Abstract
We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the first- and second-class ones. In particular, in the 3+1-dimensional case, we show that the reduced phase space has two local degrees of freedom, instead of the usual four in the non-degenerate case.
Highlights
The field-theoretical formulation of general relativity (GR) is the assignment to a manifold M of an action functional depending on a Lorentzian metric, whose Euler–Lagrange equations are Einstein’s equations
We study the reduced phase space of general relativity (GR) in the coframe formulation in the case where the boundary has a lightlike induced metric
The advantages of the Kijowski and Tulczijew (KT) alternative, in which the reduced phase space is described as a reduction of the space of free boundary fields, reside principally in the simplification of the procedure that leads to the definition of the constraints starting from the restriction of the Euler–Lagrange equations in the bulk
Summary
The field-theoretical formulation of general relativity (GR) is the assignment to a manifold M of an action functional depending on a Lorentzian metric, whose Euler–Lagrange equations are Einstein’s equations. The advantages of the KT alternative, in which the reduced phase space is described as a reduction (i.e. as a quotient space) of the space of free boundary fields, reside principally in the simplification of the procedure that leads to the definition of the constraints starting from the restriction of the Euler–Lagrange equations in the bulk This construction avoids the introduction of the artificial classifications of the constraints as primary, secondary, etc. We propose a linearized version of the theory, in “Appendix B”, where we work around a reference solution of the Euler–Lagrange equation In this case, it can be shown that there is a natural isomorphism between the quotient space of the space of fields and another space where no equivalence classes are taken into account. A Hamiltonian formulation of the theory is widely considered to be one of the best starting points for the quantization of the theory
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