Table Of ContentGravitation and spatial conformal invariance
H.F. Westman∗ and T.G. Zlosnik†
January 16, 2012
Abstract
It is well-known that General Relativity with positive cosmological constant can be formulated as a
2
gaugetheorywithabrokenSO(1,4)symmetry. Thissymmetryisbrokenbythepresenceofan internal
1
space-like vector VA, A = 0,...,4, with SO(1,3) as a residual invariance group. Attempts to ascribe
0
dynamicstothefieldVA havebeenmadeintheliteraturebutsofarwithlimitedsuccess. Regardlessof
2
this issue we can take the view that VA might actually vary across spacetime and in particular become
n nullor time-like. In this paper we will study thecase where VA is null. This is shown to correspond to
a
a Lorentz violating modified theory of gravity. Using the isomorphism between the de Sitter group and
J
the spatial conformal group, SO(1,4) ≃ C(3), we show that the resulting gravitational field equations
3
are invariant underall the symmetries, but spatial translations, of theconformal group C(3).
1
] 1 Introduction
c
q
- General Relativity is often formulated using a metric tensor g and a metric-compatible and torsion free
r µν
g connection∇ . However,amorepowerfulformulation(especiallywhenitcomestocouplingtospin-1 matter
µ 2
[ fields) employs a co-tetrad eI and an so(1,3)-valued spin connection ω IJ as independent variables1. The
µ µ
1 spacetime metric gµν is recoveredusing the relation
v
5 gµν =ηIJeIµeJν (1)
2
7 where ηIJ = diag(−1,1,1,1). In the language of differential forms [1], the gravitational action takes on a
2 particularly simply form referred to as the Palatini action:
.
1
Λ
0 S (eI,ωIJ)=κ ǫ eI ∧eJ ∧RKL− eI ∧eJ ∧eK ∧eL (2)
P P IJKL
2 Z (cid:18) 6 (cid:19)
1
where ǫ is the four-dimensional Levi-Civita symbol, RI ≡ dωI +ωI ∧ωK the curvature of ωI ,
: IJKL J J K J J
v Λ the cosmological constant, and κ the gravitational constant. The Palatini action reproduces exactly
P
i
X General Relativity in the absence of fermionic fields and is slightly modified in their presence by a non-zero
torsion field TI ≡deI +ωI ∧eJ 6=0.
r J
a In this first order formulation the spin connection ωIJ is naturally thought of as a standard gauge
connection related to local Lorentz symmetry. However, it was realized by MacDowell and Mansouri [2]
(though building on ideas by Cartan [3]) that the natural group to use for General Relativity (with positive
cosmological constant) is not SO(1,3) but SO(1,4), i.e. the group of transformations that preserve η =
AB
diag(−1,1,1,1,1) where A,B = 0,1,2,3,4. As opposed to the six parameters of the group SO(1,3), the
groupSO(1,4)hastenparameters;theconnectioncoefficientone-formsA AB maybedecomposedasfollows:
µ
h I ≡ A 4I , w IJ ≡ A IJ where recall that indices I and J range from 0,1,2,3. Consequently the
µ µ µ µ
representation independent curvature F AB may be decomposed as follows:
µν
FIJ = dwIJ +wI ∧wKJ −hI ∧hJ (3)
K
F4I = dhI +h ∧wJI (4)
J
∗[email protected]
†[email protected]
1Wewillusetheconventionthatdifferentialformswillbeintroducedwithspacetimeexplicitindicesbutthereafterreferred
towiththoseindicesimplicite.geJ andeJ ≡eJdxµ
µ µ
1
2
Amorefamiliarinterpretationofthesedifferentialformsispossible: WefirstlyidentifywIJ ascorresponding
totheSO(1,3)gaugefieldωIJ. ThefieldhI,insimplybeingapartofAAB,musthavedimensionsofinverse
length if the prior identification of wIJ is appropriate. We suggestively identify hI with the co-tetrad eI
divided by a constant length scale ℓ i.e. hI = 1eI. Therefore we have that:
ℓ
1 1
FIJ =RIJ − eI ∧eJ F4I = TI. (5)
ℓ2 ℓ
where again TI is the torsion. Therefore FAB may be decomposed into tensors familiar from the SO(1,3)
perspective.
To better understand what is going on here from a geometric point of view we visualize the spacetime
manifold as embedded in a five-dimensional space. On top of any point on the manifold we imagine a sym-
metric space called a model space, which in this case is a de Sitter spacetime2, and exhibits the symmetry
SO(1,4). Foreachmodelspaceonepointis singledoutasthe contactpointofthe modelspaceandthe four
dimensional manifold. Consider now what happens when we apply an SO(1,4) symmetry transformation.
A subgroup of those transformations will be such that the contact point is left invariant while others will
changethepointofcontactonthemodelspace. Thosewhichleavethe pointofcontactinvariantweidentify
as local Lorentz transformations SO(1,3) and the corresponding gauge connection one-form is ωIJ. Those
transformationsthatchangethepointofcontactweidentifyastranslations(ormoreaccuratelytransvections
as we are dealing with a de Sitter model space and not a flat manifold), and the corresponding gauge con-
nectionone-formis hI. Thus,weunderstandthatthe co-tetradcanbe viewedasagaugeconnectionrelated
to local translations, i.e. spacetime diffeomorphisms. It should be stressed that the relationship between
local translations, generated by a contact point changing SO(1,4) transformations, and a diffeomorphism,
is perhaps not straightforwardand has been the source of some disagreement [4].
The action
S = κ ǫ FAB∧FCD =κ ǫ FIJ ∧FKL
MM MM ABCD4 MM IJKL
Z Z
1 1 1
= −2κ ǫ eI ∧eJ ∧RKL− eI ∧eJ ∧eK ∧eL− RIJ ∧RKL (6)
MMZ IJKL(cid:18)l2 2l4 2 (cid:19)
yieldsthePalatiniactionwithapositivecosmologicalconstantaswellastheboundarytermǫ RIJ∧RKL
IJKL
3. Comparisonof(6)withthePalatiniaction(2)immediatelygivestherelationsκ =−2κ /l2,Λ=3/l2.
P MM
Thus we can think of ℓ as a conversion factor between the cosmological de Sitter scale and the arbitrary
lengthunitasdefinedbythemeterstickinPariswhichisultimatelydeterminedbythephysicsofelementary
particles. However, from the current perspective it would be natural to work in units in which ℓ = 1 4 , or
equivelently to directly identify hI with the co-tetradeI. In fact, we shall do so in this paper. In such units
Λ=3.
However, note that the action (6) is not invariant under SO(1,4) gauge transformations due to the
arbitraryselectionofa‘direction’indicatedbythe4inthefivedimensionalLevi-Civitasymbol. Thissignals
the presence of a non-dynamical ‘absolute object’ [7, 8] which can be made explicit by introducing the five-
∗
dimensional ‘space-like’ vector VA (with |V|2 ≡VAVBη =1) which takes the form VA =(0,0,0,0,1)in
AB
a privileged SO(1,4) gauge. Using the ‘absolute object’ VA the action becomes
S = κ ǫ VEFAB ∧FCD. (7)
MMV MM ABCDE
Z
In order to reproduce the Palatini action it is important that we regard VA as an `a priori postulated
object that should not be varied when extremizing the action. The presence of ‘absolute objects’ (such as
the Minkowski metric in special relativity) is often taken as an indication that the theory is incomplete.
One way to proceed would be to introduce dynamics for the absolute object (which is what happened in
the transition from special relativity with the `a priori postulated non-dynamical Minkowski metric η to
µν
2AdeSitter spacetimeisthefour-dimensionalmanifolddefinedbyXAXBηAB =ℓ2.
3For a self-contained discussion of the route to General Relativity from actions such as (6) we refer the reader to the two
veryreadableaccounts [5],[6].
4Comparetothenaturalnessofusingc=1inconventional relativisticphysics.
3
general relativity with a dynamical spacetime metric g ). From this perspective it is natural to provide
µν
somedynamics forVA. Infact, ithasprovenverydifficult to provideakinetic termto VA [6,9]. Amethod
ofenforcingthe desiredbehaviourof|V|2 wouldbeto simplyenforceafixednormalconditionupon|V|2 via
the addition of a Lagrangian constraint to the action [10]. We shall not prescribe precise dynamics for VA,
but shall assume that such dynamics may exist so that the field VE itself may be considered not merely a
Lagrangemultiplier field enforcingaconstraintuponS . Itis withthis in mindthatwe willreferto (7)
MMV
as an SO(1,4) invariant action.
Given that in general VA might be expected to have some dynamics, it is conceivable that there exist
regimeswherethenorm|V|2 maybe approximatelyequaltoavalueotherthanthe space-likevaluewhichis
knownto leadto GeneralRelativity. The purpose ofthis article is to examine in detail the behaviour ofthe
‘gravitational’ action (7) when |V|2 is assumed to be null. Towards these ends we initially take what may
seem like a diversion in establishing the isomorphism between the group SO(1,4) and the conformal group
the three dimensional conformal group C(3)
2 The isomorphism between SO(1,4) and the conformal group C(3)
Recallthatthe conformalgroupC(m) maybe definedasthe setofallcoordinatetransformationsxm →x˜m
that leave invariant the m-dimensional Euclidean metric δ up to a conformal factor dependent upon x˜p.
pq
We represent this by a general x˜m = f(θ ,JΩ,xn), where the function f is taken to depend upon the
Ω
Ω= 1(m+2)(m+1) generators JΩ of C(m) and associated small parameters θ . The generators JΩ may
2 Ω
beclassifiedbytheeffectoncoordinates,expressedasinfinitesimaltransformations: moperatorsP generate
m
translation coordinate transformations x˜m =xm+am; 1m(m−1) operators s which generate rotations
2 mn
x˜m = xm +Rmxn (R = −R ) ; one operator D which generates a dilation x˜m = (1+ǫ)xm ; and m
n mn nm
operatorsk thatgeneratespecialconformaltransformationsx˜m =xm+2xnb xm−x2bm (x2 ≡δ xmxn).
m n mn
These generators satisfy the following Lie algebra:
[D,p ] = ip (8)
m m
[D,k ] = −ik (9)
m m
[k ,p ] = 2i(δ D−s ) (10)
m n mn mn
[k ,s ] = i(δ k −δ k ) (11)
m np mn p mp n
[p ,s ] = i(δ p −δ p ) (12)
m np mn p mp n
[s ,s ] = i(δ s +δ s −δ s −δ s ) (13)
mn pq np mq mq np mp nq nq mp
where recallthatlabels runfrom1 to m. We now focus exclusively onthe case m=3. This algebramay be
written in a more compact form via the following definitions:
S ≡ s (14)
np np
S ≡ D (15)
04
1
S ≡ (p −k ) (16)
0p p p
2
1
S ≡ (p +k ) (17)
4p p p
2
Or, more compactly, using the notation η =diag(−1,1,1,1,1):
AB
[S ,S ] = i(η S +η S −η S −η S ) (18)
AB CD BC AD AC BD AC BD BD AC
Though this remains the Lie algebra for C(3), we see that this is identical to the Lie algebra for the group
SO(1,4). Inthissensetheequations(14)-(17)codifytheisomorphismbetweenthegroupsC(3)andSO(1,4).
Therefore a representation of SO(1,4) is also a representation of C(3), and by implication an SO(1,4)
invariantactionsuchas(7)isalsoinvariantunderC(3)transformations. Therefore,GeneralRelativitywith
apositivecosmologicalconstantmaybe seenasa theoryofbrokenspatialconformalinvariance. Recallthat
4
in the case of Macdowell-Mansourigravity, the SO(1,4) invariance was broken down to a residual SO(1,3)
invariance preserving the form VA = (0,0,0,0,1). We can see immediately from equations (14) to (17)
that from the C(3) perspective this form for VA is not invariant under dilations, or combined (p +k )
p p
transformations. The residual SO(1,3) gauge invariance of General Relativity (with a positive cosmological
constant) may be seen as invariance under the generators s and (p −k ) of the group C(3).
np p p
However, as we shall see, there are alternative forms for VA which secure more natural residual gauge
invariance from the C(3) perspective. A motivating feature behind the Macdowell-Mansouri approach was
to essentially be able to invoke behaviour of an object such as VA to separate in the gauge field AAB the
co-tetrad from the spin connection (hence the residual SO(1,3) symmetry).
3 The case where V A is null
Under an SO(1,4) (or equivalently, C(3)) transformation, the components of VA(xµ) transform as follows:
V˜C =(e−2iΘAB(xµ)SAB)C VD (19)
D
Acleargeometricinterpretationofthisisarotationinafivedimensionalinternalspacewithmetricη .
AB
As such, it is always possible to locally choose ‘coordinates’ such that a null vector VA has the following
components 5:
V4 = φ(xµ) V0 =±φ(xµ) Vi =0 (20)
where henceforth lowercase latin indices i,j,k,l will vary from 1 to 3, labeling space-like directions in the
internal space. One may now ask what residual gauge invariance preserves the form (20). Using the
transformation law (19), the representation of the generators S given by (74), and the relations (14)
AB
to (17), it is readily seen that the form (20) represents two possibilities. The case VA = (φ,0,0,0,−φ)
is invariant under transformations generated by s and k , and transforms homogeneously under D. The
ij i
case VA = (φ,0,0,0,φ) meanwhile is invariant under transformations generated by s , p , and transforms
ij i
homogeneouslyunderD,butwithoppositeweighttotheformercasei.e. foradilationassociatedwithasmall
parameterδ we haveV˜A →(1−δ)VA for the caseV0 =−V4 andV˜A →(1+δ)VA for the case V0 =+V4.
Therefore the null VA case displays residual gauge invariance and covariance more cleanly interpreted in
termsofC(3)variablesthaninthespace-likecase. Wemaystillvisualizethespacetimemanifoldasembedded
in a five-dimensional space. However, the functional form (20) now implies a preferred temporal direction
in this space. The residual gauge freedom for the form (20) is indeed not SO(1,3) invariance and so on the
spacetimemanifoldviolationsoflocalLorentzinvariancearetobeexpected. Wenowexpandtheconnection
coefficient one-form AAB in terms of the conformal group generators:
1
A C = Ei (p )C +Fi (k )C +C (D)C + Wij(s )C (21)
µ D µ i D µ i D µ D 2 µ ij D
1
= hi (S )C +ht (S )C +bi (S )C + Wij(s )C (22)
µ 4i D µ 4t D µ ti D 2 µ ij D
where we have introduced the spacetime fields Ei, Fi, C and Wij, the physical interpretation of which
µ µ µ µ
will be later apparent. Furthermore we decompose all fields into their time and spatial components:
Ei = (Ei,Ei ) Fi =(Fi,Fi ) C =(C,C ) Wik =(Ωik,ωik) (23)
µ m µ m µ m m
and
hi = (ǫi,hi ) ht =−C bi =(βi,bi ) (24)
µ m µ µ µ m
5Wewillassumethroughoutthatφ6=0. ThepossibilityVA=0mustbeconsideredseparately.
5
We will use the lowercaseLatin letters m,n,p,q exclusively to refer to spatial components onthe spacetime
manifold. The action(7) is constructed from three distinct quantities: ǫ , VA, and FAB. We concen-
ABCDE
trate firstly on the former two quantities. We first assume that VA takes the functional form givenby (20);
to reflect this, we introduce the following ‘internal forms’ T ≡δ0 and S ≡δ4, and hence:
A A A A
VE =φSE ±φTE (25)
where indices have been raised with ηAB, e.g. T0 = −1 Given the quantities S and T we may define
E E
a ‘spatial’ Levi-Civita symbol ε (which is non-vanishing only for indices values 1..3) via the following
ABC
relation:
ǫ =5·4T ε S (26)
ABCDE [A BCD E]
The quantity that appears in the action (7) is ǫ VE, and this may be developed as follows:
ABCDE
ǫ VE = 5·4T ε S VE (27)
ABCDE [A BCD E]
= 5·4T ε S (φSE ±φTE) (28)
[A BCD E]
= 4φT ε ±4φS ε (29)
[A BCD] [A BCD]
Explicitly, the action (7) may then be written as:
κ ǫ VEFAB∧FCD = 4κ φ T ε ±4φS ε FAB ∧FCD (30)
MM ABCDE MM [A BCD] [A BCD]
Z Z
(cid:0) (cid:1)
For simplicity, we now concentrate on the case where VA = (φ,0,0,0,−φ). As we shall see, this case
has a rather clear interpretation. Following this we shall compare this to the other case VA =(φ,0,0,0,φ).
Due tothe presenceofa preferredtemporaldirectionimpliedby (20), we wouldliketo decomposeAAB and
FAB into parts the contain time components and parts that do not. Then, to locally choose time to be the
preferred time on the spacetime manifold should lead to a simplification of the equations of motion.
Accordingly, we decompose AAB into temporal forms αAB and spatial forms aAB: AAB = αAB +aAB
i.e.
αAB = AABdt (31)
t
aAB = AABdxm (32)
m
Additionally we decompose the exterior derivative operator d as follows: d=d +(3)d i.e.
t
d (Y dxµ∧dxν ∧..) = (∂ Y )dt∧dxµ∧dxν ∧.. (33)
t µν.. t µν..
3d(Y dxµ∧dxν ∧..) = (∂ Y )dxm∧dxµ∧dxν ∧.. (34)
µν.. m µν..
Then we immediately obtain
FAB = dAAB +AAC ∧ADBη (35)
CD
= (d +(3)d)(αAB +aAB)+(αAC +aAC)∧(αDB +aDB)η (36)
t CD
= d aAB +(3)dαAB +(3)daAB +(αAC ∧aDB+aAC ∧αDB +aAC ∧aDB)η (37)
t CD
= d aAB +(3)dαAB +(αAC ∧aDB+aAC ∧αDB)η +(3)daAB+aAC ∧aDBη (38)
t CD CD
f AB fAB
t
| {z } | {z }
= f AB +fAB (39)
t
Therefore, for some ‘internal four form’ y we have:
ABCD
6
y FAB∧FCD = y f AB +fAB ∧ f CD+fCD (40)
ABCD ABCD t t
= y (cid:0)fAB∧fCD+(cid:1) fA(cid:0)B∧fCD (cid:1) (41)
ABCD t t
= 2y (cid:0)fAB∧fCD (cid:1) (42)
ABCD t
The action (7) then becomes:
κ ǫ VEFAB ∧FCD = 8κ φ T ε fAB∧fCD+S ε fAB∧fCD (43)
MM ABCDE MM [A BCD] t [A BCD] t
Z Z
(cid:0) (cid:1)
= κ φ 2ǫ fIJ ∧fKL+8S ε fAB∧fCD (44)
MMZ IJKL t [A BCD] t
(cid:0) (cid:1)
= 4κ φε f4i+fti ∧fjk+fij ∧ f4k+ftk (45)
MMZ ijk(cid:16)(cid:0) t t (cid:1) t (cid:0) (cid:1)(cid:17)
where 4T ε = ǫ ≡ ǫ and recall that indices I,J,K,L only go from 0 to 3. Following from
[A BCD] IJKL IJKL4
this, one may use the explicit form for the curvature contributions to (45) implied by (38) along with the
comparative decomposition given by (21) to present the action in terms of conformal variables of (22),
yielding the following result:
S = 4κ φε Ki∧R¯jk+Kij ∧T¯k (46)
CC MM ijk
Z
(cid:0) (cid:1)
where
Ki ≡ d Ei+(3)dEi+El∧ω i+El∧Ω i+C∧Ei+C∧Ei (47)
t l l
T¯i ≡ (3)dEi+El∧ωi+C∧Ei (48)
l
Kji ≡ fji =d ωji+(3)dΩji+Ωjl∧ωi+ωjl∧Ωi−F[j ∧Ei]−F[j ∧Ei] (49)
t t l l
R¯ji ≡ fji =(3) dωji+ωjl∧ωi−F[j ∧Ei] (50)
l
We will refer to the action (46) as Cartan Conformal Gravity, stemming as it does from a connection
AAB which includes, collectively, generators of translations on a model space as well as ‘point of contact’
preservingrotationandconformaltransformations. Theutilityofthenullnormcondition|V|2 =0istoallow
separation in the connection of the the ‘triad’ field Ei from the gauge fields of the rotation and conformal
transformations. In General Relativity plus a positive cosmologicalconstant, the model space was de Sitter
space. ForCartanConformalGravity,themodelspaceisinfactthesetofconformallyflatthreedimensional
spaces, as shall be indicated by the solution presented in Section 4.
For this theory, the only fields appearing with time derivatives are the fields Ei and ωji. The conformal
gaugefieldsCandFi appearalgebraicallyintheaction. Althoughthepresenceoftheseauxiliaryfieldsmay
seem unusual, it should be remembered that in the Palatini action the co-tetrad eI itself appears entirely
without derivatives.
The fields T¯i and R¯ji are suggestively named as they correspond respectively to the torsion and Rie-
mannian curvature with corrective terms in conformal gauge fields C and Fj which ensure the required
transformation properties under conformal transformations so as to preserve the action’s invariance under
them. However, it should be noted that the triad Ei is not identified simply with spatial parts of the
co-tetrad ei; rather it is proportional to a sum of ei and Lorentz boost fields bi.
Finally, it may be checked using the results of Appendix B that the functional form of the action (46)
is, as expected, invariant under dilation and special conformal transformations as well as three dimensional
rotations.
4 Equations of motion of Cartan Conformal Gravity
We mayrecoverthe equationsofmotionforCartanConformalGravitybythe standardmethodofrequiring
stationarityofthe action(46)under smallvariationsoffields. Notethat inthe constructionofS we have
CC
7
assumed that the null condition is satisfied at the level of the action, and so the stationary property of the
action must apply also with respect to small variations of φ when the equations of motion are satisfied.
Variation with respect to Ei, Ωij, Fi, C, Fj, C, ωmn, Em, and φ respectively yields:
3d φR¯jk
0 = ε (C∧R¯jk+Fj ∧T¯k)−ε ω m∧R¯jk−ε (51)
ijk mjk i ijk (cid:0)φ (cid:1)
3d φT¯k
0 = ε E ∧R¯mk−ωm ∧T¯k −ε ω m∧T¯k−ε (52)
jmk i i imk j ijk (cid:0)φ (cid:1)
(cid:0) (cid:1)
0 = ε Ej ∧T¯k (53)
ijk
0 = ε Ei∧R¯jk (54)
ijk
0 = ε Ki∧Ek+ε Ej ∧T¯k (55)
imk mjk
0 = ǫ Kij ∧Ek+ε Ei∧R¯jk (56)
ijk ijk
d φT¯k 3d φKi
0 = −ε t −ε −ε Ki∧ω k+ε Ki∧ωj +ε Ωi ∧T¯k
mnk (cid:0)φ (cid:1) imn (cid:0)φ (cid:1) imk n ijn m ink m
−ε Ω j ∧T¯k+ε Kij ∧E +ε E ∧R¯jk (57)
mjk n ijn m njk m
d φR¯jk 3d φKij
0 = ε t −ε −ε Ω i∧R¯jk+ε C∧R¯jk−ε Ki∧Fj
mjk (cid:0)φ (cid:1) ijm (cid:0)φ (cid:1) ijk m mjk ikm
−ε Fi∧T¯k−ε Kij ∧ω k+ε Kij ∧C (58)
imk ijk m ijm
0 = ε Ki∧R¯jk+Kij ∧T¯k (59)
ijk
(cid:0) (cid:1)
4.1 A simple solution
By inspection a particular solution to (51)-(59) is:
Ei = (0,δi ) Fi =(0,0) C =(0,0) Wik =(0,0) φ=1 (60)
µ m µ µ
which implies that
R¯jk = 0 T¯j =0 Ki =0 Kij =0 (61)
An analogous situation in the Macdowell-Mansouri case is the solution FIJ = 0 which corresponds to
RIJ = 1eI ∧eJ i.e. the Macdowell-Mansouriprescription singles out de Sitter space as a particular simple
l2
solution where the actual spacetime is identical to the model spacetime. Similarly in Cartan Conformal
Gravity, the equations of motion allow a simple solution wherein the ‘triad’ field Ei corresponds to that
m
of three dimensional flat Euclidean space up to a space and time dependent conformal factor. As may be
expected in a theory violating Lorentz invariance, the picture of the gravitational field is one of spatial
rather than spacetime structure. A suggestiveindicator as to the generality of this behaviour comes via the
SO(1,4) and C(3) invariant tensor g , defined as follows:
µν
g ≡ l2η D VAD VB (62)
µν AB µ ν
where D VA ≡∂ VA−iAA VB is the covariant derivative of VA.
µ µ B
For the case where |V|2 =1 we have that:
g(MM) = η eIeJ (63)
µν IJ µ ν
i.e. in this Macdowell-Mansouri case, g reduces to the familiar spacetime metric tensor g . In the case
µν µν
of Cartan Conformal Gravity, we have that:
g(CC) = φ2δ Ei Ej (64)
µν ij µ ν
(cid:0) (cid:1) (cid:0) (cid:1)
8
Therefore in the preferred frame, corresponding to Ei = 0, the ‘spacetime metric’ g has signature
µν
(0,+,+,+). Interestingly,thepresenceofadegeneratespacetimemetrichasmuchincommonwithgeometric
formulations of nonrelativistic gravitational theories, of which the Newton-Cartan theory is an example
[11, 12, 13, 14]. The ultimate role of a field such as g depends on the manner in which matter is coupled
µν
to AAB and VA. We now discuss in slightly more detail the scope for coupling to matter and providing
dynamics for the field VA.
5 The dynamics of V A
One may expect VA to be described by an action with ‘kinetic term’ and some accompanying mechanism
to preferentially drive the quantity |V|2 to a particular vacuum expectation value. One may then seek to
construct an SO(1,4) invariant action from VA, and DVA as well as the covariant ‘internal tensors’ η ,
AB
ǫ . The simplest possibility is:
ABCDE
S = ǫ VEDVA∧DVB ∧DVC ∧DVD (65)
V1 ABCDE
Z
Toget a senseof the ‘dynamics’implied by this action,we firstconsider the case where VA is space-like.
Onemayassumeaconstant‘background’valueforVA andconsiderperturbationsvA(xµ)aroundthis value
for fixed AC . The perturbations are taken to be small in the sense that we expand the action (65) to
D
at-most quadratic order in vA. We choose to work in a gauge where VA = (1+v4)δA, which is always
4
accessible if VA is space-like. Recalling the explicit form for the generators of SO(1,4) we have that
D VA = −(1+v4)eA+∂ vA (66)
µ (MM) µ µ
Therefore we have that:
S ∝ ǫ (1+v4)eA∧eB∧eC ∧eD ! (67)
V1(MM) ABCD4
Z
The underlying reason for this simple result is the fact that ǫ VE is forced to be proportional to
ABCDE
ǫ ;contractionsoverA,B,C,Darethennecessarilyoverindices0..3which,bythegaugechoice,involve
ABCD4
noderivativesofvA. Therefore,theaction(65)intheregimeofspace-likeVA seemsnotadequatetodescribe
kinetictermsofV4 nora‘potential’thatmightlenditselftowardsanon-vanishingvacuumexpectationvalue
for |V|2. Rather, the combined action (7) and (67) is algebraic and linear in v4, and so essentially acts as a
Lagrangemultiplier,enforcingaconstraintbetweenǫ FIJ∧FKLof(7)andtermsofǫ eI∧eJ∧eK∧eL
IJKL IJKL
of (67). The field v4 itself would be recoverable from the equations δ(S +S )/δAab. The presence of
V1 MMV
the new constraintandpossiblevariationofv4 awayfrom0 makesthe similarityofS +S toGeneral
V1 MMV
Relativity unclear.
Wenotethatthecovariantderivative’sabilitytoapproximatethe co-tetradaroundcertainbackgrounds,
as evident in (69) and (67), makes it useful for constructing realistic matter Lagrangians in an SO(1,4)
invariant manner [15]. For example, it may be shown that a Lagrangian of the following form corresponds
to a Klein-Gordon Lagrangian for appropriate κ (assumed to be formed only from combinations of
ABCDE
ǫ , η , VA) and D VA ∼−eA:
ABCDE AB µ (MM) µ
DVA∧DVB ∧DVC ∧ YEκ ∧DYD ∝ eI ∧eJ ∧eK ∧ YEκ DYD (68)
ABCDE IJKDE
(cid:0) (cid:1) (cid:0) (cid:1)
The component Y4 ultimately plays the role of a Klein-Gordon scalar field, whilst Y is representative of
J
its spacetime derivatives according to the co-tetrad (itself defined by appropriate behaviour of VA). Note
that the action (68) requires no basic counterpart to the inverse-tetrad field eµ at the level of the action.
J
Actions such as (68) are reminiscent of so-called first order DKP formulations of spin-0 and spin-1 fields
[16, 17, 18]. For the case that YA happens to be VA itself, the only non-trivial possibility for κ is
ABCDE
ǫ [15] which, as we have seen, is in severalsenses unsatisfactory. See [19] for an alternative approach
ABCDE
toaccommodatingtheVA fieldandmatterinMacdowell-Mansourigravityviatheintroductionofapreferred
volume element.
9
For the case of Cartan Conformal Gravity we have
1
D VA = −φEA−(C − ∂ φ)VA (69)
µ (CC) µ µ φ µ
From this it may be checked that in the Cartan Conformal Gravity case, the integral (65) is identically
zero, the reason being that it now contains a contraction ǫ VDVE. The action S does not seem
ABCDE V1
suitable for prescribing dynamics of VA, and it seems difficult to envision a suitable action built from AAB
and VA. If these fields cannot collectively define a realistic dynamics amongst themselves other than by
simply constraining the desired norm |V|2 via the use of a Lagrange multiplier [10], then it would seem
necessary to introduce more complicated structure into the theory.
Thusfar we have considered gravitation from the perspective of an SO(1,4) (or C(3)) gauge field and a
field VA valued in its matrix representation. However,the difficulty with constructing dynamics ofVA may
suggest that an alternative is called for. One possibility is that it is more appropriate to regard gravity as
belonging to the four dimensionalrepresentationof Spin(1,4)(the double coverof SO(1,4), and sometimes
referred to as its ‘Spin-1/2’ representation), for which the explicit form for the generators S is given in
AB
Appendix A.2. It may be shown that an action giving equivalent physics to (6) is given by [6]:
S = κ Tr γ5F ∧F (70)
MM(SP) MM(SP)
Z
(cid:0) (cid:1)
It has been shown by Randono [9] that a spinor multiplet with suitably constrained invariants can play
the role of the γ5 matrix in (70) and allow for more generalgravitationalactions than (6). Moreoverit may
be checkedthatthe LorentzviolatingcaseofCartanConformalGravitymaypresumablyby aconfiguration
of fields reducing also by tensor product to the Dirac matrix γ0. However, as in the case of the field VA, it
is not yet clear how to construct suitable kinetic terms for these fields.
As discussed in [19], it is possible that the internal group could be substantially larger than SO(1,4).
Conceivably a counterpart of VA valued in this larger group may be able, along with the larger groups
connection coefficient one-forms, to account for their own dynamics.
6 Discussion
The isomorphism between the group SO(1,4) and the conformal group C(3) has been seen to be useful
in the interpretation of the behaviour of the gravitational action (7) for the case where |V|2 is null. The
resulting theory, termed Cartan Conformal Gravity, has been seen to be more readily interpreted in terms
of spatial rather than spacetime structure, it’s field equations describing the evolution of what may be
termed conformal spatial Riemmanian curvature and torsion. However, the spatial triad Ei involved in
these quantities is not simply the spatial spacetime co-tetrad ei but is rather a sum of this field with the
spacetime-boost field bi, which in a more familiar setting corresponds to the extrinsic curvature of constant
time slices in the Palatini formulation of General Relativity; this points towards the possibility that other
degrees of freedom such as VA in the gravitational sector may lead to very different relationships between
geometry and the gravitationalfield in different regimes. Moreover,the interpretation of Cartan Conformal
Gravityintermsofconformalspace-likegeometryisentirelyasaresultofthechoiceofapositivecosmological
constant. The appropriate group for General Relativity plus a negative cosmological constant is SO(2,3),
whichisisomorphictoC(1,2)i.e. thegroupoftransformationsconformallypreservingtheMinkowskimetric
of signature (−,+,+), suggesting that the counterpart Cartan Conformal Gravity would take the form a
theoryofspacetimestructurebutofalowerdimensionalityandvaryingalongapreferredspace-likedirection.
Returning to the SO(1,4) case, one may yet choose to use C(3) variables for the case where |V|2 is
a space-like constant i.e. General Relativity with a positive cosmological constant. The possible utility
of such variables is the subject of ongoing investigation. In any case it is to be expected that geometric
interpretation of the gravitational field is more appropriately expressed in terms of eJ in this case, much
as it appears more appropriate to use Ei in the Cartan Conformal Gravity case. However, we note that
interpretationofthe theory ofGeneralRelativity, withorwithout cosmologicalconstant,in termsofspatial
conformalinvariance(thoughthe interpretationofthe termdiffers fromthatusedin this article)has beena
10
subject of much recent interest [20, 21, 22, 23, 24, 25] and it may be useful to investigate possible parallels
between approaches.
Due to the difficulty in constructing dynamics for the field VA (or indeed whatever field may play a
similar role to VA), the general consequences of Cartan Conformal Gravity are as yet unclear, though the
form of the action (46) suggests something markedly different to General Relativity. It would seem that a
necessarynextstepthenis toseewhetheranalternativetoVA maybe found. Itshouldbeemphasizedthat
not only is it desirable to construct dynamics for the role here played by VA, it is necessary also to cast
familiar matter fields in terms of actions invariant under a group larger than SO(1,3). Though in the case
of constant space-like |V|2 it appears possible to recover familiar classical equations of motion for matter
[15], much as one can recover General Relativity, it is not clear if there are differences in the behaviour of
matter beyond this. Of particular interested is the relationship between the scale l and the measured value
of the cosmologicalconstant.
Finally, recall that Cartan Conformal Gravity follows from the case V0 = −V4. Is the alternative pos-
sibility V0 = V4 a physically different theory? By inspection it would appear that the resulting theory
is identical, so long as one allows for the roles of Fi and Ei to be interchanged, and the attribution of a
different conformal weight for the field φ and the new ‘triad’ Fi. This equivalence would appear to hold
even though the interpretation of invariance properties under internal transformations on the model space
differssignificantly. Thereasonforthis is the similarmannerinwhichthe generatorsoftranslationsp and
m
special conformal translations k appear in the Lie algrebra of the group C(3). Notably this ambiguity in
m
the relationship between gravitationalfields and particular connection coefficients is present in attempts to
construct spacetime gauge theories for the four dimensional conformal group C(4) [26].
Acknowledgements: We would like to thank Florian Girelli and Tim Koslowski for useful discussions.
This researchwas supported by the Perimeter Institute-Australia Foundations (PIAF) program.
A Representations of SO(1,4)
ThegroupSO(1,4)consistsofalllineartransformationsthatpreservethelineelementη =diag(−1,1,1,1,1).
AB
The Lie algebra for SO(1,4) is:
[S ,S ]=−i(η S −η S −η S +η S ). (71)
AB CD AC BD AD BC BC AD BD AC
A field ψ in a particular linear representationof SO(1,4) transforms under an infinitesimal gauge trans-
formation as ψ˜→Uψ, where U is defined as follows:
ψ˜Ω =(e−iΘABSAB)Ω ψΓ (72)
Γ
Given this, one may define the covariant derivative D ψΩ = ∂ ψΩ − iA Ω ψΓ which transforms as
µ µ µ Γ
D˜µψ˜Ω → (e−iΘABSAB)ΩΓDµψΓ (i.e. it transforms homogeneously) as long as the connection spacetime
one-form A Ω transforms as:
µ Γ
A˜=UAU−1−i(dU)U−1 (73)
The abstract ‘indices’ Ω,Γ refer to a general representation of SO(1,4). Of particular use will be the
so-called Spin-1 and Spin-1/2 representations of the group which will be briefly discussed as follows.
A.1 Spin-1 representation of SO(1,4)
In this representation, the generators S take the form of 5-dimensional matrices. In index notation the
AB
generators take the following form:
(S )C =−i η δ C −η δ C (74)
AB D AD B BD A
(cid:0) (cid:1)
