The symplectic 2-form and Poisson bracket of null canonical gravity

TL;DR

The paper develops a covariant canonical framework for vacuum general relativity on a null two-surface, defining a phase space of free null initial data and a Peierls bracket on diffeomorphism-invariant observables. It then constructs an auxiliary bracket on the initial data by treating the presymplectic 2-form as the central object and inverting it in a generalized sense, yielding closed-form expressions for the brackets among the free data. While the resulting initial-data bracket is a pre-Poisson structure that does not satisfy the Jacobi identity, it reproduces the Peierls brackets of all observables, connecting the null initial-data formulation to the ADM Poisson structure on spacelike hypersurfaces. The work introduces a new pair of degrees of freedom at the intersection S_0 and develops an extended phase-space description to manage gauge and boundary issues, highlighting potential routes toward quantization and holographic interpretations in a null canonical setting.

Abstract

It is well known that free (unconstrained) initial data for the gravitational field in general relativity can be identified on an initial hypersurface consisting of two intersecting null hypersurfaces. Here the phase space of vacuum general relativity associated with such an initial data hypersurface is defined; a Poisson bracket is defined, via Peierls' prescription, on sufficiently regular functions on this phase space, called ``observables''; and a bracket on initial data is defined so that it reproduces the Peierls bracket between observables when these are expressed in terms of the initial data. The brackets between all elements of a free initial data set are calculated explicitly. The bracket on initial data presented here has all the characteristics of a Poisson bracket except that it does not satisfy the Jacobi relations (even though the brackets between the observables do). The initial data set used is closely related to that of Sachs. However, one significant difference is that it includes a ``new'' pair of degrees of freedom on the intersection of the two null hypersurfaces which are present but quite hidden in Sachs' formalism. As a step in the calculation an explicit expression for the symplectic 2-form in terms of these free initial data is obtained.

Figures (4)

• Figure 1: The upper diagram shows an initial data surface like ${\cal N}$ embedded in a 2+1 dimensional spacetime. In 3+1 dimensional spacetimes each of the components ${\cal N}_L$, ${\cal N}_R$, and $S_0$, of course have one dimension more than shown in this diagram. This is shown in the lower diagram, which depicts a three dimensional hypersurface ${\cal N}$ corresponding to a $3+1$ spacetime. Unlike the upper diagram, the lower diagram represents only the intrinsic differential topology of ${\cal N}$, and does not indicate it's embedding in spacetime.
• Figure 2: The domain of dependence of a pair of plane null rectangles in $2+1$ dimensional Minkowski space is shown. The straight lines ruling the rectangles are the generators. The domain of dependence is bounded to the future by the light cones of the two ends of $S_0$ and the null surfaces orthogonal to $S_R$ and $S_L$.
• Figure 3: An example of caustics and intersections of generators. $S_0$ is a spacelike curve in $2 + 1$ dimensional Minkowski spacetime having the shape of a half racetrack) - a semicircle extended at each end by a tangent straight line. The congruence of null geodesics normal to $S_0$ and directed to the future and inward - the generators shown in the diagram - sweep out a null surface having the form of a ridge roof, terminated by a (half) cone over the semicircle. The generators from the semicircle form a caustic at the vertex of the cone. There neighbouring generators intersect. On the other hand generators from the two straight segments of $S_0$ cross on a line (the ridge of the roof) starting at the caustic, but the generators that cross there are not neighbours at $S_0$.
• Figure 4: The diagram shows schematically (or literally in $1 + 1$ dimensional spacetime) the initial data hypersurface ${\cal N}$, its maximal Cauchy development (minus future Cauchy horizon) $\underline{\cal D}$, a boundaryless manifold $M$ containing $\underline{\cal D}$ and ${\cal N}$, a Cauchy surface $\Sigma_+$ of the interior of $\underline{\cal D}$, the domain $Q$ between ${\cal N}$ and $\Sigma_+$, and the domain of sensitivity $s_A$ of an observable $A$. The support in $\underline{\cal D}$ of the retarded and advanced perturbations of the metric generated by $A$ are also indicated - as white regions where the vertical or diagonal hatching is absent.

Theorems & Definitions (2)

• Proposition 2.1
• Proposition 2.2