Editing Teleparallelism

{{Short description|Theory of gravity}}
{{too technical|date=May 2019}}
'''Teleparallelism''' (also called '''teleparallel gravity'''), was an attempt by [[Albert Einstein]]<ref>{{cite journal | title=Riemann-Geometrie mit Aufrechterhaltung des Begriffes des Fernparallelismus | first=Albert | last=Einstein | journal=Preussische Akademie der Wissenschaften, Phys.-math. Klasse, Sitzungsberichte | volume=1928 | date=1928 | pages=217–221}}</ref> 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 [[Connection (vector bundle)|linear connection]] in conjunction with a [[metric tensor]] field, both defined in terms of a dynamical [[Cartan connection applications|tetrad]] field.

==Teleparallel spacetimes==
The crucial new idea, for Einstein, was the introduction of a [[Cartan connection applications|tetrad]] field, i.e., a set {{math|{X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub>, X<sub>4</sub>}<nowiki/>}} of four [[vector field]]s defined on ''all'' of {{mvar|M}} such that for every {{math|''p'' ∈ ''M''}} the set {{math|{X<sub>1</sub>(''p''), X<sub>2</sub>(''p''), X<sub>3</sub>(''p''), X<sub>4</sub>(''p'')}<nowiki/>}} is a [[Basis (mathematics)|basis]] of {{math|''T<sub>p</sub>M''}}, where {{math|''T<sub>p</sub>M''}} denotes the fiber over {{mvar|p}} of the [[tangent vector bundle]] {{mvar|TM}}. Hence, the four-dimensional [[spacetime]] manifold {{mvar|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 the '''[[Roland Weitzenböck|Weitzenböck]] connection''') {{math|{X<sub>''i''</sub>{{)}}}} to be the [[Connection (vector bundle)|linear connection]] {{math|∇}} on {{mvar|M}} such that<ref>{{cite book| last1=Bishop| first1=R. L. |last2=Goldberg|first2=S. I.|title=Tensor Analysis on Manifolds| url=https://archive.org/details/tensoranalysison0000bish| url-access=registration|date=1968|page=[https://archive.org/details/tensoranalysison0000bish/page/223 223]}}</ref>

<math display="block">\nabla_v\left(f^i\mathrm X_i\right)=\left(vf^i\right)\mathrm X_i(p),</math>

where {{math|''v'' ∈ ''T<sub>p</sub>M''}} and {{math|''f''{{isup|''i''}}}} are (global) functions on {{mvar|M}}; thus {{math|''f''{{isup|''i''}}X''<sub>i</sub>''}} is a global vector field on {{mvar|M}}. In other words, the coefficients of '''Weitzenböck connection''' {{math|∇}} with respect to {{math|{X<sub>''i''</sub>{{)}}}} are all identically zero, implicitly defined by:

<math display="block">\nabla_{\mathrm{X}_i} \mathrm{X}_j = 0,</math>

hence

<math display="block">{W^k}_{ij} = \omega^k\left(\nabla_{\mathrm{X}_i} \mathrm{X}_j\right)\equiv 0,</math>

for the connection coefficients (also called Weitzenböck coefficients) in this global basis. Here {{math|''ω<sup>k</sup>''}} is the dual global basis (or coframe) defined by {{math|''ω<sup>i</sup>''(X<sub>''j''</sub>) {{=}} ''δ''{{su|p=''i''|b=''j''}}}}.

This is what usually happens in {{math|'''R'''<sup>''n''</sup>}}, in any [[affine space]] or [[Lie group]] (for example the 'curved' sphere {{math|'''S'''<sup>3</sup>}} but 'Weitzenböck flat' manifold).

Using the transformation law of a connection, or equivalently the {{math|∇}} properties, we have the following result.

<blockquote>'''Proposition'''. In a natural basis, associated with local coordinates {{math|(''U'', ''x<sup>μ</sup>'')}}, i.e., in the holonomic frame {{math|∂<sub>''μ''</sub>}}, the (local) connection coefficients of the Weitzenböck connection are given by:

<math display="block">{\Gamma^{\beta}}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu} h^{i}_{\mu},</math>

where {{math|X<sub>''i''</sub> {{=}} ''h''{{su|p=''μ''|b=''i''}}∂<sub>''μ''</sub>}} for {{math|''i'', ''μ'' {{=}} 1, 2,… ''n''}} are the local expressions of a global object, that is, the given tetrad.</blockquote>

The '''Weitzenböck connection''' has vanishing [[Riemann curvature tensor|curvature]], but – in general – non-vanishing [[torsion tensor|torsion]].

Given the frame field {{math|{X<sub>''i''</sub>{{)}}}}, 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 {{mvar|g}} of [[metric signature|signature]] (3,1) by

<math display="block">g\left(\mathrm{X}_i,\mathrm{X}_j\right)=\eta_{ij},</math>

where

<math display="block">\eta_{ij}=\operatorname{diag}(-1,-1,-1,1).</math>

The corresponding underlying spacetime is called, in this case, a [[Roland Weitzenböck|Weitzenböck]] spacetime.<ref>{{cite journal | doi=10.12942/lrr-2004-2 | doi-access=free | title=On the History of Unified Field Theories | date=2004 | last1=Goenner | first1=Hubert F. M. | journal=Living Reviews in Relativity | volume=7 | issue=1 | page=2 | pmid=28179864 | bibcode=2004LRR.....7....2G | pmc=5256024 }}</ref>

These 'parallel vector fields' give rise to the metric tensor as a byproduct.

==New teleparallel gravity theory==
'''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:

{{ordered list|list_style_type=upper-alpha
|Underlying spacetime is the Weitzenböck spacetime, 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.
|The [[equivalence principle]] is valid only in classical physics.
|Gravitational field equations are derivable from the action principle.
|The field equations are partial differential equations in the field variables of not higher than the second order.}}

In 1961 [[Christian Møller]]<ref name=moeller/> revived Einstein's idea, and Pellegrini and Plebanski<ref name=pellegrini /> found a Lagrangian formulation for ''absolute parallelism''.

=== Møller tetrad theory of gravitation ===
In 1961, Møller<ref name=moeller>{{cite journal|last=Møller|first=Christian|date=1961|title=Conservation laws and absolute parallelism in general relativity|journal=Mat. Fys. Dan. Vid. Selsk.|volume=1|pages=1–50|issue=10}}</ref><ref>{{cite journal| last=Møller |first=Christian |date=1961 |title=Further remarks on the localization of the energy in the general theory of relativity|journal=Ann. Phys.|volume=12|pages=118–133| doi=10.1016/0003-4916(61)90148-8|issue=1|bibcode = 1961AnPhy..12..118M }}</ref> showed that a [[Frame fields in general relativity|tetrad]] description of gravitational fields allows a more rational treatment of the [[Stress–energy–momentum pseudotensor|energy-momentum complex]] than in a theory based on the [[Metric tensor (general relativity)|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.<ref>{{Cite journal| last1=Abedi|first1=Habib| last2=Salti|first2=Mustafa| date=2015-07-31| title=Multiple field modified gravity and localized energy in teleparallel framework|journal=General Relativity and Gravitation| language=en | volume=47|issue=8|pages=93|doi=10.1007/s10714-015-1935-z| issn=0001-7701| bibcode=2015GReGr..47...93A | s2cid=123324599 }}</ref>

==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.

==Non-gravitational contexts==
There exists a close analogy of [[Riemannian geometry|geometry]] of spacetime with the structure of defects in crystal.<ref>{{cite book| title = Gauge Fields in Condensed Matter Vol II|first = Hagen| last = Kleinert| author-link = Hagen Kleinert| pages = 743–1440| date = 1989| url = http://users.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents2.html}}</ref><ref>{{cite book| title = Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation| first = Hagen| last = Kleinert| author-link = Hagen Kleinert| pages = 1–496| date = 2008| bibcode = 2008mfcm.book.....K| url = http://users.physik.fu-berlin.de/~kleinert/b11/psfiles/mvf.pdf}}</ref> [[Dislocations]] are represented by torsion, [[disclination]]s by curvature. These defects are not independent of each other. A dislocation is equivalent 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.<ref>{{cite journal| title = New Gauge Symmetry in Gravity and the Evanescent Role of Torsion| first = Hagen| last = Kleinert| author-link = Hagen Kleinert| volume = 24| pages = 287–298| date = 2010| journal = Electron. J. Theor. Phys.| doi = 10.1142/9789814335614_0016 | arxiv = 1005.1460| bibcode = 2011pchm.conf..174K| isbn = 978-981-4335-60-7| s2cid = 17972657| url = http://users.physik.fu-berlin.de/~kleinert/385/385.pdf}}</ref>

A further application of teleparallelism occurs in quantum field theory, namely, two-dimensional [[non-linear sigma model]]s with target space on simple geometric manifolds, whose renormalization behavior is controlled by a [[Ricci flow]], which includes [[Torsion tensor|torsion]]. This torsion modifies the Ricci tensor and hence leads to an [[infrared fixed point]] for the coupling, on account of teleparallelism ("geometrostasis").<ref>{{Cite journal | last1 = Braaten | first1 = E. | last2 = Curtright | first2 = T. L. | last3 = Zachos | first3 = C. K. | doi = 10.1016/0550-3213(85)90053-7 | title = Torsion and geometrostasis in nonlinear sigma models | journal = Nuclear Physics B | volume = 260 | issue = 3–4 | pages = 630 | year = 1985 |bibcode = 1985NuPhB.260..630B }}</ref>

==See also==
* [[Classical theories of gravitation]]
* [[Gauge gravitation theory]]
* [[Geometrodynamics]]
* [[Kaluza-Klein theory]]

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book|last1=Aldrovandi|first1=R.|last2=Pereira|first2=J. G. | title = Teleparallel Gravity: An Introduction| publisher=Springer: Dordrecht | date=2012|isbn=978-94-007-5142-2}}
* {{cite book|last1=Bishop|first1=R. L.|author-link=Richard L. Bishop|last2=Goldberg|first2=S. I.|title=Tensor Analysis on Manifolds|publisher=Macmillan|date=1968|edition=First Dover 1980|isbn=978-0-486-64039-6|url-access=registration | url=https://archive.org/details/tensoranalysison00bish}}
* {{cite book|last1 = Weitzenböck|first1=R.|author-link=Roland Weitzenböck |title=Invariantentheorie|date = 1923|publisher = Groningen: Noordhoff}}

==External links==
*[http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/selected_papers_on_teleparallelism.pdf ''Selected Papers on Teleparallelism'', translated and edited by D. H. Delphenich]
* {{nlab|id=teleparallel+gravity|title=Teleparallel gravity}}
*[http://www.phy.olemiss.edu/~luca/Topics/grav/teleparallel.html Teleparallel Structures and Gravity Theories by Luca Bombelli]

{{Albert Einstein}}
{{theories of gravitation|state=expanded}}

[[Category:History of physics]]
[[Category:Theories of gravity]]