Weyl gravity as general relativity
Conformal gauge theories of gravity
Midwest Relativity Meeting 2013
James T Wheeler
Work done in collaboration with Juan Trujillo and Jeff Hazboun
Auxiliary conformal gauge theory: arXiv:1310.0526
(Weyl gravity)
Biconformal gauge theory: arXiv:1305.6972
jim.wheeler@usu.edu
Utah State University
Logan, UT 84318
Weyl gravity as general relativity: Connection variation in conformal gravity
There are several different gauge theories of gravity with solutions conformal
to metrics satisfying the Einstein equation.
{e2f g0ab | all f }
where g0ab satisfies the vacuum Einstein equation.
As long as the conformal transformations are integrable, we may interpret such
conformal classes of metrics as different choices of local units.
Some of these theories, e.g., Weyl gravity, have received considerable debate
over the last decade.
Our main result:
Conformal gravity, with all connection fields varied,
reduces to scale-invariant general relativity
The quotient method
General methods exist for constructing gauge theories of specified
local symmetry and spatial, spacetime or other manifold. The
quotient method (Cartan, Kobayashi and Nomizu, Ne’eman and
Regge) begins with a Lie group G and a Lie subgroup H of G.
The quotient M = H / G is a manifold.
The cosets lead to a principal fiber bundle with Cartan connection.
Suitably generalizing the Cartan connection curves the manifold.
The result is a curved manifold M with local symmetry given by the
subgroup H .
Gauge theories of (scale invariant) general relativity
Group, G
Subgroup, H (local
symmetry)
Simplest Base Manifold
M = G/H
Poincare
Lorentz
SO(n-1,1)
Rn
Inhomogeneous
Weyl
Homogeneous Weyl
SO(n-1,1) x R+
Rn
Conformal
Inhomogeneous Weyl
Rn
Conformal
Homogeneous Weyl
SO(n-1,1) x R+
R2n
also, deSitter
anti-deSitter
Any of these may lead to general relativity (see, for starters, Ivanov and Niederle)
I will discuss the first conformal theory; Jeff Hazboun will give results from the second
Auxiliary conformal gauging (Weyl gravity)
There is one gauge vector (as 1-forms) for each generator of the conformal group:
ea =
wab =
W =
fa =
ema dxm solder form
wabmdxm spin connection
Wmdxm Weyl vector (as a 1-form)
famdxm gauge field of special conformal transformation
There is a component of the SO(4,2) curvature 2-forms for each generator:
Rab = Rabmn dxm ^ dxn
Ta = Tamn dxm ^ dxn
W = Wmn dxm ^ dxn
Sa = Samn dxm ^ dxn
Lorentz curvature
Torsion
Dilatational curvature
Special conformal curvature
Action and variation
We take the Bach form of the action:
S =
aRabcdRabcd + bWabWab
Two variations are possible:
1. Constrain the Weyl vector, special conformal field, and the connection
Gmab = ½ gmn(gna,b + gnb,a  gab,n)
fab = ebm fam = 1/2 ( Rab – 1/6 R hab)
Wa = 0
and vary the metric only.
2. Vary all connection forms with no constaints.
These give different results.
1. Constrained Metric Variation
The constraints, substituted into the action, give the Weyl form
S =
aCambnCambn
where Cambn is the Weyl conformal curvature of a Riemannian
geometry.
Variation of the metric, using the Gauss-Bonnet form of the Euler
character to eliminate the Riemann-squared term from the action,
leads to the Bach equation (Bach, 1921):
DaDbCambn - ½ RabCambn =
0
Bach goes on to find the static, spherically symmetric solutions,
rederived nearly 7 decades later and used to justify the anomalous
galactic rotation curves (see Mannheim, 2006).
2. Unconstrained variation of the full connection
Varying all 15 gauge vectors gives additional equations.
It is well-known (Kaku, Townsend van Nieuwenhuizen, 1978;
Crispim-Romao, 1978) that the variation of the special conformal
field leads to its elimination as an auxiliary field,
fab = ebm fam = -1/2 ( Rab – 1/6 R hab)  Rab
We note that Rab = 0 iff Rab = 0. This is the same condition that leads to
Rabmn  Cabmn,
giving the Weyl action.
The same field equation, together with a Bianchi identity, shows that the
dilatational curvature vanishes, giving a trivial Weyl geometry:
• There is a conformal gauge where the Weyl vector vanishes, so
the spacetime is appears to be Riemannian
There are two remaining equations which together imply the Bach equation.
2. Unconstrained variation: additional field equations
With Wab = 0, the two remaining equations are:
RabCambn =
0
DaCambn =
0
The first of these expresses the vanishing of all energy-momentum tensors
for the curvatures (Trujillo and JTW, in preparation)
The second requires some care.
The first step is to choose the Riemann gauge, in which the Weyl vector
vanishes and the geometry appears Riemannian.
This is possible because Wab = 0.
The second step is to use the second Bianchi to show that DaCambn = 0
if and only if
Ra[b;c] = 0
2. Unconstrained variation: a new basis
The integrability condition for a spacetime to be conformal to a Ricci-flat
spacetime was first derived by Brinkmann (Brinkmann, 1924). It is the
existence of f such that
Ra[b;c] + f;dCdabc= 0
which is clearly different from the final field equation, Ra[b;c] = 0. However, if
we change the orthonormal basis while staying in the Riemannian gauge:
ea = e c ea
while imposing the same structure equations for the connection and curvature, the
equation takes the form
Ra[b;c] + c;dCdabc= 0
This is the integrability condition.
We conclude that the new basis is conformal to a Ricci-flat basis, and
therefore, the original basis is too.
2. Unconstrained variation: special cases
There is an important subtlety.
We know that the original basis is conformal to a Ricci flat one, so in that
basis we must have the integrability condition
Ra[b;c] + z;dCdabc= 0
for some function z. However, in this basis we also know that Ra[b;c] = 0 so
we must also have
z;dCdabc = 0
This is only possible for nonvanishing z;d if the spacetime is Petrov type O or
N.
Generically, z;d = 0, and z is constant. This means that except for Petrov O
or N spacetimes, the Ricci tensor and the Weyl vector vanish in the same
(original) gauge.
2. Unconstrained variation: counterexample?
From the field equation in the form
Ra[b;c] = 0
there seems to be an easy counterexample: Einstein spaces, for which
Rab + Lgab = 0
clearly solve the equation.
It is important to understand that solutions to conformal gravity theories
are metric equivalence classes, {e2f gab | all f }. Metrics conformal to the
metrics of Einstein spaces are easily shown to satisfy
Ra[b;c] + c;dCdabc + 2Lha[bc;c] = 0
Instead of the conformal equivalent to the actual field equation,
Ra[b;c] + c;dCdabc= 0
so while Einstein spaces satisfy the field equation in one gauge, the only
conformal class of Einstein spaces giving solutions are those with vanishing
cosmological constant, L = 0.
2. Unconstrained variation: generality
Now return to a general gauge, where we may write Rab for the trivial Weyl
geometry as
Rab =
R 0ab - W(a;b) - WaWb + ½ W2 hab
Except for Petrov O or N spacetimes,
Rab =
0
This is a conformally invariant statement that there exists a gauge in which
the spacetime satisfies the vacuum Einstein equation.
Type O and N spacetimes are conformal to Ricci-flat spacetimes, but the
Weyl vector may not vanish in the Ricci-flat gauge.
Conclusions and conjectures
While the fourth-order treatment of Weyl gravity has solutions which are not
conformally Einstein, full variation leads to scale-invariant general relativity.
The full connection variation is bound to be the quantum solution, since
fluctuations then make the metric and connection independent. This is
supported by recent tree-level quantum calculations (Maldacena, 2013).
We do not yet know how to add matter to this formulation.
While general relativity is ghost-free, it is expected that Weyl gravity is
renormalizable. Optimally, there is a formulation which has both advantages.