- Pseudo-Riemannian manifold
In
differential geometry , a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of aRiemannian manifold . It is one of many things named afterBernhard Riemann . The key difference between the two is that on a pseudo-Riemannian manifold themetric tensor need not bepositive-definite . Instead a weaker condition of nondegeneracy is imposed.Introduction
Manifolds
"Main articles:
Manifold ,differentiable manifolds "In
differential geometry adifferentiable manifold is a space which is locally similar to aEuclidean space . In an -dimensional Euclidean space any point can be specified by real numbers. These are called thecoordinate s of the point.An -dimensional differentiable manifold is a generalisation of -dimensional Euclidean space. In a manifold it may only be possible to define coordinates "locally". This is achieved by defining
coordinate patch es: subsets of the manifold which can be mapped into -dimensional Euclidean space.See
Manifold ,differentiable manifold ,coordinate patch for more details.Tangent spaces and metric tensors
"Main articles:
Tangent space ,metric tensor "Associated with each point in an -dimensional differentiable manifold is a
tangent space (denoted ). This is an -dimensionalvector space whose elements can be thought of asequivalence class es of curves passing through the point .A
metric tensor is anon-degenerate , smooth, symmetric,bilinear map which assigns areal 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 in the manifold then we have
*
* for some real number .That is
non-degenerate means there are no non-zero such that for all .Metric signatures
"Main article:
Metric signature "For an -dimensional manifold the metric tensor (in a fixed coordinate system) has
eigenvalue s. If the metric is non-degenerate then none of these eigenvalues are zero. The signature of the metric denotes the number of positive and negative eigenvalues, this quantity is independent of the chosen coordinate system bySylvester's rigidity theorem and locally non-decreasing. If the metric has positive eigenvalues and negative eigenvalues then the metric signature is . For a non-degenerate metric .Definition
A pseudo-Riemannian manifold is a
differentiable manifold equipped with a non-degenerate, smooth, symmetricmetric tensor which, unlike aRiemannian metric , need not bepositive-definite , but must be non-degenerate. Such a metric is called a pseudo-Riemannian metric and its values can be positive, negative or zero.The signature of a pseudo-Riemannian metric is where both and are non-negative.
Lorentzian manifold
A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (or sometimes , see
sign convention ). Such metrics are called Lorentzian metrics. They are named after the physicistHendrik Lorentz .Applications in physics
After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of
general relativity .A principal assumption of
general relativity is thatspacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (or equivalently (1,3)). Unlike Riemannian manifolds with positive-definite metrics, a signature of or allows tangent vectors to be classified into "timelike", "null" or "spacelike" (seeCausal structure ).Properties of pseudo-Riemannian manifolds
Just as
Euclidean space can be thought of as the modelRiemannian manifold ,Minkowski space with the flatMinkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature is with the metric:Some basic theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the
fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well. This allows one to speak of theLevi-Civita connection on a pseudo-Riemannian manifold along with the associatedcurvature 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 pseudo-Riemannian metric of a given signature; there are certain topological obstructions. Furthermore, asubmanifold of a pseudo-Riemannianmanifold need not be a pseudo-Riemannian manifold.See also
*
Riemannian manifold
*Causal structure
*Metric (mathematics)
*Metric signature
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