Sachs’ free data in real connection variables

• DOI: 10.1007/jhep11(2017)205
• Publication Date: 2017-12-01T10:09:47Z
• Authors: Elena De Paoli; Simone Speziale
• Keywords: High Energy Physics - Theory; coupling: nonminimal; fermion; FOS: Physical sciences; QC770-798; General Relativity and Quantum Cosmology (gr-qc); algebra; 01 natural sciences; General Relativity and Quantum Cosmology; Nuclear and particle physics. Atomic energy. Radioactivity; 0103 physical sciences; Models of Quantum Gravity; general relativity; Bianchi identity; [PHYS.GRQC] Physics [physics]/General Relativity and Quantum Cosmology [gr-qc]; mathematical methods; stability; boundary condition; Hamiltonian; space-time; foliation; High Energy Physics - Theory (hep-th); twist; field theory: vector; [PHYS.HTHE] Physics [physics]/High Energy Physics - Theory [hep-th]; Classical Theories of Gravity; geodesic

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.