General relativity on the multiverse and nature’s hierarchies

Abstract

We define ``third-derivative'' general relativity by promoting the integration measure in Einstein-Hilbert action to be an arbitrary 4-form field strength. We project out its local fluctuations by coupling it to another 4-form field strength. This ensures that the gravitational sector contains only the usual massless helicity-2 propagating modes. Adding the charges to these 4-forms allows for discrete variations of the coupling parameters of conventional general relativity: ${G}_{N},\mathrm{\ensuremath{\Lambda}},{H}_{0}$, and even $⟨\mathsf{Higgs}⟩$ are all variables which can change by jumps. Hence, de Sitter spacetime is unstable to membrane nucleation. Using this instability, we explain how the cosmological constant problem can be solved. The scenario utilizes the idea behind the irrational axion, but instead of an axion it requires one more 4-form field strength and corresponding charged membranes. When the membrane charges satisfy the constraint $\frac{2{\ensuremath{\kappa}}_{\mathsf{eff}}^{2}{\ensuremath{\kappa}}^{2}|{\mathcal{Q}}_{i}|}{3{\mathcal{T}}_{i}^{2}}<1$, the theory which ensues exponentially favors a huge hierarchy $\mathrm{\ensuremath{\Lambda}}/{M}_{\mathrm{Pl}}^{4}\ensuremath{\ll}1$ instead of $\mathrm{\ensuremath{\Lambda}}/{M}_{\mathrm{Pl}}^{4}\ensuremath{\simeq}1$. The discharges produce the distribution of the values of $\mathrm{\ensuremath{\Lambda}}$ described by the saddle point approximation of the Euclidean path integral.