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| {{Context|date=October 2009}}
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| {{Merge to|tetrad formalism|date=September 2011}}
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| : ''This page covers applications of the '''Cartan formalism'''. For the general concept see [[Cartan connection]].''
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| The '''vierbein''' or '''[[tetrad (general relativity)|tetrad]]''' theory much used in theoretical physics is a special case of the application of [[Cartan connection]] in four-dimensional [[manifold]]s. It applies to metrics of any signature. (See [[metric tensor]].) This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like '''triad''', '''pentad''', '''zweibein''', '''fünfbein''', '''elfbein''' etc. have been used. '''Vielbein''' covers all dimensions. (In German, ''vier'' stands for four and ''viel'' stands for many.)
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| For a basis-dependent index notation, see [[tetrad (index notation)]].
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| ==The basic ingredients==
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| Suppose we are working on a [[differential manifold]] ''M'' of dimension ''n'', and have fixed natural numbers ''p'' and ''q'' with
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| :''p'' + ''q'' = ''n''.
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| Furthermore, we assume that we are given a [[generalized special orthogonal group|SO(''p'', ''q'')]] [[principal bundle]] ''B'' over ''M'' and a [[vector bundle|SO(''p'', ''q'')-vector bundle]] ''V'' associated to ''B'' by means of the natural ''n''-dimensional [[representations of Lie groups/algebras|representation]] of SO(''p'', ''q''). Equivalently, ''V'' is a rank ''n'' real vector bundle over ''M'',
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| equipped with a [[Metric (mathematics)|metric]] η with [[metric signature|signature]] (''p'', ''q'') (aka non degenerate [[quadratic form]]).<ref name=spin(p,q)>A variant of the construction uses reduction to a [[spin group|Spin(''p'', ''q'')]] principal [[spin bundle]]. In that case, the principal bundle contains more information than the bundle ''V'' together with the metric η, which is needed to construct [[spinor]]ial fields.</ref>
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| The basic ingredient of the Cartan formalism is an [[invertible]] [[linear map]] <math>e\colon{\rm T}M\to V</math>, between [[vector bundle]]s over ''M'' where T''M'' is the [[tangent bundle]] of ''M''. The invertibility condition on ''e'' is sometimes dropped. In particular if ''B'' is the trivial bundle, as we can always assume locally,
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| ''V'' has a basis of orthogonal sections
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| <math>f_a = f_1 \ldots f_n</math>. With respect to this basis
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| <math> \eta_{ab} = \eta(f_a, f_b) = {\rm diag}(1,\ldots 1, -1, \ldots, -1)</math> is a constant matrix. For a choice of local coordinates <math>x^\mu = x^{-1}, \ldots, x^{-n}</math> on ''M'' (the negative indices are only to distinguish them from the indices labeling the <math>f_a</math>) and a corresponding local frame
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| <math>\partial_\mu = \frac{\partial}{\partial x^\mu}</math> of the tangent bundle, the map ''e'' is determined by the images
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| of the basis sections
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| :<math> e_a := e(f_a) := e^\mu_a \partial_\mu.</math>
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| They determine a (non coordinate) basis of the tangent bundle (provided ''e'' is invertible and only locally if B is only locally trivialised). The matrix <math>e^\mu_a , \mu = -1, \dots, -n, a = 1, \dots, n</math> is called the [[tetrad (general relativity)|tetrad, vierbein, vielbein etc.]].
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| Its interpretation as a local frame crucially depends on the implicit choice of local bases.
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| Note that an isomorphism
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| <math>V \cong {\rm T}M</math> gives a [[principal bundle#reduction|reduction]] <math>B \to {\rm Fr}(M)</math> of the [[frame bundle]], the principal bundle of the tangent bundle. In general, such a reduction is impossible for topological
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| reasons. Thus, in general for continuous maps ''e'', one cannot avoid that ''e'' becomes degenerate at some points of ''M''.
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| ==Example: general relativity==
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| {{Main|Frame fields in general relativity}}
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| We can describe geometries in [[general relativity]] in terms of a tetrad field instead of the usual metric tensor field. The metric tensor <math>g_{\alpha\beta}\!</math> gives the [[inner product]] in the [[tangent space]] directly:
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| :<math> \langle \mathbf{x},\mathbf{y} \rangle = g_{\alpha\beta} \, x^{\alpha} \, y^{\beta}.\, </math>
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| The tetrad <math>e_{\alpha}^i</math> may be seen as a (linear) map from the tangent space to Minkowski space that 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:
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| :<math> \langle \mathbf{x},\mathbf{y} \rangle = \eta_{ij} (e_{\alpha}^i \, x^{\alpha}) (e_{\beta}^j \, y^{\beta}).\, </math>
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| Here <math>\alpha</math> and <math>\beta</math> range over tangent-space coordinates, while <math>i</math> and <math>j</math> range over Minkowski coordinates. The tetrad field <math>e_{\alpha}^i(\mathbf{x})</math> defines a metric tensor field via the pullback <math>g_{\alpha\beta}(\mathbf{x}) = \eta_{ij} \, e_{\alpha}^i(\mathbf{x}) \, e_{\beta}^j(\mathbf{x})</math>.
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| ==Constructions==
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| A ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian metric]] is defined over ''M'' as the [[pullback (differential geometry)|pullback]] of η by ''e''. To put it in other words, if we have two sections of T''M'', '''X''' and '''Y''',
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| :''g''('''X''','''Y''') = η(''e''('''X'''), ''e''('''Y''')).
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| A [[connection form|connection]] over ''V'' is defined as the unique connection '''A''' satisfying these two conditions:
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| * ''d''η(a,b) = η(''d''<sub>'''A'''</sub>''a'',''b'') + η(''a'',''d''<sub>'''A'''</sub>''b'') for all differentiable sections ''a'' and ''b'' of ''V'' (i.e. ''d''<sub>'''A'''</sub>η = 0) where d<sub>'''A'''</sub> is the [[covariant exterior derivative]]. This implies that '''A''' can be extended to a [[connection form|connection]] over the SO(''p'',''q'') [[principal bundle]].
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| * ''d''<sub>'''A'''</sub>''e'' = 0. The quantity on the left hand side is called the [[torsion coefficient|torsion]]. This basically states that <math>\nabla</math> defined below is torsion-free. This condition is dropped in the [[Einstein-Cartan theory]], but then we cannot define '''A''' uniquely anymore.
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| This is called the '''spin connection'''.
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| Now that we have specified '''A''', we can use it to define a connection ∇ over T''M'' via the [[isomorphism]] ''e'':
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| :''e''(∇'''X''') = ''d''<sub>'''A'''</sub>''e''('''X''') for all differentiable sections '''X''' of T''M''.
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| Since what we now have here is a SO(''p'',''q'') [[gauge theory]], the curvature '''F''' defined as <math>\bold{F}\ \stackrel{\mathrm{def}}{=}\ d\bold{A}+\bold{A}\wedge\bold{A}</math> is pointwise gauge covariant. This is simply the [[Riemann tensor|Riemann curvature tensor]] in a different form.
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| 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]] <math>d_A</math> as ''D''.
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| ==The Palatini action==
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| In the tetrad formulation of general relativity, the [[action (physics)|action]], as a [[functional (mathematics)|functional]] of the vierbein e and a [[connection form]] <math>\omega</math>, with an associated field strength <math>\Omega = D\omega = d\omega + \omega \wedge \omega</math>, over a four-dimensional [[differentiable manifold]] ''M'' is given by
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| :<math>S\ \stackrel{\mathrm{def}}{=}\ M^2_{pl}\int_M \epsilon_{abcd}( e^{a} \wedge e^{b} \wedge \Omega^{cd}) = M^2_{pl}\int_M d^4x \epsilon^{\mu \nu \rho \sigma} \epsilon_{abcd} e^a_{\mu} e^b_{\nu} R^{cd}_{\rho \sigma}[\omega] </math> | |
| :<math> = M^2_{pl}\int |e| d^4 x \frac{1}{2} e^{\mu}_a e^{\nu}_b R^{ab}_{\mu \nu} </math>
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| :<math> = \frac{c^4}{16 \pi G} \int d^4x \sqrt{-g} R[g]</math>
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| where <math>\Omega_{\mu \nu} ^{ab} = R_{\mu \nu} ^{ab}</math> is the [[gauge curvature]] [[2-form]], <math>\epsilon_{abcd}</math> is the antisymmetric [[Levi-Civita symbol]], and that <math>|e| = \epsilon^{\mu \nu \rho \sigma} \epsilon_{abcd} e^a_{\mu}e^b_{\nu}e^c_{\rho}e^d_{\sigma}</math> is the determinant of <math>e_{\mu}^a</math>. Here we see that the differential form language leads to an equivalent action to that of the normal [[Einstein–Hilbert action]], using the relations <math> |e| = \sqrt{-g}</math> and <math> R^{\lambda \sigma}_{\mu \nu}= e^{\lambda}_a e^{\sigma}_b R^{ab}_{\mu \nu} </math>. Note that in terms of the Planck mass, we set <math> \hbar = c =1</math>, whereas the last term keeps all the SI unit factors.
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| Note that in the presence of [[spinor field]]s, the Palatini action implies that <math>d\omega</math> is nonzero. So there's a non-zero [[torsion tensor| torsion]], i.e. that <math> \hat{\omega}^{ab}_{\mu} = \omega^{ab}_{\mu} + K^{a b}_{\mu}</math>. See [[Einstein-Cartan theory]].
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| == Notes ==
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| <references />
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| {{tensors}}
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| {{DEFAULTSORT:Cartan Formalism (Physics)}}
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| [[Category:Differential geometry]]
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| [[Category:Mathematical methods in general relativity]]
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| [[Category:Connection (mathematics)]]
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