Teleparallelism

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Teleparallelism (also called teleparallel gravity), was an attempt by

The crucial new idea, for Einstein, was the introduction of a

In fact, one can define the connection of the parallelization (also called the Weitzenböck connection) {X_i} to be the linear connection ∇ on M such that^[2]

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

where v ∈ T_pM and fⁱ are (global) functions on M; thus fⁱX_i is a global vector field on M. In other words, the coefficients of Weitzenböck connection ∇ with respect to {X_i} are all identically zero, implicitly defined by:

$${\displaystyle \nabla _{\mathrm {X} _{i}}\mathrm {X} _{j}=0,}$$

hence

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

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

This is what usually happens in Rⁿ, in any affine space or Lie group (for example the 'curved' sphere S³ 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:

$${\displaystyle {\Gamma ^{\beta }}_{\mu \nu }=h_{i}^{\beta }\partial _{\nu }h_{\mu }^{i},}$$

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

The Weitzenböck connection has vanishing curvature, but – in general – non-vanishing torsion.

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 a

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

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

New teleparallel gravity theory (or new 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 1961 Christian Møller^[4] revived Einstein's idea, and Pellegrini and Plebanski^[5] found a Lagrangian formulation for absolute parallelism.

In 1961, Møller

More precisely, let π : M → M be the Minkowski fiber bundle over the spacetime manifold M. For each point p ∈ M, the fiber M_p is an affine space. In a fiber chart (V, ψ), coordinates are usually denoted by ψ = (x^μ, x^a), where x^μ are coordinates on spacetime manifold M, and x^a are coordinates in the fiber M_p.

Using the abstract index notation, let a, b, c,… refer to M_p and μ, ν,… refer to the tangent bundle TM. In any particular gauge, the value of x^a at the point p is given by the section

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

The covariant derivative

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

is defined with respect to the connection form B, a 1-form assuming values in the Lie algebra of the translational abelian group R⁴. Here, d is the exterior derivative of the ath 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^a,

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

and

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

and so, the covariant derivative of x^a = ξ^a(p) is

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

which is a

arises as the nonlinear translational gauge field with ξ^a interpreted as the Goldstone field describing the spontaneous breaking of the translational symmetry.

A crude analogy: Think of M_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 a

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

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

is

We can always choose the gauge where x^a is zero everywhere, although M_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 the

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

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