Sachs' free data in real connection variables

TL;DR

This work develops a Hamiltonian formulation of general relativity on a null foliation using real connection variables and clarifies how Sachs' constraint-free data arise as connection components linked to null rotations. It shows that in the torsionless case these connection components reduce to the shear of a null congruence and that torsion modifies these data; the propagating Einstein equations on the initial null surface are encoded as tertiary constraints connected to a Bianchi identity, linking bulk dynamics to asymptotic radiative data. The Bondi gauge is employed to establish the equivalence between the connection-based and metric-based symplectic potentials, with radiative data at future null infinity interpretable as a shear aligned to I^+. The results point to a group structure for the connection degrees of freedom (ISO(2)) and open avenues for loop quantum gravity approaches and BMS-centered quantization in a null framework.

Abstract

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs' constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs' propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

Figures (2)

• Figure 1: Left: Set-up of the characteristic $2+2$ initial-value problem. Two null hypersurfaces intersect on a space-like 2d surface $S_0$. When the two null hypersurfaces are intersecting light cones, as in the picture, $S_0$ has topology of a sphere. The (partial) Bondi gauge is such that $(\theta,\phi)$ are constant along $\partial_r$, and $\partial_r$ is null for all values of $u$. On the other hand, $\partial_u$ is null at at most one value of $r$, unless the spacetime has special isometries. Right: Further requiring suitable regularity conditions one can consider also a local $3+1$ foliation of light-cones generated by a time-like world-line.
• Figure 2: Characteristic initial-value problem at $\cal I^+$. One prescribes data on a chosen $u_0$ hypersurface of the foliation attached to future null infinity, plus the asymptotic transverse shear $-s^0(u,\theta,\phi)$. Thanks to the algebraic Bianchi identity (\ref{['Totti']}), this can also be understood as prescribing a certain shear for the non-geodetic asymptotic null vector $\partial_u-\partial_r$ in the physical spacetime.