Cartan connection applications

Cartan connection applications

This page covers applications of the Cartan formalism. For the general concept see Cartan connection.

Vierbeins, "et cetera"

The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifold. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, funfbein, elfbein etc. have been used. Vielbein covers all dimensions. (In German, "vier" stands for four and "viel" stands for many.)

If you're looking for a basis-dependent index notation, see tetrad (index notation).

The basic ingredients

Suppose we are working on a differential manifold "M" of dimension "n", and have fixed natural numbers "p" and "q" with

:"p" + "q" = "n".

Furthermore, we assume that we are given a SO("p", "q") principal bundle "B" over "M" (called the frame bundle) This can be turned into a Spin("p","q") principal spin bundle via the associated bundle construction if there are spinorial fields.] , and a vector SO("p", "q")-bundle "V" associated to "B" by means of the natural "n"-dimensional representation of SO("p", "q").

The basic ingredients are: η that is a SO("p", "q")-invariant metric with signature ("p", "q") over "V"; and an invertible linear map "e" between vector bundles over "M", ecolon{ m T}M o V, where T"M" is the tangent bundle of "M".

Example: general relativity

We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field. The metric tensor g_{alphaeta}! gives the inner product in the tangent space directly:

: langle mathbf{x},mathbf{y} angle = g_{alphaeta} , x^{alpha} , y^{eta}.,

The tetrad e_{alpha}^i may be seen as a (linear) map from the tangent space to Minkowski space which preserves the inner product. This lets us find the inner product in the tangent space by mapping our two vectors into Minkowski space and taking the usual inner product there:

: langle mathbf{x},mathbf{y} angle = eta_{ij} (e_{alpha}^i , x^{alpha}) (e_{eta}^j , y^{eta}).,

Here alpha and eta range over tangent-space coordinates, while i and j range over Minkowski coordinates. The tetrad field is less general than the metric tensor field: given any tetrad field e_{alpha}^i(mathbf{x}) there is an equivalent metric tensor field g_{alphaeta}(mathbf{x}) = eta_{ij} , e_{alpha}^i(mathbf{x}) , e_{eta}^j(mathbf{x}), but a metric tensor field cannot be expressed using tetrads unless it defines a Minkowskian inner product. Normally this is no limitation because we require solutions of general relativity to be locally Minkowskian everywhere.


A (pseudo-)Riemannian metric is defined over "M" as the pullback of η by "e". To put it in other words, if we have two sections of T"M", X and Y,:"g"(X,Y) = η("e"(X),"e"(Y)).A connection over "V" is defined as the unique connection A satisfying these two conditions:

* "d"η(a,b) = η("d"A"a","b") + η("a","d"A"b") for all differentiable sections "a" and "b" of "V" (i.e. "d"Aη = 0) where dA is the covariant exterior derivative. This implies that A can be extended to a connection over the SO("p","q") principal bundle.
* "d"A"e" = 0. The quantity on the left hand side is called the torsion. This basically states that abla defined below is torsion-free. This condition is dropped in the Einstein-Cartan theory, but then we can't define A uniquely anymore.

This is called the spin connection.

Now that we've specified A, we can use it to define a connection ∇ over T"M" via the isomorphism "e"::"e"(∇X) = "d"A"e"(X) for all differentiable sections X of T"M".

Since what we now have here is a SO("p","q") gauge theory, the curvature F defined as old{F} stackrel{mathrm{def{=} dold{A}+old{A}wedgeold{A} is pointwise gauge covariant. This is simply the Riemann curvature tensor in a different form.

An alternate notation writes the connection form A as ω, the curvature form F as Ω, the canonical vector-valued 1-form "e" as θ, and the exterior covariant derivative d_A as "D".

The Palatini action

In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection form A over a four dimensional differential manifold M is given by

:S stackrel{mathrm{def{=} frac{1}{2}int_M epsilon(F wedge e wedge e)

where F is the gauge curvature 2-form and ε is the antisymmetric intertwiner of four "vector" reps of SO(3,1) normalized by η.

Note that in the presence of spinor fields, the Palatini action implies that dAe is nonzero, that is, have torsion. See Einstein-Cartan theory.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Cartan connection — In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the …   Wikipedia

  • Connection (mathematics) — In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of… …   Wikipedia

  • Élie Cartan — Infobox Person name = Élie Joseph Cartan image size = 200px caption = Professor Élie Joseph Cartan birth date = birth date|1869|4|9 birth place = Dolomieu, Savoie, France death date = death date and age|1951|5|6|1869|4|9 death place = Paris,… …   Wikipedia

  • Spin connection — In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the Levi Civita connection. It can also be regarded as the gauge field generated by local Lorentz… …   Wikipedia

  • Affine connection — An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development. In the branch of mathematics called differential geometry, an… …   Wikipedia

  • Moving frame — The Frenet Serret frame on a curve is the simplest example of a moving frame. In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… …   Wikipedia

  • Tetrads in general relativity — In physics, a tetrad or vierbein is a set of four basis vector fields which can be written in terms of a local coordinate basis by means of tetrad components. For more details see Cartan connection applications, tetrad (index notation).ee also*… …   Wikipedia

  • General relativity — For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. General relativity Introduction Mathematical formulation Resources …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”