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In
differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
with a
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
that is everywhere
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler to) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The d ...
. This is a generalization of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T'p'M'' at each point ''p''. A common convention is to ta ...
in which the requirement of positive-definiteness is relaxed. Every
tangent space In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the ...
of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as Causal structure, timelike, null, and spacelike.


Introduction


Manifolds

In
differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
, a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional 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 ''n''-dimensional Euclidean space. See ''Manifold'', ''Differentiable manifold'', ''Coordinate patch'' for more details.


Tangent spaces and metric tensors

Associated with each point p in an n-dimensional differentiable manifold M is a
tangent space In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the ...
(denoted T_pM). This is an n-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point p. A
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
is a non-degenerate, 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 g we can express this as :g : T_pM \times T_pM \to \mathbb. The map is symmetric and bilinear so if X,Y,Z \in T_pM are tangent vectors at a point p to the manifold M then we have * \,g(X,Y) = g(Y,X) * \,g(aX + Y, Z) = a g(X,Z) + g(Y,Z) for any real number a\in\mathbb. That g is non-degenerate means there are no non-zero X \in T_pM such that \,g(X,Y) = 0 for all Y \in T_pM.


Metric signatures

Given a metric tensor ''g'' on an ''n''-dimensional real manifold, the quadratic form associated with the metric tensor applied to each vector of any orthogonal basis produces ''n'' real values. By Sylvester's law of inertia#Law of inertia for quadratic forms, 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 Metric signature, signature of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has and the signature may be denoted (''p'', ''q''), where .


Definition

A pseudo-Riemannian manifold (M,g) is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
M equipped with an everywhere non-degenerate, smooth, symmetric
metric tensor In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a ...
g. Such a metric is called a pseudo-Riemannian 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 pseudo-Riemannian metric is , where both ''p'' and ''q'' are non-negative. The non-degeneracy condition together with continuity implies that ''p'' and ''q'' remain unchanged throughout the manifold (assuming it is connected).


Lorentzian manifold

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the Metric signature, signature of the metric is (equivalently, ; see ''Sign convention''). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz.


Applications in physics

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications of general relativity. A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature or, equivalently, . Unlike Riemannian manifolds with positive-definite metrics, an indefinite signature allows tangent vectors to be classified into ''timelike'', ''null'' or ''spacelike''. With a signature of or , the manifold is also locally (and possibly globally) time-orientable (see ''Causal structure'').


Properties of pseudo-Riemannian manifolds

Just as Euclidean space \mathbb^n can be thought of as the model
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T'p'M'' at each point ''p''. A common convention is to ta ...
, Minkowski space \mathbb^ with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p, q) is \mathbb^ with the metric :g = dx_1^2 + \cdots + dx_p^2 - dx_^2 - \cdots - dx_^2 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 the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated Riemann curvature tensor, 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 pseudo-Riemannian metric of a given signature; there are certain topology, topological obstructions. Furthermore, a submanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any Minkowski space#Causal structure, light-like curve. The Clifton–Pohl torus provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds., p. 193.


See also

*Causality conditions *Globally hyperbolic manifold *Hyperbolic partial differential equation *Orientable manifold *Spacetime


Notes


References

* * * * *.


External links

*{{Commonscatinline, Lorentzian manifolds Lorentzian manifolds, * Smooth manifolds Bernhard Riemann