Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian mani-fold[1][2] (also called a semi-Riemannian manifold) isa generalization of a Riemannian manifold in which themetric tensor need not be positive-definite. Instead aweaker condition of nondegeneracy is imposed on themetric tensor.The tangent space of a pseudo-Riemannian manifoldis a pseudo-Euclidean space described by an isotropicquadratic form.A special case of great importance to general relativityis a Lorentzian manifold, in which one dimension hasa sign opposite to that of the rest. This allows tangentvectors to be classified into timelike, null, and spacelike.Spacetime can be modeled as a 4-dimensional Lorentzianmanifold.

1 Introduction

1.1 Manifolds

Main articles: Manifold and Differentiable manifold

In differential geometry, a differentiable manifold is aspace which is locally similar to a Euclidean space. Inan n-dimensional Euclidean space any point can be spec-ified by n real numbers. These are called the coordinatesof the point.An n-dimensional differentiable manifold is a generalisa-tion of n-dimensional Euclidean space. In a manifold itmay only be possible to define coordinates locally. Thisis achieved by defining coordinate patches: subsets of themanifold which can be mapped into n-dimensional Eu-clidean space.See Manifold, differentiable manifold, coordinate patchfor more details.

1.2 Tangent spaces and metric tensors

Main articles: Tangent space and metric tensor

Associated with each point p in an n -dimensional dif-ferentiable manifold M is a tangent space (denoted TpM

). This is an n -dimensional vector space whose elementscan be thought of as equivalence classes of curves passingthrough the point p .

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

g : TpM × TpM → R.

The map is symmetric and bilinear so if X,Y,Z∈TpM aretangent vectors at a point p to the manifold M then wehave

• g(X,Y ) = g(Y,X)

• g(aX + Y,Z) = ag(X,Z) + g(Y, Z)

for any real number a∈R .That g is non-degenerate means there are no non-zeroX ∈ TpM such that g(X,Y ) = 0 for all Y ∈ TpM.

1.3 Metric signatures

Main article: Metric signature

Given a metric tensor g on an n-dimensional real mani-fold, the quadratic form q(x) = g(x, x) associated with themetric tensor applied to each vector of any orthogonalbasis produces n real values. By Sylvester’s law of iner-tia, the number of each positive, negative and zero valuesproduced in this manner are invariants of the metric ten-sor, independent of the choice of orthogonal basis. Thesignature (p, q, r) of the metric tensor gives these num-bers, shown in the same order. A non-degenerate metrictensor has r = 0 and the signature may be denoted (p, q),where p + q = n.

2 Definition

A pseudo-Riemannian manifold (M, g) is adifferentiable manifold M equipped with a non-degenerate, smooth, symmetric metric tensor g .Such a metric is called a pseudo-Riemannian metricand 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.

1

2 7 REFERENCES

3 Lorentzian manifold

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

3.1 Applications in physics

After Riemannian manifolds, Lorentzian manifolds formthe most important subclass of pseudo-Riemannian man-ifolds. They are important in applications of general rel-ativity.A principal basis of general relativity is that spacetimecan be modeled as a 4-dimensional Lorentzian manifoldof signature (3, 1) or, equivalently, (1, 3). Unlike Rie-mannianmanifolds with positive-definitemetrics, a signa-ture of (p, 1) or (1, q) allows tangent vectors to be classi-fied into timelike, null or spacelike (see Causal structure).

4 Properties of pseudo-Riemannian manifolds

Just as Euclidean spaceRn can be thought of as themodelRiemannian manifold, Minkowski spaceRn−1,1 with theflat Minkowski metric is the model Lorentzian mani-fold. Likewise, themodel space for a pseudo-Riemannianmanifold of signature (p, q) is Rp,q with the metric

g = dx21 + · · ·+ dx2

p − dx2p+1 − · · · − dx2

p+q

Some basic theorems of Riemannian geometry can begeneralized to the pseudo-Riemannian case. In particu-lar, the fundamental theorem of Riemannian geometryis true of pseudo-Riemannian manifolds as well. Thisallows one to speak of the Levi-Civita connection on apseudo-Riemannian manifold along with the associatedcurvature tensor. On the other hand, there are many the-orems in Riemannian geometry which do not hold in thegeneralized case. For example, it is not true that everysmooth manifold admits a pseudo-Riemannian metric ofa given signature; there are certain topological obstruc-tions. Furthermore, a submanifold does not always in-herit the structure of a pseudo-Riemannian manifold; forexample, the metric tensor becomes zero on any light-like curve. The Clifton–Pohl torus provides an exam-ple of a pseudo-Riemannian manifold that is compact butnot complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds.[3]

5 See also• Spacetime

• Hyperbolic partial differential equation

• Causality conditions

• Globally hyperbolic manifold

6 Notes[1] Benn & Tucker (1987), p. 172.

[2] Bishop & Goldberg (1968), p. 208

[3] O'Neill (1983), p. 193.

7 References• Benn, I.M.; Tucker, R.W. (1987), An introduction toSpinors and Geometry with Applications in Physics(First published 1987 ed.), Adam Hilger, ISBN 0-85274-169-3

• Bishop, Richard L.; Goldberg, Samuel I. (1968),Tensor Analysis on Manifolds (First Dover 1980ed.), The Macmillan Company, ISBN 0-486-64039-6

• Chen, Bang-Yen (2011), Pseudo-Riemannian Ge-ometry, [delta]-invariants and Applications, WorldScientific Publisher, ISBN 978-981-4329-63-7

• O'Neill, Barrett (1983), Semi-Riemannian Geome-try With Applications to Relativity, Pure and Ap-plied Mathematics 103, Academic Press, ISBN9780080570570

• Vrănceanu, G.; Roşca, R. (1976), Introduction toRelativity and Pseudo-Riemannian Geometry, Bu-carest: Editura Academiei Republicii SocialisteRomânia.

3

8 Text and image sources, contributors, and licenses

8.1 Text• Pseudo-Riemannian manifold Source: https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold?oldid=673398563 Contributors:Patrick, Michael Hardy, Notheruser, Raven in Orbit, Charles Matthews, Dcoetzee, Vancouverguy, Phys, Tobias Bergemann, Tosha, Giftlite,Fropuff, Rgdboer, Kevin Lamoreau, Guy Harris, Linas, Mpatel, JATerg, Isnow, Ketiltrout, Mmoroz, Bgwhite, Roboto de Ajvol, Yurik-Bot, Hillman, Grafen, Modify, SmackBot, Incnis Mrsi, Chlewbot, Ignirtoq, William Ackerman, Ryan Roos, Thijs!bot, Konradek, Rob-Har, Escarbot, David Eppstein, Jacksonwalters, Haseldon, Squids and Chips, StevenJohnston, Hotcrocodile, Addbot, Luckas-bot, RedBot,RA0808, Quondum, ClueBot NG, Mgvongoeden, HrvaviHrtkovci, Bychen, Brad7777, Cantseethewoodforthetrees and Anonymous: 21

8.2 Images

8.3 Content license• Creative Commons Attribution-Share Alike 3.0