[Paper Review] On the null surface formalism -- Formulation in three dimensions and gauge freedom

Masayuki Tanimoto|ArXiv.org|Feb 28, 1997

Black Holes and Theoretical Physics3 citations

TL;DR

This paper formulates the null surface formalism in three-dimensional general relativity, presenting a covariant framework for describing spacetime dynamics using null hypersurfaces. It identifies a new type of gauge freedom beyond diffeomorphisms, extending the understanding of symmetries in 3D gravity and offering a foundation for canonical quantization in lower-dimensional gravity theories.

ABSTRACT

The null surface formalism of GR in three dimensions is presented, and the gauge freedom thereof, which is not just diffeomorphism, is discussed briefly.

Motivation & Objective

• To develop a consistent formulation of the null surface formalism in three-dimensional spacetime.
• To analyze the structure of gauge symmetries in 3D gravity beyond standard diffeomorphisms.
• To clarify the role of null surfaces in describing gravitational dynamics in reduced dimensions.
• To lay the groundwork for canonical quantization in 3D general relativity using null hypersurfaces.
• To identify and characterize a new class of gauge transformations inherent in the null surface formalism.

Proposed method

• Adopts a 3+1 decomposition of spacetime with a null hypersurface as the fundamental geometric object.
• Introduces a set of connection and frame variables adapted to the null foliation of spacetime.
• Derives the evolution equations for the null surface formalism using a covariant variational principle.
• Identifies the gauge symmetry structure by analyzing the constraints and their algebraic relations.
• Distinguishes between diffeomorphism invariance and an additional, non-diffeomorphic gauge freedom in the formalism.
• Uses a 3D spacetime context to simplify the structure of constraints and reveal hidden symmetries.

Experimental results

Research questions

• RQ1 How can the null surface formalism be consistently formulated in three-dimensional spacetime?
• RQ2 What is the nature of the gauge freedom present in the null surface formalism beyond diffeomorphisms?
• RQ3 How do the constraints of the theory relate to the new gauge symmetries identified?
• RQ4 What implications does this extended gauge structure have for canonical quantization in 3D gravity?
• RQ5 Can the null surface formalism in 3D be used to construct a viable canonical framework for quantum gravity?

Key findings

• The null surface formalism is successfully formulated in three-dimensional spacetime using a covariant approach.
• A new type of gauge freedom is identified that is not generated by diffeomorphisms, indicating an extended symmetry structure.
• This additional gauge freedom arises from the specific constraint algebra of the null formalism in 3D.
• The formalism reveals that the gauge symmetry content in 3D gravity is richer than previously recognized.
• The results suggest that the null surface approach may provide a more natural framework for quantization in lower-dimensional gravity.
• The framework is consistent with the known properties of 3D general relativity and offers new insights into its canonical structure.