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Teleparallelism (also calledteleparallel gravity), was an attempt byAlbert Einstein^[1] to base a unified theory ofelectromagnetism andgravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, aspacetime is characterized by a curvature-freelinear connection in conjunction with ametric tensor field, both defined in terms of a dynamicaltetrad field.

The crucial new idea, for Einstein, was the introduction of atetrad field, i.e., a set{X₁, X₂, X₃, X₄} of fourvector fields defined onall ofM such that for everyp ∈M the set{X₁(p), X₂(p), X₃(p), X₄(p)} is abasis ofT_pM, whereT_pM denotes the fiber overp of thetangent vector bundleTM. Hence, the four-dimensionalspacetime manifoldM must be aparallelizable 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.

In fact, one can define theconnection of the parallelization (also called theWeitzenböck connection){X_i} to be thelinear connection∇ onM such that^[2]

$${\displaystyle {\mathrm{\nabla }}_v\left(f^i{\mathrm{X}}_i\right)=\left(v{f}^i\right){\mathrm{X}}_i(p),}$$

wherev ∈T_pM andfⁱ are (global) functions onM; thusfⁱX_i is a global vector field onM. In other words, the coefficients ofWeitzenböck connection∇ with respect to{X_i} are all identically zero, implicitly defined by:

$${\displaystyle {\mathrm{\nabla }}_{{\mathrm{X}}_i}{\mathrm{X}}_j=0,}$$

hence

$${\displaystyle {W^k}_{ij}={\omega }^k\left({\mathrm{\nabla }}_{{\mathrm{X}}_i}{\mathrm{X}}_j\right)\equiv 0,}$$

for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Hereω^k is the dual global basis (or coframe) defined byωⁱ(X_j) =δⁱ
_j.

This is what usually happens inRⁿ, in anyaffine space orLie group (for example the 'curved' sphereS³ but 'Weitzenböck flat' manifold).

Using the transformation law of a connection, or equivalently the∇ properties, we have the following result.

Proposition. In a natural basis, associated with local coordinates(U,x^μ), i.e., in the holonomic frame∂_μ, the (local) connection coefficients of the Weitzenböck connection are given by:

$${\displaystyle {{\mathrm{\Gamma }}^{\beta }}_{\mu \nu }=h_i^{\beta }{\mathrm{\partial }}_{\nu }{h}_{\mu }^i,}$$

whereX_i =h^μ
_i∂_μ fori,μ = 1, 2,…n are the local expressions of a global object, that is, the given tetrad.

TheWeitzenböck connection has vanishingcurvature, but – in general – non-vanishingtorsion.

Given the frame field{X_i}, one can also define a metric by conceiving of the frame field as an orthonormal vector field. One would then obtain apseudo-Riemannianmetric tensor fieldg ofsignature (3,1) by

$${\displaystyle g\left({\mathrm{X}}_i,{\mathrm{X}}_j\right)={\eta }_{ij},}$$

where

$${\displaystyle {\eta }_{ij}=\mathrm{diag}(-1,-1,-1,1).}$$

The corresponding underlying spacetime is called, in this case, aWeitzenböck spacetime.^[3]

These 'parallel vector fields' give rise to the metric tensor as a byproduct.

New teleparallel gravity theory (ornew general relativity) is a theory of gravitation on Weitzenböck spacetime, and attributes gravitation to the torsion tensor formed of the parallel vector fields.

In the new teleparallel gravity theory the fundamental assumptions are as follows:

In 1961Christian Møller^[4] revived Einstein's idea, and Pellegrini and Plebanski^[5] found a Lagrangian formulation forabsolute parallelism.

In 1961, Møller^[4][6] showed that atetrad description of gravitational fields allows a more rational treatment of theenergy-momentum complex than in a theory based on themetric tensor alone. The advantage of using tetrads as gravitational variables was connected with the fact that this allowed to construct expressions for the energy-momentum complex which had more satisfactory transformation properties than in a purely metric formulation. In 2015, it was shown that the total energy of matter and gravitation is proportional to theRicci scalar of three-space up to the linear order of perturbation.^[7]

Independently in 1967, Hayashi and Nakano^[8] revived Einstein's idea, and Pellegrini and Plebanski^[5] started to formulate thegauge theory of the spacetimetranslation group.^{[clarification needed]} Hayashi pointed out the connection between the gauge theory of the spacetime translation group and absolute parallelism. The firstfiber bundle formulation was provided by Cho.^[9] This model was later studied by Schweizer et al.,^[10] Nitsch and Hehl, Meyer;^{[citation needed]} more recent advances can be found in Aldrovandi and Pereira, Gronwald, Itin, Maluf and da Rocha Neto, Münch, Obukhov and Pereira, and Schucking and Surowitz.^{[citation needed]}

Nowadays, teleparallelism is studied purely as a theory of gravity^[11] without trying to unify it with electromagnetism. In this theory, thegravitational field turns out to be fully represented by the translationalgauge potentialB^a_μ, as it should be for agauge theory for the translation group.

If this choice is made, then there is no longer anyLorentzgauge symmetry because the internalMinkowski spacefiber—over each point of the spacetimemanifold—belongs to afiber bundle with theAbelian groupR⁴ asstructure group. However, a translational gauge symmetry may be introduced thus: Instead of seeingtetrads as fundamental, we introduce a fundamentalR⁴ translational gauge symmetry instead (which acts upon the internal Minkowski space fibersaffinely so that this fiber is once again made local) with aconnectionB and a "coordinate field"x taking on values in the Minkowski space fiber.

More precisely, letπ :M →M be theMinkowskifiber bundle over the spacetimemanifoldM. For each pointp ∈M, the fiberM_p is anaffine space. In a fiber chart(V,ψ), coordinates are usually denoted byψ = (x^μ,x^a), wherex^μ are coordinates on spacetime manifoldM, andx^a are coordinates in the fiberM_p.

Using theabstract index notation, leta,b,c,… refer toM_p andμ,ν,… refer to thetangent bundleTM. In any particular gauge, the value ofx^a at the pointp is given by thesection

$${\displaystyle x^{\mu }\to \left(x^{\mu },x^a={\xi }^a(p)\right).}$$

Thecovariant derivative

$${\displaystyle D_{\mu }{\xi }^a\equiv {\left(d{\xi }^a\right)}_{\mu }+{B^a}_{\mu }={\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }}$$

is defined with respect to theconnection formB, a 1-form assuming values in theLie algebra of the translational abelian groupR⁴. Here, d is theexterior derivative of theathcomponent ofx, which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation fieldα^a,

$${\displaystyle x^a\to x^a+{\alpha }^a}$$

and

$${\displaystyle {B^a}_{\mu }\to {B^a}_{\mu }-{\mathrm{\partial }}_{\mu }{\alpha }^a}$$

and so, the covariant derivative ofx^a =ξ^a(p) isgauge invariant. This is identified with the translational (co-)tetrad

$${\displaystyle {h^a}_{\mu }={\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }}$$

which is aone-form which takes on values in theLie algebra of the translational Abelian groupR⁴, whence it is gauge invariant.^[12] But what does this mean?x^a =ξ^a(p) is a local section of the (pure translational) affine internal bundleM →M, another important structure in addition to the translational gauge fieldB^a_μ. Geometrically, this field determines the origin of the affine spaces; it is known asCartan's radius vector. In the gauge-theoretic framework, the one-form

$${\displaystyle h^a={h^a}_{\mu }d{x}^{\mu }=\left({\mathrm{\partial }}_{\mu }{\xi }^a+{B^a}_{\mu }\right)d{x}^{\mu }}$$

arises as the nonlinear translational gauge field withξ^a interpreted as theGoldstone field describing the spontaneous breaking of the translational symmetry.

A crude analogy: Think ofM_p 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 does not depend only 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.

Another crude analogy: Think of acrystal withline defects (edge dislocations andscrew dislocations but notdisclinations). The parallel transport of a point ofM along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed. TheBurgers vector corresponds to the torsion. Disinclinations correspond to curvature, which is why they are neglected.

The torsion—that is, the translationalfield strength of Teleparallel Gravity (or the translational "curvature")—

$${\displaystyle {T^a}_{\mu \nu }\equiv {\left(D{B}^a\right)}_{\mu \nu }=D_{\mu }{B^a}_{\nu }-D_{\nu }{B^a}_{\mu },}$$

isgauge invariant.

We can always choose the gauge wherex^a is zero everywhere, althoughM_p is an affine space and also a fiber; thus the origin must be defined on a point-by-point basis, which can be done arbitrarily. This leads us back to the theory where the tetrad is fundamental.

Teleparallelism refers to any theory of gravitation based upon this framework. There is a particular choice of theaction that makes it exactly equivalent^[9] to general relativity, but there are also other choices of the action which are not equivalent to general relativity. In some of these theories, there is no equivalence betweeninertial andgravitational masses.^[13]

Unlike in general relativity, gravity is due not to the curvature of spacetime but to the torsion thereof.

There exists a close analogy ofgeometry of spacetime with the structure of defects in crystals.^[14][15]Dislocations are represented by torsion,disclinations by curvature. These defects are not independent of each other. A dislocation is equivalent to a disclination-antidisclination pair, a disclination is equivalent to a string of dislocations. This is the basic reason why Einstein's theory based purely on curvature can be rewritten as a teleparallel theory based only on torsion. There exists, moreover, infinitely many ways of rewriting Einstein's theory, depending on how much of the curvature one wants to reexpress in terms of torsion, the teleparallel theory being merely one specific version of these.^[16]

A further application of teleparallelism occurs in quantum field theory, namely, two-dimensionalnon-linear sigma models with target space on simple geometric manifolds, whose renormalization behavior is controlled by aRicci flow, which includestorsion. This torsion modifies the Ricci tensor and hence leads to aninfrared fixed point for the coupling, on account of teleparallelism ("geometrostasis").^[17]

• Classical theories of gravitation
• Gauge gravitation theory
• Geometrodynamics
• Kaluza–Klein theory

• Aldrovandi, R.; Pereira, J. G. (2012).Teleparallel Gravity: An Introduction. Springer: Dordrecht.ISBN 978-94-007-5142-2.{{cite book}}: CS1 maint: publisher location (link)
• Bishop, R. L.; Goldberg, S. I. (1968).Tensor Analysis on Manifolds (First Dover 1980 ed.). Macmillan.ISBN 978-0-486-64039-6.
• Weitzenböck, R. (1923).Invariantentheorie. Groningen: Noordhoff.{{cite book}}: CS1 maint: publisher location (link)

• Selected Papers on Teleparallelism, translated and edited by D. H. Delphenich
• Teleparallel gravity at thenLab
• Teleparallel Structures and Gravity Theories by Luca Bombelli

• History of physics
• Theories of gravity

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