pseudo-Riemannian manifold
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In
mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, 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 vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
with a metric tensor that is everywhere nondegenerate. This is a generalization of a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
in which the requirement of positive-definiteness is relaxed. Every tangent space 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, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
is a four-dimensional Lorentzian manifold for modeling
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
, 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. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
is a space that is locally similar to a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. 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 that can be mapped into ''n''-dimensional Euclidean space. See ''
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
'', ''
Differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
'', '' 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 (denoted T_pM). This is an n-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
whose elements can be thought of as equivalence classes of curves passing through the point p. A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 is 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, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
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 A signature (; from , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has and the signature may be denoted , where .


Definition

A pseudo-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 vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
''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 , 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''). 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 General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. A principal premise of general relativity is that
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
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, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (''c ...
'').


Properties of pseudo-Riemannian manifolds

Just as
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\mathbb^n can be thought of as the local model of a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, Minkowski space \mathbb^ 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 In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vect ...
\mathbb^, for which there exist coordinates ''x''''i'' such that : g = dx_1^2 + \cdots + dx_p^2 - dx_^2 - \cdots - dx_^2 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 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 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 S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any light-like
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
. 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.


See also

*
Causality conditions Causality conditions are classifications of Lorentzian manifolds according to the types of causal structures they admit. In the study of spacetimes, there exists a hierarchy of causality conditions which are important in proving mathematical theor ...
* Globally hyperbolic manifold *
Hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
* Orientable manifold *
Spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
* The Clockwork Rocket


Notes


References

* * * * *


External links

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