The metric in a static cylindrical elastic medium and in an empty rotating frame as solutions of Einstein's field equations

Using the Weyl-type canonical coordinates, an integration of Einstein's field equations in the cylindrosymmetric case considered by Kurunolu is reexamined. It is made clear that the resulting metric is not describing the spacetime in a rotating frame, but in astatic cylindrical elastic medium. The conclusion of Kurunolu that for an observer on a rotating disk there is no way of escape from a curved spacetime is therefore not valid. The metric in an empty rotating frame is found as a solution of Einstein's field equations, and is not orthogonal. It is shown that the corresponding orthogonal solution represents spacetime in an inertial frame expressed in cylindrical coordinates. Introducing a noncoordinate basis, the metric in a rotating frame is given the static form of Kurunolu's solution. The essential role played by the nonvanishing structure coefficients in this case is made clear.     

On the hyperbolicity of Einstein's and other gauge field equations

It is shown that Einstein's vacuum field equations (respectively the conformal vacuum field equations) in a frame formalism imply a symmetric hyperbolic system of reduced propagation equations for any choice of coordinate system and frame field (and conformal factor). Certain freely specifiable gauge source functions occurring in the reduced equations reflect the choice of gauge. Together with the initial data they determine the gauge uniquely. Their choice does not affect the isometry class (conformal class) of a solution of an initial value problem. By the same method symmetric hyperbolic propagation equations are obtained from other gauge field equations, irrespective of the gauge. Using the concept of source functions one finds that Einstein's field equation, considered as second order equations for the metric coefficients, are of wave equation type in any coordinate system.Work supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft     

Post-Newtonian approximation for an accelerated,rotating frame of reference

The field equations of general relativity are solved to post-Newtonian order for a frame of reference having an arbitrary time-dependent, translational acceleration and an arbitrary time-dependent angular velocity. The derivation is based on a new 3+1 decomposition of the Einstein field equations and geodesic equation of motion. The resulting space-time metric and equation of motion contain gravitational terms, inertial terms, and coupled gravitational-inertial terms. These effects are expressed explicitly in terms of the Newtonian potential and standard post-Newtonian scalar and vector potentials. The physical meaning of the formulas derived is illustrated by application to a system of point-like gravitating masses. These results should be useful for the investigation of general relativistic effects in the analysis of real experimental measurements made with respect to a noninertial frame of reference, such as the surface of the rotating earth or an accelerated spacecraft.     

A Metric for an Evans-Vigier Field

In this paper we present an example of a specific metric which geometrizes explicitly a light-like four-vector potential field (Evans-Vigier field). We define the concepts of semilocal and complete geometrization and show that a light-like vector field has the same geometrical structure as a gravitational Kerr field. With this background in mind we discuss a theoretical proposition that a rotating body generates, besides a special gravitational field, a magnetic-type gauge field which might be identified with a geometrized Evans-Vigier field. We finally present a discussion which inform us that a classical Evans-Vigier field represents a novel type of field because we cannot identify it with any of the known electromagnetic fields.     

Heuristic viewpoint concerning the thermal ambience relative to an accelerated frame

A linear wave field in its Minkowski ground state is analyzed heuristically by two observers, one inertial, the other accelerating in a linear and uniform way. Relative to the accelerated observer, the zero-point oscillations of each Minkowski plane wave mode have an unusual Fourier spectrum. Its intensity is (i) the same for all plane wave modes, (ii) isotropic, and (iii) that of a thermal ambience (relative to the accelerated frame). The temperature of this ambience is given by the Davies-Unruh formula kT=(/2)(g/c). The existence of this ambience is incompatible with the requirement that the laws of nature be formulated in terms of tensors.     

Quantum fog and the degradation of information by the gravitational field

In this paper we discuss how information transferred optically through a gravitational field is degraded as the quanta interact with the madium (vacuum state). We quantify information by means of Shannon's entropy, and consider information carriers that are quanta of some field. Next, we obtain the quantum noise (quantum fog) produced by the gravitational field and derive the appropriate channel capacity formula, which quantifies the maximum amount of information that can be transmitted per pulse, in the face of this noise. We show that the channel capacity formula vanishes if the source of information is a space-time singularity because a very intense noise is produced in the vicinity of the singularity. In other words, space-time singularities are hidden behind a very intense quantum fog and cannot be optically observed. A second consequence is that information is degraded as anisotropies (lumpiness) develop in the universe.     

Use of Lorentz Transformations for Noninertial Frames of Reference

By analogy with the calculation of the path of a mass point in terms of the integral of the point velocity with respect to time, such that the point has a constant velocity V(t _i) within a time interval dt _i, then changes this velocity stepwise by V(t _i+1), moves with this velocity within a time interval dt _i+1, etc., an accelerated motion of an observer with a clock is represented by alternating states of rest in a sequence of inertial frames of reference and instantaneous jumps from one frame of reference into another. Lorentz transformations are used to calculate the readings of a resting clock observed from a noninertial frame of reference represented in this manner, during the rest of a noninertial observer in a next-in-turn inertial frame of reference belonging to the mentioned sequence, and upon a jump. For the observation from a noninertial frame of reference, the relation of the time interval counted by the resting clock to the time interval counted by the accelerated clock and to the acceleration has been obtained.     

The physical interpretation of a painlevé transcendent spacetime

We construct a nondiagonalizable solution of the stationary axially symmetric vacuum Einstein equations involving Painlevé transcendents III(V). From the asymptotic behaviour of this solution we identify the corresponding Newtonian potential as that of a modulated line mass at=0, and we identify our transcendental solution as corresponding to a very special differentially rotating modulated line source.     

Inertial reference frames in Einstein's theory of gravitation

A physical definition of the inertial reference frame (IRF) is given, and the properties of solutions of the Einstein equation (with cosmological constant), which admit an IRF (IRF solutions) are investigated. Their Petrov type is uniquely determined by the viscous stress tensor. Only the typesI, D or 0 are possible. The unique vacuum IRF solution is the Minkowski space-time. The unique IRF solution belonging to a perfect fluid is the Einstein universe. is of special importance. For=0, the only physically admissible IRF solution is the Minkowski space-time. For0, only interior solutions with strong restrictions for density and pressure are possible.     

Should There Be a Spin-Rotation Coupling for a Dirac Particle?

It was argued by Mashhoon that a spin-rotation coupling term should add to the Hamiltonian operator in a rotating frame, as compared with the one in an inertial frame. For a Dirac particle, the Hamiltonian and energy operators H and E in a given reference frame were recently proved to depend on the tetrad field. We argue that this non-uniqueness of H and E really is a physical problem. We show that a tetrad field contains two informations about local rotation, which usually do not coincide. We compute the energy operator in the inertial and the rotating frame, using three different tetrad fields. We find that Mashhoon’s term is there if the spatial triad rotates as does the reference frame—but then it is also there in the energy operator for the inertial frame. In fact, if one uses the same given tetrad field, the Dirac Hamiltonian operators in two reference frames in relative rotation differ only by the angular momentum term. If the Mashhoon effect is to exist for a Dirac particle, the tetrad field must be selected in a specific way for each reference frame.     

Some Logical Implications of the Two Einstein Assumptions Used in His Analysis of Geometry on a Relativistically Rotating Disc

We show that with the help of the two well known Einstein assumptions used by him in the treatment of geometry on relativistically rotating disc, and elementary logical analysis only, two different and opposite models of rotating discs can be constructed: In one of them the disc, set rotating from the stationary state, changes neither its radius nor circumference, as measured from stationary inertial frame; while in the other both the radius and the circumference contract by the same Lorentz factor. We also show that one can construct discs whose contraction factors vary continuously from unity (no contraction) to the full Lorentz factor.     

Correction to Einstein's perihelion precession formula from a traceless,anisotropic vacuum energy

Quantum field theory in curved space-time implies that the strong equivalence principle is violated outside a spherically symmetric, static star. Here we assume that quantum gravity effects restore the strong equivalence principle. Together with the assumption that the effective vacuum polarization energy-momentum tensor is traceless, this leads to a specific algebraic form of the energy-momentum tensor for which an exact solution of Einstein's field equations is found. The solution gives the post-Newtonian parameters=1 and=1+3, where is a dimensionless constant which determines the energy density of the anisotropic vacuum. The vacuum energy changes the perihelion precession by a factor of 1-.     

Equations of electrodynamics in a rotating solid dielectric

The equations of electrodynamics in a rotating isotropic homogeneous dielectric are obtained in a covariant form in coordinates of a reference frame that accompanies the rotation of the dielectric. It is found from these equations, which have variable coefficients, that the medium of the rotating dielectric is anisotropic and inhomogeneous. To derive tensors of the electromagnetic field in a rotating reference frame (RRF), the fields and inductions of a virtual inertial reference frame (IRF) that instantaneously accompanies the motion of one of the points of the dielectric are used twice. Initially, using instantaneous local relations, they are expressed in terms of real fields and inductions of the rotating medium, and then they are transformed into fields and inductions of a stationary IRF, in which they are used as components of the tensors of the electromagnetic field. Thus, the electromagnetic field tensors in the IRF are determined taking into account a priori unknown real inhomogeneous permittivity \(\bar \varepsilon \) and permeability \(\bar \varepsilon \) of the rotating medium. At the final stage, the tensors in the RRF are obtained by transformation rules for covariant and contravariant tensor components in accordance with known analytical relationships of fixed and rotating coordinates. The properties of modes of a rotating ring resonator in the form of homogeneous TE waves that travel along and against the direction of rotation and, in particular, their normal frequencies are examined. The contribution of inhomogeneous properties of the medium of a rotating dielectric to the difference between the normal frequencies of the counterpropagating waves (to the Sagnac effect) is determined. In a solid material with known elastic and striction characteristics, its density and dielectric permittivity depend on the radial coordinate. These dependences are caused by the action of the centrifugal force and changes in the polarization and magnetization of the medium because of the rotational motion of charged particles. The coordinate dependences of permittivity \(\bar \varepsilon \) and permeability \(\bar \varepsilon \) make additional contributions to the inhomogeneous properties of the medium of the rotating dielectric and to the Sagnac effect.     

De Broglie waves and the nature of mass

In this paper an attempt is made to interpret inertial mass as a consequence of the invariant periodicity associated with physical de Broglie waves. In the case of a free particle, such waves, observed from an arbitrary reference frame, would exhibit the velocity-dependent wavelength given by de Broglie's relation; and it is conjectured that the inertial and additive properties of mass (or, more precisely, the conservation of momentum and energy) can be related to nonlinear interference effects occurring between the de Broglie waves for different particles. This picture could throw light on the physical meaning of quantization and suggests the possibility of reformulating classical and quantum mechanics in terms of a quasi-classical nonlinear field theory in which both inertial and quantization effects result essentially from the periodicity of de Broglie waves.     

The York map as a Shanmugadhasan canonical transformation in tetrad gravity and the role of non-inertial frames in the geometrical view of the gravitational field

A new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity. This allows to identify the two pairs of physical tidal degrees of freedom (the Dirac observables of the gravitational field have to be built in term of them) and 14 gauge variables. These gauge quantities, whose role in describing generalized inertial effects is clarified, are all configurational except one, the York time, i.e. the trace of the extrinsic curvature of the instantaneous 3-spaces (corresponding to a clock synchronization convention) of a non-inertial frame centered on an arbitrary observer. In the Dirac Hamiltonian is the sum of the weak ADM energy (whose density is coordinate-dependent, containing the inertial potentials) and of the first-class constraints. The main results of the paper, deriving from a coherent use of constraint theory, are: (i) The explicit form of the Hamilton equations for the two tidal degrees of freedom of the gravitational field in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. The simplest such gauge is the 3-orthogonal one, but other gauges are discussed and the Hamiltonian interpretation of the harmonic gauges is given. This frame-dependence derives from the geometrical view of the gravitational field and is lost when the theory is reduced to a linear spin 2 field on a background space-time. (ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: (a) the explicit form of the solution of the super-momentum constraint and then of the Dirac Hamiltonian; (b) the determination of the shift functions and then of the lapse one. (iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable , determining the convention of clock synchronization. The associated relativistic inertial effects, absent in Newtonian gravity and implying inertial forces changing from attractive to repulsive in regions with different sign of , are completely unexplored and may have astrophysical relevance in the interpretation of the dark side of the universe.     

Extended inflation in higher-dimensional spacetime with a Brans-Dicke scalar field

Starting from an assumption of homogeneity of matter-energy tensor and Brans-Dicke (BD) scalar field we obtain a Robertson-Walker type of metric form in five-dimensional spacetime with the essential difference that our model is spatially inhomogeneous. The model exhibits an interesting feature in that as we approach the centre of symmetry the compact dimension becomes very large, with the implication that the Kaluza-Klein excitations become very light when located there and that the origin may represent a singular concentration of matter with motion in the extra dimension. Following Wesson the effective 4D properties of matter from the 5D vacuum solutions are also briefly discussed. Assuming particular functional relationships between and as also between the scale factor and scalar field, we obtain exact solutions which may be of relevance to the early universe and its extended inflation in the BD type of theory. We also discuss very briefly rollover time immediately after tunneling to the true vacuum state to explore if dimensionality has any marked influence on the situation.     

Explicit inertial range renormalization theory in a model for turbulent diffusion

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter, which measures the strength of the infrared divergence: for<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.     

The physical basis of astronomical aberration

The mechanism of stellar aberration was explained and formulated by Bradley in terms of the existence of a unique reference frame for light propagation. However, Einstein's denial of the existence of such a frame appears to undermine Bradley's interpretation of the phenomenon. It is suggested that the recent evidence for a cosmologically-based inertial reference frame provides a new physical basis for Bradley's explanation in a manner consistent with the requirements of special relativity. It is shown that a delay effect is also involved in our observation of the direction of nearby sources, and that the aberration and delay effects exactly compensate for nearby sources comoving with a terrestrial observer.     

Modulation of the vacuum field in the beam splitter and mirror system

We present modulation of the vacuum field with a single mirror placed close to an unused port of a beam splitter. The electromagnetic field modes of the vacuum fluctuation near the mirror are modified with respect to those in free space, with their amplitudes having a sinusoidal spatial dependence upon the distance from the mirror. If we combine this spatially amplitude-modulated vacuum field mode and a coherent local oscillator with a beam splitter, we may obtain squeezed states of light in a homodyne detection scheme. We show that the amplitude fluctuation of the combined light can be reduced by a factor of below that of the coherent light and the fluctuations of the intensity difference of the two beams can be also reduced.     

Quantum relativistic action at a distance

A well-known relativistic action at a distance interaction of two unequal masses is altered so as to yield purely Newtonian radial forces with fixed particle rest masses in the system center-of-momentum inertial frame. Although particle masses experience no kinematic mass increase in this frame, speeds are naturally restricted to less than the speed of light. We derive a relation between the center-of-momentum frame total Newtonian energy and the composite rest mass. In a new proper time quantum formalism, we obtain an L²(R⁴ R⁴, C) Hilbert space by varying individual particle rest masses. We propose the use of density operators, recognizing that the auxiliary proper time parameter is not an observable. The quantum formalism is applied to our altered version of the relativistic harmonic oscillator. Our generalized coherent states yield four-dimensional wave packets which follow the correct classical world lines. Appendices contain reviews of classical Hamiltonian reparametrization (incorporating our notion of manifest covariance), and a comparison of this work with the literature.