Emergence of Einstein gravity from Weyl gravity

It is shown that in a quadratic gravity based on Weyl's conformal geometry, not only the Einstein-Hilbert action emerges but also a Weyl gauge field becomes massive in the Weyl gauge condition, $\tilde R = k$, for a Weyl gauge symmetry within the framework of the BRST formalism. We also consider a more general gravitational theory with a scalar field in the Weyl geometry and see that the Einstein-Hilbert action can be induced from spontaneous symmetry breakdown of the Weyl gauge symmetry. Thus, it turns out that Weyl's conformal gravity is quantum mechanically equivalent to Einstein's general relativity plus a massive Weyl gauge field, and the Weyl geometry is free from an infamous ``second clock problem'' in quantum regime.