Leaf of leaf foliation and Beltrami parametrization in d > 2 dimensional gravity

Abstract

This work establishes the existence of a covariant “Beltrami vielbein” in dimensions $d>2$, generalizing the well-known $d=2$ case. The definition of this vielbein is motivated by a sub-foliation structure of the Arnowitt–Deser–Misner (ADM) slices of a d-dimensional Lorentzian manifold $M_d$, namely ${\Sigma }_{d-1}^{ADM}={\Sigma }_{d-3}\times {\Sigma }_2$. Here, ${\Sigma }_2$ denotes dynamically varying Riemann surfaces that foliate $M_d$ within each ADM slice. The covariant “Beltrami vielbein” associated with any generic d-bein of $M_d$ is systematically determined via a covariant gauge-fixing of the $M_d$ Lorentz symmetry. It is parametrized by $\frac{d(d+1)}{2}$ independent fields, which fall into two distinct categories, each bearing a specific interpretation. Importantly, the Beltrami parametrization is not restricted to the ADM framework, it is a general feature that holds in any local coordinate system. The Weyl-invariant sector of the Beltrami d-bein precisely selects the $\frac{d(d-3)}{2}$ physical local bulk degrees of freedom of gravity for $d>3$, while retaining some flexibility depending on the specific physical problem considered in the gravitational bulk and boundary. The components of the Beltrami d-bein correspond one-to-one with those of the associated Beltrami d-dimensional metric. The Beltrami parametrization of the spin connection and the Einstein–Hilbert action leads to compact and suggestive expressions, which may facilitate, for instance, the search for new Ricci-flat solutions classified by the genus of the submanifolds ${\Sigma }_2$. More subtle issues arise in the Carrollian case, where the Lorentzian metric becomes degenerate and the correspondence between the spin connection and the Beltrami vielbein is no longer one-to-one. These aspects will be studied elsewhere, potentially opening interesting avenues for further investigation. Gravitational “Beltrami physical gauges” can be introduced to exploit the geometrical and intuitive advantages of the Beltrami parametrization in specific well-defined physical contexts. Further constraints simplify the Beltrami d-bein expression when $M_d$ possesses a given holonomy group, as exemplified by the case of eight-dimensional Lorentzian manifolds with $G_2\subset SO(1,7)$ holonomy. The results obtained in Lorentzian signature can be straightforwardly extended to the Euclidean case, as well as to situations where the foliation is taken along null coordinates, albeit with some subtleties specific to the null case.