The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set {X1, X2, X3, X4} of four vector fields defined on all of M such that for every p ∈ M the set {X1(p), X2(p), X3(p), X4(p)} is a basis of TpM, where TpM denotes the fiber over p of the tangent vector bundle TM.
His attempt failed because there was no Schwarzschild solution in his simplified field equation.
where v ∈ TpM and fi are (global) functions on M; thus fiXi is a global vector field on M. In other words, the coefficients of Weitzenböck connection ∇ with respect to {Xi} are all identically zero, implicitly defined by:
Here ωk is the dual global basis (or coframe) defined by ωi(Xj) = δij.
This is what usually happens in Rn, in any affine space or Lie group (for example the 'curved' sphere S3 but 'Weitzenböck flat' manifold).
where Xi = hμi∂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.
[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[4][6] showed that a tetrad description of gravitational fields allows a more rational treatment of the energy-momentum complex than in a theory based on the metric 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 the Ricci 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 the gauge theory of the spacetime translation group.
[clarification needed] Hayashi pointed out the connection between the gauge theory of the spacetime translation group and absolute parallelism.
[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.
If this choice is made, then there is no longer any Lorentz gauge symmetry because the internal Minkowski space fiber—over each point of the spacetime manifold—belongs to a fiber bundle with the Abelian group R4 as structure group.
However, a translational gauge symmetry may be introduced thus: Instead of seeing tetrads as fundamental, we introduce a fundamental R4 translational gauge symmetry instead (which acts upon the internal Minkowski space fibers affinely so that this fiber is once again made local) with a connection B and a "coordinate field" x taking on values in the Minkowski space fiber.
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).
which is a one-form which takes on values in the Lie algebra of the translational Abelian group R4, whence it is gauge invariant.
Geometrically, this field determines the origin of the affine spaces; it is known as Cartan’s radius vector.
A crude analogy: Think of Mp as the computer screen and the internal displacement as the position of the mouse pointer.
The change in the internal displacement as we move the mouse about a closed path on the mousepad is the torsion.
The parallel transport of a point of M along a path is given by counting the number of (up/down, forward/backwards and left/right) crystal bonds transversed.
We can always choose the gauge where xa is zero everywhere, although Mp 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.
[13] Unlike in general relativity, gravity is due not to the curvature of spacetime but to the torsion thereof.
There exists a close analogy of geometry of spacetime with the structure of defects in crystal.
[16] 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").