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In this theory of general relativity, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy, and momentum (or stress-energy) within it. The relationship between stress-energy and the curvature of spacetime is governed by the Einstein field equations. The motion of objects being influenced solely by the geometry of spacetime (inertial motion) occurs along special paths called timelike and null geodesics of spacetime.
In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. The key difference between the two is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.
Arguably, the most important type is pseudo-Riemannian manifold is a Lorentzian manifold. Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime. Just as Riemannian manifolds may be thought of a being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.