Joshua N. Goldberg

Spin‐s Spherical Harmonics and ð

Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm(θ, φ) to the spherical harmonics of R4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

Degeneracy in loop variables

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekars connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.

Canonical general relativity on a null surface with coordinate and gauge fixing

We use the canonical formalism developed together with David Robinson to study the Einstein equations on a null surface. Coordinate and gauge conditions are introduced to fix the triad and the coordinates on the null surface. Together with the previously found constraints, these form a sufficient number of second-class constraints so that the phase space is reduced to one pair of canonically conjugate variables: and . The formalism is related to both the Bondi - Sachs and the Newman - Penrose methods of studying the gravitational field at null infinity. Asymptotic solutions in the vicinity of null infinity which exclude logarithmic behaviour require the connection to fall off like after the Minkowski limit. This, of course, gives the previous results of Bondi - Sachs and Newman - Penrose. Introducing terms which fall off more slowly leads to logarithmic behaviour which leaves null infinity intact, allows for meaningful gravitational radiation, but the peeling theorem does not extend to in the terminology of Newman - Penrose. The conclusions are in agreement with those of Chrusciel, MacCallum and Singleton. This work was begun as a preliminary study of a reduced phase space for quantization of general relativity.

On Hamiltonian systems with first‐class constraints

Some new results are presented on the theory of Hamiltonian systems with first‐class constraints. In these systems it is possible to separate the physical part from the gauge part by transforming to canonical coordinates in which the constraints are a subset of the new momenta; this construction is accomplished by algebraic methods and the use of a set of Hamilton–Jacobi‐like equations. Finally, the problem and meaning of evolution in systems with weakly vanishing Hamiltonian is commented on.

Null hypersurfaces and new variables

Following Ashtekars (1987) recently revised version of the standard canonical theory, the construction of a new variables canonical formalism for Einsteins theory of gravity is investigated when the time parameter has level sets which are null hypersurfaces. The configuration space variables are the components of a tetrad and the self-dual components of a connection. Because a null time parameter is used, the Hamiltonian formalism has second-class constraints as well as the first-class constraints which are associated with the invariance of the theory under diffeomorphisms and local gauge transformations. The first-class constraint algebra is discussed and reality conditions which relate the complex formalism to real general relativity are displayed.

Newman‐Penrose Constants and Their Invariant Transformations

The origin and significance of the Newman‐Penrose (N‐P) constants of the motion are examined from the point of view that constants of the motion generate invariant transformations. Here the calculation makes use of a generalization of Greens theorem to a situation applicable to the coupled Einstein‐Maxwell fields in general relativity. One finds strictly electromagnetic constants generated by an incoming electromagnetic shock wave, with dipole symmetry, at future null infinity. The gravitational constants contain an admixture from the electromagnetic field. They are generated by an incident quadrupole gravitational shock wave at future null infinity (J+). Both the electromagnetic dipole field and the gravitational quadrupole field behave like linearized fields at J+. All higher‐multipole fields do not uncouple from the nonlinear corrections induced by the self‐coupling of the gravitational field and its coupling with the electromagnetic field. It is shown that the gravitational constants are related to t...

Generalization of Green's Theorem

For a system of field equations which is derivable from a Lagrangian whose density is (i) homogeneous quadratic in the first derivatives of the field variables yA,μ and (ii) homogeneous of degree n in the undifferentiated field variables yA, one has the identity (n+1)zALA(y)−yAMA(y,z)≡tρ,ρ, where MA(y, z) is the first‐order change in the field equations LA(y) = 0 under the mapping yA → yA + zA. The specific example of general relativity is discussed.

Invariant Transformations and Newman‐Penrose Constants

The relationship between constants of the motion and invariant transformations is discussed. Particular emphasis is placed on strong conservation laws which are of the form U∧ ,μνμν≡0. The existence of such a law leads to constants of the motion which are surface integrals and therefore generally do not generate an invariant transformation. However, when there is an associated weak conservation law, such that tμ,μ ≡ −δyALA (LA = 0 are the field equations andδyA the invariant change in field variables), a nontrivial invariant transformation exists. These results are applied to the discussion of the Newman‐Penrose constants for the electromagnetic field. The conclusion arrived at isδyA = 0, which suggests that the invariant transformation generated by the Newman‐Penrose constants is trivial.

Null Surface Canonical Formalism

More than 20 years ago, Peter Bergmann together with one of us (JNG) considered the problem of constructing the canonical formalism for general relativity on a null cone. The motivation for doing so came from the difficulty in constructing the observables which could then become the basic operators in a quantum theory of gravity. The analysis of Bondi1, Sachs2, and Newman-Unti3 of the Einstein equa-tions in the vicinity of null infinity indicated that the constraint equations were easy to integrate and that the data to be specified freely was easily recognizable. On the outgoing null cone, the geometry of the 2-surface foliation is given and at null infinity one gives the news function at all times and the mass aspect and the dipole aspect at one time. This result gave hope that in the canonical formalism one would be able to recognize the appropriate variables for the quantum theory.

Dirac brackets for general relativity on a null cone

The Hamiltonian for the Einstein equations is constructed on a outgoing null cone with the help of the usual null tetrad. The resulting null surface constraints are shown to be second class in the terminology of Dirac. These second class constraints are eliminated by use of the “starring” procedure of Bergmann and Komar.