| Lorentzian manifold | |
|---|---|
| Name | Lorentzian manifold |
| Field | Differential geometry |
| Introduced | 20th century |
| Applications | General relativity, Quantum field theory, Quantum gravity |
Lorentzian manifold
A Lorentzian manifold is a smooth manifold equipped with a nondegenerate, symmetric metric tensor of signature (-,+,+,+) (or the opposite sign convention) that distinguishes timelike, spacelike, and null directions. It provides the geometric arena for General relativity and underpins formulations of Quantum field theory on curved spacetime and approaches to Quantum gravity. In quantum contexts, Lorentzian manifolds determine causal structure, propagation of quantum fields, and the semiclassical limits of candidate theories.
A Lorentzian manifold (M,g) is a pair where M is a smooth n-dimensional manifold and g is a smooth symmetric 2-tensor of signature (1,n-1) or (n-1,1). The metric g assigns lengths and causal character to tangent vectors and induces the light cone at each point. Fundamental structures include the Levi-Civita connection ∇ determined by g, the Riemann curvature tensor R, the Ricci curvature Ric and the scalar curvature R_g. These tensors satisfy differential identities such as the Bianchi identities and enter the Einstein field equations when coupled to stress–energy tensors such as those of Quantum field theory in semiclassical approximations. The manifold may be oriented and time-oriented; the existence of time orientation is a topological constraint often assumed in physical models.
Causality conditions classify Lorentzian manifolds by their admissible causal behavior. Standard causal properties include chronological, causal, strongly causal, and globally hyperbolic conditions. Global hyperbolicity ensures well-posedness of hyperbolic partial differential equations like the Klein–Gordon equation and the Dirac equation for quantum fields, and underlies existence of Cauchy surfaces. Time orientation selects a continuous choice of future-directed timelike vectors, enabling definition of retarded and advanced Green functions used in quantum propagators. Violations of causality, such as closed timelike curves, appear in solutions like the Gödel metric and raise deep issues in quantum causal structure and unitarity studied by researchers at institutions like Princeton University and CERN.
Key examples serve as models and testbeds. Minkowski space is the flat Lorentzian manifold underlying special relativity and free quantum field theory; it supports the Poincaré group and canonical quantization methods developed by figures such as Paul Dirac and Richard Feynman. The Schwarzschild metric describes the exterior of a nonrotating black hole and is central to studies of Hawking radiation, pioneered by Stephen Hawking and explored using semiclassical methods at Cambridge University and Harvard University. Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes model homogeneous cosmology and are used in studies of cosmological perturbations and inflationary quantum fluctuations by researchers affiliated with programs like Institute for Advanced Study. Each metric provides distinct causal horizons, Killing symmetries, and curvature properties relevant to quantum field behavior.
Quantum field theory on Lorentzian manifolds generalizes flat-space QFT by replacing global Poincaré invariance with local geometric structure. The algebraic approach (algebraic quantum field theory) defines local operator algebras and states satisfying the Hadamard condition to yield renormalizable stress–energy expectation values. Construction of Green functions, Feynman propagators, and vacuum states depends on the manifold's causal structure; important works include those by Rudolf Haag and Stephen Fulling. Techniques from spectral geometry, microlocal analysis (e.g., Hörmander), and the DeWitt–Schwinger expansion are used to compute anomalies and effective actions. Seminal problems studied include particle creation in expanding FLRW universes and the Unruh effect for accelerated observers.
Lorentzian manifolds play differing roles across quantum gravity programs. In canonical quantization and the ADM formalism developed by Richard Arnowitt, Stanley Deser, and Charles Misner, the Lorentzian metric yields Hamiltonian constraints and a phase space for quantization. In path integral approaches inspired by Richard Feynman, one integrates over Lorentzian metrics or performs a Wick rotation to Riemannian metrics; debates about the validity of Euclideanization appear in work by Stephen Hawking and G. W. Gibbons. In loop quantum gravity the causal structure emerges from discrete spin networks, while in string theory Lorentzian backgrounds such as anti-de Sitter space are central to the AdS/CFT correspondence studied at Institute for Theoretical Physics groups worldwide. Causal set theory takes the causal order of Lorentzian manifolds as fundamental, exemplified by research from the Perimeter Institute.
Geodesics of Lorentzian manifolds describe worldlines of free particles and light rays; completeness of timelike geodesics relates to singularity theorems by Roger Penrose and Stephen Hawking, which use energy conditions and global hyperbolicity to prove existence of singularities. Stability analyses examine perturbations of metrics and the nonlinear stability of solutions like Minkowski space (proved by Demetrios Christodoulou and Sergiu Klainerman). In quantum contexts, backreaction of quantum stress–energy can affect singularity formation and cosmic censorship, topics pursued at Caltech and in semiclassical gravity literature.
Isometries of a Lorentzian manifold are diffeomorphisms preserving g; their infinitesimal generators are Killing vector fields, which yield conserved quantities for particle trajectories and stress–energy flux via Noether's theorem. Symmetry groups such as the Poincaré group of Minkowski space or the isometry group of anti-de Sitter space determine selection rules and classification of particle states in quantum theories. In curved backgrounds lacking global symmetries, one relies on local conservation laws and approximately conserved quantities, central to scattering theory on curved spacetimes developed by mathematical physicists in institutions like University of Cambridge and Princeton University.
Category:Differential geometry Category:General relativity Category:Mathematical physics