Null dust in canonical gravity

TL;DR

This paper develops a Hamiltonian framework for null dust in canonical gravity, describing its unique degrees of freedom and coupling to geometry, and explores its quantization and differences from ordinary dust.

Contribution

It introduces a novel canonical formulation of null dust with six potentials, highlighting its distinct degrees of freedom and algebraic structure compared to ordinary dust.

Findings

Null dust has three degrees of freedom per space point.

The Hamiltonian includes energy and momentum densities of null dust.

Constraints form a true Lie algebra, enabling quantization.

Abstract

We present the Lagrangian and Hamiltonian framework which incorporates null dust as a source into canonical gravity. Null dust is a generalized Lagrangian system which is described by six Clebsch potentials of its four-velocity Pfaff form. The Dirac--ADM decomposition splits these into three canonical coordinates (the comoving coordinates of the dust) and their conjugate momenta (appropriate projections of four-velocity). Unlike ordinary dust of massive particles, null dust therefore has three rather than four degrees of freedom per space point. These are evolved by a Hamiltonian which is a linear combination of energy and momentum densities of the dust. The energy density is the norm of the momentum density with respect to the spatial metric. The coupling to geometry is achieved by adding these densities to the gravitational super-Hamiltonian and supermomentum. This leads to appropriate Hamiltonian and momentum constraints in the phase space of the system. The constraints can be rewritten in two alternative forms in which they generate a true Lie algebra. The Dirac constraint quantization of the system is formally accomplished by imposing the new constraints as quantum operator restrictions on state functionals. We compare the canonical schemes for null and ordinary dust and emhasize their differences.