Teleparallelism - New Translation Teleparallel Gauge Theory of Gravity

New Translation Teleparallel Gauge Theory of Gravity

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.

Nowadays, people study teleparallelism purely as a theory of gravity without trying to unify it with electromagnetism. In this theory, the gravitational field turns out to be fully represented by the translational gauge potential, as it should be for a gauge theory for the translation group.

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 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.

More precisely, let be the Minkowski fiber bundle over the spacetime manifold M. For each point, the fiber is an affine space. In a fiber chart, coordinates are usually denoted by, where are coordinates on spacetime manifold M, and xa are coordinates in the fiber .

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

The covariant derivative

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 αa,

and

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

(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 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 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, i.e., the translational field strength of Teleparallel Gravity (or the translational "curvature"),

is gauge invariant.

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