Sachs’ free data in real connection variables

Abstract

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.

A gauge-invariant symplectic potential for tetrad general relativity

“…A symplectic potential for tetrad gravity giving vanishing internal Lorentz charges can also be obtained when the 2d corner hinges between a space-like and a null hypersurface [23,24]. We explored the Hamiltonian structure of Einstein-Cartan gravity on null hypersurfaces in [45,46], and we plan in future work to look at the covariant phase space perspective on them.…”

Static compact objects in Einstein-Cartan theory

“…Since the Riemann tensor also depends on the torsion tensor, the latter must also be invariant under the action of G. Therefore, from Eqs. (35) and (36), the tensor field s α , Eq. (37), must be null.…”

“…Now, an LRS space-time is said to be of class I (LRSI) if the congruence of the curves associated with vector field e -defined to have the same direction as the axis of symmetry -is hypersurface orthogonal. 3 If the congruence of curves associated with the vector field u is 2 It should be remarked here that the presence of a generic torsion tensor field affects the definition of the kinematical quantities [35][36][37][38]. See the Appendix A 4 for further details.…”

“…As discussed in Refs. [35][36][37][38], the presence of a generic torsion field will affect the definition of the kinematical quantities that characterize a congruence of curves, such that, θ, σ αβ and ω αβ , Eqs. (A8) -(A10), in general, do not represent the actual geometric -physical -expansion, shear and vorticity of the time-like congruence to which u is tangent.…”

Raychaudhuri and optical equations for null geodesic congruences with torsion

“…Hence, although it drops out from the optical equations if one restricts to orthogonal connecting vectors, it plays an important dynamical role when the full set of Einstein's equations is considered, since it is one of Sachs' constraint-free data at the 2d corner between two null hypersurfaces [17] (see also [18] and references therein).…”