Metric Tensor Research Articles

ON CERTAIN RIEMANN SPACE WITH METRICAL TENSOR WITH SEPARABLE COORDINATES AND ITS APPLICATIONS

ON CERTAIN RIEMANN SPACE WITH METRICAL TENSOR WITH SEPARABLE COORDINATES AND ITS APPLICATIONS

Phase Transition at the Metric Elastic Universe

At the last time the concept of the curved space-time as the some medium with stress tensor σαβon the right part of Einstein equation is extensively studied in the frame of the Sakharov - Wheeler metric elasticity(Sakharov (1967), Wheeler (1970)). The physical cosmology pre- dicts a different phase transitions (Linde (1990), Guth (1991)). In the frame of Relativistic Theory of Finite Deformations (RTFD) (Gusev (1986)) the transition from the initial stateof the Universe (Minkowskian's vacuum, quasi-vacuum(Gliner (1965), Zel'dovich (1968)) to the final stateof the Universe(Friedmann space, de Sitter space) has the form of phase transition(Gusev (1989) which is connected with different space-time symmetry of the initial and final states of Universe(from the point of view of isometric groupGnof space). In the RTFD (Gusev (1983), Gusev (1989)) the space-time is described by deformation tensorof the three-dimensional surfaces, and the Einstein's equations are viewed as the constitutive relations between the deformations ∊αβand stresses σαβ. The vacuum state of Universe have the visible zero physical characteristics and one is unsteady relatively quantum and topological deformations (Gunzig & Nardone (1989), Guth (1991)). Deformations of vacuum state, identifying with empty Mikowskian's space are described the deformations tensor ∊αβ, wherethe metrical tensor of deformation state of 3-geometry on the hypersurface, which is ortogonaled to the four-velocityis the 3 -geometry of initial state,is a projection tensor.

Scalar fields in a seven-dimensional manifold behaving as Lorentz-covariant spinor fields in space-time

A Lorentz-invariant model of vacuum is given in the form of a 7-dimensional manifold endowed with a statistical metrical tensor. Certain scalar fields on this manifold behave then as spinor fields when viewed from their space-time projection. This paper generalizes previous work fromSO(3)-covariance to Lorentz-covariance.

Natural origin of SO3-spinors in an eight-dimensional Kaluza theory exhibiting an additional U1�SU2 internal symmetry

The higher-dimensional spaceM4×S1×S3 (M4 = Minkowski space) is assumed. The symmetry group ofS3 isSO4 ≃SU2×SU2 “Gravitational waves” produce a symmetry breaking so thatSO3, rotations inM4 are linked with one of theSU2. Certain components of the metrical tensor ofM4×S1×S3, appear then as spinor fields.

On a coordinate-invariant description of Riemannian manifolds

A coordinate-invariant description of a Riemannian manifold is known to be furnished by the curvature tensor and a finite number of its covariant derivatives relative to a field of orthogonal frames. These tensors are closer to measurements than the metrical tensor is. The present article discusses this description's usefulness in general relativity and the redundancy among the curvature tensor and its derivatives.

Particle creation and vacuum polarisation in an isotropic universe

The particle interpretation of quantised fields in homogeneous isotropic space-time is given which is based on the diagonalisation of the Hamiltonian constructed via the metrical stress-energy tensor (SET). This interpretation and the regularisation procedure of Zeldovich and Starobinsky (1971) allows one to obtain the total renormalised vacuum expectation values of SET which include both vacuum polarisation and non-local terms describing the creation of particles. The case of the explicitly soluble cosmological model with initial singularity which is asymptotically static in the future is considered. The results of the calculation of the total SET in realistic Friedman cosmological models are presented.

Torsion contributions to the magnetic field of a spinning charge

In fact , even if torsion does no t di rect ly in terac t wi th the e lect romagnet ic field, there is a coupl ing between torsion and the v i r tua l pairs of fermions, produced according to the v a c u u m polar izat ion effect, which gives rise to the photontorsion in terac t ion (1) displayed in (1) (notice tha t we have assumed a flat metr ic tensor, since we wan t to eva lua te the specific torsion contr ibut ions , apar t f rom the usual R ieman iann terms (2)). We like to stress, in this paper , t ha t an electric charge, in te rac t ing wi th a cons tan t torsionic background, can produce a s tat ic magnet ic field even in its rest system. Consider in fact a poin t l ike charge distr ibut ion, ~ ( x ) ~ e~(x), placed in an ex te rna l torsionic background which we assume generated, according to the Einste inCaf tan theory, by a cons tant semi-classical spin d is t r ibut ion (a) at rest wi th the charge density. We have then Qkuk~0, where u k is the 4-veloci ty vector of the charged part icle.

Vacuum Stress‐Energy Tensor and Particle Creation in Isotropic Cosmological Models

AbstractQuantum theory of quantized scalar and spinor fields in homogeneous isotropic space‐time is considered. Particle interpretation of quantized field is given which is based on diagonalization of the Hamiltonian constructed by means of the metrical stress‐energy tensor (SET). This interpretation allows to define a normal ordering prescript which respects the conservation property of SET. With the help of Zeldovich‐Starobinsky regularization procedure [3] this allows to obtain the total renormalized vacuum expectation values of SET which include both nonlocal terms corresponding to the particle creation and local ones describing the vacuum polarization. Estimates for the total SET of scalar and spinor field on various stages of the evolution for realistic cosmological models are obtained. Also the effect of spontaneous breakdown of gauge symmetry for selfinteracting scalar field in a hyperbolic Friedman model is considered.

On the generally invariant lagrangians for the metric field and other tensor fields

The Krupka and Trautman method for the description of all generally invariant functions of the components of geometrical object fields is applied to the invariants of second degree of the metrical field and other tensor fields. The complete system of differential identities fulfilled by the invariants mentioned is found and it is proved that these invariants depend on the tensor quantities only.

Strong gravity and the Yukawa potential

A number of recent papers, seemingly based on very different premises (~-3), have ~uggested tha t the physical d~er ip t ions of ma t t e r on the cosmological level and in the ex t reme microrealm a re s tr ikingly similar. This curious s i tuat ion results from the ut i l izat ion of the formalism of general re la t iv i ty in the analysis of hadronie interactions. Thus the region of space-t ime in the vic ini ty of hadronie ma t t e r is viewed as being ,characterized locally by high curvature, in marked contras t to the t rad i t ional assumption tha t the various part icle interactions occur in the fiat Minkowski background. The coupling of geometry and mat te r effected in Einstein 's equations through the Newtonian gravi ta t ional constant produces, for typical par t ic le mass densities, radi i of curvature (for example) whioh are far too large to be meaningfully in terpre ted in terms of a particle concept. The afore-mentioned works, however, propose the unconventional hypothesis tha t in the region of the hadronie influence the re levant coupling pertains to the strong forces. Wi th this assumption a var ie ty of propert ies of the hadrons can be successfully deduced from the general-relat ivist ic field equations (~-3). One impact of the s t rong-gravi ty model is tha t the strong interact ions should be mediated by massive 2 + mesons (s), quanta of the metr ic tensor field (~) ],~, which satisfies the Einstein equation (~)

Scalar differential concomitants of the second order of the metrical tensor in three-dimensional Riemann spaces

Scalar differential concomitants of the second order of the metrical tensor in three-dimensional Riemann spaces

Théorie euclidienne linéaire du champ de gravitation et tests expérimentaux

The consequences of the energy-momentum conservation law (using the « eulerian metrical tensor ») in a phenomenological linear theory of gravitation are investigated. It is shown that it is possible to give account of the three classical tests : motion of the perihelia, deflection of light and redshift.

Identités et lois du mouvement déduites d’un lagrangien. Application à la définition de l’impulsion-énergie dans une théorie euclidienne de la gravitation

In a phenomenological linear theory of gravitation, following the lagrangian method, the formulation of the enery-momentum conservation law, by means of the « eulerian metrical tensor » (rather than the usual « canonical tensor ») leads to the same formal results that general relativity: principle of geodesics, and mass conservation. An expression for the « eulerian metrical tensor » is given.

Separation of Motions of Many-Body Systems into Dynamically Independent Parts by Projection onto Equilibrium Varieties in Phase Space. II

A condensed notation in phase space that facilitates calculations is developed. With the aid of this notation, the results are given a simple geometrical interpretation in phase space by introducing a certain canonically invariant metrical tensor O/sub ij/. This tensor does ot yield the usual orthogonal or pseudo-orthogonal metric, but rather, what is called a symplectic metric. One then sees that the projections that are made are orthogonal, in the symplectic sense, to the equilibrium varieties. Likewise, one can see quite generally, that the entire canonical formalism, including the Poisson brackets and the hamiltonian equations of motion, reduces to simple geometrical relations in phase space, the form of which is suggestive for possible (urther developments, especially with regard to the treatment in higher approximations. The ideas are applied to the electron gas, and the dynamics of the plasma are illustrated with the aid of a comparison with a simple two-dimensional model, possessing all the essential features described above. In this way, one can understand many of the basic features of the plasma motions, in terms of concepts such as the generalization of the notion of centrifugal force and Coriolis force to phase space. By going over to a local geodeticmore » frame in the equilibrium variety, one is led in a natural way to the concept of a set of quasiparticles for the plasma. If the number of collective oscillatory coordinates is s, then there will be 3N-s of these quasiparticle coordinates. The latter do not represent any of the actual original particles out of which the system is constituted, but rather, they represent effective pulse-like distributions of charge, which move together in a correlated way so as to resemble an actual particle in many respects. (auth)« less

Mikroinstabilitäten vom Spiegeltyp in inhomogenen Plasmen

The canonical momentum in the direction of the electrical current or an other ordered motion is a constant of motion in the case of mirror type perturbations of plasma configurations in which the current and the magnetic field are perpendicular to each other and in which all the macroscopic quantities depend only on one coordinate in a rectangular system of coordinates, of which the metrical tensor depends also on that coordinate only. This fact enables one, to discuss rather simply the onset of instabilities of that kind of perturbations within the frame of a microscopic collisionless theory. Several examples like plane and cylindrical z- and Θ-pinches and plane periodic configurations without space charges or without currents are treated in this way. A special result is, that at least a very wide class of the nonlinear electrostatic waves, found by BERNSTEIN, GREEN and KRUSKAL 1 are unstable.

Contribution to Stephenson-Kilmister’s unified theory of gravitation and elektromagnetism

A geometrical interpretation for Stephenson-Kilmister’s unified theory of gravitation and electromagnetism is given on the base of Finslerian geometry. It is known that the correct equations of motion for a charged test body moving in the Eiemannian space under the pon-deromotorical influence of the electromagnetic field can be deduced from a variationsl principle with the Lagrangian: ℱ(x, dx) = Aµ(x)dxµ + +gµv(x) dxµ dxv1/2, whereAµ(x) andgµv(x) are the components of the potential of the electromagnetic field and the components of the metrical ground tensor of the original Riemannian space, respectively. Considering ℱ(x, dx ) as the ground function of a Pinsler’s space a reformulation of Stephenson-Kilmister’s idea is suggested.

On a Set of Conform-Invariant Equations of the Gravitational Field

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

The Relativity of Size

A uniform change of size of all space-time intervals could not be detected, since the measuring instruments would be changed too. Therefore, a principle of similarity is proposed requiring that the equations of physics be invariant under such changes. This is shown to correspond to the principle that all units of interval are equivalent for expressing physical laws. It is pointed out that a given metrical tensor in the usual tensor calculus defines a specific unit of interval, so that the principle of similarity requires an enlargement of the tensor formalism and a corresponding change in the underlying geometry.

LXXXVII. Non-symmetric stress-energy-momentum tensor and spin-density

Abstract The question of the physical meaning of a non-symmetric stress-energy-momentum tensor is discussed. It is shown that an antisymmetric part of this tensor arises necessarily if there are particles with spin which must be considered as point particles. Some formulae connecting the anti-symmetric part of T ik and the spin density are derived. It is shown that the unified field theory of Einstein and Straus with non-symmetric metrical tensor does not lead to an explanation of geomagnetism.

On Continued Gravitational Contraction

When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.