Utah State University · 2009. 7. 2. · Contents 1 Basic conformal relations 2 1.1 Structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents1 Basic conformal relations 2

1.1 Structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Killing metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Lorentz-invariant subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.1 Bianchi identities of conformal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.2 Bianchi identities in Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.3 Bianchi identities in Weyl geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Relationship between the Riemann, Weyl, Ricci and Eisenhart and dilatation tensors . . . . . . . . . . 81.6.1 Curvature relations in Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6.2 Curvature relations in Weyl geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7 Transformation of the Weyl and Eisenhart tensors under conformal transformation . . . . . . . . . . . 131.8 Symmetries of the Weyl curvature with nonzero dilatation . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Poincaré gauge theory 152.1 The structure equations and Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The Einstein-Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 The field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Conformal Ricci flatness in Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Necessary condition: Exterior derivative of the Eisenhart tensor . . . . . . . . . . . . . . . . 172.4.2 Sufficient condition: Integrability for conformal Ricci flatness . . . . . . . . . . . . . . . . . 19

3 Weyl (homothetic) gauge theory 203.1 Conformal Ricci flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 An action for General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Auxiliary conformal gauging 254.1 Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Inconsistency of the linear action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 An invariant action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Quadratic action theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.1 Varying the Weyl vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Varying the spin connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.3 Varying the special conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . 304.3.4 Varying the solder form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.5 Summary of the field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4 The auxiliary field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 The remaining field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5.1 A little digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5.2 Combine the Lorentz field equation and the Bianchi identity . . . . . . . . . . . . . . . . . . 37

4.6 Structures and new features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Biconformal gauging 375.1 New structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1.1 Uniform linear action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.2 Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1.3 Natural metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1.4 SL(2,R) and complex structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Summary of biconformal solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 General relativity on biconformal spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 The existence of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.1 Canonical, orthogonal submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1

5.4.2 New coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.4.3 Integrability of the new coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4.4 The signature of the induced metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A Variation of the modified linear action 46

1 Basic conformal relations

1.1 Structure equationsThe Maurer-Cartan equations for the conformal group are

dωab = ωcbω

ac +2∆

acdbωcω

d

dωa = ωcωac +ωωa

dωa = ωcaωc +ωaωdω = ωaωa

where the geometric interpretation of each set of generators is

ωab Lorentz trans f ormationωa Translationωa Special con f ormal trans f ormationω Dilatation

This particular breakdown of the generators is important because each of these four sets of transformations is invariantunder the homothetic group. We discuss this below.

If we set the special conformal transformations to zero, ωa = 0, the remaining subgroup is the inhomogeneousWeyl group, described by the structure equations.

dωab = ωcbω

ac

dωa = ωcωac +ωωa

dω = 0

This has further subgroups. If we set the Weyl vector to zero, ω = 0, so that there are no dilatations, then we have theMaurer-Cartan equations of the Poincaré group

dωab = ωcbω

ac

dωa = ωcωac

If, instead of eliminating dilatations, we set the translational gauge vector to zero, ωa = 0, then we have the homoge-neous Weyl group,

dωab = ωcbω

ac

dω = 0

comprised of Lorentz transformations and dilatations only. Finally, setting both translatiosn and dilatations to zero,ωa = ω = 0, only the Maurer-Cartan structure equations for the Lorentz group remain:

dωab = ωcbω

ac

1.2 Killing metricThe Killing metric of the conformal group, given by

KΣΛ = λc ∆ΞΣ cΞ

∆Λ

2

With a suitable choice of the constant λ this becomes

KΣΛ =

12 ∆

acdb 0 0 0

0 0 δ ab 00 δ ba 0 00 0 0 1

While the conformal Killing metric is itself non-degenerate, its restriction to the various subgroups may or may notinduce a natural, non-degenerate metric.

The restrictions of KΣΛ to the inhomogeneous Weyl group and the Poincaré group, respectively, give

KAB =

12 ∆acdb 0 00 0 00 0 1

and

KAB =( 1

2 ∆acdb 0

0 0

)where the bold zero, 0, denotes the vanishing n×n translation sector. When gauged, these groups do not have naturalmetrics induced from the Killing metric.

The homogeneous Weyl and Lorentz groups fare better, both having non-degenerate Killing metrics

KAB =( 1

2 ∆acdb 0

0 1

)and

KAB =( 1

2 ∆acdb

)respectively.

1.3 Gauge transformationsA Lie group acts on its connection 1-forms according to

ω̃ = gωg−1−dg g−1

Expanding this infinitesimally for the conformal group using the 6-dim representation,

gAB =

Λab Λa4 Λa5Λ4b Λ44 Λ45Λ5b Λ

54 Λ

55

=

Λab Λa4 ηacΛ4cΛ4b Λ44 0ηbcΛc4 0 −Λ44

where Λ44 generates a dilatation, Λ

ab generates a Lorentz transformation, Λ

a4 generates a translation and Λ

4a generates

a special conformal transformation. Let Λ̄AB denote the inverse transformations, and in particular, Λ̄44 =1

Λ44. Then we

have for the Lorentz piece,

ω̃ab = ΛaBω

BCΛ̄

Cb−dΛaC Λ̄Cb

= ΛacωcdΛ̄

db +Λ

acω

c4Λ̄

4b +Λ

acω

c5Λ̄

5b

+Λa4ω4dΛ̄

db +Λ

a4ω

44Λ̄

4b +Λ

a5ω

5dΛ̄

db +Λ

a5ω

55Λ̄

5b

−dΛac Λ̄cb−dΛa4 Λ̄4b−dΛa5 Λ̄5b= Λacω

cdΛ̄

db +Λ

acω

c4Λ̄

4b +Λ

acη

ceω4eηbdΛ̄d4

+Λa4ω4dΛ̄

db +Λ

a4ω

44Λ̄

4b +η

acΛ4cωe4ηedΛ̄

db−ηacΛ4cω44ηbdΛ̄d4

−dΛac Λ̄cb−dΛa4 Λ̄4b−ηacdΛ4c ηbdΛ̄d4

3

For translations,

ω̃a4 = ΛaBω

BCΛ̄

C4−dΛaC Λ̄C4

= ΛacωcdΛ̄

d4 +Λ

acω

c4Λ̄

44

+Λa4ω4dΛ̄

d4 +Λ

a4ω

44Λ̄

44

+Λa5ω5dΛ̄

d4

−dΛac Λ̄c4−dΛa4 Λ̄44= Λacω

cdΛ̄

d4 +Λ

acω

c4Λ̄

44

+Λa4ω4dΛ̄

d4 +Λ

a4ω

44Λ̄

44 +η

acΛ4cωe4ηedΛ̄

d4

−dΛac Λ̄c4−dΛa4 Λ̄44

For special conformal transformations,

ω̃4a = Λ4Bω

BCΛ̄

Ca−dΛ4C Λ̄Ca

= Λ4cωcdΛ̄

da +Λ

4cω

c4Λ̄

4a +Λ

4cω

c5Λ̄

5a

+Λ44ω4dΛ̄

da +Λ

44ω

44Λ̄

4a

+Λ45ω5dΛ̄

da +Λ

45ω

55Λ̄

5a

−dΛ4c Λ̄ca−dΛ44 Λ̄4a= Λ4cω

cdΛ̄

da +Λ

4cω

c4Λ̄

4a +Λ

4cη

cdω4dηacΛ̄c4

+Λ44ω4dΛ̄

da +Λ

44ω

44Λ̄

4a

−dΛ4c Λ̄ca−dΛ44 Λ̄4a

and dilatations,

ω̃44 = Λ4Bω

BCΛ̄

C4−dΛ4C Λ̄C4

= Λ4cωcdΛ̄

d4 +Λ

4cω

c4Λ̄

44

+Λ44ω4dΛ̄

d4 +Λ

44ω

44Λ̄

44

−dΛ4c Λ̄c4−dΛ44 Λ̄44

Lie-algebra valued tensors transform in the same way except for the inhomogeneous −dΛAC Λ̄CB terms.Now that we have the right transformations, we can simplify the notation by dropping all A = 4 labels, and setting

Λ̄ = 1Λ for dilatations. Collecting the results:

ω̃ab = Λacω

cdΛ̄

db +Λ

acω

cΛ̄b +ΛacηceωeηbdΛ̄d

+ΛaωdΛ̄db +ΛaωΛ̄b +ηacΛcωeηedΛ̄db−ηacΛcωηbdΛ̄d

−dΛac Λ̄cb−dΛa Λ̄ b−ηacdΛc ηbdΛ̄d

ω̃a = ΛacωcdΛ̄

d +ΛacωcΛ̄+ΛaωdΛ̄d

+ΛaωΛ̄+ηacΛcωeηedΛ̄d−dΛac Λ̄c−dΛa Λ̄ω̃a = ΛcωcdΛ̄

da +Λcω

cΛ̄a +ΛcηcdωdηacΛ̄c

+ΛωdΛ̄da +ΛωΛ̄a−dΛc Λ̄ca−dΛ Λ̄aω̃ = ΛcωcdΛ̄

d +ΛcωcΛ̄+ΛωdΛ̄d

+ΛωΛ̄−dΛ c Λ̄c−dΛ Λ̄

These relations specialize immediately to subgroups of the conformal group.

1.4 Lorentz-invariant subgroupsThe conformal group acts on its own Lie algebra,

G̃A = gGAg−1

4

and since we expect, at a minimum, homothetic symmetry of our final gauged space, it is useful to have a breakdownof the conformal transformations into homothetically invariant subgroups. It is sufficient to consider an infinitesimalhomothetic transformation, given by an arbitrary linear combination of the Lorentz and dilatational generators,

δ =12

wabMba +λD

It is easy to see from the Lie algebra,

[Ma b,Mc

d ] = −12(δ cb δ

af δ

ed −ηbdηceδ af −ηacηb f δ ed +δ ad ηceηb f

)M f e

[Ma b,Pc] =12

(δ ac δ

db −ηbcηad

)Pd

[Ma b,Kc] = −1

2(δ cb δ

ad −ηbdηac)Kd[

Pa,Kb]

= −δ ba D−(

ηacηbd−δ da δ bc)

Mc d

[D,Pa] = −δ ba Pb[D,Ka] = δ ab K

b

that

[δ ,Ma b] ∼ Ma b[δ ,Pa] ∼ Pa[δ ,Ka] ∼ Ka

[δ ,D] = 0

Moreover, since

[D,Pa] = −Pa[D,Ka] = Ka

we may assign conformal weight−1 to the generator, Pa, of translations and weight +1 to the generator, Ka, of specialconformal transformations. Combining this with the behavior under Lorentz transformations, we may summarize theirreducible homothetic parts:

Ma b weight 0(

11

)tensor

Pa weight −1(

01

)tensor

Ka weight +1(

10

)tensor

D weight 0 scalar

By combining these in various ways and exponentiating we readily identify all homothetically invariant subgroupsof the conformal group by checking whether the commutators close. First, notice that since[

Pa,Kb]

=−δ ba D+2Mb a

we cannot have any proper subgroup with both Pa and Ka among the generators, since it must necessarily also includethe D and Ma b generators. Denoting homothetically invariant subgroups by the subset of generators that generatethem, we have the four single-sets,

{Ma b} ,{Pa} ∼={

Kb}

,{D}

Since the n-dimensional translation groups generated by {Pa} and{

Kb}

are isomorphic, this gives three groups. Thereare five closed double-sets,

{Ma b,Pa} ∼= {Ma b,Ka} ,{Ma b,D} ,{Pa,D} ∼= {Ka,D}

5

(but not the final pair, ,{

Pa,Kb}

!), with isomorphisms reducing these to three independent homothetically invariantsubgroups. There are only two triples:

{Ma b,Pa,D} ∼= {Ma b,Ka,D}

and these generate isomorphic (inhomogeneous, homothetic) groups. Finally the improper subgroup generated by allfour sets,

{Ma b,Pa,K

a,D}

. There are therefore seven homothetically invariant proper subgroups of the conformalgroup.

We note for future reference that only two of these subgroups contain the homothetic group – the one generatedby{

Ma b,D}

is the homothetic group itself, while the group generated by{

Ma b,Ka,D}

is the inhomogeneoushomothetic group.

Also, notice that neither of these gaugings contain Pa in the subgroup. This means that the translational symmetrywill be broken when we generalize the base manifold geometry. The gauge transformation of the torsion then becomestensorial, so setting the torsion to zero is a gauge invariant specification. We may therefore consider torsion-free gaugetheories.

1.5 Bianchi identitiesWe find the Bianchi identities in Riemannian, Weyl and conformal geometries.

1.5.1 Bianchi identities of conformal geometry

The conformal Cartan equations with vanishing torsion are

dωab = ωcbω

ac +2∆

acdbωcω

d +Ωabdωa = ωcωac +ωω

a

dωa = ωcaωc +ωaω +Ωadω = ωaωa +Ω

Taking the exterior derivative of each equation and substituting for the derivatives leads to

0 = d2ωab= dωcbω

ac−ωcbdωac +2∆acdbdωcωd−2∆acdbωcdωd +dΩab

= Ωcbωac−ωcbΩac +2∆acdbΩcωd +dΩab

= DΩab +2∆acdbΩcω

d

0 = d2ωa

= dωcωac−ωcdωac +dωωa−ωdωa

= −ωcΩac +Ωωa

0 = d2ωa= dωcaωc−ωcadωc +dωaω−ωadω +dΩa= Ωcaωc−ωcaΩc +Ωaω−ωaΩ+dΩa= DΩa +Ωcaωc−ωaΩ

0 = d2ω= dωaωa−ωadωa +dΩ= dΩ−ωaΩa

Collecting terms, we have

DΩab +2∆acdbΩcω

d = 0ωcΩac−Ωωa = 0

DΩa +Ωcaωc−ωaΩ = 0dΩ−ωaΩa = 0

6

The Bianchi identities for Weyl and Riemannian geometries follow immediately as

DΩab = 0ωcΩac−Ωωa = 0

dΩ = 0

in a Weyl geometry and

DRab = 0ecRac = 0

in Riemannian.In the scale-invariant geometries, i.e., conformal or Weyl, the torsion-free condition leads to an algebraic constraint

relating the dilatational and Lorentz curvatures,

ωbΩab−Ωωa = 0

Expanding this in components, we have

0 = Ωa[bcd]−δa[bΩcd]

=13

(Ωabcd +Ωacdb +Ω

adbc−δ ab Ωcd−δ ac Ωdb−δ ad Ωbc)

Now contract on the ac components,

0 = Ωcbcd +Ωccdb +Ω

cdbc−Ωbd−nΩdb−Ωbd

so that the antisymmetric part of the trace of the curvature is proportional to the dilatation:

Ωcbcd−Ωcdcb =−(n−2)Ωbd

The trace of the curvature is symmetric if and only if the dilatation vanishes.

1.5.2 Bianchi identities in Riemannian geometry

The first Bianchi identity is

0 = ebRab= eb

(Cab +2∆

acdbRce

d)

= ebCab +Rbebea

= ebCab

The second curvature Bianchi identity is

DRab = 0Rab[cd;e] = 0

Substititing

Rabcd = Cabcd−δ ac Rbd +δ ad Rbc +ηbcRad−ηbdRac

for the curvature,

0 = DRab= DCab +2∆

acdbDRce

d

7

0 = Rabcd;e +Rabde;c +R

abec;d

= Cabcd;e−δ ac Rbd;e +δ ad Rbc;e +ηbcRad;e−ηbdRac;e+Cabde;c−δ ad Rbe;c +δ ae Rbd;c +ηbdRae;c−ηbeRad;c+Cabec;d−δ ae Rbc;d +δ ac Rbe;d +ηbeRac;d−ηbcRae;d

and take the trace on ae,

0 = Cabcd;a−Rbd;c +Rbc;d +ηbcRad;a−ηbdRac;a−Rbd;c +nRbd;c +ηbdRaa;c−Rbd;c−nRbc;d +Rbc;d +Rbc;d−ηbcRaa;d

= Cabcd;a− (n−3)Rbc;d +(n−3)Rbd;c +ηbd(Raa;c−Rac;a

)−ηbc

(Raa;d−Rad;a

)and contract again on bd,

0 = 2(n−2)(Raa;c−Rac;a

)so,

Cabcd;a = (n−3)(Rbc;d−Rbd;c)

1.5.3 Bianchi identities in Weyl geometry

The torsion-free structure equations are:

dωab = ωcbω

ac +R

ab

dea = ecωac +ωea

dω = Ω

Taking the exterior derivative of each, we have

DRab = dRab +R

cbω

ac−ωcbRac = 0

ecΩac = Ωea

dΩ = 0

In components the second becomes

Ωabcd +Ωacdb +Ω

adbc = δ

ab Ωcd +δ

ac Ωdb +δ

ad Ωbc

with traceΩcbcd−Ωcdcb =−(n−2)Ωbd

In contrast to Riemannian geometry, the trace of the curvature tensor has an antisymmetric part.

1.6 Relationship between the Riemann, Weyl, Ricci and Eisenhart and dilatation tensorsWe consider these relationships in Riemannian, Weyl, and conformal geometries. We consider only torsion-freegeometries.

1.6.1 Curvature relations in Riemannian geometry

The structure equations are:

dωab = ωcbω

ac +R

ab

dωa = ωcωac

8

Riemann, Weyl and Ehrenfest The Weyl curvature is given by

Cabcd = Rabcd−

1n−2

(δ ac Rbd−δ ad Rbc−ηbcRad +ηbdRac)+1

(n−1)(n−2)R(δ ac ηbd−δ ad ηbc)

Check the trace,

Ccbcd = Rcbcd−

1n−2

(nRbd−Rbd−Rbd +ηbdR)+1

(n−1)(n−2)R(n−1)ηbd

= Rbd−Rbd−1

n−2ηbdR+

1n−2

Rηbd

= 0

The Eisenhart tensor is defined in terms of the Ricci tensor as

Rab =−1

n−2

(Rab−

12(n−1)

Rηab)

This is invertible,

Rab = −(n−2)Rab−RηabR = −2(n−1)R

Check,

Rab = −1

n−2

(Rab−

12(n−1)

Rηab)

= − 1n−2

(−(n−2)Rab−Rηab−

12(n−1)

(−2(n−1)R)ηab)

= Rab

The relationship between the curvatures may be rewritten as

Cabcd = Rabcd

− 1n−2

δ ac

(Rbd−

12(n−1)

Rηbd)

+1

n−2δ ad

(Rbc−

12(n−1)

Rηbc)

+1

n−2ηbc(

Rad−1

2(n−1)Rδ ad

)− 1

n−2ηbd(

Rac−1

2(n−1)Rδ ac

)− 1

2(n−1)(n−2)R(δ ac ηbd−δ ad ηbc−ηbcδ ad +ηbdδ ac )+

1(n−1)(n−2)

R(δ ac ηbd−δ ad ηbc)

= Rabcd +δac Rbd−δ ad Rbc−ηbcRad +ηbdRac

= Rabcd +δac Rbd−δ ad Rbc−ηbcRad +ηbdRac

Then in terms of curvature 2-forms,

Cab = Rab +

12

(δ ac Rbd−δ ad Rbc−ηbcRad +ηbdRac)eced

= Rab +12

(2δ ac δeb −2ηbcηae)Redeced

= Rab−2∆adcb Rdec

so that finally,Rab = C

ab +2∆

acdbRce

d

9

Divergence of the Weyl curvature We may use this relationship to write the divergence of the Weyl curvature interms of the curl of the Ehrenfest tensor.

Starting from the algebraic Bianchi identity,


abec;d

we substitute

Rabcd = Cabcd−δ ac Rbd +δ ad Rbc +ηbcRad−ηbdRac

for the curvature to find


abec;d

= Cabcd;e−δ ac Rbd;e +δ ad Rbc;e +ηbcRad;e−ηbdRac;e+Cabde;c−δ ad Rbe;c +δ ae Rbd;c +ηbdRae;c−ηbeRad;c+Cabec;d−δ ae Rbc;d +δ ac Rbe;d +ηbeRac;d−ηbcRae;d

Taking the trace on ae,

0 = Cabcd;a−Rbd;c +Rbc;d +ηbcRad;a−ηbdRac;a−Rbd;c +nRbd;c +ηbdRaa;c−Rbd;c−nRbc;d +Rbc;d +Rbc;d−ηbcRaa;d

= Cabcd;a− (n−3)Rbc;d +(n−3)Rbd;c +ηbd(Raa;c−Rac;a

)−ηbc

(Raa;d−Rad;a

)and contracting again on bd,

0 = 2(n−2)(Raa;c−Rac;a

)we find

Cabcd;a = (n−3)(Rbc;d−Rbd;c)

1.6.2 Curvature relations in Weyl geometry

We find the corresponding results for Weyl geometry.

Riemann, Weyl, Ehrenfest and dilatational curvatures In Weyl geometry, the Bianchi identities show that thetrace of the curvature has an antisymmetric part,

Ωcbcd−Ωcdcb =−(n−2)Ωbdso that we no longer have Ωab = Cab +2∆

acdbRce

d . Instead, let

Ωabac = Rbc−12

(n−2)ΩbcΩac ac = η

acRac = R

where Rab = Rba. Then we may still write the traceless part of the curvature as

Cabcd = Ωabcd−

1(n−2)

(δ ac Ωebed−δ ad Ωebec−ηbcΩea ed +ηbdΩea ec)

+1

(n−1)(n−2)Ωe f e f (δ

ac ηbd−δ ad ηbc)

= Ωabcd−1

(n−2)(δ ac Rbd−δ ad Rbc−ηbcRad +ηbdRac)

+12

(δ ac Ωbd−δ ad Ωbc−ηbcΩad +ηbdΩac)

+1

(n−1)(n−2)R(δ ac ηbd−δ ad ηbc)

= Ωabcd +(δac Rbd−δ ad Rbc−ηbcRad +ηbdRac)

+12


10

or, using differential forms, we may decompose the full curvature 2-form as

Ωab = Cab +2∆

acdb

(Rc +

12

Ωceωe)

ωd

that is, the usual decomposition in terms of the Weyl and Eisenhart tensors, together with a dilatation term. TheEisenhart tensor is still defined in terms of the (symmetric) Ricci tensor as

Rab =−1

n−2

(Rab−

12(n−1)

Rηab)

and the Ricci scalar is unchanged.

Symmetries of the Weyl curvature We find the symmetries of the Weyl curvature in a Weyl geometry. From

Cab = Ωab−2∆acdb

(Rc +

12

Ωceωe)

ωd

we have the decomposition of the full curvature,

Ωab = Cab +2∆

adcb

(Rd +

12

Ωdeωe)

ωc

= Rab +∆adcb Ωdω

c

where Rab has the usual symmetries of the Riemann tensor. Therefore, we have antisymmetry on both the first andsecond pairs of indices. The triple antisymmetry usually follows from the Bianchi identity. Here we have:

Ωabcd = Rabcd +∆

aecbΩed−∆aedbΩec

Ωa[bcd] = Ra[bcd] +

13

(∆aecbΩed−∆aedbΩec)+13

(∆aedcΩeb−∆aebcΩed)+13

(∆aebdΩec−∆aecdΩeb)

=13

(∆aecbΩed−∆aedbΩec +∆aedcΩeb−∆aebcΩed +∆aebdΩec−∆aecdΩeb)

=16

(δ ac δeb Ωed−δ ad δ eb Ωec +δ ad δ ec Ωeb−δ ab δ ec Ωed +δ ab δ ed Ωec−δ ac δ ed Ωeb)

−16

(ηaeηbcΩed−ηaeηbdΩec +ηaeηcdΩeb−ηaeηbcΩed +ηaeηbdΩec−ηaeηcdΩeb)

= −13

(δ ad Ωbc +δab Ωcd +δ

ac Ωdb)

Then

Ωa[bcd] = −δa[bΩcd]

Writing the curvature in covariant form,


aedbΩec−∆aecbΩed

Ωabcd = Rabcd +

12

δ ad Ωbc−12

ηaeηbdΩec−12

δ ac Ωbd +12

ηaeηbcΩed

Ωabcd = Rabcd +12

ηadΩbc−12

ηbdΩac−12

ηacΩbd +12

ηbcΩad

we also have

Ωabcd−Ωcdab = Rabcd−Rcdab +12

ηadΩbc−12

ηbdΩac−12

ηacΩbd +12

ηbcΩad

−12

ηcbΩda +12

ηdbΩca +12

ηacΩdb−12

ηdaΩcb

=12

ηadΩbc−12

ηdaΩcb−12

ηbdΩac +12

ηdbΩca−12

ηacΩbd +12

ηacΩdb +12

ηbcΩad−12

ηcbΩda= ηadΩbc−ηbdΩac−ηacΩbd +ηbcΩad

11

Therefore, the new symmetries are:

Ωabcd = −Ωbacd =−ΩabdcΩa[bcd] = δ

a[bΩcd]

Ωabcd−Ωcdab = ηadΩbc−ηbdΩac−ηacΩbd +ηbcΩad

Divergence of the Weyl curvature As before, we start with the Bianch identity

DΩab = dΩab +Ω

cbω

ac−ωcbΩac = 0

where

Ωabcd = Cabcd− (δ ac Rbd−δ ad Rbc−ηbcRad +ηbdRac)

−12


Combining these, we have

0 = Ωabcd;e +Ωabde;c +Ω

abec;d

= Cabcd;e +Cabde;c +C

abec;d

−(δ ac Rbd;e−δ ad Rbc;e−ηbcRad;e +ηbdRac;e

)− 1

2(δ ac Ωbd;e−δ ad Ωbc;e−ηbcΩad;e +ηbdΩac;e

)−(δ ad Rbe;c−δ ae Rbd;c−ηbdRae;c +ηbeRad;c

)− 1

2(δ ad Ωbe;c−δ ae Ωbd;c−ηbdΩae;c +ηbeΩad;c

)−(δ ae Rbc;d−δ ac Rbe;d−ηbeRac;d +ηbcRae;d

)− 1

2(δ ae Ωbc;d−δ ac Ωbe;d−ηbeΩac;d +ηbcΩae;d

)Now contract ae,

0 = Cabcd;a

−(Rbd;c−Rbc;d−ηbcRad;a +ηbdRac;a

)− 1

2(Ωbd;c−Ωbc;d−ηbcΩad;a +ηbdΩac;a

)−(Rbd;c−nRbd;c−ηbdRaa;c +Rbd;c

)− 1

2(Ωbd;c−nΩbd;c +Ωbd;c)

−(nRbc;d−Rbc;d−Rbc;d +ηbcRaa;d

)− 1

2(nΩbc;d−Ωbc;d−Ωbc;d)

Simplify,

Cabcd;a = (n−3)Rbc;d− (n−3)Rbd;c +ηbcRaa;d−ηbcRad;a +ηbdRac;a−ηbdRaa;c

+12((n−3)Ωbc;d− (n−3)Ωbd;c +ηbdΩac;a−ηbcΩad;a

)One further trace, on bd, gives

0 = 2(n−2)(

Rac;a−R;c +12

Ωac;a

)so that Rac;a = R;c− 12 Ω

ac;a

Cabcd;a = (n−3)Rbc;d− (n−3)Rbd;c +ηbcR;d−ηbcR;d +12

ηbcΩad;a +ηbdR;c−12

ηbdΩac;a−ηbdR;c

+12((n−3)Ωbc;d− (n−3)Ωbd;c +ηbdΩac;a−ηbcΩad;a

)= (n−3)

(Rbc;d−Rbd;c +

12

Ωbc;d−12

Ωbd;c)

12

Now cycle and combine:

Cabcd;a = (n−3)(

Rbc;d−Rbd;c +12

Ωbc;d−12

Ωbd;c)

Cacdb;a = (n−3)(

Rcd;b−Rcb;d +12

Ωcd;b−12

Ωcb;d)

−Cadbc;a = (n−3)(−Rdb;c +Rdc;b−

12

Ωdb;c +12

Ωdc;b)

Sum:

Cabcd;a +Cacdb;a−Cadbc;a = (n−3)

(Rbc;d−Rbd;c +Rcd;b−Rcb;d−Rdb;c +Rdc;b +

12

Ωbc;d−12

Ωbd;c +12

Ωcd;b−12

Ωcb;d−12

Ωdb;c +12

Ωdc;b)

= (n−3)(2Rcd;b−2Rbd;c +Ωbc;d)

Now useCabcd +C

acdb +C

adbc = 0

to write this as

Cabcd;a = (n−3)(

Rbc;d−Rbd;c−12

Ωcd;b)

1.7 Transformation of the Weyl and Eisenhart tensors under conformal transformationThe structure equations for a Riemannian geometry are

dωab = ωcbω

ac +R

ab

dea = ecωac

Changing the solder form ea by a conformal factor, ea→ eϕ ea changes connection to, say, αab, and the second equationgives

eϕ dea + eϕ dϕea = eϕ ebαabThis is solved by setting αab = ω

ab +2∆

acdbϕ,ce

d , where ωab is the original spin connection, since then

eϕ dea + eϕ dϕea = eϕ ebαab= eϕ eb

(ωab +2∆

acdbϕ,ce

d)

= eϕ ebωab +2∆acdbϕ,ce

ϕ ebed

= eϕ ebωab + eϕ ϕ,bebea−ηacηbdϕ,ceϕ ebed

eϕ dea + eϕ dϕea = eϕ ebωab + eϕ dϕea

dea = ebωab

Then the first equation gives the change in curvature,

R̃ab = dαab−αcbαac

= d(

ωab +(δad δ

cb −ηacηdb)ϕ,ced

)−(

ωcb +(δcd δ

eb −ηceηdb)ϕ,eed

)(ωac +

(δ ag δ

fc −ηa f ηcg

)ϕ, f eg

)= dωab +dϕ,be

a−ηacηdbdϕ,ced +ϕ,bdea−ηacηdbϕ,cded

−ωcbωac−ϕ,becωac +ηceηdbϕ,eedωac−ωcbϕ,cea +ωcbηa f ηcgϕ, f eg

−ϕ,becϕ,cea +ϕ,becηa f ηcgϕ, f eg +ηceηdbϕ,eedϕ,cea−ηceηdbϕ,eedηa f ηcgϕ, f eg

13

Simplify,

R̃ab = Rab +(dϕ,b−ϕ,cωcb

)ea +ϕ,b (dea− ecωac)

−ηacηbd (dϕ,c−ϕ,eωec)ed−ηacϕ,cηbd(

ded− eeωde)

+ϕ,beadϕ−ηa f ηbdϕ, f eddϕ +(ηceϕ,cϕ,e)ηbdedea

= Rab +(δad δ

cb −ηacηbd)

(Dϕ,c−ϕ,cdϕ +

12(η f gϕ, f ϕ,g

)ηceee

)ed

Then using

R̃ab = C̃ab +2∆

acdbR̃cẽ

d

= C̃ab +2∆acdbR̃ce

φ ed

= Cab +2∆acdb

(Rc +Dϕ,c−ϕ,cdϕ +

12(η f gϕ, f ϕ,g

)ηceee

)ed

we have

C̃ab = Cab

R̃c = e−φ(

Rc +Dϕ,c−ϕ,cdϕ +12(η f gϕ, f ϕ,g

)ηceee

)

1.8 Symmetries of the Weyl curvature with nonzero dilatationIn scale-invariant geometries, there may exist a nonvanishing dilational curvature. We find the symmetries of the Weylcurvature when this dilatational curvature is non-zero. From

Cab = Ωab−2∆acdb

(Rc +

12

Ωceωe)

ωd

we have the decomposition of the full curvature,

Ωab = Cab +2∆

adcb

(Rd +

12

Ωdeωe)

ωc

= Rab +∆adcb Ωdω

c

where Rab has the usual symmetries of the Riemann tensor. Therefore, we have antisymmetry on both the first andsecond pairs of indices. The triple antisymmetry usually follows from the Bianchi identity. Here we have:


aecbΩed−∆aedbΩec

Ωa[bcd] = Ra[bcd] +

13

(∆aecbΩed−∆aedbΩec)+13

(∆aedcΩeb−∆aebcΩed)+13

(∆aebdΩec−∆aecdΩeb)

=13

(∆aecbΩed−∆aedbΩec +∆aedcΩeb−∆aebcΩed +∆aebdΩec−∆aecdΩeb)

=16

(δ ac δeb Ωed−δ ad δ eb Ωec +δ ad δ ec Ωeb−δ ab δ ec Ωed +δ ab δ ed Ωec−δ ac δ ed Ωeb)

−16

(ηaeηbcΩed−ηaeηbdΩec +ηaeηcdΩeb−ηaeηbcΩed +ηaeηbdΩec−ηaeηcdΩeb)

= −13

(δ ad Ωbc +δab Ωcd +δ

ac Ωdb)

Then

Ωa[bcd] = −δa[bΩcd]

Writing the curvature in covariant form,


aedbΩec−∆aecbΩed

14

Ωabcd = Rabcd +

12

δ ad Ωbc−12

ηaeηbdΩec−12

δ ac Ωbd +12

ηaeηbcΩed

Ωabcd = Rabcd +12

ηadΩbc−12

ηbdΩac−12

ηacΩbd +12

ηbcΩad

we also have

Ωabcd−Ωcdab = Rabcd−Rcdab +12

ηadΩbc−12

ηbdΩac−12

ηacΩbd +12

ηbcΩad

−12

ηcbΩda +12

ηdbΩca +12

ηacΩdb−12

ηdaΩcb

=12

ηadΩbc−12

ηdaΩcb−12

ηbdΩac +12

ηdbΩca−12

ηacΩbd +12

ηacΩdb +12

ηbcΩad−12

ηcbΩda= ηadΩbc−ηbdΩac−ηacΩbd +ηbcΩad

Therefore, the new symmetries are:

Ωabcd = −Ωbacd =−ΩabdcΩa[bcd] = δ

a[bΩcd]

Ωabcd−Ωcdab = ηadΩbc−ηbdΩac−ηacΩbd +ηbcΩad

2 Poincaré gauge theoryWe begin our set of gauge theories of general relativity with the gauging of the Poincaré group, taking the quotient ofthe Pioncaré group by its Lorentz subgroup. The base manifold is interpreted as spacetime and the quotient bundlegives local Lorentz symmetry.

2.1 The structure equations and Bianchi identitiesThe Maurer-Cartan structure equations, extended to the Cartan equations with the addition of horizontal curvatures,are

dωab = ωcbω

ac +R

ab

dea = ecωac +Ta

with Bianchi identities

0 = DRab = dRab +R

cbω

ac +ω

cbR

ac

0 = ecRac +DTa = ecRac +dT

a +Tcωac

or, in components,

Rab[cd;e] = 0

Ra[bcd] +Ta[bc;d] = 0

2.2 The Einstein-Hilbert actionAfter eliminating the translational gauge transformations, the gauge transformations of the connection reduce to

ω̃ab = Λacω

cdΛ̄

db−dΛac Λ̄cb

ẽa = Λacec

so the solder form now transforms as a tensor. For the curvatures we have

R̃ab = ΛacR

cdΛ̄

db

T̃a = ΛacTc

15

so the torsion and Lorentz components of the Poincaré curvature do not mix. The full set of available tensors istherefore

{ηab,eabcd ,ea,Ta,Rab

}. The most general action we can build from these, up to terms linear in the curvature,

isS =

∫ (Rabec . . .ed +Λeaebec . . .ed

)eabc...d

where Λ is constant. Torsion terms can enter only at second order.

2.3 The field equationsVarying the connection, we have

0 = δS

=∫ (

δRabec . . .ed +(n−2)Rabec . . .δed +nΛeaebec . . .δed)

eabc...d

=∫ (

Dδωabec . . .edee +(n−2)Rabec . . .edδee +nΛeaebec . . .edδee)

eabc...de

=∫ (

(n−2)δωabDec . . .edee +(n−2)Rabec . . .edδee +nΛeaebec . . .edδee)

eabc...de

Since each component of the connection is varied independently we have

0 = (n−2)Deced . . .eeeabcd...e0 =

((n−2)Rabec . . .ed +nΛeaebec . . .ed

)eabc...de

We consider these in turn.In the first field equation, we recognize the covariant exterior derivative of the solder form, Dec, as the torsion, so

(for n > 2)

0 = Tced . . .eeeabcd...e

=12

T cf gef eged . . .eeeabcd...e

Take the wedge product of this equation with one further arbitrary solder form, eh, then define the convenient volumeform

Φ =1n!

ea...bea . . .eb

so thatea . . .eb =−ea...bΦ

We may then write

0 = −12

T cf geh f gd...eeabcd...eΦ

Taking the Hodge dual and discarding the overall constant this becomes

0 = T cf geh f gd...eeabcd...e

= T cf gδh f gabc

where δ h f gabc = δ[ha δ fb δ

g]c . Expanding, while relying on the antisymmetry of the f g indices on the torsion,

0 = T cf gδh f gabc

=13

T cf g(

δ ha δf

b δgc +δ

fa δ

gb δ

hc +δ

ga δ

hb δ

fc

)=

13

(T cbcδ

ha +T

cabδ

hc +T

ccaδ

hb

)16

Contract on the hb indices,

0 =13

(T cac +Tc

ac +nTc

ca)

0 = −13

(n−2)T cac

so the trace of the torsion must be zero and the full field equation reduces to vanish torsion, T cab = 0.The second field equation is

0 =((n−2)Rabec . . .ed +nΛeaebec . . .ed

)eabc...de

0 =(

12

(n−2)Rab f g +nΛδ af δ bg)

e f egec . . .edeabc...de

Taking the wedge product with one further solder form and using the volume form as before leads to

0 =(

12


ehe f egec . . .edeabc...de

0 = −(−1)n−1(

12


eh f gc...deeabc...dΦ

and therefore, taking the dual,

0 =(

12

(n−2)Rab f g +12

nΛ(

δ af δbg −δ ag δ bf

))δ h f geab

=16

((n−2)Rab f g +nΛ

(δ af δ

bg −δ ag δ bf

))(δ he δ

fa δ

gb +δ

fe δ

ga δ

hb +δ

ge δ

ha δ

fb

)=

16

((n−2)

(Rab abδ

he +R

ahea +R

hbbe

)+nΛ

(n2δ he −nδ he +δ he −nδ he +δ he −nδ he

))=

16

((n−2)

(Rδ he −2Rh e

)+n(n−1)(n−2)Λδ he

)Lowering the upper index this becomes the Einstein equation with cosmological constant, Λ̃ = 12 n(n−1)Λ,

Rab−12

Rηab− Λ̃ηab = 0

2.4 Conformal Ricci flatness in Riemannian geometryWe find the necessary and sufficient conditions for a Riemannian geometry to be conformally related to a Ricci-flatRiemannian geometry.

2.4.1 Necessary condition: Exterior derivative of the Eisenhart tensor

We compute the covariant exterior derivative of the Eisenhart tensor when the metric is conformal to a Ricci flat metric.Suppose we have

dωab = ωcbω

ac +R

ab

dea = ecωac

where the Ricci and Eisenhart tensors of Rab vanish, Rab = 0; Ra = 0, and let

ẽa = eϕ ea

The corresponding spin connection is given by

dẽa = ẽcω̃acdϕeϕ ea + eϕ dea = eϕ ecω̃ac

dϕea + ecωac = ecω̃ac

17

Letω̃ac = ω

ac +2∆

aedcϕ,ee

d

Then

ecω̃ac = ec(

ωac +2∆aedcϕ,ee

d)

= ecωac + ec (δ ad δ

ec −ηaeηdc)ϕ,eed

= ecωac +dϕea

as desired.We wish to compute D̃R̃a,

D̃R̃a = dR̃a− ω̃caR̃c= dR̃a−

(ωca +2∆

cedaϕ,ee

d)

R̃c

= DR̃a−2∆cedaφ,eedR̃c

= D(

e−ϕ(

Dϕ,a−ϕ,adϕ +12(η f gϕ, f ϕ,g

)ηaeee

))−2∆cedaϕ,eed

(e−ϕ

(Dϕ,c−ϕ,cdϕ +

12(η f gϕ, f ϕ,g

)ηcbeb

))= −e−ϕ Dϕ

(Dϕ,a−ϕ,adϕ +

12(η f gϕ, f ϕ,g

)ηaeee

)+e−ϕ

(DDϕ,a−D(ϕ,adϕ)+

12

D(η f gϕ, f ϕ,g

)ηaeee

)−2∆cedaϕ,eed

(e−φ

(Dϕ,c−ϕ,cdϕ +

12(η f gϕ, f ϕ,g

)ηcbeb

))Then

eϕ D̃R̃a = −dϕDϕ,a−12(η f gϕ, f ϕ,g

)dϕηaeee

+DDϕ,a−D(ϕ,adϕ)+12

D(η f gϕ, f ϕ,g

)ηaeee

−ϕ,aecDϕ,c +φ,aecϕ,cdϕ−12(η f gϕ, f ϕ,g

)ϕ,aecηcbeb

+ηceηdaϕ,eedDϕ,c− (ηceφ,eϕ,c)ηdaeddϕ +12(η f gϕ, f ϕ,g

)ηdaeddϕ

and so

eϕ D̃R̃a = DDϕ,a +η f gϕ, f Dϕ,gηaeee−ϕ,aD(dϕ)+ηceηdaϕ,eedDϕ,c

+12(η f gϕ, f ϕ,g

)(ηdaeddϕ +ηaeeedϕ−2ηdaeddϕ

)= DDϕ,a +

(η f gϕ, f Dϕ,g−ηceϕ,eDϕ,c

)(ηaded

)= DDϕ,a

Finally, the Ricci identity gives

DDϕ,a = D(

dϕ,a−ϕbωba)

= d(

dϕ,a−ϕbωba)

+(

dϕ,c−ϕbωbc)

ωca

= −ϕb(

dωba−ωcaωbc)

= −ϕbRba= −ϕbCba= −ϕbC̃ba

18

Thus,

eϕ D̃R̃a +ϕbC̃ba = 0D̃R̃a−

(e−ϕ),b C̃

ba = 0

This condition must hold in any geometry related to a Ricci flat geometry by a conformal transformation.However, notice that our original choice of ϕ was fully arbitrary. Therefore, this equation holds regardless of ϕ .

We can choose eϕ to be any gauge factor, and an equation of this form holds. What this means is that there is one gaugein which ϕ = 0; this is the Ricci-flat gauge. In general, when this equation is satisfied, the Ricci and Eisenhart tensorsare constructed from derivatives of ϕ . The curl of the Eisenhart tensor doesn’t vanish, but ends up being (e−ϕ),b C̃ba.

2.4.2 Sufficient condition: Integrability for conformal Ricci flatness

We have shown that if a spacetime is conformal to a Ricci flat spacetime, then we necessarily have

D̃R̃a−(e−ϕ),b C̃

ba = 0

We now seek a sufficient conditon.Start with a spacetime with curvatures Cab,Rc, and perform a conformal transformation. The new curvatures are

given by

C̃ab = Cab

R̃c = e−ϕ(

Rc +Dϕ,c−ϕ,cdϕ +12(η f gϕ, f ϕ,g

)ηceee

)and the new spacetime will be Ricci flat if and only if R̃c = 0. We therefore require the existence of a scalar ϕsatisfying

0 = Rc +Dϕ,c−ϕ,cdϕ +12(η f gϕ, f ϕ,g

)ηceee

= Rc +dϕ,c−ϕ,aωac−ϕ,cdϕ +12(η f gϕ, f ϕ,g

)ηceee

Write this as

dϕ,c = ϕ,aωac +ϕ,cdϕ−12(η f gϕ, f ϕ,g

)ηceee−Rc

Then the integrability condition is given by the Poincaré lemma, d2ϕ,c = 0. We therefore require

0 = d(

ϕ,aωac +ϕ,cdϕ−12(η f gϕ, f ϕ,g

)ηceee−Rc

)= dϕ,aωac +ϕ,adω

ac +dϕ,cdϕ−

(η f gϕ, f dϕ,g

)ηceee−

12(η f gϕ, f ϕ,g

)ηcedee−dRc

Now substitute for dϕ,c,

0 =(

ϕ,bωba +ϕ,adϕ−12(η f gϕ, f ϕ,g

)ηaeee−Ra

)ωac +ϕ,adω

ac

+(

ϕ,aωac +ϕ,cdϕ−12(η f gϕ, f ϕ,g

)ηceee−Rc

)dϕ

−(

η f gϕ, f(

ϕ,aωag +ϕ,gdϕ−12

(ηhkϕ,hϕ,k

)ηgded−Rg

))ηceee

−12(η f gϕ, f ϕ,g

)ηcedee−dRc

19

Simplify,

0 = ϕ,a(

dωac−ωbcωab)

+ϕ,adϕωac−Rcdϕ− (dRc−ωacRa)

+ϕ,aωacdϕ−(ϕ, f ϕ,aωa f

)ηceee +η f gϕ, f Rgηceee

−12(η f gϕ, f ϕ,g

)(ηceeedϕ +2dϕηceee−dϕηceee +ηac (dea− eeωae))

and so

0 = ϕ,a(

dωac−ωbcωab)

−(

dRc−ωacRa +Rb(

δ bc δda −ηbdηca

)ϕ,dea

)= ϕ,aRac−

(dRc−

(ωbc +2∆

bdca ϕ,de

a)

Rb)

= ϕ,aRac−(

dRc− ω̃bcRb)

= ϕ,aRac− D̃Rc

Try to separate out the ϕ-dependence,

0 = ϕ,aRac−DRc +2∆bdca ϕ,deaRb

Expand the curvature,Rab = C

ab +2∆

acdbRce

d

so that

0 = ϕ,aCac +ϕ,a2∆aedcRee

d−DRc +2∆bdca ϕ,deaRb= ϕ,aCac−DRc +2∆bdca ϕ,d (Rbea + eaRb)= ϕ,aCac−DRc

Therefore, a sufficient condition for the space to be conformally Ricci flat is the existence of a scalar field such that

ϕ,aCac−DRc = 0

This is the usual form of the condition. Unfortunately, the condition still depends on the scalar field.This condition is therefore necessary and sufficient.How do we use this condition to solve for the geometry? Here’s the program: Having this condition tells us that

there exists a gauge in which the Ricci tensor vanishes. Choose that gauge and solve for the metric. Now, gauge againby an arbitrary conformal factor. This gives the conformal equivalence class of metrics which solve the problem. Eachchoice of gauge will satisify ϕ,aCac −DRc = 0 with its appropriate conformal factor, Weyl curvature and Eisenharttensor.

The usefulness of the condition is that it is the conformal equivalent of the Einstein equation. It is not particularlyuseful for solving for the metric – the whole point is that we can still use the Einstein equation for that, but thenhave the freedom to choose a gauge. But knowing that ϕ,aCac −DRc = 0 is the conformal Einstein equation lets usinterpret that expression when it occurs in other gravity theories. Any gravity theory that leads to this condition is onexperimentally sound footing – all data that support general relativity support such a theory as well, and the freedomto choose a local scale makes it epistemologically more sound.

3 Weyl (homothetic) gauge theoryWe ask corresponding questions in a Weyl geometry. The overall symmetry is that of the Poincaré group together withdilatations. We take the quotient of the inhomogeneous Weyl group by the homogeneous Weyl group to get an n-dimbase manifold with local homothetic (Lorentz plus dilatations) symmetry.

20

The structure equations are now

dωab = ωcbω

ac +Ω

ab

dea = ecωac +ωea

dω = Ω

and setting ω = Waea, the connection may be written in the form

ωac = αac−2∆adbcWdeb

where αac is the usual spin connection for ea, since then

dea = ecωac +ωea

= ecαac−2∆adbcWdeceb +ωea

= ecαac−Wcecea +ωea

= ecαac

Notice that under conformal transformation, the connection changes by

ẽa = eϕ ea

ω̃ = ω +dϕω̃ab = ω

ab

Notice that the spin connection is unchanged (this happens because dilatations and Lorentz transformations commute)and therefore the curvature is unchanged:

Ω̃ab = Ωab

C̃ab = Cab

R̃a = Ra

As a result, it does not make sense to ask for a conformally related metric with vanishing Ricci tensor – the Ricci tensoris conformally invariant. We now consider the homothetic equivalent of the vanishing of the Ricci tensor, namely,

Ωabac = 0

We now show that this is the necessary and sufficient condition for the existence of a gauge in which the Riemannianpart of the curvature has vanishing Ricci tensor.

3.1 Conformal Ricci flatnessWe now consider the necessary and sufficient conditions under which the Ehrenfest tensor, Ra, vanishes in a Weylgeometry. Since the vanishing of the Ehrenfest tensor is equivalent to vanishing Ricci tensor, this condition solves thevacuum Einstein equation.

We have seen that the Lorentz curvature is given by

Ωab = Rab−2∆acdb

(DWc +Wcω−

12

ηce(η f gWfWg

)ee)

ed

Suppose the trace of this curvature vanishes,Ωabac = 0

Then, writing Ωab in components,

Ωabcd = Rabcd−2∆aedb

(We;c +WeWc−

12

ηce(η f gWfWg

))eced

21

we must have

0 = Ωcbcd= Rbd +(n−2)Wb;d +ηceηbdWe;c

+(n−2)WbWd +ηceηbdWeWc−ηbd (n−1)

(η f gWfWg

)The trace gives

0 = R+2(n−1)ηbdWb;d−(n−1)(n−2)

(η f gWfWg

)ηbdWb;d =

12(n−1)

(−R+(n−1)(n−2)

(η f gWfWg

))Substituting,

0 = Rbd−1

2(n−1)ηbdR

+(n−2)Wb;d +(n−2)WbWd−12

(n−2)ηbd(η f gWfWg

)or,

Rbd = −1

n−2

(Rbd−

12(n−1)

ηbdR)

= Wb;d +WbWd−12

ηbd(η f gWfWg

)This has symmetric and antisymmetric parts,

Rbd = W(b;d) +WbWd−12

ηbd(η f gWfWg

)0 = W[b;d]

The antisymmetric part is just half the dilatation. Therefore, Ωcbcd = 0 if and only if

Rbd = Wb;d +WbWd−12

ηbd(η f gWfWg

)Ωbd = 0

Solution for the Weyl vector We treat this as a differential equation for the Weyl vector. Contract with ed to writeit as a 1-form equation,

0 = DWb +Wbω−12

ηbdW 2ed−Rb

dWb = Wcαcb−Wbω +12

ηbdW 2ed +Rb

The integrability condition is then

0 = d2Wb

= dWcαcb +Wcdαcb−dWbω−Wbdω +ηbdW cdWced +

12

ηbdW 2ded +dRb

= dRb +(

Weαec−Wcω +12

ηcdW 2ed +Rc)

αcb +Wcdαcb

−(

Wcαcb−Wbω +12

ηbdW 2ed +Rb)

ω−Wbdω

+ηbdW c(

Weαec−Wcω +12

ηcdW 2ed +Rc)

ed +12

ηbdW 2ded

22

Now collect terms,

0 = dRb−αcbRc−Rbω +Wcdαcb−Weαcbαec +ηbdW cRced

+12

ηbdW 2ded−12

ηcbW 2edαcd−12

ηbdW 2edω−ηbdW 2ωed +12

ηbdW 2Weeeed

−Wbdω

and therefore,

0 = D(α)Rb−WbΩ+WcRcb−Rbω +ηbdW cRced

Now write

−Rbω +ηbdW cRced = We (−δ ed δ cb +ηbdηec)Rced

= −2We∆ecdbRced

so we have

0 = D(α)Rb−WbΩ+Wc(

Rcb−2∆cedbReed)

= D(α)Rb−WbΩ+WcCcb

together withΩ = 0

Since the dilatation vanishes, the Weyl vector is a gradient, so there exists a function ϕ such that

0 = D(α)Rb−WbΩ+WcCcb= D(α)Rb +ϕ,cCcb

Of course, this is the condition in the underlying Riemannian geometry for conformal Ricci flatness.Conversely, suppose the underlying Riemannian geometry is conformally related to a Ricci flat geometry. Then

the condition above holds and we can run the calculation backwards to show that the integrability condition hold. This,in turn, means that we can solve

0 = DWb +Wbω−12

ηbdW 2ed−Rb

Such a solution givesΩcbcd = 0

Therefore, the Riemannian geometry underlying a Weyl geometry is conformal to a Ricci flat geometry if and onlyif Ωcbcd = 0.

3.2 An action for General RelativityUsually, scale invariant actions for gravity theories are taken to be quadratic in the curvature. There is a simple reasonfor this: while the curvatures, Ωab, Ra, and Ω are conformally invariant, it is not obvious how to build a scale invariantaction because the volume element has conformal dimension +4. Thus, although the functional

S1 =∫

Ωabecedeabcd

will lead to the field equation we desire, the integrand has conformal weight +2. The usual solution is to write anaction quadratic in the curvature, such as

S2 =∫

Ωab∗Ωbaeabcd

This leads field equations that contain second derivatives of the connection, and are therefore of third order for themetric. Such higher derivatives are often found to lead to negative norm states – ghosts – in the quantum theory.However, the situation is not actually as bad as that, for if we use the quadratic action the field equation,

D∗Ωab = 0

23

may be related to the integrability condition for the conformal Einstein equation. But there is a still better solution.An often overlooked fact of scale-invariant geometry is that dimensionful “constants” may have nontrivial evolu-

tion. To see this, note that the homothetically covariant derivative of a tensor with dimensions of (length)n is givenby

DaT b...c = ∂aT b...c +T e...cΓbea + . . .+Tb...eΓcea +nWaT

b...c

and tensors are labeled by both Lorentz type,(

pq

), and by conformal weight, n. We may therefore have tensors, κ ,

of Lorentz type(

00

), but conformal weight n, and these will have covariant derivative

Daκ = ∂aκ +nWaκ

even though in a Riemannian spacetime κ would be scalar or even constant. The physical meaning of this is clear.Suppose κ represents the length of a meter stick. Then, if we use global units with Wa = 0, that length is unchanging,

∂aκ0 = 0

and the length is constant. But suppose we choose a local length standard given by a set of springs. The length ofthose springs, ls, relative the the meter stick, changes from place to place and time to time. If the meter stick were ourstandard, then the length of the spring at spacetime position

(t,xi)

is a function,

ls(t,xi)

= ls0eϕ

However, if we take the length of the springs as our length standard, it is the length of the meter stick which changes,κ = κ0ls = κ

(t,xi): it is the dimensionless ratio,

κls0

=κ0ls

= e−ϕ

which is the measured quantity. With the springs as standards, the Weyl vector is given by Wa = ∂aϕ . We still regardκ as constant because it satisfies the condition

Daκ = ∂aκ +Waκ= ∂aκ +∂aϕκ= ∂aκ−κ∂a ln(κ/ls0)

= ∂aκ−∂a (κ/ls0)(κ/ls0)

κ

= 0

Notice that the existence of global constants of this type, that is, one or more quantities satisfying Dκ = 0, we mustsatisfy the integrability condition

0 = d2κ= dωκ−ωdκ= dωκ−ωωκ= dωκ

If the constant κ is not to vanish, then the dilatation must be zero, Ω = dω = 0. A weaker condition is possible: wecan demand the existence of a congruence of curves along which κ remains constant. In simple applications, thesecurves turn out to be the classical paths of motion. See Gauging Newton’s law.

Now, if we insert a constant of the appropriate conformal weight into the functional S1, we have a suitable action,

S =∫

κΩabecedeabcd

24

We require κ to have units of (length)−2 in order for the integrand to be of weight zero. If we use the gravitationalconstant,

[G] =m3

kg · sec2

together with the speed of light and Planck’s constant,[G}c3

]=

m3

kg · sec2· sec

3

m3· kg ·m

2

sec

= m2

then we may set

κ =c3

G}Dirac, in his large numbers paper, takes this action a step further by adding a kinetic term for κ . By allowing

solutions with slowly changing κ , he accounts for the relative sizes of certain fundamental quantities. This approachmay be taken in any dimension, n, with

S =∫

κ(n−2)/2Ωabec · · ·edeabc···d +Kinetic term

Notice that the form of the action depends on the dimension of the space.

4 Auxiliary conformal gaugingThe auxiliary conformal gauging has the structure equations

dωab = ωcbω

ac +2∆

adcb ωdω

c +Ωab (1)dωa = ωbωab +ωω

a +Ωa (2)dωa = ωba ωb +ωaω +Ωa (3)dω = ωaωa +Ω (4)

where the horizontal directions are spanned by the gauge fields of translations, ωa. The quotient is{

Mab,Pa,Ka,D}

/{

Mab,Ka,D}

,resulting in an n-dimensional manifold which we identify with spacetime. The solder forms, ωa, span the base mani-fold, and all forms and curvatures are horizontal, for example,

Ωab =12

Ωabcdωcωd

Each of the connection 1-forms is a function of n coordinates, xα , and is a linear combination of dxα ,

ωA = ωAα(

xβ)

dxα

4.1 Bianchi identitiesIn the next sub-Section, we show that it is consistent to set the torsion, Ωa, to zero. We have found the Bianchiidentities for any torsion-free conformal gauging to be

DΩab +2∆acdbΩcω

d = 0ωcΩac−Ωωa = 0

DΩa +Ωcaωc−ωaΩ = 0dΩ−ωaΩa = 0

Because the torsion vanishes, the second of these is algebraic:

0 = δ ab Ωcd +δac Ωdb +δ

ad Ωbc−Ωabcd−Ωacdb−Ωadbc

25

The trace on ac,

0 = Ωbd +nΩdb +Ωbd−Ωcbcd−Ωcdbc= −(n−2)Ωbd−Ωcbcd +Ωcdcb

so thatΩcbcd−Ωcdcb =−(n−2)Ωbd

then shows that, just as in the Weyl case, there is an antisymmetric part to the trace of the curvature, which is propor-tional to the dilatation.

The first Bianchi expands to give

0 = DΩab +2∆a feb Ω f ω

e

=12

(Ωabcd;e +2∆

a feb Ω f cd

)ωcωdωe

Take the trace on ac,

0 = Ωcbcd;e +Ωcbde;c +Ω

cbec;d

+2∆c febΩ f cd +2∆c fcbΩ f de +2∆

c fdbΩ f ec

= Ωcbcd;e +Ωcbde;c +Ω

cbec;d

+(n−3)Ωbde +ηdbηc f Ω f ce−ηebηc f Ω f cdTake an additional trace, on bd,

0 = Ωcd cd;e +Ωcd

ec;d−Ωcd ed;c+2(n−2)Ωdde

so we have the trace of the co-torsion,

Ωdde =1

2(n−2)

(Ωcd ed;c−Ωcd ec;d−Ωcd cd;e

)=

12(n−2)

(Ωdc de;c +Ω

dcde;c−Ωcd cd;e

)=

12(n−2)

(2Ωdc de;c−Ωdc dc;e

)Substitute back to find

0 = Ωcbcd;e +Ωcbde;c +Ω

cbec;d

+(n−3)Ωbde +1

2(n−2)ηdb(2Ωac ae;c−Ωac ac;e

)− 1

2(n−2)ηeb(2Ωac ad;c−Ωac ac;d

)and we may solve for the entire co-torsion,

Ωbde = −1

n−3(Ωcbcd;e +Ω

cbde;c−Ωcbce;d

)− 1

2(n−2)(n−3)ηdb(2Ωac ae;c−Ωac ac;e

)+

12(n−2)(n−3)

ηeb(2Ωac ad;c−Ωac ac;d

)Skipping the third, the final identity is

dΩ−ωaΩa = 0

Expanding,

Ω[ab;c]−Ω[cab] = 0

or

Ωab;c +Ωbc;a +Ωca;b−Ωcab−Ωabc−Ωbca = 0

When combined with the result from the first identity, this relates the Lorentz curvature and the derivative of thedilatation.

26

4.2 Inconsistency of the linear action4.2.1 An invariant action

To write an action for the auxiliary conformal gauging, we first find the available tensors. Setting the translationalgauge transformations to zero, Λa4 = 0, the gauge transformations of the connection forms become

ω̃ab = Λacω

cdΛ̄

db +Λ

acω

cΛ̄b+ηacΛcωeηedΛ̄db−dΛac Λ̄cb

ω̃a = ΛacωcΛ̄

ω̃a = ΛcωcdΛ̄da +Λcω

cΛ̄a+ΛωdΛ̄da +ΛωΛ̄a−dΛc Λ̄ca−dΛ Λ̄a

ω̃ = ω +ΛcωcΛ̄−dΛ Λ̄

so the solder form, ωa, becomes a tensor. Notice that a special conformal transformation, Λa, can be chosen such thatthe Weyl vector vanishes, ω̃ = 0. The curvatures gauge in the same way except for the derivative terms,

Ω̃ab = ΛacΩ

cdΛ̄

db +Λ

acΩ

cΛ̄b+ηacΛcΩeηedΛ̄db

Ω̃a = ΛacΩcΛ̄

Ω̃a = ΛcΩcdΛ̄da +ΛcΩ

cΛ̄a+ΛΩdΛ̄da +ΛΩΛ̄a

Ω̃ = Ω+ΛcΩcΛ̄

Only one of the irreducible curvature components is a tensor – the Lorentz curvature, special conformal curvature anddilatational curvature mix with one another. This makes it difficult to write an invariant action. However, the situationis considerably improved if we set the torsion to zero,

Ωa = 0

This is a gauge invariant constraint because the torsion, at least, is a tensor. The constraint is also consistent with ourexpectation that ωa is the spacetime solder form, so that to recover general relativity we will want vanishing torsion.

Dropping all torsion terms, the curvatures transform as

Ω̃ab = ΛacΩ

cdΛ̄

db

Ω̃a = ΛacΩcΛ̄ = 0

Ω̃a = ΛcΩcdΛ̄da +ΛΩdΛ̄

da +ΛΩΛ̄a

Ω̃ = Ω

so now the Lorentz and dilatational curvatures transform as independent tensors, both of conformal weight zero.For the action, the simplest choice is similar to that for Weyl geometry, by introducing a dimensionful “constant”

or field,S =

∫κ2(

Ωab +Λeaeb)

ec · · ·edeabc···d +Dκ∗Dκ +m2κ∗κ

Look at the kinetic term

Dκ∗Dκ +m2κ∗κ =1

(n−1)!DaκDbκgbcecd···eeaed · · ·ee +

1n!

m2κ2ecd···eeced · · ·ee

=1

(n−1)!DaκDbκgbcecd···eead···eΦ+

1n!

m2κ2ecd···eecd···eΦ

= −(

gabDaκDbκ +m2κ2)

Φ

27

Since the conformal weight of Φ is n, and both m2 and gab have conformal weight−2, we require κ to have conformalweight − n−22 , and the covariant derivative is given by

Daκ = ∂aκ−(

n−22

)Waκ

Notice that we must also haveDam = ∂am−mWa = 0

if the mass is to be considered constant.The cosmological constant Λ, written as we have with a factor of κ2, has conformal weight −2.It quickly becomes clear that we must modify the linear action, because the varying the special conformal gauge

field leads to a contradiction,

δS = δ∫

κ2Ωabec · · ·edeabc···d

=∫

κ2(−2ηbh∆aef hδωee f

)ec · · ·edeabc···d

Then

0 = 2ηbh∆aef hege f ec · · ·edeabc···d

0 = 2ηbh∆aef h(

δ ga δf

b −δgb δ

fa

)=

(δ af δ

eh −ηaeη f h

)(ηbhδ ga δ

fb −η

bhδ gb δf

a

)= ηge−nηge−nηge +ηge

= −2(n−1)ηge

and since we have n > 2 and κ 6= 0, this is incorrect. We must therefore modify the action by adding one or moreadditional terms.

We may try to fix this problem by adding additional terms. Since the torsion vanishes, the only possibilities atquadratic order are the square of the dilatation or the square of the Lorentz curvature.

The simplest term to add is one quadratic in the dilatation, Ω∗Ω. Then we have

S =∫

κ2(

Ωab +Λeaeb)

ec · · ·edeabc···d +κ2βΩ∗Ω+Dκ∗Dκ +m2κ∗κ

The conformal weight of β is given by expanding the dilatation term,

βΩ∗Ω = βΩabΩcdgcegd f ee f g···heaebeg · · ·eh

The solder forms give a weight of n, the inverse metrics −4, the factor κ2 a weight of −n+2, and the components ofthe dilatation −4. Therefore, the conformal weight of β must be −6.

However, when we vary the special conformal gauge field, we again find a contradiction. We have

δS =∫

κ2δΩabec · · ·edeabc···d +2κ2βδΩ∗Ω

=∫

κ2(−2ηbe∆acdeδωced

)e f · · ·egeab f ···g +2κ2β (−ecδωc)∗Ω

=∫

2δωcκ2(−ηbe∆acdeede f · · ·egeab f ···g +βec∗Ω

)and therefore,

0 = −ηbe∆acdeede f · · ·egeab f ···g +1

(n−2)!βecΩabηadηbeede f ···ge f · · ·eg

0 = −ηbe∆acdeehede f · · ·egeab f ···g +1

(n−2)!βΩabηadηbeede f ···gehece f · · ·eg

28

0 = −ηbe∆acdeehd f ···geab f ···g +1

(n−2)!βΩabηadηbeede f ···gehc f ···g

0 = (n−2)!ηbe∆acde(

δ ha δdb −δ da δ hb

)−βΩabηadηbe

(δ hd δ

ce −δ cd δ he

)=

12

(n−2)!(

ηch−nηch−nηhc +ηch)−β

(Ωhc−Ωch

)= −(n−1)!ηch−2βΩhc

so that this generalization does not solve the problem: the antisymmetric part forces the dilatation to vanish, while thesymmetric part is inconsistent.

Exercise: Find the remaining variational field equations from this action

Answer: See Appendix A Returning to the central problem, we consider a fully quadratic action.

4.3 Quadratic action theoryConsider the quadratic action:

S =∫

αΩab∗Ωba +βΩ

∗Ω

We find the resulting field equations.

4.3.1 Varying the Weyl vector

Varying the action with respect to the Weyl vector,

0 = δω S

= δω∫

αΩab∗Ωba +βΩ

∗Ω

=∫

2βδω Ω ∗Ω

since the Lorentz curvature is independent of ω . The structure equation for dilatations now gives

Ω = dω−ωaωaδω Ω = d(δω)

Therefore, integrating by parts,

0 =∫

V2βd(δω) ∗Ω

=∫

V2β (d((δω) ∗Ω)+δω d∗Ω)

= 2β∫

δV(δω) ∗Ω+

∫V

2βδω d∗Ω

=∫

V2βδω d∗Ω

since the variation vanishes on the boundary, δV . Since the variation is arbitrary, the field equation is

βd ∗Ω = 0

29

4.3.2 Varying the spin connection

Next, varying the spin connection, we have a similar calculation,

0 = δωab S

= δωab

∫αΩba

∗Ωab +βΩ∗Ω

= 2α∫

δωab Ωba∗Ωab

since the dilatation is independent of ωab. From the Lorentz structure equation

Ωab = dωab−ωcbωac−2∆acdbωcωd

δωcΩba = dδω

ba−δωcaωbc−ωcaδωbc

= Dδωbawhere D is the covariant exterior derivative. Substituting and integrating by parts,

0 = 2α∫

δωba Ωba∗Ωab

=∫

V2βDδωba

∗Ωab

=∫

V2βD

(δωba

∗Ωab)

+2βδωba D∗Ωab

=∫

V2βd

(δωba

∗Ωab)

+2βδωba D∗Ωab

=∫

δV2βδωba

∗Ωab +2β∫

Vδωba D

∗Ωab

= 2β∫

Vδωba D

∗Ωab

Notice that D(δωba ∗Ωab

)= d

(δωba ∗Ωab

)since the expression in parentheses is a scalar. The variation vanishes on

the boundary and is arbitrary inside the volume of interest, V , so the field equation is

αD ∗Ωab = 0

4.3.3 Varying the special conformal transformations

The variation with respect to the special conformal gauge vector is simpler. We have

0 = δωc S

= δωc∫

αΩba∗Ωab +βΩ

∗Ω

= 2∫

αδωcΩba∗Ωab +βδωcΩ

∗Ω

From the expressions for the curvature, the variations are:

Ωab = dωab−ωcbωac−2∆acdbωcωd

δωcΩba = −2∆bcdaδωcωd

Ω = dω−ωaωaδωcΩ = −ωaδωa

so substituting,

0 = 2∫

αδωcΩba∗Ωab +βδωcΩ

∗Ω

= 2∫

α(−2∆bcdaδωcωd

)∗Ωab +β (−ωaδωa) ∗Ω

= −2∫

δωc(

2α∆bcdaωd ∗Ωab−βωc ∗Ω

)30

Using the antisymmetry of the Lorentz curvature, the final field equation is

2αωd ∗Ωcd−βωc ∗Ω = 0

4.3.4 Varying the solder form

The final variation, with respect to the solder form, ωa, is complicated by the dependence of the volume element onthe solder form. To make these terms explicit, we expand the dual form in the action,

S =∫

αΩba∗Ωab +βΩ

∗Ω

=18

∫ (αΩbaαβ Ω

abλτ +βΩαβ Ωλτ

)gλρ gτσ

√−gερσ µν dxα dxβ dxµ dxν

Now, variation of the solder form leads to

0 = δωk S

=14

∫ (αδωk Ω

baαβ Ω

abλτ +βδωk Ωαβ Ωλτ

)gλρ gτσ


+18



)(2δωk g

λρ gτσ − 12

gλρ gτσ gθϕ δωk gθϕ)√−gερσ µν dxα dxβ dxµ dxν

Next, we need the curvature and metric variations:

12

δΩabαβ dxα dxβ = δ

(ωabβ ,α −ω

cbα ω

acβ −2∆

acdbωcα ω

dβ

)dxα dxβ

= −2∆acdbωcα δω dβ dxα dxβ

δΩabαβ = 2∆acdbωcβ δω

dα −2∆acdbωcα δω dβ

12

δΩαβ dxα dxβ = δ(ωβ ,α −ω aα ωaβ

)dxα dxβ

12

δΩαβ dxα dxβ = −δω aα ωaβ dxα dxβ

δΩαβ = δω aβ ωaα −δωa

α ωaβ

and (this will be clearer if we denote the solder form by ω aβ = eaβ from here on):

δgαβ = δ(

gαρ gβσ gρσ)

= 2δgαρ gβσ gρσ +gαρ gβσ δgρσ= 2δgαβ +gαρ gβσ δgρσ

δgαβ = −gαρ gβσ δgρσ

= −gαρ gβσ δ(

ηabe aρ ebσ

)= −gαρ gβσ ηab

(δe aρ e

bσ + e

aρ δe

bσ

)= −gαρ

(ηcde βc e

σd

)ηabδe aρ e

bσ −

(ηcde αc e

ρd

)gβσ ηabe aρ δe

bσ

= −gαρ(

ηcde βc)

ηadδe aρ − e αc gβσ δe cσ

= −gαρ e βa δe aρ − e αc gβσ δe cσ

= −(

gασ e βc + eαc g

βσ)

δe cσ

The variation becomes

0 =14

∫ (αδωk Ω

baαβ Ω

abλτ +βδωk Ωαβ Ωλτ

)gλρ gτσ


31

+18



)(2δωk g

λρ gτσ − 12

gλρ gτσ gθϕ δωk gθϕ)√−gερσ µν dxα dxβ dxµ dxν

=14

∫ (α(

4∆bcdaωcβ δωd

α

)Ωabλτ +2βδω

aβ ωaα Ωλτ

)gλρ gτσ


−18



)(2δ λθ δ

ρϕ gτσ −

12

gλρ gτσ gθϕ

)((gξ θ e ϕc + e

θc g

ϕξ)

δe cξ)√−gερσ µν dxα dxβ dxµ dxν

=14

∫ (α(

4∆bdca ωdβ δξα

)Ωabλτ +2βδ

ξβ ωcα Ωλτ

)gλρ gτσ

√−gερσ µν dxα dxβ dxµ dxν δω cξ

−18



)(2δ λθ δ

ρϕ gτσ −

12

gλρ gτσ gθϕ

)(gξ θ e ϕc + e

θc g

ϕξ)√−gερσ µν dxα dxβ dxµ dxν δe cξ

Then the field equation is

0 =14

(α(


)Ωabλτ +2βδ

ξβ ωcα Ωλτ

)gλρ gτσ ερσ µν dxα dxβ dxµ dxν

−18

(αΩbaαβ Ω


)(2δ λθ δ

ρϕ gτσ −

12

gλρ gτσ gθϕ

)(gξ θ e ϕc + e

θc g

ϕξ)

ερσ µν dxα dxβ dxµ dxν

Taking the dual,

0 =14

(α(


)Ωabλτ +2βδ

ξβ ωcα Ωλτ

)gλρ gτσ δ αβρσ

−18

(αΩbaαβ Ω


)(2δ λθ δ

ρϕ gτσ −

12

gλρ gτσ gθϕ

)(gξ θ e ϕc + e

θc g

ϕξ)

δ αβρσ

0 = 4α∆bdca ωdβ Ωa ξ βb +2βωcα Ω

αξ

−2α(

Ωbaρσ Ωb ξ σa e

ρc −

14

Ωbaρσ Ωa ρσb e

ξc

)−2β

(Ωρσ Ωξ σ e ρc −

14

Ωρσ Ωρσ e ξc

)Finally, we rewrite the final equation in the orthonormal basis by multiplying by the solder form, e eξ and replacing

coordinate (Greek) contractions with orthonormal (Latin) contractions:

0 = 2α∆bdca ωd f Ωa e fb +βωcaΩ

ae

−α(

Ωbac f Ωb e fa −

14

Ωba f gΩa f gb δ

ec

)−β(

ΩcdΩed−14

ΩabΩabδ ec

)Now lower the e index, and use ∆bdca Ω

a fbe = Ω

d fce ,

0 = 2αωd f Ωd fce +βωcaΩae

−α(

Ωbac f Ωb fae −

14

ηceΩba f gΩa f gb

)−β(

ΩcdΩ de −14

ηceΩabΩab)

4.3.5 Summary of the field equations

Collecting the results, the conformal field equations are:

αD ∗Ωab = 0βd ∗Ω = 0

2αωd ∗Ωcd−βωc ∗Ω = 02αωd f Ωd fce +βωcaΩ

ae = αΘce +βTce

32

where

Θab = ΩdcaeΩc e

db −14

Ωdce f Ωc e f

d ηab

Tab = ΩacΩ cb −14

ηabΩcdΩcd

This time, all four equations form a sensible set. In particular, the equation from the variation of the special conformalgauge field may now be consistently solved to eliminate that field.

The two fields, Θab and Tab have the form we typically expect for the energy-momentum tensor of a Yang-Millsfield. Here, however, they depend on the curvature and dilatation, and may not have such a straightforward physicalmeaning.

4.4 The auxiliary fieldConsider the field equation from the variation of the special conformal variation. Because the co-torsion, Ωa, does notappear in the action, this equation is algebraic in the curvatures. This means that we can use algebraic techniques tosolve for the curvatures, rather than having to integrate.

We begin by choosing a special conformal gauge. Recall that the gauge transformation for the Weyl vector is givenby

ω̃ = ω +ΛcωcΛ̄−dΛ Λ̄

and we may choose the arbitrary gauge, Λc, so that ω +ΛcωcΛ̄ = 0. Then the Weyl vector is pure (conformal) gauge,

ω̃ = −dΛ Λ̄= −dϕ

For the remainder of the calculation we choose this gauge, while leaving the conformal factor undetermined. In thisgauge, let ωa = Wabωb.

Now consider the special conformal field equation. To write this in components first expand the duals:

0 = 2αωd ∗Ωcd−βωc ∗Ω

=1

(n−2)!

(2αΩc e fd ee f g...h−βδ

cd Ω

e f ee f g...h)

ωdωg . . .ωh

Now wedge this with one additional solder form and take the dual:

0 =1

(n−2)!

(2αΩc e fd ee f gh−βδ

cd Ω

e f ee f g...h)

ωbωdωg . . .ωh

0 =1

(n−2)!

(2αΩc e fd −βδ

cd Ω

e f)

ee f g...hebdg...h

=(

2αΩc e fd −βδcd Ω

e f)

δ bde f

= 2αΩc bdd −βΩbc

Lowering the bc indices and rearranging gives,

2αΩabac = −βΩbc

However, the Bianchi identity for the solder form for torsion-free conformal geometry also relates these twocurvatures,

ωcΩac−Ωωa = 0

which, as we have shown, gives

Ωabad−Ωadab = −(n−2)Ωbd

33

Now compare the antisymmetric part of the field equation to the Bianchi identity:

α (Ωcacb−Ωcbca) = −βΩabΩcacb−Ωcbca = −(n−2)Ωab

The difference between the first, and α times the second gives

0 = (β − (n−2)α)Ωab

so that generically (i.e., unless β = (n−2)α), the dilatational curvature must vanish. Rather than tracking two cases,we set β = (n−2)α in the field equation. Then, whether the dilatational curvature vanishes or not, we must have

Ωcacb =−12

(n−2)Ωab

and we simply remember that except for a special choice of α,β in the original action, the dilatation vanishes. Wewill see below that the dilatation vanishes in the special case as well.

Now define

Rab = dωab−ωcbωac

ωa = Wabωb

where ωcb is the Weyl spin connection, satisfying

dωa = ωbωab +ωωa

We may now express the special conformal gauge field in terms of the trace, Rcbcd , of Rab. Notice that R

abcd is the

curvature of a Weyl connection, so that its trace satisfies

Rcbcd−Rcdcb =−(n−2)Ωbd

Let Rab be the symmetric part of the trace of Rabcd , so that

Rcbcd +Rcdcb = 2Rbd

Then adding,

Rcbcd = Rbd−12

(n−2)Ωbd

Substituting these definitions into the structure equation for the Lorentz curvature:

Ωab = Rab−2∆adcb ωdωc

Ωabcd = Rabcd +2∆

aecbWed−2∆aedbWec

= Rabcd +δac Wbd−ηaeηcbWed−δ ad Wbc +ηdbηaeWec

we impose the field equation:

−12

(n−2)Ωbd = Ωcbcd

= Rbd−12

(n−2)Ωbd +nWbd−Wbd−Wbd +ηbdηecWec

= Rbd−12

(n−2)Ωbd +(n−2)Wbd +ηdb (ηceWce)

so that0 = Rbd +(n−2)Wbd +ηdb (ηceWce)

34

Take one further contraction, and solve for the trace of Wab,

0 = R+(n−2)ηabWab +nηceWce

ηabWab = −1

2(n−1)R

substituting, we have

0 = Rab +(n−2)Wab−1

2(n−1)Rηab

Solving for Wab,

Wab = −1

n−2

(Rab−

12(n−1)

Rηab)

= Rab

so that the special conformal gauge field is given by the Eisenhart tensor of the Weyl connection. This solves com-pletely for the special conformal field, Wab, in terms of the spin connection and its derivatives. We may thereforeremove the special conformal gauge field from the rest of the problem.

4.5 The remaining field equationsSummarizing the results of the preceeding section, we have the remaining field equations, where the Weyl 1-form andauxiliary field satisfy

dω = 0ωa = Rabωb = Rab

Rab = −1

n−2

(Rab−

12(n−1)

Rηab)

Rab = dωab−ωcbωac

We also have the structure equations, which may now be simplified by eliminating the special conformal gauge field.Starting with the dilatation equation, we have

Ω = dω−ωaωa= d2ϕ−ωaωbRab= 0

since the Eisenhart tensor is symmetric. This shows that, regardless of the value of the constants α and β in theoriginal action, the dilatation vanishes.

The structure equation for the special conformal transformations simply shows that the “new” curvature, Ωa, isdetermined from the Riemann curvature as the covariant exterior derivative of the Eisenhart tensor,

Ωa = dωa−ωcaωc−ωaω= DRa

Finally, the Lorentz equation simplifies,

Ωab = dωab−ωcbωac−2∆adcb ωdωc

= Rab−2∆adcb Rdωc

= Cab +2∆adcb

(Rd +

12

Ωdeωe)

ωc−2∆adcb Rdωc

= Cab +12

2∆adcb Ωdeωeωc

= CabCollecting these results we have the following:

35

Connection The connection is that of a trivial Weyl geometry

dωa = ωbωab +ωωa

dω = 0

Field equations The field equations, including results from the algebraic equations, are

Cabcd;a = 0

2RabCacbd = Θcd

where Cabcd is the Weyl curvature tensor of Rab = dω

ab−ωcbωac .

Bianchi identities We have the Bianchi identities:

Ca[bcd] = 0

0 = Cabde; f +Cabe f ;d +C

abe f ;d

+2∆aceb(Rcd; f −Rc f ;d

)+2∆acf b (Rce;d−Rcd;e)+2∆acdb

(Rc f ;e−Rce; f

)DDRa = 0

with the consequent trace relations,

Cabde;a = (n−3)(Rbd;e−Rbe;d)

4.5.1 A little digression

It is possible that the third Bianchi identity is related to the remaining algebraic field equation. This third relation,which follows from the special conformal transformations, expands to give

0 = DDRa= D

(dRa−ωbaRb−Raω

)= d

(dRa−ωbaRb−Raω

)−ωca

(dRc−ωbcRb−Rcω

)+(

dRa−ωbaRb−Raω)

ω

= −dωbaRb +ωbadRb−dRaω−Radω−ωcadRc +ωcaωbcRb +ωcaRcω+dRaω−ωbaRbω−Raωω

= −(

dωba−ωcaωbc)

Rb

= −RbaRb

Expanding the curvature in terms of the Weyl curvature and the Eisenhart tensor, this becomes,

0 = RbaRb= CbaRb +2∆

bcdaRcω

dRb

= CbaRb +RaωdRd−ηdaηbcRcωdRb

= CbaRb

In components

0 = CabcdRae +CabdeRac +CabecR

ad

The traces vanish identically, so this identity is distinct from the condition required by the remaining algebraic fieldequation:

RacCabcd =12

Tbd

36

4.5.2 Combine the Lorentz field equation and the Bianchi identity

The structure equations are those of a trivial Weyl geometry, while the curvature field equation

Cabcd;a = (n−3)(Rbc;d−Rbd;c) = 0

provides the integrability condition for the Ricci tensor to be built purely from a conformal factor. Therefore, solutionsto these equations include the conformal equivalence class of Ricci flat spacetimes. However, we have two constraintson the allowed curvatures:

CbaRb = 0

RacCabcd =12

Tbd

While they are built purely from the conformal factor and its derivatives, the Ricci tensor and the Eisenhart tensor are,in general, nonvanishing. Therefore, these remaining constraints are nontrivial: the class of spacetimes satisfying therestrictions is a probably a proper subset of the class of Ricci flat spacetimes.

4.6 Structures and new featuresThere are no new features in the auxiliary conformal gauging. The Weyl vector is pure gauge, and the special conformalgauge field, ωa, has been eliminated – hence the name auxiliary. The Killing metric of the conformal group, restrictedto the base manifold, vanishes entirely, and we must introduce the original metric of the representation space, ηab, byhand.

Numerous other treatments of this gauging, with a variety of choices for the action, are found in the literature. Areview of these is given in the accompanying papers (Wheeler; Wehner and Wheeler).

5 Biconformal gaugingWe now consider same condition in the biconformal gauging.

dωab = ωcbω

ac +2∆

adcb ωdω

c +Ωabdωa = ωbωab +ωω

a +Ωa

dωa = ωba ωb +ωaω +Ωadω = ωaωa +Ω

We take the quotient{

Mab,Pa,Ka,D}

/{

Mab,D}

, resulting in a 2n-dimensional manifold. We expect n of these di-mensions to correspond to spacetime, with the remaining n to remaining to be interpreted. The horizontal directionsare spanned by the gauge fields of translations, ωa, and special conformal transformations, ωa, (often, in this context,called co-translations). Horizontal curvatures are now expanded in all 2n gauge forms, for example,

Ωab =12

Ωabcdωcωd +Ωacbdωcω

d +12

Ωacdb ωcωd

Each of the connection 1-forms is a function of 2n coordinates,(xα ,yβ

), and is a linear combination of

(dxα ,dyβ

),

ωab = ωabα

(xβ ,yσ

)dxα +ωaαb

(xβ ,yσ

)dyα

5.1 New structuresIn sharp constrast to the previous gauge theories, the biconformal gauging leads to new structures:

1. Uniform linear action in any dimension

2. Symplectic structure

37

3. Natural metric

4. SL(2,R) and complex structure

These structures, which follow entirely from the properties of the conformal group, lead to further results. Most im-portantly, the presence of a symplectic form tells us how to interpret the extra dimensions. Unlike higher dimensionalgravity theories which require compactification or other dimensional reduction to make physical comparisons, thebiconformal gauging gives a type of phase space. Physical questions may be addressed directly in this phase space,or on any suitable configuration subspace, without compactification. New, geometric insights into both Hamiltonianmechanics and quantum mechanics are revealed when they are formulated on these biconformal spaces.

Two further results which we explore in more detail in the next sub-Section are:

1. The linear action leads to general relativity on an n-dimensional submanifold.

2. By combining the metric and symplectic structures, we can derive the existence of time in initially Euclideanmodels.

5.1.1 Uniform linear action

We find the most general action linear in the biconformal curvatures.The gauge transformations of the connection forms are simpler in the biconformal gauging because we no longer

have either translational or special conformal transformations. Therefore, Λa = Λa = 0, and the connection transformsas

ω̃ab = Λacω

cdΛ̄

db−dΛac Λ̄cb

ω̃a = ΛacωcΛ̄

ω̃a = ΛωdΛ̄daω̃ = ω−dΛ Λ̄

Notice that the basis forms, (ωa,ωb) transform as tensors under the remaining Lorentz and dilatational symmetries.Each of the curvatures now transforms as a separate tensor:

Ω̃ab = ΛacΩ

cdΛ̄

db

Ω̃a = ΛacΩcΛ̄

Ω̃a = ΛΩdΛ̄daΩ̃ = Ω

It is therefore consistent to set the torsion to zero,

Ωa = 0 (5)

This is consistent with our usual expectation for spacetime, and also guarantees the existence of a momentum subman-ifold of biconformal space.

Second, we assume the minimum condition consistent with the existence of a spacetime submanifold. This min-imum condition will be discussed below. We show that under these conditions the low-energy field equations reduceto the conformal Einstein equation. The proof is lengthy and will only be summarized here. Details will be found inanother set of notes, [?].

As with the Poincaré and Weyl gauge theories, and unlike the auxiliary conformal gauge theory, it is possible touse an action linear in the biconformal curvatures. The most general curvature-linear action in n-dimensions is

S =∫

ε be··· fac···d (αΩab +βδ

ab Ω+ γω

aωb)ωc · · ·ωdωe · · ·ω f

There can be no linear torsion or co-torsion (Ωa) terms.

38

Exercise: Show that the resulting field equations, with vanishing torsion, are:

0 = β(

Ω ddb−Ω dbd)

0 = Ω baa0 = 2α∆c febδ

abdg Ω

dac

0 = αΩabac +βΩbc0 = 2(αΩeccd +βΩ

ed)δ

adeb +Λ

ab

0 = αΩbaca +βΩbc

0 = 2(αΩcedc +βΩed)δ

adeb +Λ

ab

Answer: See Notes: Solution to the biconformal field equations

5.1.2 Symplectic structure

The structure equations of the conformal group show that almost all biconformal spaces have a symplectic form. Thedilatation equation is

dω = ωaωa +Ω

In all known solutions to the field equations, the dilatation takes the form

Ω =−κωaωa

so that the structure equation becomesdω = (1−κ)ωaωa

The 2-form ωaωa is necessarily non-degenerate, and the equality shows that it is also exact. Therefore, ωaωa is aclosed, non-degenerate 2-form. This is the requirement for it to be symplectic.

The presence of a symplectic form tells us that the 2n-dimensional base manifold should be understood as a phasespace (in classical or quantum mechanical applications) or a co-tangent bundle when we build a gravity theory. Thisturns out to work extremely well. For classical mechanics or quantum mechanics, we take the action functional to beproportional to the integral of the Weyl vector,

S =∫

ω

Then the dilatational gauge freedom provides the usual ability to change the Lagrangian by a total derivative. With theNewtonian assumption of universal time, ω becomes

ω = H(xi, p j, t

)dt− pidxi

and we immediately get Hamilton’s equations as the extrema. Along these extrema, no physical size change is everobservable.

5.1.3 Natural metric

As we have seen, the Killing metric of the conformal group is

KΣΛ =

12 ∆

acdb 0 0 0

0 0 δ ab 00 δ ba 0 00 0 0 1

When KΣΛ is restricted to the base manifold it now remains non-degenerate:

KΣΛ =(

0 δ abδ ba 0

)

39

We therefore have the inner products 〈ωa,ωb

〉= 0

〈ωa,ωb〉 = δ ab〈ωa,ωb〉 = 0

This is in sharp contrast to all of the other gauge theories of general relativity that we have examined. In the othergauge theories, the induced Killing metric vanishes entirely. It is only by keeping both translation sectors of theconformal group (i.e., translations of the origin and translations of the point at infinity) on the base manifold that theKilling metric is nontrivial.

5.1.4 SL(2,R) and complex structure

If we introduce of two copies of the metric, ηab, of the original n-dimensional space as an additional structure on thebase manifold, we get a conformal equivalence class of matrices,

GAB =(

e2ϕ ηabe−2ϕ ηab

)since the original metric depends on the conformal factor. If we combine this, and its inverse, with the induced Killingmetric and the symplectic form,

KAB =(

δ abδ ab

)ΩAB =

(−δ ab

δ ab

)we readily build the mixed tensors

KA B = GACKCB

(δ ab

δ ab

)=

(e−2ϕ ηab

e2ϕ ηab

)ΩA B = G

ACKCB

=(

−e−2ϕ ηabe2ϕ ηab

)These commute to give

Q = [K,Ω] = 2(

δ ab−δ ba

)and we easily see that K,Ω,Q generate SL(2,R). Finally, notice that

ΩA CΩC

B =−δ AB

This gives biconformal spaces complex structure. The complex structure should allow us to establish an isomorphismbetween the biconformal space and twistor space. However, without the conformal gauging, we do not readily see thesymplectic and metric structures.

It is well known that twistor space admits both Lorentzian and Euclidean submanifolds. However, using thesymplectic form and Killing metric to form orthogonal configuration and momentum submanifolds, the relationshipbetween this different signature submanifolds becomes unique.

40

5.2 Summary of biconformal solutionSolving the field equations and structure equations is lengthy (see Notes). After certain gauge and coordinate choices,the connection takes the final form

ωab = αab −2∆acdbWced (6)

ωa = ea(x) (7)ωa = a(fa+ba) (8)ω = Wcec =−yβ dxβ (9)

where αab is the usual Lorentzian spin connection compatible with the solder form, ea,

dea = ebαaband

a = (1−κ)−1 (10)

κ = − 1(n−1)

(1+

γn2

(α (n−1)−β )

)(11)

fa = e βa (x) dyβ (12)

ba = Ra− e νa yµ Γµν − yaycec +

12

ηac(

ηghygyh)

ec (13)

It is important to notice that all dependence on yα is explicit: the only undetermined field is the solder form, ea(x),and this depends only on xα . The full 2n-dimensional geometry is determined by the solution on an n-dimensionalsubspace.

Notice that the strange looking Γµν = Γµνα dxα term in ba may be written α-covariantly as

Dν yµ =∂

∂xνyµ − yα Γαµν =−yα Γαµν

Dyµ = dx yµ − yα Γαµ =−yα Γαµwith the understanding that the µ index of Dyµ labels n different functions, but does not transform as a vector. Thereis a sense in which yµ does transform as a vector? The submanifolds spanned by yµ are flat Riemannian geometries.If we assume the corresponding manifolds are Rn, then the coordinate yα doubles as a vector. Then this term is fullycovariant.

We also have

Rab ≡−1

(n−2)

(Rab−

12(n−1)

Rηab)

(14)

The curvatures are then

Ωab =12

Ωabcdeced−2κ∆acdbωced (15)

= Rab−2∆acdbRced−2aκ∆acdbfced (16)= Cab−2aκ∆acdbfced (17)

Ωa = 0 (18)�

Documents

Utah State University · 2009. 7. 2. · Contents 1 Basic conformal relations 2 1.1 Structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .