Editing Teleparallelism (section)

==New translation teleparallel gauge theory of gravity==
Independently in 1967, Hayashi and Nakano<ref>{{cite journal|title=Extended Translation Invariance and Associated Gauge Fields| first1=K. |last1=Hayashi| first2=T.|last2=Nakano|journal=Prog. Theor. Phys.| volume=38| number=2| date=1967| pages=491–507| doi=10.1143/ptp.38.491|bibcode = 1967PThPh..38..491H |doi-access=free}}</ref> revived Einstein's idea, and Pellegrini and Plebanski<ref name=pellegrini>{{cite journal|title=Tetrad fields and gravitational fields| first1=C.|last1=Pellegrini| first2=J.|last2=Plebanski|journal=Mat. Fys. SKR. Dan. Vid. Selsk.| volume=2| number=4| date=1963| pages=1–39}}</ref> started to formulate the [[gauge theory]] of the spacetime [[translation group]].{{clarify|date=September 2023}} Hayashi pointed out the connection between the gauge theory of the spacetime translation group and absolute parallelism. The first [[fiber bundle]] formulation was provided by Cho.<ref name=cho>{{cite journal| title=Einstein Lagrangian as the translational Yang–Mills Lagrangian|first=Y.-M. |last=Cho|journal=Physical Review D |volume=14 |issue=10 |date=1976 |page=2521 | doi=10.1103/physrevd.14.2521 |bibcode = 1976PhRvD..14.2521C}}</ref> This model was later studied by Schweizer et al.,<ref>{{cite journal|title=Postnewtonian generation of gravitational waves in a theory of gravity with torsion| first1=M.| last1=Schweizer |first2=N.| last2=Straumann |first3=A.|last3=Wipf|journal=Gen. Rel. Grav.|volume=12 | issue=11|date=1980|pages=951–961|doi=10.1007/bf00757366|bibcode = 1980GReGr..12..951S | s2cid=121759701|arxiv=2305.01603}}</ref> Nitsch and Hehl, Meyer;{{cn|date=September 2023}} 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.{{cn|date=September 2023}}

Nowadays, teleparallelism is studied purely as a theory of gravity<ref>{{cite journal |last1=Arcos |first1=H. I. |first2=J. G. |last2=Pereira |date=January 2005 |title=Torsion Gravity: a Reappraisal |doi=10.1142/S0218271804006462 |volume=13 |issue=10 |pages=2193–2240 |journal=Int. J. Mod. Phys. D |arxiv=gr-qc/0501017|bibcode = 2004IJMPD..13.2193A |s2cid=119540585 }}</ref> without trying to unify it with electromagnetism. In this theory, the [[gravitational field]] turns out to be fully represented by the translational [[gauge potential]] {{math|''B<sup>a</sup><sub>μ</sub>''}}, as it should be for a [[gauge theory]] for the translation group.

If this choice is made, then there is no longer any [[Hendrik Lorentz|Lorentz]] [[gauge symmetry]] because the internal [[Minkowski space]] [[Fiber bundle|fiber]]—over each point of the spacetime [[manifold]]—belongs to a [[fiber bundle]] with the [[Abelian group]] {{math|'''R'''<sup>4</sup>}} as [[structure group]]. However, a translational gauge symmetry may be introduced thus: Instead of seeing [[Frame fields in general relativity|tetrads]] as fundamental, we introduce a fundamental {{math|'''R'''<sup>4</sup>}} translational gauge symmetry instead (which acts upon the internal Minkowski space fibers [[Affine space|affinely]] so that this fiber is once again made local) with a [[connection (mathematics)|connection]] {{mvar|B}} and a "coordinate field" {{mvar|x}} taking on values in the Minkowski space fiber.

More precisely, let {{math|''π'' : {{mathcal|M}} → ''M''}} be the [[Minkowski space|Minkowski]] [[fiber bundle]] over the spacetime [[manifold]] {{mvar|M}}. For each point {{math|''p'' ∈ ''M''}}, the fiber {{math|{{mathcal|M}}<sub>''p''</sub>}} is an [[affine space]]. In a fiber chart {{math|(''V'', ''ψ'')}}, coordinates are usually denoted by {{math|''ψ'' {{=}} (''x<sup>μ</sup>'', ''x<sup>a</sup>'')}}, where {{mvar|x<sup>μ</sup>}} are coordinates on spacetime manifold {{mvar|M}}, and {{mvar|x<sup>a</sup>}} are coordinates in the fiber {{math|{{mathcal|M}}<sub>''p''</sub>}}.

Using the [[abstract index notation]], let {{math|''a'', ''b'', ''c'',…}} refer to {{math|{{mathcal|M}}<sub>''p''</sub>}} and {{math|''μ'', ''ν'',…}} refer to the [[tangent bundle]] {{mvar|TM}}. In any particular gauge, the value of {{mvar|x<sup>a</sup>}} at the point ''p'' is given by the [[Section (fiber bundle)|section]]

<math display="block">x^\mu \to \left(x^\mu,x^a = \xi^a(p)\right).</math>

The [[covariant derivative]]

<math display="block">D_\mu \xi^a \equiv \left(d \xi^a\right)_\mu + {B^a}_\mu = \partial_\mu \xi^a + {B^a}_\mu</math>

is defined with respect to the [[connection form]] {{mvar|B}}, a 1-form assuming values in the [[Lie algebra]] of the translational abelian group {{math|'''R'''<sup>4</sup>}}. Here, d is the [[exterior derivative]] of the {{mvar|a}}th ''component'' of {{mvar|x}}, which is a scalar field (so this isn't a pure abstract index notation). Under a gauge transformation by the translation field {{mvar|α<sup>a</sup>}},

<math display="block">x^a\to x^a+\alpha^a</math>

and

<math display="block">{B^a}_\mu\to {B^a}_\mu - \partial_\mu \alpha^a</math>

and so, the covariant derivative of {{math|''x<sup>a</sup>'' {{=}} ''ξ<sup>a</sup>''(''p'')}} is [[gauge invariant]]. This is identified with the translational (co-)tetrad

<math display="block">{h^a}_\mu = \partial_\mu \xi^a + {B^a}_\mu</math>

which is a [[one-form]] which takes on values in the [[Lie algebra]] of the translational Abelian group {{math|'''R'''<sup>4</sup>}}, whence it is gauge invariant.<ref>{{cite journal|title=Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance|first1=F. W.|last1=Hehl| first2=J. D.|last2=McCrea|first3=E. W.|last3=Mielke|first4=Y.|last4=Ne’eman|journal=Phys. Rep. |volume=258 |issue=1 |date=1995 | pages=1–171|doi=10.1016/0370-1573(94)00111-F|arxiv = gr-qc/9402012 |bibcode = 1995PhR...258....1H |s2cid=119346282 }}</ref> But what does this mean? {{math|''x<sup>a</sup>'' {{=}} ''ξ<sup>a</sup>''(''p'')}} is a local section of the (pure translational) affine internal bundle {{math|{{mathcal|M}} → ''M''}}, another important structure in addition to the translational gauge field {{math|''B<sup>a</sup><sub>μ</sub>''}}. Geometrically, this field determines the origin of the affine spaces; it is known as [[Élie Cartan|Cartan]]’s radius vector. In the gauge-theoretic framework, the one-form

<math display="block">h^a = {h^a}_\mu dx^\mu = \left(\partial_\mu \xi^a + {B^a}_\mu\right)dx^{\mu}</math>

arises as the nonlinear translational gauge field with {{math|''ξ<sup>a</sup>''}} interpreted as the [[Goldstone boson|Goldstone field]] describing the spontaneous breaking of the translational symmetry.

A crude analogy: Think of {{math|{{mathcal|M}}<sub>''p''</sub>}} 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 [[crystal]] with [[line defect]]s ([[edge dislocation]]s and [[screw dislocation]]s but not [[disclination]]s). 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 neglected.

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

<math display="block">{T^a}_{\mu\nu} \equiv \left(DB^a\right)_{\mu\nu} = D_\mu {B^a}_\nu - D_\nu {B^a}_\mu,</math>

is [[gauge invariant]].

We can always choose the gauge where {{mvar|x<sup>a</sup>}} is zero everywhere, although {{math|{{mathcal|M}}<sub>''p''</sub>}} 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 [[action (physics)|action]] that makes it exactly equivalent<ref name=cho /><!-- perhaps, the action may be made equivalent to Hilbert's one, but what about space-time topology? can you made a Schwarzschild black hole or a Kerr black hole in Minkowski space? --> to general relativity, but there are also other choices of the action which are not equivalent to general relativity. In some of these theories, there is no equivalence between [[Mass#Inertial mass|inertial]] and [[gravitational mass]]es.<ref>{{cite journal|title=Is teleparallel gravity really equivalent to general relativity?| first1=L. |last1=Combi| first2=G.E.|last2=Romero|journal=Annalen der Physik |volume=530 | number=1 |date=2018 |pages=1700175 |doi=10.1002/andp.201700175| arxiv=1708.04569 | bibcode=2018AnP...53000175C | hdl=11336/36421| s2cid=119509267 | hdl-access=free}}</ref>

Unlike in general relativity, gravity is due not to the curvature of spacetime but to the torsion thereof.