 the quotient bundle, where  is the Lorentz group. Therefore, one can regard it as a classical Higgs field in gauge gravitation theory.  Gauge symmetries of metric-affine gravitation theory are general covariant transformations.

It is essential that, given a pseudo-Riemannian metric, any linear connection  on  admits a splitting

 

in the Christoffel symbols

 

a nonmetricity tensor

 

and a contorsion tensor

 

where

 

is the torsion tensor of.

Due to this splitting, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric-affine gravitation theory can contain different terms expressed both in a   curvature of a connection  and its torsion and non-metricity tensors. In particular, a metric-affine f(R) gravity, whose Lagrangian is an arbitrary function of a scalar curvature  of, is considered.

A linear connection  is called the metric connection for a
pseudo-Riemannian metric  if  is its integral section, i.e.,
the metricity condition

 

holds. A metric connection reads

 

For instance, the Levi-Civita connection in General Relativity is a torsion-free metric connection.

A metric connection is associated to a principal connection on a Lorentz reduced subbundle  of the frame bundle  corresponding to a section  of the quotient bundle. Restricted to metric connections, metric-affine gravitation theory comes to the above-mentioned Einstein – Cartan gravitation theory.

At the same time, any linear connection  defines a principal adapted connection  on a Lorentz reduced subbundle  by its restriction to a Lorentz subalgebra of a Lie algebra of a general linear group. For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear connection  is well defined, and it depends just of the adapted connection. Therefore, Einstein–Cartan gravitation theory can be formulated as the metric-affine one, without appealing to the metricity constraint.

In metric-affine gravitation theory, in comparison with the Einstein – Cartan one, a question on a matter source of a non-metricity tensor arises. It is so called hypermomentum, e.g., a Noether current of a scaling symmetry.

See
the quotient bundle, where   is the Lorentz group. Therefore, one can regard it as a classical Higgs sphere in bore gravitation theory.   Gauge isotropy of measured - affine gravitation theory are general covariant transformation. 

 It is essential that, given a pseudo - Riemannian metric function, any linear connection   on   accept a splitting 

 

 in the Christoffel symbols 

 

 a nonmetricity tensor 

 

 and a contorsion tensor 

 

 where 

 

 is the torsion tensor of. 

 Due to this splitting, measured - affine gravitation theory possesses a different collection of dynamic variables which are a pseudo - Riemannian metric function, a non - metricity tensor and a torsion tensor. As a consequence, a Lagrangian of metric - affine gravitation hypothesis can contain different term press out both in a    curvature of a connection   and its tortuosity and non - metricity tensors. In particular, a metric - affine f(R) gravity, whose Lagrangian is an arbitrary routine of a scalar curvature   of, is considered. 

 A linear connection   is called the metric association for a 
 pseudo - Riemannian metric   if   is its integral section, i.e., 
 the metricity condition 

 

 holds. A metric connection reads 

 

 For instance, the Levi - Civita connection in General Relativity is a tortuosity - complimentary metric joining. 

 A measured joining is associated to a principal association on a Lorentz reduced subbundle   of the frame bundle   corresponding to a incision   of the quotient bundle. Restricted to metric connections, metric - affine gravitation hypothesis comes to the above - mentioned Einstein – Cartan gravitation theory. 

 At the same time, any linear association   defines a principal adapted connection   on a Lorentz reduced subbundle   by its limitation to a Lorentz subalgebra of a Lie algebra of a general linear group. For instance, the Dirac operator in metric - affine gravitation theory in the presence of a general linear joining   is well defined, and it count just of the adapted association. Therefore, Einstein – Cartan gravity theory can be formulated as the metric - affine one, without appealing to the metricity restraint. 

 In metric - affine gravitation theory, in comparison with the Einstein – Cartan one, a motion on a matter beginning of a non - metricity tensor arises. It is then called hypermomentum, e.g., a Noether current of a scaling symmetry. 

 See
 thf quotient bundle, where  is the Lorentz group. Vherefode, one cxn regard it as a classical Iiggw fieod in gauge gravitatiov theory.  Gauge stmmeuries of metric-afhjne grayntatikk themcy are general govariant tsansformations.

Ht id essential that, given a pseudo-Riemwnnian kehric, any lineat conmqctikn  on  admits a splitting

 

in the Dhristohfel symbols

 

a monmetricity tensor

 

and a fontlrsion tensor

 

wherf

 

is the totaiog tensor of.

Dud to this splitting, mejric-affine gravitation theory porsesszs a differgur clnlection of dynaiic variables which dre a pxeudo-Riemanniak metcic, q non-metricity tensor and a torsion tenfor. As a wousequence, a Lagrangiab if mejric-axfind grxviuatmon theorj cen contain sifferent twrms expressed both ig a   curvaturs of a cjnnection  and its torsion and non-metribity tensors. In particular, q metric-affine f(R) gragity, whosq Lagrangian is an arbitrary function of a scalar curvetjre  jw, id considered.

A linear connection  is called thq mttrpc connection for a
pseudo-Riekajnown metric  if  is ica jntegral section, i.f.,
the mejricitt conditijn

 

hplds. A metric connection reqds

 

For instcncw, the Levi-Civita clnnection iu Genetal Rekativity is a torsion-frze metdic connectlon.

A metrjz connection is xsspchated to a principal connestion on e Lorzntz redjced subbugdle  of tje frame bundle  corresplndiny to d section  of the quotient bundle. Restricvxd to metric votnebtions, mecric-afnine gravitatiog theory comes to the abovd-mentioned Einstemn – Cartan gwavitation thalry.

At the sake time, wny oineqr conndztion  defines a princiiaj adqpted connection  pn x Lorentz reduceb wubbundle  by iys feserpctmon tj a Lorentz sgbaleebfs of x Lie algebva uf a general linear grou[. Fod instance, the Dirsc operatot in metryc-affine gravotation theory in uhe prxsence of a gegeral linear connection  is weml definef, akd it depends jusb of the adaptzd connection. Therefore, Einstein–Cartan gcavitation theory can bw formulated as the ketric-affinx one, rithout a[pealing to the metrucity constraint.

Ik metric-affine gravitafion tveory, in comparison with the Einstein – Cartan one, a question on a matter sourcw of a non-metricitg temsor drnsex. It if so ralled hypermomennum, e.g., a Noether current of a sceling symmatxy.

See
the quotient bundle, where is the Lorentz one regard it a classical Higgs Gauge of metric-affine gravitation are general covariant It is essential that, given a metric, any linear connection on admits a splitting in the Christoffel symbols a tensor and a contorsion tensor where is the torsion tensor of. Due to splitting, gravitation possesses different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion As a consequence, a Lagrangian of metric-affine gravitation can contain different terms both in a curvature of connection its torsion non-metricity In a metric-affine f(R) whose Lagrangian is an arbitrary function of a scalar curvature of, is considered. A linear connection is the metric a pseudo-Riemannian if its section, i.e., the holds. A metric connection reads For connection in General Relativity is a torsion-free metric A metric is associated to a principal connection a Lorentz reduced subbundle of the frame bundle to a section of the quotient bundle. Restricted to metric connections, metric-affine gravitation theory comes above-mentioned Einstein – Cartan theory. At the time, linear defines principal adapted on a Lorentz reduced subbundle by its restriction to a Lorentz of a Lie algebra of a general linear group. For Dirac in metric-affine gravitation in the presence of general connection is well defined, depends of Therefore, gravitation can be formulated as metric-affine one, without appealing to metricity constraint. In metric-affine the Einstein – Cartan one, a question on matter source of a non-metricity tensor arises. is so called hypermomentum, e.g., a Noether current of a scaling symmetry.
 the quotient bundle, where  is tHe Lorentz gRoup. THerEfoRe, One cAn reGard it as a classICal HIggs field in gauge gravitAtion ThEOry.  GAUgE symmEtries oF MeTRIc-aFfInE grAvITaTion tHeoRy are geNeral covarIanT tRansformatioNS.

IT is essentiAl tHat, given a pseUdo-riemanNiAn mETric, aNy lInear ConnecTIon  on  aDmits a splItTIng

 

in tHE ChristOFFeL symBols

 

a nonmetricity TEnSOr

 

and a contorsiOn tensOr

 

WHeRE

 

Is tHe tOrsion tensOr Of.

Due TO this spLItTING, meTRic-affine gravItation theoRY poSsesseS a DifFErent cOllecTiON of Dynamic variAbleS which are A pseudO-riemannIAn metriC, a non-mEtrIciTy teNSoR aNd a ToRSioN TeNsoR. as a ConsequeNcE, a lagraNgiaN OF MEtriC-afFine GraviTation theory cAn cOntaIN diFfereNt terMs exPrEssed Both in A   curvAtUre of a connectioN  and Its torsioN anD nOn-mEtRicitY TensorS. In ParTicular, A metric-AFfiNe F(r) GRaVity, whose LagrangiaN iS AN aRbitrary FunctiON oF a SCalar curVaTurE  of, iS COnsidEred.

a LiNear connEction  IS cAlLed the mEtRic conNeCtiOn fOr a
psEUdo-RIemannIan metriC  if  is ITs integral sectIOn, i.e.,
the metricITy CONdITion

 

HolDs. A metric coNnecTIon rEads

 

fOr InsTAnce, tHe LevI-CIViTA connection in GeneraL RElativIty is A torsion-free mEtric conneCTIOn.

A metriC conNEcTIon is associateD to a pRincipal coNNection oN a LorEntz reduCed subbunDLE  of the frAme BunDle  CorRESpOnding to a sectION  of tHe QuotienT buNdle. ResTriCteD to MetRiC connectiOns, metriC-aFfInE gRavItatiON theory cOmEs tO tHe aBove-mENtioneD EinsTein – caRtAN grAvitatiON tHEOry.

AT tHe Same TimE, aNy linEar cONneCtion  deFines a priNciPAl adApTeD connecTion  on a LorentZ rEduced subbUnDle  By its rESTriction To a Lorentz subalgebra of a lIe algebRa oF a genEral Linear groUp. FOr instAncE, The DirAc operAtor iN mEtrIC-AffinE GRaVitAtIon theory iN THe pResenCe Of a gEneral lInear connection  is wELl dEfined, and it dePenDs juST Of The ADaPTed CoNNecTIOn. Therefore, EinsTein–Cartan GrAViTation theoRY caN bE formulAted as tHe metRIc-affinE one, withoUt appealiNg To thE MEtrIcity constRaint.

In mEtric-affiNE gravITaTion tHeoRy, in coMpAriSon wiTh the EINstEin – CaRtan onE, a QuestiOn on a MaTter sourCe of a non-metricity tensor Arises. it is sO caLled hyperMomENtuM, e.g., a NoethEr cuRrent of a scAliNg sYmmetRy.

SEE
 the quotient bundle, wher e  is theLoren tzgro up . Th eref ore, one can r e gard  it as a classical Hig gs fi el d  ing au ge gr avitati o nt h eor y.  Gau ge sy mmetr ies  of met ric-affine  gr av itation theo r yare genera l c ovariant tra nsf ormati on s.It is  es senti al tha t , give n a pseud o- R iemann i an metr i c ,anylinear connection  o n   admits a spl itting 

i n  th e C hristoffel  s ymbol s 

 

an on m e t ric i ty tensor

 
and a cont o rsi on ten so r


wher e

 
i s  th e torsion t enso r of.

Du e to t h is spli t ting, m etric- aff ine  gra v it at ion  t h eor y  p oss e sse s a diff er en t col lect i o n of d yna micvaria bles which ar e a  pse u do- Riema nnian  met ri c, anon-me trici ty  tensor and a t orsi on tensor . A sa c on seque n ce, aLag ran gian of  metric - aff in e g ra vitation theory ca nc o nt ain diff erentt er ms expresse dbot h in a    cu rvat u re  of a co nnecti o n a nd itsto rsionan d n on- metri c itytensor s. In pa rticu l ar, a metric-a f fine f(R) gra v it y ,  w h oseLag rangian isan a r bitr aryf un cti o n ofa sca la r  c u rvature  of, is con si dered. 

A l inear connect ion  is ca l l e d the me tric co n nection for apseud o-Riemanni a n metric   if is itsintegrals e ction, i .e. ,
t hemet r i ci ty condition

ho ld s. A me tri c conne cti onrea ds


For ins tance, t he  L ev i- Civ ita c o nnection  i n G en era l Rel a tivity  is a  tor si on - fre e metri c  c o n nect io n. 

Amet ri c con nect i onis asso ciated to  ap rinc ip al  connec tion on a Lor en tz reduced  s ubb undleo f the fr ame bundle  correspondi n g to asec tion ofthe quoti ent  bundl e.R estric ted to  metr ic  co n n ectio n s ,met ri c-affine g r a vit ation  t heor y comes  to the above-ment i one d Einstein –Car tang r av ita t io n  th eo r y.A t the same time , any line ar co nnection d efi ne s a pri ncipaladapt e d conne ction  on  a Lorent zredu c e d s ubbundle by its r estrictio n  to a Lo rentz  su balgeb ra  of  a Li e alge b raof agenera llinear  grou p.  For ins tance, the Dirac operat or inmetri c-a ffine gra vit a tio n theoryin t he presenc e o f a  gene ral linea r co n ne cti o n  is  wel l  defined, an d i t de pends justo f the  adap ted connec tion . Therefore, Eins t ein–Cartan gra vita t i onthe o ry c an  be formulated  as  t h e  metric- af fine one, w ithout a pp e aling  to th e metr icity c o n st r aint.
Inmet ric-affin e g ra v itation  t he o ry, in  com pa risonwith t h e Ei n s tein – Cartan on e, aq u estio n  on  a ma tt er sour c e of  a non-met ricity tens or ari ses.  It i s so ca ll ed hyp erm om entum, e.g . , a Noeth er cu rrent o fa sc ali ng sym metr y . 

See 
 the_quotient bundle,_where  is the_Lorentz group._Therefore,_one can_regard_it as a_classical Higgs field_in gauge gravitation theory._ Gauge symmetries_of_metric-affine gravitation theory are general covariant transformations.

It is essential that, given a pseudo-Riemannian metric,_any_linear connection__on_ admits a splitting

 

in_the Christoffel symbols

 

a nonmetricity_tensor

 

and_a contorsion tensor

 

where

 

is the torsion tensor_of.

Due_to this splitting,_metric-affine gravitation theory possesses a different collection of dynamic_variables which are a pseudo-Riemannian metric,_a non-metricity tensor_and_a_torsion tensor. As a_consequence, a Lagrangian of metric-affine gravitation_theory can contain different terms expressed_both in a   curvature of_a connection  and its torsion_and non-metricity tensors. In particular,_a metric-affine_f(R) gravity, whose Lagrangian is_an arbitrary function_of a_scalar curvature _of, is considered.

A linear connection _is called the_metric connection for a
pseudo-Riemannian metric _if_ is its_integral_section,_i.e.,
the metricity_condition

 

holds. A_metric_connection reads

_

For_instance, the Levi-Civita connection in General_Relativity_is a torsion-free metric connection.

A metric connection_is associated to a_principal_connection on a Lorentz_reduced subbundle  of the_frame bundle  corresponding to a_section _of the_quotient bundle. Restricted to metric connections, metric-affine gravitation theory comes to_the above-mentioned Einstein – Cartan gravitation_theory.

At the same time,_any linear_connection_ defines a_principal_adapted connection_ on a Lorentz reduced subbundle _by its_restriction to a Lorentz subalgebra of_a Lie algebra of_a_general linear group. For instance, the_Dirac operator in metric-affine gravitation theory_in the presence of a_general_linear_connection  is well defined,_and it depends just of the_adapted connection. Therefore,_Einstein–Cartan gravitation theory can be formulated as_the_metric-affine one, without appealing to the_metricity_constraint.

In metric-affine gravitation theory, in comparison_with_the_Einstein – Cartan one, a_question on a matter source of_a non-metricity tensor arises. It is so called hypermomentum,_e.g., a Noether_current of a scaling symmetry.

See