Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold ^[1]^[2] (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the metric tensor need not be positive-definite. Instead a weaker condition of nondegeneracy is imposed.

1 Introduction
2 Definition
3 Lorentzian manifold
- 3.1 Applications in physics
4 Properties of pseudo-Riemannian manifolds
5 See also
6 Notes
7 References

Introduction

Manifolds

Main articles: Manifold and Differentiable manifold

In differential geometry, a differentiable manifold 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

Main articles: Tangent space and metric tensor

Associated with each point $\scriptstyle p$ in an $\scriptstyle n$ -dimensional differentiable manifold $\scriptstyle M$ is a tangent space (denoted $\scriptstyle T_pM$ ). This is an $\scriptstyle n$ -dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point $\scriptstyle p$ .

A metric tensor is a non-degenerate, smooth, symmetric, bilinear map which assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by $\scriptstyle g$ we can express this as

$g�: T_pM \times T_pM \to \mathbb{R}$ .

The map is symmetric and bilinear so if $\scriptstyle X,Y,Z \in T_pM$ are tangent vectors at a point $\scriptstyle p$ to the manifold $\scriptstyle M$ then we have

$\,g(X,Y) = g(Y,X)$
$\,g(aX %2B Y, Z) = a g(X,Z) %2B g(Y,Z)$

for any real number $\scriptstyle a\in\mathbb{R}$ .

That $\scriptstyle g$ 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

Main article: Metric signature

For an n-dimensional manifold the metric tensor (in a fixed coordinate system) has n eigenvalues. If the metric is non-degenerate then none of these eigenvalues are zero. The signature of the metric denotes the number of positive and negative eigenvalues, this quantity is independent of the chosen coordinate system by Sylvester's rigidity theorem and locally non-decreasing. If the metric has p positive eigenvalues and q negative eigenvalues then the metric signature is (p, q). For a non-degenerate metric $p %2B q = n$ .

Definition

A pseudo-Riemannian manifold $\,(M,g)$ is a differentiable manifold $\,M$ equipped with a non-degenerate, smooth, symmetric metric tensor $\,g$ which, unlike a Riemannian metric, need not be positive-definite, but must be non-degenerate. Such a metric is called a pseudo-Riemannian metric and its values can be positive, negative or zero.

The signature of a pseudo-Riemannian metric is (p, q) where both p and q are non-negative.

Lorentzian manifold

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, n−1) (or sometimes (n−1, 1), see sign convention). Such metrics are called Lorentzian metrics. They are named after the physicist Hendrik Lorentz.

Applications in physics

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important because of their physical applications to the theory of general relativity.

A principal assumption of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positive-definite metrics, a signature of (p, 1) or (1, q) allows tangent vectors to be classified into timelike, null or spacelike (see Causal structure).

Properties of pseudo-Riemannian manifolds

Just as Euclidean space $\mathbb{R}^n$ can be thought of as the model Riemannian manifold, Minkowski space $\mathbb{R}^{n-1,1}$ with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo-Riemannian manifold of signature (p, q) is $\mathbb{R}^{p,q}$ with the metric

$g = dx_1^2 %2B \cdots %2B dx_p^2 - dx_{p%2B1}^2 - \cdots - dx_{p%2Bq}^2$

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 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 become zero on any light-like curve.

Notes

^ Benn, I.M.; Tucker, R.W. (1987), p. 172
^ Bishop, R.L.; Goldberg, S.I. (1968), p. 208

References

Benn, I.M.; Tucker, R.W. (1987), An introduction to Spinors and Geometry with Applications in Physics (First published 1987 ed.), Adam Hilger, ISBN 0-85274-169-3
Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6
G. Vrănceanu & R. Roşca (1976) Introduction to Relativity and Pseudo-Riemannian Geometry, Bucarest: Editura Academiei Republicii Socialiste România.