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| '''Teleparallelism''' (also called '''teleparallel gravity'''), was an attempt by [[Albert Einstein|Einstein]] <ref>{{cite journal|title=Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus|author=A. Einstein|journal=Preussische Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte|volume=1928|year=1928|pages=217–221}}</ref> to base a unified theory of [[electromagnetism]] and [[gravity]] on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a [[spacetime]] is characterized by a curvature-free [[linear connection]] in conjunction with a [[metric tensor]] field, both defined in terms of a dynamical [[Cartan connection applications|tetrad]] field.
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| ==Teleparallel spacetimes==
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| The crucial new idea, for Einstein, was the introduction of a [[Cartan connection applications|tetrad]] field, i.e., a set <math>\{\mathrm X_1, \dots,\mathrm X_4\}</math> of four vector fields defined on ''all'' of <math>M\,</math> such that for every <math>p\in M\,</math> the set <math>\{\mathrm X_1(p), \dots,\mathrm X_4(p)\}</math> is a [[Basis (mathematics)|basis]] of <math>T_pM\,</math>, where <math>T_pM\,</math> denotes the fiber over <math>p\,</math> of the [[tangent vector bundle]] <math>TM\,</math>. Hence, the fourdimensional [[spacetime]] manifold <math>M\,</math> must be a [[parallelizable manifold]]. The tetrad field was introduced to allow the distant comparison of the direction of tangent vectors at different points of the manifold, hence the name distant parallelism. His attempt failed because there was no Schwarzschild solution in his simplified field equation.
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| In fact, one can define the '''connection of the parallelization''' (also called '''[[Roland Weitzenböck|Weitzenböck]] connection''') <math>\{\mathrm X_{i}\}</math> to be the [[linear connection]] <math>\nabla\,</math> on <math>M\,</math> such that <ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I.|title=Tensor Analysis on Manifolds|year=1968|page=223}}
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| </ref>
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| :<math>\nabla_{v}(f^{i}\mathrm X_{i})=(vf^{i})\mathrm X_{i}(p)\,</math>,
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| where <math>v\in T_pM\,</math> and <math>f^{i}\,</math> are (global) functions on <math>M\,</math>; thus <math>f^{i}X_{i}\,</math> is a global vector field on <math>M\,</math>. In other words, the coefficients of '''Weitzenböck connection''' <math>\nabla\,</math> with respect to <math>\{X_{i}\}</math> are all identically zero, implicitly defined by:
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| :<math>\nabla_{\mathrm{X}_i} \mathrm{X}_j = 0 \, ,</math>
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| hence <math>W^k{}_{ij}=\omega^k(\nabla_{\mathrm{X}_i} \mathrm{X}_j)\equiv0 \, ,</math> for the connection coefficients (also called Weitzenböck coefficients) —in this global base. Here <math>\omega^k\,</math> is the dual global base (or co-frame) defined by <math>\omega^i(\mathrm{X}_j)=\delta^i_j\,</math>.
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| This is what usually happens in ''R''<sup>n</sup>, in any [[affine space]] or [[Lie group]] (for example the 'curved' sphere ''S''<sup>3</sup> but 'Weitzenböck flat' manifold).
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| '''Weitzenböck connection''' has vanishing [[Riemann curvature tensor|curvature]], but —in general— non-vanishing [[torsion tensor|torsion]].
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| Given the frame field <math>\{X_{i}\}</math>, one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain a [[pseudo-Riemannian]] [[metric tensor]] field <math>g\,</math> of [[metric signature|signature]] (3,1) by
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| :<math>g(X_{i},X_{j})=\eta_{ij}\,</math>,
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| where
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| :<math>\eta_{ij}={\mathrm {diag}}(-1,-1,-1,1)\,</math>.
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| The corresponding underlying spacetime is called, in this case, a [[Roland Weitzenböck|Weitzenböck]] spacetime.<ref>[http://relativity.livingreviews.org/open?pubNo=lrr-2004-2&page=articlese10.html On the History of Unified Field Theories]</ref>
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| It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor as a by-product.
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| ==New teleparallel gravity theory==
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| '''New teleparallel gravity theory''' (or '''new general relativity''') is a theory of gravitation on Weitzenböck space-time, and attributes gravitation to the torsion tensor formed of the parallel vector fields.
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| In the New teleparallel gravity theory the fundamental assumptions are as follows: (A) Underlying space-time is the Weitzenböck space-time, which has a quadruplet of parallel vector fields as the fundamental structure. These parallel vector fields give rise to the metric tensor as a by-product. All physical laws are expressed by equations that are covariant or form invariant under the group of general coordinate transformations. (B) The [[equivalence principle]] is valid only in classical physics. (C) Gravitational
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| field equations are derivable from the action principle. (D) The field equations are partial differential equations in the field variables of not higher than the second order.
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| In 1961 Møller<ref>{{cite journal|title=|author=C. Møller|journal=K. Dan. Vidensk. Selsk. Mat. Fys. Skr.|volume=1|number=10|year=1961|page=1}}</ref> revived Einstein’s idea, and Pellegrini e Plebanski<ref>{{cite journal|title=|author=C. Pellegrini and J. Plebanski|journal=K. Dan. Vidensk. Selsk. Mat. Fys. Skr.|volume=2|number=2|year=1962|page=1}}</ref> found a Lagrangian formulation for ''absolute parallelism''.
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| ==New translation teleparallel gauge theory of gravity==
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| In 1967, quite independently, Hayashi and Nakano started to formulate the gauge theory of the space-time translation group. Hayashi pointed out the connection between the gauge theory of space-time translation group and absolute parallelism.
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| Nowadays, people study teleparallelism purely as a theory of gravity <ref>{{cite journal |last=Arcos |first=H.I. |coauthors=J.G. Pereira |date=January 2005 |title=Torsion Gravity: a Reappraisal |doi=10.1142/S0218271804006462 |volume=13 |issue=10 |pages=2193–2240 |journal=Int.J.Mod.Phys. D |arxiv=gr-qc/0501017|bibcode = 2004IJMPD..13.2193A }}</ref> without trying to unify it with electromagnetism. In this theory, the [[gravitational field]] turns out to be fully represented by the translational [[gauge potential]] <math>B^a{\!}_\mu</math>, as it should be for a [[gauge theory]] for the translation group.
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| If this choice is made, then there is no longer any [[Hendrik Lorentz|Lorentz]] [[gauge symmetry]] because the internal [[Minkowski space]] [[Fiber bundle|fiber]]—over each point of the spacetime [[manifold]]—belongs to a [[fiber bundle]] with the abelian '''R'''<sup>4</sup> as [[structure group]]. However, a translational gauge symmetry may be introduced thus: Instead of seeing [[Frame fields in general relativity|tetrads]] as fundamental, we introduce a fundamental '''R'''<sup>4</sup> translational gauge symmetry instead (which acts upon the internal Minkowski space fibers [[affine]]ly so that this fiber is once again made local) with a [[connection (mathematics)|connection]] '''B''' and a "coordinate field" '''x''' taking on values in the Minkowski space fiber.
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| More precisely, let <math>\pi\colon{\mathcal M}\to M</math> be the [[Minkowski]] [[fiber bundle]] over the spacetime [[manifold]] M. For each point <math>p\in M</math>, the fiber <math>{\mathcal M}_p</math> is an [[affine space]]. In a fiber chart <math>(V,\psi)\,</math>, coordinates are usually denoted by <math>\psi = (x^\mu,x^a)\,</math>, where <math>x^{\mu}\,</math> are coordinates on spacetime manifold M, and '''x<sup>a</sup>''' are coordinates in the fiber <math>{\mathcal M}_p\,</math>.
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| Using the [[abstract index notation]], let ''a'', ''b'', ''c'', ... refer to <math>{\mathcal M}_p</math> and ''μ'', ''ν'', ... refer to the [[tangent bundle]] <math>TM</math>. In any particular gauge, the value of '''x'''<sup>''a''</sup> at the point ''p'' is given by
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| :<math>x^a(p).</math>
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| The [[covariant derivative]]
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| :<math>D_\mu x^a \equiv (dx^a)_\mu + B^a{\!}_\mu = \partial_\mu x^a + B^a{\!}_\mu</math>
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| is defined with respect to the [[connection form]] B, a 1-form assuming values in the [[Lie algebra]] of the translational abelian group '''R'''<sup>4</sup>. Here, d is the [[exterior derivative]] of the a<sup>th</sup> ''component'' of x, which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation field '''α<sup>a</sup>''',
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| :<math>x^a\rightarrow x^a+\alpha^a</math>
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| and
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| :<math>B^a{\!}_\mu\rightarrow B^a{\!}_\mu - \partial_{\mu}\alpha^a</math>
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| and so, the covariant derivative of '''x'''<sup>''a''</sup> is [[gauge invariant]]. This is identified with the tetrad
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| :<math>e^a{\!}_\mu = \partial_{\mu}x^a + B^a{\!}_\mu</math>
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| (which is a [[one-form]] which takes on values in the vector [[Minkowski space]], not the affine Minkowski space, which is why it's gauge invariant). But what does this mean? '''x<sup>a</sup>''' is somewhat like a coordinate function, giving an internal space value to each point p. The [[holonomy]] associated with B specifies the displacement of a path according to the internal space.
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| A crude analogy: Think of <math>{\mathcal M}_p</math> as the computer screen and the internal displacement as the position of the mouse pointer. Think of a curved mousepad as spacetime and the position of the mouse as the position. Keeping the orientation of the mouse fixed, if we move the mouse about the curved mousepad, the position of the mouse pointer (internal displacement) also changes and this change is path dependent; i.e., it doesn't only depend upon the initial and final position of the mouse. The change in the internal displacement as we move the mouse about a closed path on the mousepad is the torsion.
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| Another crude analogy: Think of a [[crystal]] with [[line defect]]s ([[edge dislocation]]s and [[screw dislocation]]s but not [[disclination]]s). The parallel transport of a point of <math>{\mathcal M}</math> along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed. The [[Burgers vector]] corresponds to the torsion. Disinclinations correspond to curvature, which is why they are left out.
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| The torsion, i.e., the translational [[field strength]] of Teleparallel Gravity (or the translational "curvature"),
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| :<math>T^a{\!}_{\mu\nu} \equiv (DB^a)_{\mu\nu} = D_\mu B^a{\!}_\nu - D_\nu B^a{\!}_\mu,</math>
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| is gauge invariant.
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| Of course, we can always choose the gauge where '''x<sup>a</sup>''' is zero everywhere (a problem though; <math>{\mathcal M}_p</math> is an affine space and also a fiber and so, we have to define the "origin" on a point by point basis, but this can always be done arbitrarily) and this leads us back to the theory where the tetrad is fundamental.
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| Teleparallelism refers to any theory of gravitation based upon this framework. There is a particular choice of the [[action (physics)|action]] which makes it exactly equivalent <ref>{{cite journal|title=Einstein Lagrangian as the translational Yang–Mills Lagrangian|author=Y.M. Cho|journal=Physical Review|volume=D 14|year=1976|page=2521}}</ref><!-- perhaps, the action may be made equivalent to Hilbert’s one, but what about space-time topology? can you made a Schwarzschield black hole or a Kerr black hole in Minkowski space? --> to general relativity, but there are also other choices of the action which aren't equivalent to GR. In some of these theories, there is no equivalence between [[inertial mass|inertial]] and [[gravitational mass]]es.
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| Unlike GR, gravity is not due to the curvature of spacetime. It is due to the torsion.
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| ===Remark===
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| There exists a close analogy of [[Riemannian geometry|geometry]] of spacetime with the structure of defects in crystal.<ref>
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| {{cite journal
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| | title = Gauge Fields in Condensed Matter Vol II
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| | author = [[Hagen Kleinert|H. Kleinert]]
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| | pages = 743–1440
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| | year = 1989
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| | url = http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html
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| }}</ref><ref>
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| {{cite journal
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| | title = Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation
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| | author = [[Hagen Kleinert|H. Kleinert]]
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| | pages = 1–496
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| | year = 2008
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| | url = http://users.physik.fu-berlin.de/~kleinert/b11/psfiles/mvf.pdf
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| }}</ref> [[Dislocations]] are represented by torsion, [[disclination]]s by curvature.
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| These defects are not independent of each other.
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| A dislocation is equaivalent to a disclination-antidisclination pair, a disclination is equivalent to a string of dislocations.
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| This is the basic reason why Einstein's theory based purely on curvature can be rewritten as
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| a teleparallel theory based only on torsion.
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| There exists, moreover, infinitely many ways of rewriting
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| Einstein's theory, depending on how much of the curvature one wants to reexpress
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| in terms of torsion, the teleparallel theory being merely one specific version of these.<ref>
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| {{cite journal
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| | title = New Gauge Symmetry in Gravity and the Evanescent Role of Torsion
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| | author = [[Hagen Kleinert|H. Kleinert]]
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| | pages = 287–298
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| | year = 2010
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| | url = http://users.physik.fu-berlin.de/~kleinert/385/385.pdf
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| }}</ref>
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| ==See also==
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| *[[Classical theories of gravitation]]
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| *[[Gauge gravitation theory]]
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| ==References==
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| <references/>
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| ==Books==
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| * {{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds| publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
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| * {{citation|last1 = Weitzenböck|first1=R.|authorlink=Roland Weitzenböck |title=Invariantentheorie|year = 1923|publisher = Groningen: Noordhoff}}
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| ==External links==
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| *[http://www.phy.olemiss.edu/~luca/Topics/grav/teleparallel.html Teleparallel Structures and Gravity Theories by Luca Bombelli]
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| *[http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/selected_papers_on_teleparallelism.pdf ''Selected Papers on Teleparallelism'', translated and edited by D. H. Delphenich]
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| {{theories of gravitation}}
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| [[Category:History of physics]]
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| [[Category:Theories of gravitation]]
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