General relativity 


In differential geometry, a pseudoRiemannian manifold,^{[1]}^{[2]} also called a semiRiemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positivedefiniteness is relaxed.
Every tangent space of a pseudoRiemannian manifold is a pseudoEuclidean vector space.
A special case used in general relativity is a fourdimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ndimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.
An ndimensional differentiable manifold is a generalisation of ndimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into ndimensional Euclidean space.
See Manifold, Differentiable manifold, Coordinate patch for more details.
Associated with each point in an dimensional differentiable manifold is a tangent space (denoted ). This is an dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point .
A metric tensor is a nondegenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by we can express this as
The map is symmetric and bilinear so if are tangent vectors at a point to the manifold then we have
for any real number .
That is nondegenerate means there are no nonzero such that for all .
Given a metric tensor g on an ndimensional real manifold, the quadratic form q(x) = g(x, x) associated with the metric tensor applied to each vector of any orthogonal basis produces n real values. By Sylvester's law of inertia, the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis. The signature (p, q, r) of the metric tensor gives these numbers, shown in the same order. A nondegenerate metric tensor has r = 0 and the signature may be denoted (p, q), where p + q = n.
A pseudoRiemannian manifold is a differentiable manifold equipped with an everywhere nondegenerate, smooth, symmetric metric tensor .
Such a metric is called a pseudoRiemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.
The signature of a pseudoRiemannian metric is (p, q), where both p and q are nonnegative. The nondegeneracy condition together with continuity implies that p and q remain unchanged throughout the manifold (assuming it is connected).
A Lorentzian manifold is an important special case of a pseudoRiemannian manifold in which the signature of the metric is (1, n−1) (equivalently, (n−1, 1); see Sign convention). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz.
After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudoRiemannian manifolds. They are important in applications of general relativity.
A principal premise of general relativity is that spacetime can be modeled as a 4dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positivedefinite metrics, an indefinite signature allows tangent vectors to be classified into timelike, null or spacelike. With a signature of (p, 1) or (1, q), the manifold is also locally (and possibly globally) timeorientable (see Causal structure).
Just as Euclidean space can be thought of as the model Riemannian manifold, Minkowski space with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudoRiemannian manifold of signature (p, q) is with the metric
Some basic theorems of Riemannian geometry can be generalized to the pseudoRiemannian case. In particular, the fundamental theorem of Riemannian geometry is true of pseudoRiemannian manifolds as well. This allows one to speak of the LeviCivita connection on a pseudoRiemannian manifold along with the associated curvature tensor. On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is not true that every smooth manifold admits a pseudoRiemannian metric of a given signature; there are certain topological obstructions. Furthermore, a submanifold does not always inherit the structure of a pseudoRiemannian manifold; for example, the metric tensor becomes zero on any lightlike curve. The Clifton–Pohl torus provides an example of a pseudoRiemannian manifold that is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds.^{[3]}