Teleparallel gravity

Teleparallelism

Teleparallelism (also called distant parallelism and teleparallel gravity), was an attempt by Einstein to unify electromagnetism and gravity. The idea is to use a geometry with a pseudo-Riemannian metric of signature (3,1), vanishing curvature, and non-vanishing torsion, and to use tetrads, rather than the metric, as basic variables.

Nowadays, people study teleparallelism purely as a theory of gravity without trying to unify it with electromagnetism.

Mathematical details

General relativity is usually formulated as a diffeomorphism covariant theory of a metric tensor. Associated with this metric tensor, there is a Levi-Civita connection which is torsionless but has a nonzero curvature form (Riemann tensor) in general.

In the alternative but equivalent tetrad formalism, tetrads and spin connections are fundamental and, as before, there are diffeomorphism symmetries, but there is also a Lorentz gauge symmetry (i.e. a Spin(3,1) gauge symmetry). The standard assumption is to make the spin connection torsionless, but it can equally well be chosen to have a nonzero torsion but zero curvature form. The latter choice leads to the Weitzenböck connection. The zero curvature condition means that there is a local moving frame, provided the spacetime is simply connected, since the parallel transport of the tetrad is path independent. There is a global moving frame provided the spacetime is a parallelizable manifold.

If this choice is made, then there is no longer any Lorentz gauge symmetry and the internal Minkowski space fiber is now global. 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.

More precisely, let the Minkowski space fiber be M. This is an affine space. Using the abstract index notation, let a, b, c,... refer to M and μ, ν, ... refer to the tangent bundle. In any particular gauge, the value of xa at the point p is given by

x^a(p).

The covariant derivative

D_mu x^a equiv (dx^a)_mu + B^a_mu

is defined with respect to the connection form B. 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 αa,

x^arightarrow x^a+alpha^a

and

B^a_murightarrow B^a_mu - (dalpha^a)_mu

and so, the covariant derivative of xa is gauge invariant. This is identified with the tetrad eaμ (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? xa is sort of 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.

A crude analogy: Think of M 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.

Another crude analogy: Think of a crystal with line defects (edge dislocations and screw dislocations but not disclinations). 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. The Burgers vector corresponds to the torsion. Disinclinations correspond to curvature, which is why they are left out.

The torsion,

T^a_{munu} equiv (dB^a)_{munu},

is gauge invariant.

Of course, we can always choose the gauge where xa is zero everywhere (a problem though; M 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.

Teleparallelism refers to any theory of gravitation based upon this framework. There is a particular choice of the action which makes it exactly equivalent 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 and gravitational masses.

Unlike GR, gravity is not due to the curvature of spacetime. It is due to the torsion.

See also

References

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