LeCosPA 2 at NTU, 2015-12-14

Gravity: a gauge theory perspective

James M. NesterNational Central U & LeCosPa NTU

working with C.M. Chen

present address:Morningside Center of Mathematics, Chinese Academy of Sciences,

Beijing

email: nester@phy.ncu.edu.tw, file 2LeCosPA151214.tex

1

Abstract

The evolution of a generally covariant theory is under-determined.100 years ago such dynamics had never before been considered;its ramifications were perplexing, its future important role for all thefundamental interactions under the name gauge principle couldnot be foreseen. We recount some history regarding Einstein,Hilbert, and Klein, and the novel features of gravitational energythat led to Noether’s two theorems. Under-determined evolution isbest revealed in the Hamiltonian formulation. We developed acovariant Hamiltonian formulation. The Hamiltonian boundaryterm gives covariant expressions for the quasi-local energy,momentum and angular momentum. Gravity can be consideredas a gauge theory of the local Poincare group. The dynamicalpotentials of the Poincare gauge theory of gravity are the frameand the connection. The spacetime geometry has in general bothcurvature and torsion. Torsion naturally couples to spin; it couldhave a significant magnitude and yet not be noticed, except on acosmological scale where it could have significant effects.

2

outline

◮ Einstein, Hilbert, Noether, Klein, gravitational energy & gauge theory

◮ Noether’s results

◮ energy-momentum pseudotensors

◮ the covariant Hamiltonian and the role of its boundary term

◮ spacetime geometry and its local gauge symmetries

◮ Riemann-Cartan geometry and torsion

◮ the Poincare gauge theory of gravity (PG)

◮ PG and GR quasi-local energy-momentum

◮ PG cosmology

The focus will be on the less well-known aspects

GR with general covariance was the premier gauge theory. Theconsequences of this, especially regarding the nature of gravitational energyand under-determined evolution, were long perplexing. The work of Hilbertand Noether has been under-appreciated. The Hamiltonian approachclarifies these issues. Gravity can be understood as a gauge theory of thelocal Poincare symmetries of spacetime.

3

some historical background

There have long been disputes about all of the principles used by Einstein forhis gravity theory.

◮ The principle of Relativity: what does General Relativity mean?

◮ Mach’s principle

◮ The equivalence principle

◮ The principle of general covariance

Krestschmann in 1917 argued that general covariance has no real physicalcontent and no connection to an extension of the principle of relativity. seeJ. D. Norton “General covariance and the foundations of general relativity:eight decades of dispute” Rep. Prog. Phys. 56, 791–858 (1993) pp 791–861.

Here, looking from a different perspective, general covariance will be seen tohave deep fundamental ramifications.

4

David Hilbert, Emmy Noether and Felix Klein

In a letter of December 4, 1915 Hilbert wrote: “. . . also themathematical-physical developments (Einsteins theory of gravitation, theoryof space and time) are presently moving towards an unforeseen point ofculmination; and in this matter Miss Noether is my most successfulcollaborator” pp 561–562 in T. Sauer (1999) [physics/9811050]

◮ Hilbert gave talks on his “Foundations of physics” November 16 and 20

◮ he submitted his first note on Nov. 19

◮ Einstein spoke on Nov. 4, 11, 18 & 25, published 1 week later.

◮ The last had his generally covariant eqns with energy conservation.

5

Hilbert: The foundations of physics

THEOREM I (Leitmotiv1). In the system of n Euler-Lagrange differentialequations in n variables obtained from a generally covariant variationalintegral such as in Axiom I, 4 of the n equations are always a consequence ofthe other n− 4 in the sense that 4 linearly independent combinations of the nequations and their total derivatives are always identically satisfied.

Hilbert’s work has mainly been viewed from the Einstein GR perspective.For Hilbert’s agenda see Brading & Ryckman, (2008).

A main aim was to reconcile the tension between general covariance and itsinevitable consequence: a lack of unique determinism,this is the essence of gauge theory.[Note: The issue had caused much trouble for Einstein (the Hole argument)].

Dynamical equations obtained from a variational principle had always haddeterministic Cauchy initial value problems, but for GR there was a differentialidentity connecting the evolution equations, so they were not independentand could not give uniquely determined evolution.

Much later it was found that the best way to cope with this is the Hamiltonianapproach developed by Dirac, see Earman (2003)

1guiding theme6

From Einstein’s correspondence

From http://einsteinpapers.press.princeton.edu Vol. 8.

“Highly esteemed Colleague, . . . I am sitting over your relativitypaper, . . . , and am honestly toiling over it. I do admire yourmethod, as far as I have understood it. But at certain points Icannot progress and therefore ask that you assist me with briefinstructions. . . . I still do not grasp the energy principle at all, noteven as a statement.” (Doc. 221 to Hilbert 25 May 1916)

“In your paper everything is understandable to me now exceptfor the energy theorem. Please do not be angry with me that I askyou about this again. . . . How is this cleared up? It would suffice,of course, if you would charge Miss Noether with explaining this tome.” (Doc. 223 to Hilbert 30 May 1916)

“I have succeeded in discovering the organic formation law forHilbert’s energy vector” (Doc. 588 from Klein 15 July 1918)

7

“The only thing I was unable to grasp in your paper is theconclusion at the top of page 8 that εσ was a vector.”

(Doc. 638 to Klein 22 Oct 1918)

“. . . Meanwhile, with Miss Noether’s help, I understand that theproof for the vector character of εσ from “higher principles” as Ihad sought was already given by Hilbert on pp. 6, 7 of his firstnote, . . . ” (Doc. 650 from Klein 10 Nov 1918)

Briefly, after a couple of years Klein clarified Hilbert’s energy-momentum“vector”; he related it to Einstein’s pseudotensor, but disagreed with Einstein’sphysical interpretation of divergenceless expressions.Enlisted by Hilbert and Klein, it was Emmy Noether who resolved the primarypuzzle regarding gravitational energy.

Remark: Things were not as easy then; in particular the Bianchi identity andits contracted version were not known to these people [Pais 1982, Rowe2002]; they each had to rediscover an equivalent identity on their own.

8

automatic conservation of the source & gauge theory

◮ In 1916 Einstein showed that local coordinate invariance plus his fieldequations gives material energy momentum conservation, without usingthe matter field equations.

◮ This is referred to as automatic conservation of the source (see MTWsection 17.1); it uses a Noether 2nd theorem local (gauge) symmetrytype of argument to obtain current conservation.

◮ Herman Weyl used this type of argument for electromagnetic current inhis gauge theory paper of 1918 (the name gauge come from this work)and of 1929, whereas modern field theory generally uses Noether’s 1sttheorem for current conservation.

◮ The essence of gauge theory is a local symmetry, consequently(i) a differential identity,(ii) under-determined evolution,(iii) restricted type of source coupling,(iv) automatic conservation of the source.Yang-Mills is only one special type.

◮ our gauge approach to gravity is not forcing it into the Yang-Mills mold,but simply recognizing the natural symmetries of spacetime geometry.

9

Noether’s 1918 contribution

◮ One word well describes 20th century physics: symmetry

◮ Most of the theoretical physics ideas involved symmetry.

Essentially they are applications of Noether’s theorems

◮ Noether’s 1st theorem associates conserved quantities with globalsymmetries

◮ Noether’s 2nd theorem concerns local symmetries: it is the foundationof the modern gauge theories.

Unfortunately her work was largely overlooked for about 50 years. SeeY. Kosmann-Schwarzbach, The Noether Theorems: Invariance andConservation Laws in the Twentieth Century (Springer, 2011).

Why did Noether make her investigation?To clarify the issue of gravitational energy.

Klein was looking into Einstein’s theory and the relationship betweenEinstein’s pseudotensor and Hilbert’s energy vector. Some of thecorrespondence between Hilbert and Klein was published in a paper. Wequote some excerpts

10

Klein 1918

Klein: “You know that Miss Noether advises me continually regarding mywork, and that in fact it is only thanks to her that I have understood thesequestions. When I was speaking recently to Miss Noether about my resultconcerning your energy vector, she was able to inform me that she hadderived the same result on the basis of developments of your note (and thusnot from the simplified calculations of my section 4) more than a year ago,and that she had then put all of that in a manuscript . . . ”

Hilbert 1918: “I fully agree in fact with your statements on the energytheorems: Emmy Noether, on whom I have called for assistance more than ayear ago to clarify this type of analytical questions concerning my energytheorem, found at that time that the energy components that I hadproposed—as well as those of Einstein—could be formally transformed,using the Lagrange differential equations (4) and (5) of my first note, intoexpressions whose divergence vanishes identically, . . . ”

“Indeed I believe that in the case of general relativity, i.e., in the case of thegeneral invariance of the Hamiltonian function, the energy equations which inyour opinion correspond to the energy equations of the theory of orthogonalinvariance do not exist at all; I can even call this fact a characteristic of thegeneral theory of relativity.”

11

Noether 1918: clarifies the situation and much more

Theorem I. If the integral I is invariant under a [finite continuous group with ρparameters] Gρ, then there are ρ linearly independent combinations amongthe Lagrangian expressions which become divergences—and conversely,that implies the invariance of I under a [group] Gρ. The theorem remainsvalid in the limiting case of an infinite number of parameters.

Theorem II. If the integral I is invariant under a [an infinite continuous group]G∞ρ depending on arbitrary functions and their derivatives up to order σ,then there are ρ identities among the Lagrangian expressions and theirderivatives up to order σ. Here as well the converse is valid.

(“Theorem iii” [my designation])Given I invariant under the group of translations, then the energy relationsare improper if and only if I is invariant under an infinite group which containsthe group of translations as a subgroup.

As Hilbert expresses his assertion, the lack of a proper law of energyconstitutes a characteristic of the “general theory of relativity.” For thatassertion to be literally valid, it is necessary to understand the term “generalrelativity” in a wider sense than is usual, and to extend it to theaforementioned groups that depend on n arbitrary functions.

12

Implications for gravitational energy

Her result regarding the lack of a proper law of energy applies not just toEinstein’s general relativity theory, but in fact to all geometric theories ofgravity:

As a well known textbook expresses it:

Anyone who looks for a magic formula for “local gravitationalenergy-momentum” is looking for the right answer to the wrongquestion. Unhappily, enormous time and effort were devoted inthe past to trying to “answer this question” before investigatorsrealized the futility of the enterprise.

Misner, Thorne & Wheeler Gravitation p 467.

The modern view is that energy-momentum is quasi-local, associated with aclosed 2 surface, seeL.B. Szabados Living Rev. Relativ. 12 (2009) 4.

13

energy-momentum pseudotensors

The Einstein Lagrangian differs from Hilbert’s by a total divergence

2κLE(gαβ, ∂µgαβ) := −√−ggβσΓα

γµΓγβνδ

µνασ ≡

√−gR− div

The Einstein pseudotensor is the associated canonical energy-momentum

tµEν := δµνLE − ∂LE

∂∂µgαβ

∂νgαβ.

∇µTµν = 0 and

√−gGµν = κTµ

ν =⇒ a conserved total energy-momentum:

∂µ(Tµν + t

µEν) = 0, ⇐⇒

√−gGµ

ν + tµEν = ∂λU

[µλ]ν .

The superpotential was found only years later by Freud (1939):

UµλF ν := −g

βσΓαβγδ

µλγασν .

The pseudotensors of Papapetrou, Landau-Lifshitz, Bergmann-Thompsom,Møller, Goldberg, Weinberg likewise follow from different superpotentials.They are all inherently reference frame dependent.

2 big problems: (1) which pseudotensor? (2) which reference frame?

The Hamiltonian approach has answers.

14

Pseudotensors and the Hamiltonian

with a constant vector field Zµ we find

−ZµPµ(V ) := −∫

V

Zµ(Tνµ + t

νµ)√−gd3Σν

≡∫

V

[

Zµ√−g

(

1

κGν

µ − T νµ

)

− 1

2κ∂λ

(

ZµUνλ

µ

)

]

d3Σν

≡∫

V

ZµHGRµ +

∮

S=∂V

BGR(Z) ≡ H(Z,V ).

HGRµ is just the covariant expression for the ADM Hamiltonian density. The

boundary term 2-surface integral is determined by the superpotential. Thevalue of the pseudotensor/Hamiltonian is quasi-local, determined just by thisboundary term, since by the initial value constraints the spatial volumeintegral vanishes “on shell”.

From the Hamiltonian variation one gets information that tames the ambiguityin the boundary term—namely boundary conditions. The pseudotensorvalues are values of the Hamiltonian with the associated boundary conditions(see PRL 83 (1999) 1897). Thus problem 1 is under control.

15

Currents as generators

◮ Noether’s work was Lagrangian based. Her results can be taken furtherwhen combined with the Hamiltonian formulation, which, moreover,gives a handle on the Noether current ambiguity.

◮ One key feature can be seen already in Hamiltonian mechanics. Aquantity Q conserved under the time evolution generated by aHamiltonian H = H(q, p) is not just a conserved quantity, it is also thecanonical generator of a one parameter transformation on phase space(q(λ), p(λ)) which is a symmetry of the Hamiltonian:

0 =dQ

dt= [Q,H ] =⇒ dH

dλ= [H,Q] = 0.

◮ In Hamiltonian field theory, the conserved currents are the generators ofthe associated symmetry.

◮ For spacetime translations (infinitesimal diffeomorphisms), theassociated Noether conserved current expression (i.e., theenergy-momentum density) is the Hamiltonian density—the canonicalgenerator of spacetime displacements.

◮ Because it can be varied this translation generator gives a handle onthe conserved current ambiguity. The Lagrangian formulation affords nosuch handle.

16

gauge and geometry

For the history of gauge theory seeO’Raifeartaigh The dawning of gauge theory (1997).The main milestones are(i) electromagnetism: Weyl (1918,1929),(ii) non-Abelian groups by Yang & Mills (1954) and Utiyama (1956, 1959).(iii) Gravity as a gauge theory was pioneered by

(a) Utiyama (1956,1959), using the Lorentz group and Riemanniangeometry.

(b) Sciama (1961) using the Lorentz group and Riemann-Cartan geometry(non-vanishing torsion).

(c) Kibble (1961) gauged the Poincare group, the symmetry group ofMinkowski spacetime.

For accounts of gravity as a spacetime symmetry gauge theory, see Hehl andcoworkers, Mielke (1987) and Blagojevic (2002). A comprehensive readerwith summaries, discussions, and many reprints has recently appeared:Blagojevic & Hehl Gauge Theories of Gravitation (2013).

For the observational constraints on torsion see Ni (2010).

17

To us it is rather surprising that the idea of regarding gravity as a gaugetheory is not better known. Gravity played an important role in the argumentused in both of the above mentioned seminal works of Weyl, and thus in all ofthe above—except for the Yang-Mills paper. Furthermore, in 1974 Yanghimself published a paper where he proposed a certain treatment of gravityas a gauge theory.2

According to our understanding, properly speaking, GR can be understoodas the original gauge theory. After all, it was the first physical theory wherelocal gauge freedom (in the guise of general coordinate invariance) played akey role.

It is true that the electrodynamics potentials along with their gauge freedomwere known long before GR but this gauge invariance was not seen ashaving any important role in connection with the nature of the interaction, theconservation of current, or a differential identity—until the seminal work ofWeyl, which post-dated (and was inspired by) GR.

2The aforementioned reader includes a chapter with a critical discussionof Yang’s gauge theory of gravity. In March 2015 Yang was asked, here atNTU, about his 1974 paper, he said: “I do not believe that paper is correct.”

18

◮ The conserved quantities, energy-momentum and angularmomentum/center-of-mass momentum are associated with thegeometric symmetry of Minkowski spacetime, the spacetimetranslations and Lorentz rotations, i.e., the Poincare group. And thisgroup is used to classify physical particles according to mass and spin.

◮ One should also note the developments of the concept of a connectionin differential geometry by Hessenberg, Levi-Civita, Weyl, Schouten,Cartan, Eddington, Ehresmann, and Koszul.

◮ Riemann-Cartan geometry (with a metric and a metric compatibleconnection, having both curvature and torsion) is the most appropriatefor a dynamic spacetime geometry theory: its local symmetries are justthose of the local Poincare group.

19

gauge theory of interactions

Einstein 1915, Hilbert 1916, 1917, Noether 1918, Weyl 1918,1929 Yang 1954, Utiyama 1956, Sciama 1961, Kibble 1961,Hayashi & Shirafuji 1980, Hehl 1980

20

Dynamical spacetime geometry and the Hamiltonian

◮ We considered gravity based on Riemann-Cartan geometry: with ametric and a metric compatible connection, curvature and torsion areallowed.

◮ The variational principles were developed.◮ The Noether symmetries and the associated conserved quantities and

differential identities were discussed.◮ From a 1st order Lagrangian formalism using differential forms, we

constructed a spacetime covariant Hamiltonian formalism.◮ The Hamiltonian boundary term gives appropriate expressions for the

quasi-local quantities, energy-momentum, angular momentum andcenter-of-mass momentum, as well as quasi-local energy flux.

◮ The formalism easily specializes to Einstein’s GR.◮ The Hamiltonian approach reveals certain aspects of a theory, including

the constraints, gauges, and degrees-of-freedom, as well asexpressions for energy-momentum and angular momentum. Howeverfor GR the usual ADM approach achieves this at a heavy cost: the lossof manifest 4D-covariance.

◮ Our alternative approach is complementary: a major benefit ismanifestly 4D-covariant expressions for the quasi-local quantities:energy-momentum and angular momentum/center-of-mass momentum.

21

The main ideas

◮ The Hamiltonian for physical systems and dynamic spacetime geometrygenerates the evolution of a spatial region along a vector field.

◮ It includes a boundary term which determines the boundary conditionsand supplies the value of the Hamiltonian. The Hamiltonian value givesthe quasi-local quantities: energy-momentum andangular-momentum/center-of-mass momentum.

◮ A spacetime gauge theory perspective identifies suitable geometricvariables.

◮ We found a certain preferred Hamiltonian boundary term.

◮ The Hamiltonian boundary term depends not only on the dynamicalvariables but also on their reference values; they determine the groundstate—the state with vanishing quasi-local quantities.

◮ To determine the “best matched” reference metric and connectionvalues for our preferred boundary term we propose on the boundary2-surface: (i) 4D isometric matching, and (ii) extremizing the energy.

22

Hamiltonian formalism

◮ for GR: [Pirani Shild & Skinner (1952), Dirac (1958), ADM (1959–61)]

◮ for the PG: [Blagojevic & Nicolic (1983)]

◮ For GR, the standard ADM approach, is not 4D diffeomorphisminvariant

◮ from a 1st order formulation which gives pairs of 1st order equations,with the aid of differential forms, we developed a covariant Hamiltonianformalism

◮ the Hamiltonian volume 3-form is proportional to the field equationswhich vanish on-shell.

◮ the Hamiltonian boundary term has 2 important roles:

(i) it gives the quasi-local values

(ii) it gives the boundary conditions

23

The covariant HamiltonianThe 1st order translational Noether current 3-form is the Hamiltonian

H(Z) := £Zϕ ∧ p− iZL

it satisfies the identity

−dH(Z) ≡ £Zϕ ∧ δLδϕ

+δLδp

∧£Zp;

a conserved “current” on shell (i.e., when the field equations are satisfied). itis a 3-form linear in the displacement vector plus a total differential:

H(Z) =: ZµHµ + dB(Z).

From local diffeomorphism invariance, Hµ is proportional to field eqns andvanishes on shell; hence translational Noether current conservation reducesto a differential identity between Euler-Lagrange expressions. This aninstance of Noether’s 2nd theorem, and exactly the case to which Hilbert’sremark regarding the lack of a proper energy law applies. The value isquasi-local (associated with a closed 2-surface):

−P (Z,V ) :=

∫

V

H(Z) =

∮

∂V

B(Z).

24

covariant-symplectic boundary terms

The boundary term can be adjusted to match suitable boundary conditions.We were led to a set of general boundary terms which are linear in∆ϕ := ϕ− ϕ, ∆p− p, where ϕ, p are reference values:

B(Z) := iZ

{

ϕϕ

}

∧∆p− ς∆ϕ ∧ iZ

{

pp

}

.

The associated variational Hamiltonian boundary term is

δH(Z) ∼ d

[{

iZδϕ ∧∆p

−iZ∆ϕ ∧ δp

}

+ ς

{

−∆ϕ ∧ iZδp

δϕ ∧ iZ∆p

}]

.

Here for each bracket independently one may choose either the upper orlower term, which represent essentially a choice of Dirichlet (fixed field) orNeumann (fixed momentum) boundary conditions for the space and timeparts of the fields separately.3

3There are more complicated possibilities, “mixed” choices involving somelinear combination of the upper and lower expressions.

25

asymptotics

For asymptotically flat spaces the Hamiltonian is well defined, i.e., theboundary term in its variation vanishes and the quasi-local quantities are welldefined at least on the phase space of fields satisfying Regge-Teitelboim likeasymptotic conditions:

∆ϕ ≈ O+(1/r) +O−(1/r2), ∆p ≈ O−(1/r2) +O+(1/r3).

also the formalism has natural Hamiltonian boundary term related energy fluxexpressions, see [Chen, N, Tung 2005]

26

Riemann-Cartan geometry

Dgµν ≡ 0 metric compatible connection

Rµν := dΓµ

ν + Γµλ ∧ Γλ

ν =1

2Rµ

νijdxi ∧ dxj. curvature 2-form

Tα := Dϑα := dϑα + Γαβ ∧ ϑβ =

1

2Tα

ijdxi ∧ dxj . torsion 2-form

Bianchi identities I and II:

DTα ≡ Rαβ ∧ ϑβ, DRα

β ≡ 0.

for an orthonormal frame gµν = const and Γαβ is antisymmetric

The Ricci identity

[∇µ,∇ν ] Vα = Rα

βµνVβ − T γ

µν∇γVα.

reflects the holonomy and the Lorentz and translational field strengths.

27

PG Dynamics

in Chen, Nester & Tung, Int J Mod Phys D 24 1530026 (2015)arXiv:1507.07300we have discussed in detail the PG dynamics including

◮ the Lagrangian, both 2nd and 1st order

◮ Noether symmetries, conserved currents & differential identities,

◮ the first class covariant Hamiltonian including generators of the localPoincare gauge symmetries

◮ our preferred Hamiltonian boundary term

◮ the quasi-local energy-momentum and angularmomentum/center-of-mass moment obtained therefrom

◮ the choice of reference in the boundary term

28

our preferred Hamiltonian boundary terms

preferred PG Hamiltonian boundary term:

B(Z) = iZϑατα +∆Γα

β ∧ iZραβ + DβZ

α∆ραβ,

preferred GR Hamiltonian boundary term:

B(Z) =1

2κ(∆Γα

β ∧ iZηαβ + DβZ

α∆ηαβ); ηαβ... := ∗(ϑα ∧ ϑβ ∧ · · · ).

Like many other boundary term choices, at spatial infinity it gives the ADM(1961), MTW (1973), Regge-Teitelboim (1974), Beig-O Murchadha (1987),Szabados (2003) energy, momentum, angular-momentum, center-of-massmomentum.

It has some special virtues:(i) at null infinity: the Bondi-Trautman energy & the Bondi energy flux(ii) it is covariant,(iii) it is positive—at least for spherical solutions and large spheres,(iv) for small spheres it is a positive multiple of the Bel-Robinson tensor,(v) first law of thermodynamics for black holes,(vi) for spherical solutions it has the hoop property,(vii) for a suitable choice of reference it vanishes for Minkowski space.

29

best matched reference (solving problem 2)

◮ Regarding the second ambiguity inherent in quasi-localenergy-momentum expressions: the choice of reference. Minkowskispace is the natural choice, but one needs to choose a specificMinkowski space. Recently we proposed(i) 4D isometric matching on the boundary,(ii) energy optimizationas criteria for selecting the “best matched” reference on the boundary ofthe quasi-local region.

◮ Remark: the hardest part of 4D isometric matching is the embedding ofthe 2D surface S into Minkowski space; S.T. Yau and coworkers havemade deep investigations into this

◮ For a detailed discussion of our covariant Hamiltonian boundary termsand our reference choice proposal see the Thursday MS parallelsession talkC.-M. Chen, “Covariant Hamiltonian Boundary Term: Reference andQuasi-local quantities”and G. Sun, C.-M Chen, J.-L. Liu and J.M. Nester: Chinese J. Phys. 53(2015) 110107

30

The Poincare gauge theory of gravity

◮ The translation and Lorentz gauge potentials are

the coframe ϑα and connection Γαβ .

◮ The associated field strengths are the torsion and curvature 2-forms:

Tα = Dϑα, Rαβ = DΓα

β

◮ The PG Lagrangian density

LPG ∼ 1

κ

(

Λ+ curvature + torsion2)

+1

curvature2 ,

κ := 8πG/c4 and −1 has the dimensions of action.

◮ varying ϑ,Γ gives dynamical eqns of the form

κ−1(Λ + curv +D tor + tor2) + −1curv2 = energy-momentum,

κ−1tor + −1D curv = spin.

◮ The general theory has 11 scalar plus 7 pseudoscalar parameters, butthere are 1 even parity and 2 odd total differentials, so effectively 10scalar + 5 pseudoscalar = 15 “physical” parameters.

31

dark torsion?

◮ Torsion couples to intrinsic spin, not orbital angular momentum,see Hehl, Obkhov, Peutzfeld, Phys Lett A 377 (2013) 1775.

◮ The fundamental leptons and quarks of the standard model arespin-1/2. Their spin density is totally antisymmetric. Totallyantisymmetric torsion is the important part in the material interaction,the rest of the torsion mainly interacts only indirectly through non-linearterms.

◮ Furthermore, highly polarized spin density is practically nonexistent inthe present day universe. Consequently, ordinary matter hardly excitesor responds directly to torsion. Torsion could have a significantmagnitude and yet be hardly observable: “dark torsion”.

◮ At very high densities the situation is quite different, at around 1052

gm/cm3, the nucleon spin density is comparable to the material energydensity, beyond that density the spin-torsion interaction dominatesgravity in the PG.

◮ However, while hardly noticeable on the lab, solar system, or galacticscale, as we shall illustrate (in common with the cosmological constant)torsion could well have measurable effects on the cosmological scale.

32

General PG homogeneous & isotropic cosmologiesFor such cosmologies the general theory has an effective Lagrangian.Including a = aH , one gets 6 1st order equations for a,H , the scalar andpseudoscalar curvatures R,X and the two “scalar” torsion components u, x.

−w4+6

2R − µ3−2

4X = −

[

−3a2 − w4+6R− µ3−2

2X]

u

+[

6σ2 −µ3−2

2R+ w2+3X

]

x

+w4−2[2X − 24(H − u)x]x,

−µ3−2

4R +

w2+3

2X = −

[

6σ2 −µ3−2

2R+ w2+3X

]

u

+[

12a3 + w4+6R +µ3−2

2X]

x

−w4−2(2R − 12[(H − u)2 − x2 + ka−2])x,

H − u =R

6− 2H2 + 3Hu− u2 + x2 − ka−2,

a2u =1

3(−a0R − σ2X + ρ− 3p+ 4Λ)

+a2(u2 − 3Hu)− 4a3x

2

x =X

6− 3Hx+ 2xu, .

33

material energy density

ρ = −Λ+ 3a0[(H − u)2 − x2 + ka−2]

−3

2a2(u

2 − 2Hu) + 6a3x2 + 6σ2x(H − u)

+w4+6

24

[

R2 − 12R{

(H − u)2 − x2 + ka−2}]

+µ3−2

24

[

RX − 6X{

(H − u)2 − x2 + ka−2}− 12Rx(H − u)]

−w2+3

24

[

X2 − 24Xx(H − u)]

.

34

generic PG cosmology evolution

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0HHtL

0 2 4 6 8 10-0.2

-0.1

0.0

0.1

0.2

0.3

0.4uHtL

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

xHtL

0 2 4 6 8 10

-50

0

50

Time�T0

d2aHtL�dt2

0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

1.0

1.5RHtL

0 2 4 6 8 10-3

-2

-1

0

1

2

3

XHtL

The observables H, u, x, a, R, X, for w4−2 = 1 (red tiny), 0.5 (orange long

dashed), 0 (grey solid), −0.5 (green medium dashed), −1 (blue dot-dashed).

35

remarks

◮ The above equations are the most general PG cosmologies.

◮ Generically the model has in addition to the usual scale factor,effectively 2 “scalar” torsion components carrying spin 0+ and 0−, withspecific non-linear couplings distinct from any other scalar fields.

◮ Typically they, and the Hubble expansion rate, show dampedoscillations.

◮ But the model has other types of behavior in degenerate special cases;including the Einstein-Cartan theory and GR.

For more details seeF.-H. Ho, H. Chen, J.M. Nester, H.-J. Yo: Chinese J. Phys. 53 (2015) 110109arXiv:1512.01202

36

Concluding summary

◮ some contributions of Einstein, Hilbert, Klein & Noether

◮ gravity as the premier gauge theory

◮ the important role of gravitational energy and its non-localization

◮ energy-momentum pseudotensors, their ambiguity, and Hamiltonianconnection

◮ from a 1st order approach to a covariant Hamiltonian formalism

◮ Poincare gauge theory of gravity (PG), including GR & Einstein Cartan

◮ PG and GR preferred Hamiltonian boundary terms for quasi-localenergy-momentum and angular momentum

◮ dark torsion

◮ general PG homogeneous & isotropic cosmology, where torsion couldhave significant effects

Thank you for your patience and attention.

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