General Coordinate Transformations Research Articles

THE RADIATION DAMPING ON BIBARY STARS DUE TO EMISSION OF GRAVITATIONAL WAVES

According to the general theory of relativity, The force is not an covariant quantity under general coordinate transformations. Different coordinate conditions will lead to different damping forces. It will be convenent to find out the physical coordinate system in which the calculated value of the damping effect is the same as or nearest to that observed on earth. We expect that this physical coordinate system should be free from spurious non-transverse gravitational waves at infinity. In this coordinates the change of orbital period P due to emission of gravitational waves is given by P=-6.5×10-12(m1m2)/M⊙2((m1+m2)/m⊙)-1/3.

Generalized Hamiltonian theory: Central coordinates

The basis of a generalized formalism of Hamilton-Lagrange mechanics for elementary (i.e., individual) open systems was introduced in a previous work. In the present paper a covariance theory is constructed for the proposed generalization. The result is an extended transformation theory, or geometry, for phase space, in which the behavior of generalized Hamiltonian equations under time-dependent general coordinate transformations is determined. The proposed mechanics subsumes the ordinary mechanics in a natural way. New features include generalization of "pictures" (coordinate-system classes for fixed values of the Hamiltonian) to "viewpoints," which are characterized by values of a phase-space "vorticity" tensor, that always vanishes in ordinary mechanics: analogous to extremes of Heisenberg and Schr\"odinger pictures are those of canonical and Hamiltonian viewpoints. The Liouville theorem receives a novel generalization, and the scheme possesses a (viewpoint) covariant time derivative with analogies to the covariant derivative in a Riemannian space. Viewpoint covariance in the new formalism is the natural extension of canonical covariance in the old. The role of dynamical gauge invariance, investigated in a second previous paper, is elaborated, and the example of the undriven damped oscillator worked out to illustrate the operation of the formalism, including that of a generalized Hamilton-Jacobi theory. A line-element-space (Finslerian) interpretation is given for the new scheme, and a corresponding generalization is proposed for nonlinear self-interacting systems.

Hypervirial theorems and co-ordinate transformation in quantum mechanics

The tensor virial theorem was derived in the framework of both classical and quantum mechanics (1.4). The virial theorem is a particular case of the hypervirial theorems for a special choice of the hypervirial operator and its validity is restricted to stationary states (s) as well as for scattering states (6). A method to derive the quantum virial theorem consists in using the cxtrcmum property of (/~ and a dilatation transformation (7). From this point of view, one possible application of the tensor virial theorem consists in the introduction of more than one scale factor in a trial wave function. This scaling method proposes different stretehings for the different spatial co-ordinates. Examples were given where the introduction of multiple scale factors and the imposition of the tensor virial theorem yield a better result than the usual procedure of subjecting the wave function to a single scale transformation and imposing the scalar virial theorem (3,s). But the most general co-ordinate transformation involves not only stretehings and rotations, but translations too. The purpose of the present communication is to analyse a general co-ordinate transformation through an optimization of the functional energy for a variational wave function, in which the variational parameters are associated to the geometric transformations of stretehings, rotations and translations. We shah follow the procedure of Pandres (2) and Einstein's convention of summation over repeated indices. Here, and henceforth, ~a~ is the i-th Cartesian co-ordinate of the ~-th particle of a N-particle system. We use the summation convention that repeated Greek subscripts are summed from 1 to N, while repeated Latin subscripts are summed from 1 to 3. Let the original set of Cartesian co-ordinates {~) be subject to the transformation

Calculation of Transonic Inlet Flowfields Using Generalized Coordinates

A method for calculating in viscid supercritical flowfields about axisym metric inlet cowls with centerbodies is presented. A finite-differe nce approximation to the full-potential equation is solved under a general coordinate transformation, using a numerical evaluation of the transformation matrix at each mesh point. For the present problem, a boundary conforming coordinate system was generated by a sequence of conformal and shearing transformations, but this transformation is not essential to the method. Both the quasiconservative and non- conservative forms of Jameson's rotated differencing scheme are used, and the difference equations are solved by relaxation. Numerical results for pressure distributions generally agree well with experiment. INCE the initial success of Murman and ColeJ in applying the type-dependent differencing concept to the transonic small-disturbance equation, this technique has been ex- tensively applied to compute transonic flowfields about airfoils, using both the small-disturbance and full-potential equations. For flowfields around blunt bodies, the full- potential equation is required to properly resolve the solution in the vicinity of the stagnation points. The introduction of the rotated differencing scheme by Jameson 2 enables the use of line relaxation for solving the full-potential equation. There have been many successful applications of the rotated difference scheme to compute inviscid transonic flowfields around blunt-nosed bodies, including nacelle inlets and wing- body configurations. For a flowfield around a nacelle inlet, Arlinger3 applied a conformal mapping technique to trans- form the full-potential equation to a boundary conforming coordinate system. Caughey and Jameson4 solved the same problem using a sequence of simpler transformations, and studied ways to accelerate the iteration scheme used to solve the difference equations. Reyhner5'6 solved the full-potential equation in a Cartesian mesh. This requires an interpolation scheme to accurately treat the surface boundary condition. The mapping techniques generally require transforming the governing equations under the sequence of mappings. For complex geometries, these transformations can become a lengthy and tedious process. The boundary interpolation scheme has the advantage of always differencing the equation in Cartesian coordinates. There are, however, various types of irregular boundary elements, and the effort required to keep track of surface-mesh intersections is a major drawback in the application of this scheme. Caughey and Jameson7 proposed a finite-difference scheme to compute the supercritical flowfield about wing-body configuration using a form of the

Poisson brackets in condensed matter physics

A general method of deriving nonlinear equations of hydrodynamics for both normal liquid and superfluid 4He and 3He, equations of the elasticity theory, equations for spin waves in magnets and spin glasses, liquid crystals, and so on is described. The method is based on the use of the Poisson “hydrodynamic” brackets. Hydrodynamic brackets are on the one hand, a classical limit of quantum commutators, on the other hand, Poisson brackets of certain symmetry groups inherent in the given problem: groups of general coordinate transformations for hydrodynamics and elasticity theory, groups of local spin rotations for spin waves, etc. Along with well-known examples nonlinear equations of the elasticity theory for bodies with impurities, dislocations and disclinations, and equations of motion for spin glasses and multisublattice magnets are studied.

On an OSp(1,4) renormalisable theory of supergravity with higher derivatives

Presents a supergravity Lagrangian invariant under local orthosymplectic and general coordinate transformations. It is conjectured that the Lagrangian describes a renormalisable theory. New features in this formulation are the introduction of a group-invariant metric tensor and the corresponding connection. As in other higher-derivative theories metric ghosts are present.

Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow

An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrary geometry through the use of general coordinate transformations is described. Viscous effects are optionally incorporated with a thin-layer approximation of the Navier-Stokes equations. An implicit approximate factorization technique is employed so that the small grid sizes required for spatial accuracy and viscous resolution do not impose stringent stability limitations. Results obtained from the program include transonic inviscid or viscous solutions about simple body configurations. Comparisons with existing theories and experiments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are also discussed.

Scalar quantum field in an external gravitational field

We consider a class of Lorentz metrics on R4 which are Minkowskian off a compact set. For each such metric we construct a quantum field operator which satisfies the generalization of the Klein–Gordon equation. The field has a causal commutator and transforms as a scalar under general coordinate transformations. The field determines a unitary scattering operator which is invariant under coordinate transformations.

On a quantization of Einstein-Cartan's theory with anticommuting variables in Dirac's Hamiltonian formalism

A quantization of Einstein-Cartan's theory is discussed in Dirac's Hamiltonian formalism. The spinor field is regarded as an anticommutingc-number. To find the simplest Hamiltonian formalism for this case, we choose one leg of the vierbein field to be normal to the hypersurfacex0=const. Since this theory has two kinds of gauge symmetries related to general co-ordinate transformations and to local Lorentz transformations, these gauge symmetries must be fixed. After the gauge fixing, the anticommutator for the spinor field is derived.

Useful coordinate transformations for antenna applications

General coordinate transformations which are commonly encountered in many antenna applications are presented. Neither the feed coordinates nor the far-field pattern coordinates in general coincide with the antenna coordinates. Transformations discussed allow one to relate the spherical and Cartesian components of one system to the spherical and Cartesian compoents of the other system. In particular, attempts are made to use unified notations to assist the reader in a straightforward application of the transformations.

Implications of non-linear invariances for spinor theories in curved space-time

We analyze the relationship for spinor theories in a gravitational background between the invariances of the full non-linear theory (general coordinate transformations and local vierbein rotations) and the behaviour of matrix elements ( S-matrix elements or matrix elements of composite operators) when an external graviton is longitudinal Special emphasis is laid on the massless Rarita-Schwinger field. Here we find a connection between fermionic gauge invariance and invariance under the non-linear transformations. Consequently the one-loop axial vector current here only possesses the invariances of the non-linear theory, as well as the property of having matrix elements that vanish for longitudinal external gravitons, in a completely covariant gauge.

The component formalism follows from the superspace formulation of supergravity

We identify the transformations of local supersymmetry as particular combinations of general coordinate transformations and local Lorentz transformations in superspace. Component fields for supergravity are defined as the values at θ = 0 of the supervielbein and (for the auxiliary fields) of certain components of the supertorsion. For chiral and general multiplets they are defined as the values at θ = 0 of (tangent space) spinorial covariant derivatives. We derive the transformation properties for field multiplets and for chiral density multiplets. The same method can be used for general densities. The lagrangian of supergravity in terms of component fields is obtained.

Massive supergravity from spontaneously breaking orthosymplectic gauge symmetry

We construct a Lagrangian density which is manifestly invariant under the orthosymplectic gauge group OSP(1; 4) and under general coordinate transformations. This is done by the use of two multiplets, the symmetric and antisymmetric representations of OSP(1; 4). We present the general features of OSP( m; 2 n) and, in particular, its irreducible representations. The absence of OSP(1; 4) symmetry from the ground state indicates that one of the scalar fields, which is an element of the symmetric multiplet, has a nonvanishing vacuum expectation value. A shift in the fields reveals the physical spectrum of our Lagrangian. Two Goldstone fields are present, a vector and a spinor, corresponding to the breakdown of OSP(1; 4) to the Lorentzian group. The full Lagrangian contains a graviton, a massive spin- 3 2 field, and two massive scalar fields. The generalization to OSP(2; 4) is immediate.

Gauge theory of gravity and supergravity on a group manifold

The natural arena for the physics of gravity, supergravity and their enlargements appears to be the group manifold of the Poincare group P, the graded Poincare group GP of supersymmetry, and the corresponding enlargements. The dynamics of these theories correspond to geometrical algorithms in P and GP. Differential geometry on Lie groups is reviewed and results applied to P and GP. Curvature, gauge transformations and factorization are introduced. Also reviewed is the general coordinate transformation group and a hybrid gauge transformation, the anholonomized G.C.T. gauge. A study is made of the construction of an action, including the introduction of a set of special 2 forms, the ''pseudo curvatures.'' The possibilities of factorization in supersymmetry are analyzed. The version of supergravity is present which has now become a completely geometrical theory.

Indefinite-Metric Quantum Field Theory of General Relativity

Quantum field theory of Einstein's general relativity is formulated in the indefinite-metric Hilbert space in such a way that asymptotic fields are manifestly Lorentz covariant and the physical S-matrix is unitary. The general coordinate transformation is transcribed into a q-number transformation, called the BRS transformation. Its abstract definition is presented on the basis of the BRS transformation for the Yang-Mills theory. The BRS transformation for general relativity is then explicitly constructed. The gauge-fixing Lagrangian density and the Faddeev-Popov one are introduced in such a way that their sum behaves like a scalar density under the BRS transformation. One can then proceed in the same way as in the Kugo-Ojima formalism of the Yang-Mills theory to establish the unitarity of the physical S-matrix.

Conformal extensions of the Galilei group and their relation to the Schrödinger group

Various authors have considered a conformal extension CG0 of the Galilei group which in some sense is the nonrelativistic limit of the conformal extension of the Poincaré group, and have also established an invariance group for the free-particle Schrödinger equation, the ’’Schrödinger group.’’ Here we establish the most general conformal extension CG of the Galilei group, which is found to be identical to the group of the most general coordinate transformations that permit the use of noninertial frames of reference and of curvilinear coordinates in Galilei-invariant theories, which was considered by one of us some time ago, and is a gauge group containing a number of arbitrary functions. Both CG0 and the Schrödinger group are subgroups of CG containing the Galilei group, but otherwise they do not overlap. The Hamilton–Jacobi and Schrödinger equations for particles which are free or interact via inverse-square potentials are shown to be invariant under the Schrödinger group, and a further invariance of the Hamilton–Jacobi equation is established.

Massive supergravity from non-linear realization of orthosymplectic gauge symmetry and coupling to [formula omitted] multiplet

We construct a Lagrangian density manifestly invariant under OSP(1;4) gauge symmetry and general coordinate transformations, and broken spontaneously to the Lorentz symmetry. The non-linear transformations of the Goldstone fields are obtained by using a constrained linear representation that contains these fields. The resulting Lagrangian is the usual massive supergravity Lagrangian when written in the unitary gauge. We prove explicitly that the supersymmetric Higgs effect is occurring. Finally, we present a gauge-invariant matter Lagrangian for a massive spin- 1 2 — spin-1 multiplet.

Covariant formulation of non-equilibrium statistical thermodynamics

The Fokker Planck equation is considered as the master equation of macroscopic fluctuation theories. The transformation properties of this equation and quantities related to it under general coordinate transformations in phase space are studied. It is argued that all relations expressing physical properties should be manifestly covariant, i.e. independent of the special system of coordinates used. The covariance of the Langevin-equations and the Fokker Planck equation is demonstrated. The diffusion matrix of the Fokker Planck equation is used as a contravariant metric tensor in phase space. Covariant drift vectors associated to the Langevin- and the Fokker Planck equation are found. It is shown that special coordinates exist in which the covariant drift vector of the Fokker Planck equation and the usual non-covariant drift vector are equal. The physical property of detailed balance and the equivalent potential conditions are given in covariant form. Finally, the covariant formulation is used to study how macroscopic forces couple to a system in a non-equilibrium steady state. A general fluctuation-dissipation theorem for the linear response to such forces is obtained.

Gravitational interaction of hadrons: Band-spinor representations of GL(n,R)

We demonstrate the existence of double-valued linear (infinite) spinorial representations of the group of general coordinate transformations. We discuss the topology of the group of general coordinate transformations and its subgroups GA(nR), GL(n,R), SL(nr) for n = 2,3,4, and the existence of a double covering. We present the construction of band-spinor representations of GL(n,R) in terms of Harish-Chandra modules.It is suggested that hadrons interact with gravitation as band-spinors of that type. In the metric-affine extension of general relativity, the hadron intrinsic hypermomentum is minimally coupled to the connection, in addition to the coupling of the energy momentum tensor to the vierbeins. The relativistic conservation of intrinsic hypermomentum fits the observed regularities of hadrons: SU(6) ( approximately spin independence), scaling, and complex-J trajectories. The latter correspond to volume-preserving deformations (confinement?) exciting rotational bands.

Unified Geometric Theory of Gravity and Supergravity

A unified geometric formulation of gravitation and supergravity is presented. The action for these theories is constructed out of the components of the curvature tensor for bundle spaces with four-dimensional Lorentz base manifold and structure groups Sp(4) for gravity and OSp(1,4) for supergravity. The requirement of invariance under reflections, local Lorentz transformations, and general coordinate transformations uniquely determines the action and ensures the existence of local supersymmetry in supergravity.