Letter: Slowly Rotating, Compact Fluid Sources Embedded in Kerr Empty Space-Time

Spherically symmetric static fluid sources are endowed with rotation and embedded in Kerr empty space-time up to and including quadratic terms in an angular velocity parameter using Darmois junction conditions. The boundary behaviour of the metric tensor and partial derivatives is used to develop a series solution of Einstein's equation's for the rotating fluid. The boundary of the rotating source is expressed explicitly in terms of sinusoidal functions of the polar angle. As an example of the analysis the Schwarzschild interior solution is endowed with rotation and the equation of the fluid boundary is generated together with surface behaviour of the fluid density and angular velocity.     

A rotating three component perfect fluid source and its junction with empty space-time

The Kerr solution for empty space-time is presented in an ellipsoidally symmetric coordinate system and it is used to produce generalised ellipsoidal metrics appropriate for the generation of rotating interior solutions of Einstein’s equations. It is shown that these solutions are the familiar static perfect fluid cases commonly derived in curvature coordinates but now endowed with rotation. These are also shown to be potential fluid sources for not only Kerr but also Kerr-de Sitter empty space-time. The approach is further discussed in the context of T-solutions of Einstein’s equations and the vacuum T-solution outside a rotating source is presented. The interior source for these solutions is shown not to be a perfect fluid but rather an anisotropic three component perfect fluid for which the energy momentum tensor is derived. The Schwarzschild interior solution is given as an example of the approach.     

Slowly, Rotating Non-Stationary, Fluid Solutions of Einstein's Equations and Their Match to Kerr Empty Space-Time

A general class of solutions of Einstein's equation for a slowly rotating fluid source, with supporting internal pressure, is matched using Lichnerowicz junction conditions, to the Kerr metric up to and including first order terms in angular speed parameter. It is shown that the match applies to any previously known non-rotating fluid source made to rotate slowly for which a zero pressure boundary surface exists. The method is applied to the dust source of Robertson-Walker and in outline to an interior solution due to McVittie describing gravitational collapse. The applicability of the method to additional examples is transparent. The differential angular velocity of the rotating systems is determined and theinduced rotation of local inertial frame is exhibited.     

Rotation and twist regular modes for trapped ghosts

A parameter-independent notion of stationary slow motion is formulated then applied to the case of stationary rotation of massless trapped ghosts. The excitations correspond to a rotation mode with angular momentum J ≠ 0 and twist modes. It is found that the rotation mode, which has no parity, causes excess in the angular velocity of dragged distant coordinate frames in one sheet of the wormhole while in the other sheet the angular velocity of the ghosts is that of rotating stars: 2J/r ³. As to the twist modes, which all have parity, they cause excess in the angular velocity of one of the throat’s poles with respect to the other.     

Acoustic streaming in a rotating fluid

Acoustic streaming theory is derived that is applicable to a fluid that is slow moving in a reference frame that rotates with a constant angular velocity omega. A simplified streaming equation is obtained for the special case in which the acoustic angular frequency omega is large relative to omega, and the change in fluid density due to rotation alone is negligible. For this special case it is shown that the "driving force" for the acoustic streaming is independent of omega. Thus, if no acoustic streaming is present in a fluid system that is stationary, then no steady-state acoustic streaming is predicted for a similar system that rotates with constant angular velocity. For a system in which acoustic streaming is present, the flow behavior depends on the relative magnitudes of the Coriolis forces and the viscous forces. If the Ekman number is large (that is, the viscous force dominates) then the predicted flow is identical to that which would exist in a stationary system. If, on the other hand, the Ekman number is small then the Coriolis force dominates and the component of flow in the direction of the axis of rotation can be much smaller in the rotating system than in a similar system at rest.     

Relativistic vortex dynamics in axisymmetric stationary perfect fluid configuration

Relativistic formulation of Helmholtz’s vorticity transport equation is presented on the basis of Maxwell-like version of Euler’s equation of motion. Entangled characteristics associated with vorticity flux conservation in a vortex tube and in a stream tube are displayed on basis of Greenberg’s theory of spacelike congruence of vortex lines and \(1+1+(2)\) decomposition of the gradient of fluid’s 4-velocity. Vorticity flux surfaces are surfaces of revolution about the rotation axis and are rotating with fluid’s angular velocity due to gravitational isorotation in a stationary axisymmetric perfect fluid configuration. Fluid’s angular velocity, angular momentum per baryon, injection energy, and invariant rotational potential are constant on such vorticity flux surfaces. Gravitation causes distortion of coaxial cylindrical vorticity flux surfaces in the limit of post-Newtonian approximation. The rotation of the fluid with angular velocity relative to vorticity flux surfaces generates swirl which causes the stretching of material vortex lines being wrapped on vorticity flux surfaces. Fluid helicity which is conserved in the fluid’s rest frame does not remain conserved in a locally nonrotating frame because of the existence of swirl. Vortex lines are twist free in the absence of meridional circulations, but the twisting of spacetime due to dragging effect leads to the increase in vorticity flux in a vortex tube.     

Petrov Types of Slowly Rotating Fluid Balls

Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various special Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. It is shown that Petrov type II solutions must reduce to the de Sitter spacetime in the static limit. Two space-times with a physically satisfactory energy-momentum tensor are investigated in detail. For the rotating incompressible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tolman IV space-time cannot be Petrov type D.     

Slowly rotating universes with radiating perfect fluid distribution coupled with a scalar field in general relativity

Along with the presentation of some interesting new analytic solutions, the dynamics of slowly rotating radiating perfect fluid universes coupled with a scalar field are investigated, and their physical and geometrical properties are studied from various angles. The rotational perturbations of such models are examined in detail in order to substantiate the possibility that the universe is endowed with some rotation. The nature and role of the metric rotation which is related to the local dragging of inertial frames and that of the matter rotation are studied. The effects of the radiation and the scalar fields on the rotation are discussed. The periods of physical validity for some of the models and the restrictions on the radii of the models for real astrophysical situations are found. Most of the rotating models obtained here turn out to be expanding ones as well, and may be taken as good examples of real astrophysical objects in this universe.     

Nonlinear evolution of the r-modes in neutron stars

The evolution of a neutron-star r-mode driven unstable by gravitational radiation is studied here using numerical solutions of the full nonlinear fluid equations. The dimensionless amplitude of the mode grows to order unity before strong shocks develop which quickly damp the mode. In this simulation the star loses about 40% of its initial angular momentum and 50% of its rotational kinetic energy before the mode is damped. The nonlinear evolution causes the fluid to develop strong differential rotation which is concentrated near the surface and poles of the star.     

An approximate global solution of Einstein’s equations for a rotating finite body

We obtain an approximate global stationary and axisymmetric solution of Einstein’s equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find second-order approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to check that Kerr’s metric cannot be the exterior part of that metric.     

Influence of Non-linear Radiation Heat Flux on Rotating Maxwell Fluid over a Deformable Surface: A Numerical Study

Mathematical model for Maxwell fluid flow in rotating frame induced by an isothermal stretching wall is explored numerically. Scale analysis based boundary layer approximations are applied to simplify the conservation relations which are later converted to similar forms via appropriate substitutions. A numerical approach is utilized to derive similarity solutions for broad range of Deborah number. The results predict that velocity distributions are inversely proportional to the stress relaxation time. This outcome is different from that observed for the elastic parameter of second grade fluid. Unlike non-rotating frame, the solution curves are oscillatory decaying functions of similarity variable. As angular velocity enlarges, temperature rises and significant drop in the heat transfer coefficient occurs. We note that the wall slope of temperature has an asymptotically decaying profile against the wall to ambient ratio parameter. From the qualitative view point, temperature ratio parameter and radiation parameter have similar effect on the thermal boundary layer. Furthermore, radiation parameter has a definite role in improving the cooling process of the stretching boundary.A comparative study of current numerical computations and those from the existing studies is also presented in a limiting case. To our knowledge, the phenomenon of non-linear radiation in rotating viscoelastic flow due to linearly stretched plate is just modeled here.     

Destroying superfluidity by rotating a Fermi gas at unitarity

We study the effect of the rotation on a harmonically trapped Fermi gas at zero temperature under the assumption that vortices are not formed. We show that at unitarity the rotation produces a phase separation between a nonrotating superfluid (S) core and a rigidly rotating normal (N) gas. The interface between the two phases is characterized by a density discontinuity n(N)/n(S)=0.85, independent of the angular velocity. The depletion of the superfluid and the angular momentum of the rotating configuration are calculated as a function of the angular velocity. The conditions of stability are also discussed and the critical angular velocity for the onset of a spontaneous quadrupole deformation of the interface is evaluated.     

The construction of exact solutions of the model problem on rotating fluid in domains with angular points

A new method has been developed for the construction of a two-dimensional model problem on the oscillations of rotating ideal fluid in some domains containing angular points. It is proved that these solutions correspond to the absolutely continuous component of the spectrum of the linear operator connected with the problem.     

The analysis of the flow of a micropolar fluid between two orthogonally moving porous disks with counter rotating directions

The present work investigates the unsteady, imcompressible flow of a micropolar fluid between two orthogonally moving porous coaxial disks. The lower and upper disks are rotating with the same angular speed in counter directions. The flows are driven by the contraction and the rotation of the disks. An extension of the Von Kármán type similarity transformation is proposed and is applied to reduce the governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. These differential equations with appropriate boundary conditions are responsible for the flow behavior between large but finite coaxial rotating disks. The analytical solutions are obtained by employing the homotopy analysis method. The effects of some various physical parameters like the expansion ratio, the rotational Reynolds number, the permeability Reynolds number, and micropolar parameters on the velocity fields are observed in graphs and discussed in detail.     

ROTATING RINDLER SPACE TIME WITH CONSTANT ANGULAR VELOCITY

A new space time metric is derived from Kerr metric if its mass and location approach to infinite in an appropriate way. The new space-time is an infinitesimal neighborhood nearby one of the two horizon poles of an infinite Kerr black hole. In other words, it is the second order infinitesimal neighborhood nearby one of the two horizon poles of a Kerr black hole. It is flat and has event horizon and infinite red shift surface. We prove that it is a rotating Rindler space time with constant angular velocity.     

Centrifugal instability in rotating shallow water and the problem of the spiral structure in galaxies

A new instability predicte by theory to occur in rotating shallow water in which the rotation velocity has a discontinuity, in a regime where the flow velocity exceeds the characteristics velocity of the waves, has been found experimentally. The instability develops when the radial gradient of the angular velocity across the discontinuity is negative; such an instability is likely to be responsible for the formation of the spiral structure in galaxies which have a similar rotational velocity profile.     

On the non-synchronous rotation of binary systems

During the evolution of the binary system,many physical processes occur,which can influence the orbital angular velocity and the spin angular velocities of the two components,and influence the non-synchronous or synchronous rotation of the system.These processes include the transfer of masses and angular momentums between the component stars,the loss of mass and angular momentum via stellar winds,and the deformation of the structure of component stars.A study of these processes indicates that they are closely related to the combined effects of tide and rotation.This means,to study the synchronous or non-synchronous rotation of binary systems,one has to consider the contributions of different physical processes simultaneously,instead of the tidal effect alone.A way to know whether the rotation of a binary system is synchronous or non-synchronous is to calculate the orbital angular velocity and the spin angular velocities of the component stars.If all of these angular velocities are equal,the rotation of the system is synchronous.If not,the rotation of the system is non-synchronous.For this aim,a series of equations are developed to calculate the orbital and spin angular velocities.The evolutionary calculation of a binary system with masses of 10M⊙+6M⊙shows that the transfer of masses and angular momentums between the two components,and the deformation of the components structure in the semidetached or in the contact phase can change the rotation of the system from synchronous into non-synchronous rotation.     

MAXWELL-Theorie (in Medien) und Quantentheorie in einem rotierenden Bezugssystem

On the basis of our 3-dimensional conceptions of the electromagnetic quantities [1] the MAXWELL theory is represented in a rotating frame of reference. For such a frame the MAXWELL equations obtain additional terms depending on the angular velocity (analogous to the CORIOLIS term etc. in NEWTON ian mechanics). Using these results with the help of the DIRAC equation we investigate quantum mechanics in a rotating frame of reference. Also in this case we obtain interesting additional terms depending on the angular velocity. The need for such a theory is obvious for many reasons: 1. Our Earth is such a rotating laboratory. Later the rotation effects must not be neglected. 2. In magnetic resonance physics theoretical questions with respect to rotating systems are important. 3. Interesting statements result for a rotational JOSEPHSON effect of superconductors. — All our calculations are carried out up to the first order in the angular velocity.     

A NUT-like solution with fluid matter

Stationary solutions of the Einstein equation are investigated when the source is a rigidly rotating fluid. Using the three-dimensional spin coefficient method the angular dependence of the metric tensor can be analytically calculated if the eigenray congruence is geodesic and shear free. The nonstatic solutions of this class do not describe physically realistic bodies, but, instead, bodies with NUT-type geometry.