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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
with a
metric tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
that is everywhere
nondegenerate In mathematics, a degenerate case is a limiting case (mathematics), limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of ...
. This is a generalization of a
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...
in which the requirement of positive-definiteness is relaxed. Every
tangent space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
is a four-dimensional Lorentzian manifold for modeling
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
, where tangent vectors can be classified as timelike, null, and spacelike.


Introduction


Manifolds

In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
is a space which is locally similar to a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the
coordinate In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
es: subsets of the manifold which can be mapped into ''n''-dimensional Euclidean space. See ''
Manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

Manifold
'', ''
Differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
'', ''
Coordinate patch In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
'' 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(denoted T_pM). This is an n-dimensional
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
whose elements can be thought of as
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es of curves passing through the point p. A
metric tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
is a
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ...
, smooth, symmetric,
bilinear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
that assigns a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
associated with the metric tensor applied to each vector of any
orthogonal basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
produces ''n'' real values. By
Sylvester's law of inertia Sylvester's law of inertia is a theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a ...
, 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 A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a s ...
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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
M equipped with an everywhere non-degenerate, smooth, symmetric
metric tensor In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
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 signature of the metric is (equivalently, ; see ''
Sign convention In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succes ...
''). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical met ...

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 General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. A principal premise of general relativity is that
spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
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 In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in ...
'').


Properties of pseudo-Riemannian manifolds

Just as
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
\mathbb^n can be thought of as the model
Riemannian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...
,
Minkowski space In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...
\mathbb^ with the flat
Minkowski metric In mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems i ...

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 Fundamental may refer to: * Foundation of reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary Imaginary may refer to: * Imaginary (sociology), a concept in socio ...
is true of pseudo-Riemannian manifolds as well. This allows one to speak of the
Levi-Civita connection In Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo Riemannian geometry (in particular the Lorentzian manifold, Lorentzian geometry of General Relativity, general relativity), the Levi-Civita connection is the unique affine co ...
on a pseudo-Riemannian 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 pseudo-Riemannian metric of a given signature; there are certain
topological In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topological
obstructions. Furthermore, a
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any
light-like In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
. The
Clifton–Pohl torus In geometry, the Clifton–Pohl torus is an example of a compact space, compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space sh ...
provides an example of a pseudo-Riemannian manifold that is compact but not complete, a combination of properties that the
Hopf–Rinow theorem Hopf–Rinow theorem is a set of statements about the Geodesic manifold, geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931. Statement Let (M, g) be a Connected space, ...
disallows for Riemannian manifolds., p. 193.


See also

*
Causality conditions In the study of Lorentzian manifold In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study p ...
*
Globally hyperbolic manifold In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold ...
*
Hyperbolic partial differential equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Orientable manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
*
Spacetime In physics, spacetime is any mathematical model which fuses the three-dimensional space, three dimensions of space and the one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visuali ...


Notes


References

* * * * *.


External links

* {{Riemannian geometry Bernhard Riemann Differential geometry * Riemannian geometry Riemannian manifolds Smooth manifolds