In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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 ma ...
with a
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
that is everywhere
nondegenerate. This is a generalization of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
in which the requirement of
positive-definiteness is relaxed.
Every
tangent space of a pseudo-Riemannian manifold is a
pseudo-Euclidean vector space In mathematics and theoretical physics, a pseudo-Euclidean space 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 vector , giving
q( ...
.
A special case used in
general relativity is a four-dimensional Lorentzian manifold for modeling
spacetime, where tangent vectors can be classified as
timelike, null, and spacelike.
Introduction
Manifolds
In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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 ma ...
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
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 ma ...
'', ''
Coordinate patch'' for more details.
Tangent spaces and metric tensors
Associated with each point
in an
-dimensional differentiable manifold
is a
tangent space (denoted
). This is an
-dimensional
vector space whose elements can be thought of as
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of curves passing through the point
.
A
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
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
we can express this as
:
The map is symmetric and bilinear so if
are tangent vectors at a point
to the manifold
then we have
*
*
for any real number
.
That
is
non-degenerate means there is no non-zero
such that
for all
.
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 to a ...
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 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
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 ma ...
equipped with an everywhere non-degenerate, smooth, symmetric
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
.
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).
Properties of pseudo-Riemannian manifolds
Just as
Euclidean space can be thought of as the model
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
,
Minkowski space with the flat
Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (
p,
q) 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 the
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
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 obstructions. Furthermore, a
submanifold 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 a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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 (ge ...
. 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.]
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
Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorentz t ...
.
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'').
See also
*
Causality conditions
In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 19 ...
*
Globally hyperbolic manifold
*
Hyperbolic partial differential equation
*
Orientable manifold
*
Spacetime
Notes
References
*
*
*
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*.
External links
*
{{Riemannian geometry
Bernhard Riemann
Differential geometry
*
Riemannian geometry
Riemannian manifolds
Smooth manifolds