Gravity as a Yang-Mill Gauge Theory?

1. Introduction2. Is GR a Gauge Theory of Gravity?

3. SO(4,2) Yang-Mills Gauge Theory of Gravity4. Physics of SO(4,2) GYM

5. Further Work6. Appendices

Gravity as a Yang-Mill Gauge Theory?CCGRRA-16, Simon Fraser University, 2016

J. Gegenberg (plus Gabor Kunstatter,Shohreh Rahmati,SanjeevSeahra)

University of New Brunswick

July 1, 2016

Gegenberg (slide 1 of 45)




Contents

1. Introduction

2. Is GR a Gauge Theory of Gravity?

3. SO(4,2) Yang-Mills Gauge Theory of Gravity

4. Physics of SO(4,2) GYM

5. Further Work

6. Appendices





GR inspires YM Gauge Theories of non-gravitational forces-60s, Yang, Mills, Kibble.

YM Gauge Theories of non-gravitational forces inspiresgravitational gauge theories -60s-80s, etc. plus Utiyama,Kaku, Mansouri, ...





GR inspires YM Gauge Theories of non-gravitational forces-60s, Yang, Mills, Kibble.

YM Gauge Theories of non-gravitational forces inspiresgravitational gauge theories -60s-80s, etc. plus Utiyama,Kaku, Mansouri, ...





Lightening Review of Gauge Theories 1

Gauge theory < −− > Noether Theorem.

Particle physics uncovers conservation laws: charge,momentum, angular momentum,...

Noether Theorem: Conservation laws derive from symmetriesof action: ψ(x)→ ψ′(x) = Sψ(x).

S is representation of some Lie group with constantparameters:

Phase change invariance ↔ U(1): ψ(x)→ e iαψ(x).

Loretnz invariance ↔ SO(3, 1): V µ(x)→ ΛµνV ν(x).




























Smash Global- Go Local.

For U(1): Constant α replaced by function α(x).

For SO(3,1): Later- complicated.

Action no longer gauge invariant.

So introduce compensating field Aµ(x), that is, a connection:

∂µψ(x)→ Dµψ(x) = ∂µψ(x)− Aµ(x)ψ(x). (1)




























Dynamics of gauge potential Aµ(x):

SMaxwell = κ

∫d4x√−ggµαgνβFµνFαβ. (2)

[Aµ] = L−1 ⇒ [κ] = L0.

Non-Abelian case:

DµψB(x) = ∂µψ

B(x) + f BCDAC

µ (x)ψD(x);

SYM = κ

∫d4x√−ggµαgνβhBCFB

µνFCαβ. (3)

[κ] = L0.






Dynamics of gauge potential Aµ(x):

SMaxwell = κ

∫d4x√−ggµαgνβFµνFαβ. (2)

[Aµ] = L−1 ⇒ [κ] = L0.

Non-Abelian case:

DµψB(x) = ∂µψ

B(x) + f BCDAC

µ (x)ψD(x);

SYM = κ

∫d4x√−ggµαgνβhBCFB

µνFCαβ. (3)

[κ] = L0.





So what’s the big deal?

Need dimensionless coupling constant to have a chance forrenormalizable perturbative theory of gravity.

U(1) and its non-Abelian Yang-Mills generalizations in 4Dhave dimensionless coupling constants.

Can we construct gauge gravity theory which hasdimensionless coupling constant?





Einstein-Hilbert?

Replace global Lorentz invariance of Spec. Rel. with localgauge invariance.

Huh? How to implement this?

Replace ∂µV ν(x) by ∂µV ν(x) +νµρ

(x)V ρ(x).

Not a theory of connection- theory of metric.

Einstein-Hilbert action has dimensionfull coupling constant.





Einstein-Hilbert?




(x)V ρ(x).







Einstein-Hilbert?




(x)V ρ(x).







Einstein-Hilbert?




(x)V ρ(x).







What then?

Einstein-Cartan? Still linear in curvature so [κ] = L−2, butnow theory of connection.

Chern-Simons? [κ] = L0, but hard to construct in evenspacetime dimensions.





Yang-Mills type theory? (Quadratic in curvature.)

Gauge group? Should be some group relevant to spacetime.

Back to Poincare ISO(3,1). Utiyama, Kibble, Kaku et. al.,Mansouri,...Significant tampering required, and still does not work well...

Symmetry groups relevant to other spacetimes of constantcurvature (dS, AdS)?Maybe- Huang et. al. and earlier Kaku et. al. and Hehl.





















Why SO(4,2)?

Particle physics points there!

Standard model: matter built from massless fermions.

Massless fields in Minkowski spacetime SO(4,2) invariant:Bessel-Hagen 1913, ...Mass from dynamical symmetry breaking a.k.a. Higgsmechanism.

AdS/CFT.

Massless spin 2 via linearized GR NOT SO(4,2) invariant:Drew-Gegenberg 1980.





Why SO(4,2)?

Particle physics points there!

Standard model: matter built from massless fermions.

Massless fields in Minkowski spacetime SO(4,2) invariant:Bessel-Hagen 1913, ...Mass from dynamical symmetry breaking a.k.a. Higgsmechanism.

AdS/CFT.

Massless spin 2 via linearized GR NOT SO(4,2) invariant:Drew-Gegenberg 1980.





History of SO(4,2) YM Gravity

1920: Weyl.

1950-1980: Kaku et. al. Hehl, ...

To the present: P. Mannheim (Weyl gravity), J.T. Wheeler.





And Sg [A] is what?

SO(4,2) gauge potential:

A = eaPa + ωabJab + laKa + qD

Action

S [A,Φ] = − 1

2g 2YM

∫d4x√−ggαµgβν hBC FB

αβ FCµν + Sm[A,Φ].

(4)

hBC Cartan-Killing metric (constructed from f BCD),

gµν = ηabeaµebν .

Coupling constant κ := 12g2

YMis dimensionless.

Geometric interpretation of `aα, qα?





Gauge Symmetries

Gauge transformations

δABµ = Dµε

B = ∂µεB + f B

CDACµ εD . (5)

Action not invariant under full infinitesimal SO(4,2) YMgauge transformations, because of factor

√−ggαµgβν .

Translation invariance lost? No- still have diffeo invariance.

Action is gauge invariant under 11 parameter subgroup C11 ofSO(4,2) generated by Ka, Jab,D.Invariance: C11 × diff (R4).





Gauge Symmetries

Gauge transformations

δABµ = Dµε

B = ∂µεB + f B

CDACµ εD . (5)

Action not invariant under full infinitesimal SO(4,2) YMgauge transformations, because of factor

√−ggαµgβν .

Translation invariance lost? No- still have diffeo invariance.

Action is gauge invariant under 11 parameter subgroup C11 ofSO(4,2) generated by Ka, Jab,D.Invariance: C11 × diff (R4).





Gauge Fixing

By gauge transformations of the form εAJA = λaKa we canimpose the gauge condition

qα = 0.

Residual gauge freedom: qα = 0 is preserved undertransformations parametrized by:

εAJA = −2eaα∂αΩ Ka + ΛabJab + ΩD. (6)

After fixing qα = 0, have to restrict gauge transforms toLorentz × Dilatations.





Under this, the frame fields and metric transform as

δeaα = (−Λab + 1

2 Ωδab)ebα, δgαβ = Ωgαβ. (7)

Λab generates infinitesimal Lorentz rotations of the framefields and

Ω generates infinitesimal local conformal transformations.

Our model is conformally invariant.





Preserving Zero Torsion

εAJA = −2eaα∂αΩ Ka + ΩD, (8)

generates a conformal transformation of the metric andpreserves the gauge condition qα = 0 and

The torsion tensor transforms as

δT aαβ = 1

2 T aαβΩ. (9)

So, under transformations that preserve the qα = 0 gauge, thetorsion tensor is gauge invariant, and the torsion-free sector ispreserved under this restricted group of gauge transformation.





Equations of Motion

Parametrize the matter current

jBνJB = aaνPa + baνKa + cabνJab + dνD, (10)

Define aαν := eαa aaν , etc. These are tensorized mattercurrents.

where aαβ := aαβ − 16 gαβa, a = aµµ,

Sαβ is the Schouten tensor,Bµν is the Bach tensor, Cαβγδ is the Weyl tensor,and Qαν is complicated tensor bilinear in curvatures andcurrents.

F[µν] := ea[µ`aν].





Eq. of motion:

f(αβ) = 4Sαβ − 2aαβ; (11)

0 = ∇αaαβ + cβαα − 1

2 dβ; (12)

∇[αFβν] = 19 gνβcαµµ − 1

9 gναcβµµ − 13 cαβν ; (13)

∇αFαβ = 12 dβ − 1

3 cβαα − 1

6∇βa; (14)

Bαν = −14 bαν +∇µ(∇[ν aµ]α +∇[νFµ]α) + Qαν .(15)

Note f(αβ) given algebraically.





Linearizing About Flat Spacetime

Minkowski background: Rαβγδ = 0, Fαβ = 0.

All matter fields vanish aµν |Background = 0 etc.

Perturbative matter sources turned off except aαβ, bαβ.

As in Mannheim , traceless metric perturbation Hαβ:

gαβ = ηαβ + hαβ, Hαβ = hαβ − 14ηαβh. (16)





Wave-ish Equation

− 14 ( 2

3∂α∂β∂µ∂νHµν − 2∂ν∂(αHβ)ν + 1

3ηαβ∂µ∂νHµν

+2Hαβ) = 12aTF

αβ + 14 bαβ + 1

6∂α∂βa. (17)

where, aTFαβ trace-free part of aαβ.





For static sources and fields:

Hαβ(r) =

∫d3r′

aTFαβ(r′)

2π|r − r′|+

∫d3r′|r − r′|

8πbαβ(r′). (18)





Modification of the gravitational interaction for longwavelengths.

Short and long range gravitational forces < −− > aTFαβ , as

opposed to undifferentiated aTFαβ appearing in GR, term in

wave-equation.

We can use this freedom to impose a (PPN) gauge conditionfrom which it follows that the standard PPN parameter γ is

γ = 1 +2r 2

|rarb|. (19)

Hence, in order to satisfy the Cassini constraint|γ − 1| . 10−5, we need |rarb| r 2 in the solar system.





Towards GYM Cosmology

See talk by Sanjeev Seahra, coming soon...

Input FRW geometry and matter coupled to metric andspecial conformal gauge potential.

Big Bounce instead of Big Bang (for generic values ofintegration constants).

Observationally consistent inflation, followed by radiationphase followed by matter-dominated phase.

Late time acceleration consistent with observations.





Further Work

Include torsion in cosmology.

Revisit experimental tests of relativistic gravity, includinggravitational waves.

Quantization, starting from careful constraint analysis.





Appendix 1: SO(4,2) Generator Algebra

[Jab, Jcd ] = ηadJbc + ηbcJad + ηacJdb + ηbdJca,

[Pa, Jbc ] = ηbaPc − ηcaPb, [D,Ka] = Ka,

[Ka, Jbc ] = ηbaKc − ηcaKb, [Pa,D] = Pa,

[Pa,Kb] = 2(ηabD− Jab),

where ηab is the Minkowski metric.The structure constants are defined by [JA, JB ] = f C

ABJC .





Structure constants and Cartan-Killing metric

hAB = f MAN f N

BM .

The non-trivial components of hAB are:

hab = hab = −2ηab, h14,14 = 2. (20)

h[ab][cd ] = h[cd ][ab] = −4ηa[cηd ]b. (21)





Relation to geometry

eaα are the components of an orthonormal frame field.

ωabα are connection one-forms.

Affine connection:

Γγαβ = eγa (∂αeaβ + ωacα ecβ).

Affine connection is metric compatible(Carroll 2004):

0 = ∇αgβγ ,





Appendix 2: Action in terms of spacetime tensors

The Riemann curvature and torsion tensors of M are given by:

Rµναβ = eµa eνb (dωab + ωac ∧ ωc

b)αβ, (22)

Tαβγ = eαa (dea + ωac ∧ ec)βγ . (23)

Note that we do NOT assume Tαβγ = 2Γα[βγ] = 0.





S =2

g 2YM

∫d4x√−g[(Rαβγδ − 1

2φαβγδ)2

+ 2(∇αf µβ + f µσTσαβ − f µαqβ)

× (Tµαβ + 2gµ[αqβ])+

(∂[αqβ] + Fαβ)2]

+ Sm. (24)

Here, we have defined:

fαβ := ηabeaαlbβ , Fαβ := 12 f[αβ], (25)

φαβγδ := gγ[αf β]δ − g δ[αf β]γ . (26)





Appendix 3: Equations of Motion

DµFBµν = kBν + jBν , (27)

where Dµ is the gauge covariant derivative:

DµFBµν := ∇µFBµν + f BCDAC

µFDµν , (28)

and ∇µ is the derivative operator defined from the Levi-Civitaconnection.Note that ∇α = ∇α if and only if Tα

βγ = 0.





The currents in (27) are given by:

kAνJA =τµνebµKb

2, jνC = −

g 2YM

2√−g

δ(√−gLm)

δACν

, (29)

where Lm is the Lagrangian density of the matter fields,jνC = hBC jBν and

τρσ = hAB

(FAρµFBσ

µ − 14 gρσFA

µνFBµν). (30)





Explicit equations of motion

In a gauge where qα = 0:

aαν = ∇µTαµν + Rαν − 12 (f (αν) + 1

2 fgαν), (31)

bαν = ∇µθαµν − fλµΦαλµν − f αµFµν − 1

2ταν , (32)

cαβν = ∇µΦαβµν + 12θ

[αβ]ν − 12 f [α|µ|T β]ν

µ, (33)

dν = (2∇µ +∇µ)Fµν + 12∇µ(f (µν) − gµν f )

+2FµσT [σµ]ν . (34)

Here, Rαβ is the Ricci tensor and we have defined

θµαβ := ∇αf µβ −∇βf µα + f µσTσαβ, (35)

Φαβγδ := Rαβγδ − 12 (gγ[αf β]δ − g δ[αf β]γ). (36)





Appendix 4: The tensor Qαν

:

Qαν = 12 aλµCαλµν − 1

8ταν − 1

2 (2Sλµ − aλµ + Fλµ)× (37)

(gλ[µaν]α − gα[µaν]λ + gα[µFν]λ − gλ[µFν]α (38)

+ gαλFµν). (39)





Appendix 5: The Linearized Equations of Motion

Under these assumptions and expanding to linear order, we findthat

∇[αFβν] = 0, ∇αFαβ = −16∇βa, a = 0, (40)

− 14 ( 2

3∂α∂β∂µ∂νHµν − 2∂ν∂(αHβ)ν + 1

3ηαβ∂µ∂νHµν

+2Hαβ) = 12aTF

αβ + 14 bαβ + 1

6∂α∂βa. (41)

where, aTFαβ trace-free part of aαβ.





Explicit Solution

Do not impose the transverse condition (∂αHαβ = 0).

Assume aαβ and bαβ both have δ-function support at theorigin.

Then

ds2 = −(1 + 2φ)dt2 + (1− 2ψ)(dx2 + dy 2 + dz2), (42)

φ+ ψ

2= − ra

r− r

rb. (43)

ra and rb are constants proportional to the amplitude of theδ-functions in aαβ and bαβ, respectively.

Note that ra and rb are not necessarily positive.





φ− ψ not fixed by wave-equation. This is because φ+ ψ isgauge invariant while φ− ψ is a purely gauge degree offreedom.

We can use this freedom to impose a (PPN) gauge conditionφ = −ra/r from which it follows that the standard PPNparameter γ is

γ =ψ

φ= 1 +

2r 2

|rarb|. (44)

Hence, in order to satisfy the Cassini constraint|γ − 1| . 10−5, we need |rarb| r 2 in the solar system.





Appendix 6: Cosmology Details

a acts as a source for both Fαβ and Hαβ.

Hence consider Hαβ to be the sum of contributions sourced bya and aTF

αβ .

Impose transverse gauge condition ∂αHαβ = 0.





Isotropy and homogeneity −− > aµν , bµν of the form ofperfect fluid stress tensors.

For any homogeneous and isotropic spacetime the Bach andWeyl tensors vanish identically.

Conservation of aµν and constancy of a fix the ‘density’ and‘pressure’ of aµν in terms of two integration constants and theFRW scale factor.

We fix the “ordinary” matter content of the universe to benon-interacting pressureless dust and radiation as in ΛCDM.





Boils down to ‘Friedmann equation’:

H2 =g 2YM

8

(ρm + ρr

Λ− ΠA4

)− k

r 20 A2

+Λ

3− Π

3A4, (45)

Reduces to GR+ΛCDM Friedmann in late time limit, andidentifying one of the integration constants with thecosmological constant Λ.

Probes of the expansion history of the late time universe givesus the order of magnitude of Λ:

Λ ∼ (10−3 eV)4

M2Pl

∼ (10−33 eV)2. (46)

This in turn fixes the size of Yang-Mills coupling constant tobe g 2

YM ∼ 10−120.





Cosmological dynamics

Rewrite the Friedmann equation as the equation of motion ofa zero-energy particle moving in a one-dimensional effectivepotential:

1

2

(dA

dτ

)2

+ Veff(A) = 0, (47)

where we have defined τ = H0t and

Veff(A) = − ΩΛΩm

2A(ΩΛ + ΩΠA4 )− ΩΛΩr

2A2(ΩΛ + ΩΠA4 )−Ωk

2−ΩΛA2

2+

ΩΠ

2A2.

(48)

All values of the scale factor with Veff(A) > 0 are classicallyforbidden.





Figure: Numeric solutions for the scale factor A assuming(Ωm,Ωr,ΩΛ) = (0.27, 8.24× 10−5, 0.73). All simulations show a bounceat time t = t0. The scale factor at the bounce increases with increasingΩΠ.





Inflation

Estimate how many e-folds N of exponential expansion occurafter the bounce:We see from this that we can make N arbitrarily large byselecting ΩΠ to be very small.Take cosmological parameters as their central values:

N ∼ 60− 1

4ln

ΩΠ

10−109∼ 66− 1

4ln

ΩΠ

g 2YM

. (49)

Energy scale during inflation:

Einf = (3M2PlH

2inf)

1/4 ≈ 31/4Ω1/4r Ω

1/4Λ Ω

−1/4Π M

1/2Pl H

1/20 . (50)

Taking central values for the usual density parameters andH0 ∼ 10−33 eV, we find

N ∼ 60 + ln

(Einf

1015 GeV

), (51)






Gauge potential components eaα for FRW metric.

qα = 0 gauge fixing.

Torsion-free and f[αβ] = 0 sector.

Only nontrivial sources are aµν , bµν .

Boils down to dynamics

1

2

(dA

dτ

)2

+ Veff(A) = 0, (52)

where we have defined τ = H0t






Gauge potential components eaα for FRW metric.

qα = 0 gauge fixing.

Torsion-free and f[αβ] = 0 sector.

Only nontrivial sources are aµν , bµν .

Boils down to dynamics

1

2

(dA

dτ

)2

+ Veff(A) = 0, (52)

where we have defined τ = H0t





Cosmology Summary

Observation gives gYM ∼ 10−60.

Early Universe “big bounce” that occurs when Veff(A) = 0and there is no big bang singularity in our model.

Immediately after this cosmological bounce there is a phase ofnearly dS early-time acceleration - inflation with ∼ 60 E-folds.

Two further transitions where V ′eff(A) = 0: The first transitionmarks when the acceleration in the early Universe ends andthe radiation dominated epoch starts,

Second transition occurs in the late Universe when matterdomination ends and the second acceleration epoch starts.

Latter is consistent with the observed late-time acceleration ofthe Universe.





Cosmology Summary

Observation gives gYM ∼ 10−60.

Early Universe “big bounce” that occurs when Veff(A) = 0and there is no big bang singularity in our model.

Immediately after this cosmological bounce there is a phase ofnearly dS early-time acceleration - inflation with ∼ 60 E-folds.

Two further transitions where V ′eff(A) = 0: The first transitionmarks when the acceleration in the early Universe ends andthe radiation dominated epoch starts,

Second transition occurs in the late Universe when matterdomination ends and the second acceleration epoch starts.

Latter is consistent with the observed late-time acceleration ofthe Universe.


Documents

Gravity as a Yang-Mill Gauge Theory?