Modular exponentiation

Teleparallelism (also called teleparallel gravity), was an attempt by Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In this theory, a spacetime is characterized by a curvature-free linear connection in conjunction with a metric tensor field, both defined in terms of a dynamical tetrad field.

Teleparallel spacetimes

The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e., a set $\{{\mathrm{X}}_1,\dots ,{\mathrm{X}}_4\}$ of four vector fields defined on all of $M$ such that for every $p\in M$ the set $\{{\mathrm{X}}_1(p),\dots ,{\mathrm{X}}_4(p)\}$ is a basis of $T_{p}M$, where $T_{p}M$ denotes the fiber over $p$ of the tangent vector bundle $TM$. Hence, the fourdimensional spacetime manifold $M$ must be a parallelizable 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 the connection of the parallelization (also called Weitzenböck connection) $\{{\mathrm{X}}_i\}$ to be the linear connection $\mathrm{\nabla }$ on $M$ such that

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

where $v\in T_{p}M$ and $f^i$ are (global) functions on $M$; thus $f^i{X}_i$ is a global vector field on $M$. In other words, the coefficients of Weitzenböck connection $\mathrm{\nabla }$ with respect to $\{X_i\}$ are all identically zero, implicitly defined by:

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

hence $W^k{}_{ij}={\omega }^k({\mathrm{\nabla }}_{{\mathrm{X}}_i}{\mathrm{X}}_j)\equiv 0,$ for the connection coefficients (also called Weitzenböck coefficients) —in this global base. Here ${\omega }^k$ is the dual global base (or co-frame) defined by ${\omega }^i({\mathrm{X}}_j)={\delta }_j^i$.

This is what usually happens in Rⁿ, in any affine space or Lie group (for example the 'curved' sphere S³ but 'Weitzenböck flat' manifold).

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 pseudo-Riemannian metric tensor field $g$ of signature (3,1) by

$g(X_i,X_j)={\eta }_{ij}$,

where

${\eta }_{ij}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,-1,-1,1)$.

The corresponding underlying spacetime is called, in this case, a Weitzenböck spacetime.

It is worth noting to see that these 'parallel vector fields' give rise to the metric tensor as a by-product.

New teleparallel gravity theory

New teleparallel gravity theory (or new general relativity) is a theory of gravitation on Weitzenböck space-time, 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: (A) Underlying space-time is the Weitzenböck space-time, which has a quadruplet of parallel vector fields as the fundamental structure. These parallel vector fields give rise to the metric tensor as a by-product. All physical laws are expressed by equations that are covariant or form invariant under the group of general coordinate transformations. (B) The equivalence principle is valid only in classical physics. (C) Gravitational field equations are derivable from the action principle. (D) The field equations are partial differential equations in the field variables of not higher than the second order.

In 1961 Møller revived Einstein’s idea, and Pellegrini e Plebanski found a Lagrangian formulation for absolute parallelism.

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 $B^a{}_{\mu }$, 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 $\pi :\mathcal{ℳ}\to M$ be the Minkowski fiber bundle over the spacetime manifold M. For each point $p\in M$, the fiber ${\mathcal{ℳ}}_p$ is an affine space. In a fiber chart $(V,\psi )$, coordinates are usually denoted by $\psi =(x^{\mu },x^a)$, where $x^{\mu }$ are coordinates on spacetime manifold M, and x^a are coordinates in the fiber ${\mathcal{ℳ}}_p$.

Using the abstract index notation, let a, b, c, ... refer to ${\mathcal{ℳ}}_p$ 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

$x^a(p).$

The covariant derivative

$D_{\mu }{x}^a\equiv (d{x}^a{)}_{\mu }+B^a{}_{\mu }={\partial }_{\mu }{x}^a+B^a{}_{\mu }$

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 a^th 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,

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

and

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

and so, the covariant derivative of x^a is gauge invariant. This is identified with the tetrad

$e^a{}_{\mu }={\partial }_{\mu }{x}^a+B^a{}_{\mu }$

(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? x^a is somewhat 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 ${\mathcal{ℳ}}_p$ 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 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 $\mathcal{ℳ}$ 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, i.e., the translational field strength of Teleparallel Gravity (or the translational "curvature"),

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

is gauge invariant.

Of course, we can always choose the gauge where x^a is zero everywhere (a problem though; ${\mathcal{ℳ}}_p$ 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.

Remark

There exists a close analogy of geometry of spacetime with the structure of defects in crystal. Dislocations are represented by torsion, disclinations by curvature. These defects are not independent of each other. A dislocation is equaivalent 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.

See also

• Classical theories of gravitation
• Gauge gravitation theory

Books

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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External links

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