View source for Pseudo-Riemannian manifold

{{Use American English|date = March 2019}}
{{Short description|Differentiable manifold with nondegenerate metric tensor}}
{{General relativity sidebar}}

In [[mathematical physics]], a '''pseudo-Riemannian manifold''',{{sfn|Benn|Tucker|1987|p=172|ps=}}{{sfn|Bishop|Goldberg|1968|p=208|ps=}} also called a '''semi-Riemannian manifold''', is a [[differentiable manifold]] with a [[metric tensor]] that is everywhere [[non-degenerate bilinear form|nondegenerate]].  This is a generalization of a [[Riemannian manifold]] in which the requirement of [[Positive-definite bilinear form|positive-definite]]ness is relaxed. 

Every [[tangent space]] 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 ===

{{main|Manifold|Differentiable manifold}}

In [[differential geometry]], a [[differentiable manifold]] is a space that 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 [[coordinate]]s 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 patch]]es: subsets of the manifold that can be mapped into ''n''-dimensional Euclidean space.

See ''[[Manifold]]'', ''[[Differentiable manifold]]'', ''[[Coordinate patch]]'' for more details.

=== Tangent spaces and metric tensors ===

{{main|Tangent space|Metric tensor}}

Associated with each point <math>p</math> in an <math>n</math>-dimensional differentiable manifold <math>M</math> is a [[tangent space]] (denoted <math>T_pM</math>). This is an <math>n</math>-dimensional [[vector space]] whose elements can be thought of as [[equivalence class]]es of curves passing through the point <math>p</math>.

A [[metric tensor]] 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 <math>g</math> we can express this as 
: <math>g : T_pM \times T_pM \to \mathbb{R}.</math>

The map is symmetric and bilinear so if <math>X,Y,Z \in T_pM</math> are tangent vectors at a point <math>p</math> to the manifold <math>M</math> then we have
* <math>\,g(X,Y) = g(Y,X)</math>
* <math>\,g(aX + Y, Z) = a g(X,Z) + g(Y,Z)</math>
for any real number <math>a\in\mathbb{R}</math>.

That <math>g</math> is [[non-degenerate]] means there is no non-zero <math>X \in T_pM</math> such that <math>g(X,Y) = 0</math> for all <math>Y \in T_pM</math>.

=== Metric signatures ===

{{main|Metric signature}}

Given a metric tensor ''g'' on an ''n''-dimensional real manifold, the [[quadratic form]] {{nowrap|1=''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#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]]''' {{nowrap|(''p'', ''q'', ''r'')}} of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has {{nowrap|1=''r'' = 0}} and the signature may be denoted {{nowrap|(''p'', ''q'')}}, where {{nowrap|1=''p'' + ''q'' = ''n''}}.

== Definition ==

A '''pseudo-Riemannian manifold''' {{nowrap|(''M'', ''g'')}} is a [[differentiable manifold]] ''M'' that is equipped with an everywhere non-degenerate, smooth, symmetric [[metric tensor]] ''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 {{nowrap|(''p'', ''q'')}}, 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 ==
<!--'Lorentzian manifold' and 'Lorentzian metric' redirect here-->
A '''Lorentzian manifold'''<!--boldface per WP:R#PLA--> is an important special case of a pseudo-Riemannian manifold in which the [[Metric signature|signature of the metric]] is {{nowrap|(1, ''n''−1)}} (equivalently, {{nowrap|(''n''−1, 1)}}; see ''[[Sign convention]]''). Such metrics are called '''Lorentzian metrics'''<!--boldface per WP:R#PLA-->. 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 {{nowrap|(3, 1)}} or, equivalently, {{nowrap|(1, 3)}}.  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 {{nowrap|(''p'', 1)}} or {{nowrap|(1, ''q'')}}, the manifold is also locally (and possibly globally) time-orientable (see ''[[Causal structure]]'').

== Properties of pseudo-Riemannian manifolds ==

Just as [[Euclidean space]] <math>\mathbb{R}^n</math> can be thought of as the local model of a [[Riemannian manifold]], [[Minkowski space]] <math>\mathbb{R}^{n-1,1}</math> with the flat [[Minkowski metric]] is the local model of a Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (''p'', ''q'') is [[pseudo-Euclidean space]] <math>\mathbb{R}^{p,q}</math>, for which there exist coordinates ''x''<sub>''i''</sub> such that
: <math>g = dx_1^2 + \cdots + dx_p^2 - dx_{p+1}^2 - \cdots - dx_{p+q}^2</math>

Some theorems of Riemannian geometry can be generalized to the pseudo-Riemannian case. In particular, the [[fundamental theorem of Riemannian geometry]] is true of all pseudo-Riemannian manifolds. 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 that 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.{{sfn|O'Neill|1983|p=193|ps=}}

== See also ==
* [[Causality conditions]]
* [[Globally hyperbolic manifold]]
* [[Hyperbolic partial differential equation]]
* [[Orientable manifold]]
* [[Spacetime]]

== Notes ==
{{reflist}}

== References ==
* {{citation | last1=Benn|first1=I.M.|last2=Tucker|first2=R.W. | title = An introduction to Spinors and Geometry with Applications in Physics| publisher=Adam Hilger | year=1987|edition=First published 1987|isbn=0-85274-169-3}}
* {{citation|last1=Bishop|first1=Richard L.|author1-link=Richard L. Bishop|last2=Goldberg|first2=Samuel I.|title=Tensor Analysis on Manifolds|publisher=The Macmillan Company|year=1968|edition=First Dover 1980|isbn=0-486-64039-6|url-access=registration|url=https://archive.org/details/tensoranalysison00bish}}
* {{citation | last1=Chen|first1=Bang-Yen| title = Pseudo-Riemannian Geometry, [delta]-invariants and Applications| publisher=World Scientific Publisher | year=2011|isbn=978-981-4329-63-7}}
* {{citation|title=Semi-Riemannian Geometry With Applications to Relativity|volume=103|series=Pure and Applied Mathematics|first=Barrett|last=O'Neill|publisher=Academic Press|year=1983|isbn=9780080570570|url=https://books.google.com/books?id=CGk1eRSjFIIC&pg=PA193}}
* {{citation|first1=G.|last1=Vrănceanu|first2=R.|last2=Roşca|year=1976|title=Introduction to Relativity and Pseudo-Riemannian Geometry|location=Bucarest|publisher=Editura Academiei Republicii Socialiste România}}

== External links ==
* {{Commonscatinline|Lorentzian manifolds}}

{{Manifolds}}
{{Riemannian geometry}}

[[Category:Bernhard Riemann]]
[[Category:Differential geometry]]
[[Category:Lorentzian manifolds|*]]
[[Category:Riemannian geometry]]
[[Category:Riemannian manifolds]]
[[Category:Smooth manifolds]]