Hamiltonian formulation of general relativity in terms of Dirac spinors

The Dirac spinors and matrices are used in combination with the Arnowitt-Deser-Misner formalism in order to obtain yet another formulation of Hamiltonian general relativity, together with a new form of the Gauss-Codazzi equations. The relation with Ashtekar's variables is analyzed; it is shown, for instance, that the matrices are equivalent to the electric field variable. The electric and magnetic decomposition of the gravitational field is also studie using Dirac matrices.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

Deformations of the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity

We show that the constraint algebra of Ashtekar's Hamiltonian formulation of general relativity can be nontrivially deformed by allowing the cosmological constant to become an arbitrary function of the (Weyl) curvature. Our result implies that there is not one but infinitely many (parametrized by an arbitrary function) four-dimensional generally covariant local gravity theories propagating 2 degrees of freedom.     

Improved Hamiltonian for general relativity

A Hamiltonian formalism for asymptotically flat spaces in general relativity which is manifestly covariant under Poincaré transformations at infinity is proposed and some of its implications are briefly discussed.     

Lattice formulation of general relativity

It is shown that almost the entire excitation energy acquired by the fission fragments during the descent from the saddle to the scission point comes from Landau-Zener transitions. The states tractable in the first order adiabatic approximation carry an excitation energy of a few hundred keV.     

Conformal Hamiltonian dynamics of general relativity

The General Relativity formulated with the aid of the spin connection coefficients is considered in the finite space geometry of similarity with the Dirac scalar dilaton. We show that the redshift evolution of the General Relativity describes the vacuum creation of the matter in the empty Universe at the electroweak epoch and the dilaton vacuum energy plays a role of the dark energy.     

The Hamiltonian of general relativity on a null surface

The Hamiltonian for the Einstein equations is constructed on an outgoing null cone with the help of the usual null tetrad used in the study of the asymptotical gravitational radiation field.     

The Hamiltonian constraint in the teleparallel equivalent of general relativity

In the teleparallel equivalent of general relativity the integral form of the Hamiltonian constraint contains explicitly theadm energy in the case of asymptotically flat space-times. We show that such expression of the constraint leads to a natural and straightforward construction of a Schrödinger equation for time-dependent physical states. The quantized Hamiltonian constraint is thus written as an energy eigenvalue equation. We further analyse the constraint equations in the case of a space-time endowed with a spherically symmetric geometry. We find the general functional form of the time-dependent solutions of the quantized Hamiltonian and vector constraints.     

Linear derivative Cartan formulation of general relativity

Beside diffeomorphism invariance also manifest SO(3,1) local Lorentz invariance is implemented in a formulation of Einstein gravity (with or without cosmological term) in terms of initially completely independent vielbein and spin connection variables and auxiliary two-form fields. In the systematic study of all possible embeddings of Einstein gravity into that formulation with auxiliary fields, the introduction of a “bi-complex” algebra possesses crucial technical advantages. Certain components of the new two-form fields directly provide canonical momenta for spatial components of all Cartan variables, whereas the remaining ones act as Lagrange multipliers for a large number of constraints, some of which have been proposed already in different, less radical approaches. The time-like components of the Cartan variables play that role for the Lorentz constraints and others associated to the vierbein fields. Although also some ternary ones appear, we show that relations exist between these constraints, and how the Lagrange multipliers are to be determined to take care of second class ones. We believe that our formulation of standard Einstein gravity as a gauge theory with consistent local Poincaré algebra is superior to earlier similar attempts.Received: 24 January 2005, Published online: 8 June 2005     

Canonical formulation of spin in general relativity

The present article aims at an extension of the canonical formalism of Arnowitt, Deser, and Misner from self‐gravitating point‐masses to objects with spin. This would allow interesting applications, e.g., within the post‐Newtonian (PN) approximation. The extension succeeded via an action approach to linear order in the single spins of the objects without restriction to any further approximation. An order‐by‐order construction within the PN approximation is possible and performed to the formal 3.5PN order as a verification. In principle both approaches are applicable to higher orders in spin. The PN next‐to‐leading order spin(1)‐spin(1) level was tackled, modeling the spin‐induced quadrupole deformation by a single parameter. All spin‐dependent Hamiltonians for rapidly rotating bodies up to and including 3PN are calculated.     

Analysis of the Hamiltonian formulations of linearized general relativity

The different forms of the Hamiltonian formulations of linearized General Relativity/spin-2 theories are discussed in order to show their similarities and differences. It is demonstrated that in the linear model, non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary. The Hamiltonians and the constraints are different in these two formulations but the structure of the constraint algebra and the gauge invariance derived from it are the same. It is shown that these equivalent Hamiltonian formulations are related to each other by a canonical transformation, which is explicitly given. The relevance of these results to the full theory of General Relativity is briefly discussed.     

Reparametrization-invariant Hamiltonian formalism in the general theory of relativity

A method is proposed that is appropriate for resolving the Hamiltonian constraint and which leads to a reparametrization-invariant reduced theory specified by a well-defined nonzero local Hamiltonian. This method is based on introducing a global (dependent only on time) conformal variable. The physical and geometric meaning of the variables in the reduced action functional is investigated. It is shown that, within the theory, the method of small perturbations is self-consistent. It is demonstrated that, in the theory of gravity, there are no wavelike excitations that make a negative contribution to the Hamiltonian. From an analysis of the reduced classical theory in the linear approximation, it follows that, at the first instants from the birth of the Universe, the extremely rigid equation of state appeared to be the effective equation of the state of gravity matter.     

A new Hamiltonian structure for the dynamics of general relativity

A new compact form of the dynamical equations of relativity is proposed. The new form clarifies the covariance of the equations under coordinate transformations of the space-time. On a deeper level, we obtain new insight into the infinite-dimensional symplectic geometry behind the dynamical equations, the decompositions of gravitational perturbations, and the space of gravitational degrees of freedom. Prospects for these results in studying fields coupled to gravity and the quantization of gravity are outlined.This essay was awarded the second prize for 1976 by the Gravity Research Foundation.     

Algebraic programming of Hamiltonian formalism in general relativity: Application to inhomogeneous space-times

We describe the structure and the use of a program written in the algebraic programming languagereduce 2, giving the super-Hamiltonian and supermomenta constraints, as well as Hamilton's canonical equations in terms of the canonical variables, for vacuum relativistic space-times. The program uses as input the components of the spatial metric tensor and of the corresponding canonically conjugate momenta in a coordinate or in a spatial Cartan basis. The results of the application of the program to a series of inhomogeneous (cosmological as well as noncosmological) space-times are given: in particular, the constraints, the Dirac Hamiltonian and the canonical equations are explicitly written for axisymmetric space-times, constituting the starting point for the study of the dynamics and of the canonical quantization of these configurations.     

Lagrangian formulation of a general Hamiltonian theory with constraints

A Lagrangian formulation is constructed for a general Hamiltonian theory with constraints. A modification is proposed of the standard procedure of the Hamiltonianization of a Lagrangian theory in the case when the Lagrangian theory has primary constraints. The obtained results are used to establish the Lagrangian form of infinitesimally small canonical transformations in the Hamiltonian formalism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 104–112, March, 1986.