James T Wheeler Work done in collaboration with Juan Trujillo and Jeff Hazboun

1. Weyl gravity as general relativityConformal gauge theories of gravityMidwest 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

2. There are several different gauge theories of gravity with solutions conformal to metrics satisfying the Einstein equation. {e2fg0ab| 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 Weyl gravity as general relativity: Connection variation in conformal gravity

3. 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 . The quotient method

4. Gauge theories of (scale invariant) general relativity 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

5. Auxiliary conformal gauging (Weyl gravity) There is one gauge vector (as 1-forms) for each generator of the conformal group: ea = ema dxm solder form wab = wabmdxm spin connection W = Wmdxm Weyl vector (as a 1-form) fa = 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 Lorentz curvature Ta = Tamn dxm ^dxn Torsion W = Wmn dxm ^dxn Dilatational curvature Sa = Samn dxm ^dxn Special conformal curvature

6. Action and variation We take the Bach form of the action: S = aRabcdRabcd + bWabWab • Two variations are possible: • 1. Constrainthe Weyl vector, special conformal field, and the connection • and vary the metric only. • 2. Vary all connection forms with no constaints. • These give different results. Gmab = ½ gmn(gna,b + gnb,a – gab,n) fab = ebmfam = 1/2( Rab – 1/6 R hab) Wa = 0

7. 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).

8. 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 = ebmfam = -1/2( Rab – 1/6 R hab) ºRab • 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.

9. 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

10. 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 = ec 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.

11. 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.

12. 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.

13. 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 W2hab 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.

14. Conclusions and conjectures While the fourth-order treatment of Weylgravity 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.