorems from Riemannian geometry (for example, Hodge theory) have straightforward formulations only in the positive-definite setting, many ideas generalize or adapt to the indefinite case, often with additional technical hurdles. - Related structures: The formalism interacts with other geometric frameworks, including Lorentzian manifold theory, conformal geometry, and various approaches to quantum gravity where the nature of the metric and its signature is central to physical interpretation.
Controversies and debates - Physical interpretation and alternatives: In physics, the use of a Lorentzian metric to represent spacetime is standard, but there are debates about deeper metaphysical implications of time and causality. Some contemporary approaches in quantum gravity explore frameworks where spacetime is emergent or where the metric is not fundamental, prompting mathematical inquiries into more general or alternative geometric structures. - Wick rotation and Euclidean methods: A common technique in quantum field theory is to perform a Wick rotation to pass from a Lorentzian metric to a Riemannian (positive-definite) metric for certain calculations. While this can simplify analysis and lead to powerful mathematical tools, its physical justification and the interpretation of results in the original Lorentzian setting remain topics of discussion among physicists and mathematicians. - Analytical challenges in the indefinite case: Unlike the positive-definite setting, many analytical results (such as certain elliptic estimates, Hodge theory, and spectral theory) become more delicate or fail outright in the indefinite metric context. This has led to ongoing research into semi-Riemannian analysis, with debates about the most effective methods and the proper scope of classical theorems when generalized beyond the Riemannian case. - Causality and determinism: The causal structure inherent to Lorentzian geometry imposes strong restrictions on global geometry. Some discussions emphasize how causality interacts with topology and the possibility of singularities, while others explore how alternative models of spacetime might relax or reinterpret causal notions. These debates often touch on philosophical questions about the nature of time and the limits of mathematical models in depicting physical reality.
See also - Manifold - Riemannian manifold - Pseudo-Riemannian metric - Lorentzian manifold - Minkowski space - de Sitter space - Anti-de Sitter space - Levi-Civita connection - Riemann curvature tensor - Geodesic - Causal structure - Global hyperbolicity - Cauchy surface - General relativity - Wick rotation