Central Field in Rotating Spherical Space in 5D

A t ime-like geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory and astrophysical applications are proposed. For example of such space-time in 4D spherical coordinates with the space-like fifth dimension a transition to the cylindrical frame is implemented. After this transformat ion a space is divided into two parts that is interpreted physically as pair of the universe and anti-universe. The rotation in 5D space-time exh ibits itself in 4D as action of extra force, which reveals itself with observed radial mot ion of the material particle. A lso it is considered similar (3+2)D space-time in which additional dimension is time-like and a motion of space is hyperbolic. For these space-times it is found geodesics with constant radius in 4D spherical coordinates and studied their small deviations in cylindrical frames. It is shown that model with space-like fifth coordinate conforms to the basic properties of the Pioneer-effect, namely, a) constant additional acceleration of apparatus on distance from 20 to 50 a.e., b) its increase from 5 to 20 a.e., c) observed absence of one in motion of p lanets.


Introduction
A five-dimensional model of the space-time was proposed by Nordstrom [1] and Kalu za [2] for unity of gravitation and electro magnetism. Klein [3] suggested a co mpactification mechanis m, owing to which internal space of the Planck size forms additional dimension. In his theory a motion of particle having rest mass in 4D can be described by equations of null geodesic line in 5D, which are interpretation of massless wave equation with some conditions.
In development of this model 5D space-time is considered as low energy limit of mo re h igh-dimensional theories of supersymmetry, supergravity and string theory. They admit scenario, in wh ich particle has a rest mass in 5D [4,5]. Exact solutions of Kaluza-Klein and low limit o f bosonic string theories in 5D-6D [6] with to ro idal co mpact ificat ion are equ iv alent . A nalogous conclus ion is made in [7] with co mparison space-t ime-mass theo ry based on geo met ric p ropert ies o f 5D sp ace withou t co mpact ificat ion and braneworld model. Predictions of five-d imensional model of extended space and its experimental tests are considered in [8,9]. Cos mo log ical model with mot ion o f matter in fifth dimension also is examined [10]. Astrophysical applications o f b ran ewo rld th eo r ies , in clu d in g A rkan i-Ha med -Dimopoulos-Dvali and Einstein-Maxwell models with large extra d imensions, are analyzed in [11]. EM model in 6D has b eco me fu rth er d ev elo p men t in [ 1 2] , wh ere lin ear perturbations sourced by matter on the brane are studied.
In [13] it is proposed low energy effective theory on a regularized brane in 6D gauged chiral supergravity. A possibility of the orbits of particles around the extra d imesions in (4+n)D space, periodically returning in 4D surface, is phenomenologically pred icted in ADD model [14]. Phenomena, described by one-time physics in 3+1 dimensions, appear as various "shadows" similar phenomena that occur in 4+2 dimensions with one ext ra space and one extra time dimensions (more generally, d+2) [15,16,17].
In present paper it is considered some geometrical construction in (4+1)D space-time with space-like fifth dimension and rotation in 4D spherical coordinates with transition thereupon to the standard cylindrical frame. It is studied also (3+2)D space-time with t ime-like addit ional dimension, where the motion is hyperbolic. A motion of the particle in certain do main of space in appropriate coordinates is assigned to be described with sufficient accuracy by geodesic equations. Their solutions lead to conclusion that rotation in 5D space-time exhib its itself in 4D as action of central force.
In astrophysical applications proposed model of space-time is of interest with respect to Pioneer effect. In some papers [18,19], see also review of effo rts to explain anomaly [20], presence of signal frequency bias is associated with dependence of fundamental physical parameters from time. Though it should allow for data [21], which witness independence of direction of additional acceleration fro m route of radial motion with respect to Sun. A radial Rindler-like acceleration [22,23] in itself also can't explain the Pioneer effect because it is absent in planets motion [24].
For particle with a rest mass the solutions of these equations must correspond to given by metric condition (11) Fro m the second equation of the system we obtain (12) where is . We name corresponding solutions as a solution of type I with and as a solution of type II with .
When particles move along geodesics, which are arcs of circle: (13) equations of motion have following solutions: (14) and (15) where is . Change of passage of time is defined as relation between intervals of proper time and coordinate time .
For geodesic of type I for solution (14) chosen we have (16) i.e. t ime d ilat ion is absence. With motion of particle along circular geodesic of type II (15) we obtain (17)

Representation in Cylindrical Frame
After substitutions of coordinate transformat ion being inverse to (3), namely, (18) metric (1) with (2) is rewritten as (19) Geodesic equations will be ) , , , ,    Substituting this in (37) gives (52) The same line element can be obtained by replacement and addition of (˘) in the notation of other coordinates in (19).
Geodesics equations for motion of part icle having rest mass are written as

Kaluza-Klein Model
In Kalu za-Klein theory the line element is brought in form Metrical coefficients and potentials are functions of and . Constant equals 1 for t ime-like fifth coordinate and -1, when it is space-like.
In this form metrics (19) and (52)  In more general case with 4D metrical coefficients being dependent on the relationship between and is not identical [4], but value also corresponds to .
For considering space-time after substituting components of the found five-velocity vectors of the type I (27)

Astrophysical Applications
Considering phenomenology of particles motion in 5D we assume that stationary in 3D space particles, having rest mass, move in spherical or hyperbolic frames in 5D along geodesics with constant radial coordinate: Eqs. (14), (15) and (27)- (30) or Eqs. (45), (46) and (59)-(62). It is suggested also that in cylindrical frame the matter moves along fifth coordinate in single direction, which is opposite to the antimatter mot ion.
In case of the space-like fifth dimension the function fro m metric (1) is chosen so that its meaning is continuously increasing in intervals . Since value is inad missible in cylindrical frame we must assume that function has discontinuity on the endpoints of , wh ich prescribes singularity. It can be avoided if model of binary world consisting of the universeanti-universe pair [26][27][28] is considered under the assumption that it possesses a large number of copies [29,30], in which a physical laws are identical. In bulk a space-time half put into accordance with packet of 4D anti-universes. With condition (5) the intervals , contain values of additional coordinate . Rotation of one particle with transition to cylindrical coordinates should be interpreted as motion of particle and anti-particle through opposite packets of branes, which conforms to CPT-symmet ry of the universe and anti-universe. Thus a birth of the pair particle-antipart icle is assumed to occur in points , after which they move through opposite packets of branes and annihilate, when .

Basic Properties of the Pi oneer Effect Model
Recently much attention was attracted to the Pioneer effect, which consists in additional accelerat ion of spacecrafts Pioneer 10/11 [20,24,31,32] cm s at dictances 15-45 AU directed to the Sun [33]. We will analy ze how much studying models of rotating space conform to this data. Motion of the spacecrafts and the planets will be considered in the frame of the Sun. For this analysis we must use geodesics of the type I because, as it was shown in Sec. 5, for geodesics with constant radial coordinate they correspond to the neutrally charged particles. Their proper t ime coincides with coordinate time for trajectories, wh ich are the arcs of circle (16) or hyperbola (47). Also motion of light is assumed to correspond with equation (75), i.e. a light shift along the fifth coordinate is absent.
In the Sun's gravity field motion of the particle with rest mass is described approximately by equations The accelerations correspond to Eqs. (31)-(33) or (63)-(65). By using analogy with motion of part icle in central grav ity field in 4D, we take and for in spherical coordinates or and for in hyperbolic coordinates as radial velocity and acceleration observed in 4D surface of five-d imensional space-time.

Model wi th Space-Like Fifth Coordi nate
In spherical coordinates in the neighborhood of point Eqs. (31)-(33) correspond to system

Additi onal Acceleration of Planets
In this section we will test proposed model by finding additional accelerat ion for planets of the solar system and comparing them with observations data. Further we will use following denotations: is gravity constant, is the Sun's mass, is its semimajor axis, is eccentricity, is unperturbed Keplerian mean motion, is orbital period and is eccentric anomaly. and anomalous accelerations of planets , obtained from observations are in Table 2 . Pred icted additional radial acceleration fo r Yup iter, Saturn, Uranus is with in the observation error and for asteroid Icarus it is close to upper limit of .

Model wi th Ti me -Like Fifth Coordi nate
In hyperbolic coordinates in the neighborhood of point This result doesn't conform to the Pioneer effect, so far as in accordance with this expression with increase of magnitude of radial velocity corresponding growth of acceleration will be positive.
For cylindrical coordinates in case in po int fro m Eq. (64) we obtain With condition , this equation conforms to unmodeled acceleration of Pioneer 10/11 on distance 20-50 a.e., but gives the same acceleration for Pioneer 11 on distance less than 20 a.e. and for planets of the Sun system that contradicts data of observations (Tables 1,2).

Conclusions
Proposed toy-model of space is based on idea of double manyfold Universe. It is supported by notion that closed geodesic of elementary particle having a rest mass corresponds to motion of the pair particle-antipart icle in the mirror worlds.
Found solutions of geodesics equations corresponding to the particle with a rest mass for cosiderig space-times describe mot ion in a circle with space-like fifth coordinate and hyperbolic motion with time-like fifth coordinate. Time dilation is absence for the solutions corresponded in the Kalu za-Klein model to neutrally charged particle. In this case the angular velocity is inversely proportional to the square root of radius in spherical coordinates. In astrophysical applications the center of the Sun has been chosen as the center of this motion.
Analogy with motion in central gravity field in 4D is emp loyed for determination of velocity and acceleration observed in 4D sheet for particles moving in 5D bulk. We obtain approximate solution in the neighborhood of surface with zero fifth coordinate in cy lindrical frame for geodesics deviating fro m having constant radius. With space-like fifth coordinate a body in included 4D space-time with limited velocity will have centripetal acceleration being proportional to square of radial velocity and directed towards center of rotaion of space in 5D. Th is acceleration is interpreted as action of additional force in 4D. That roughly corresponds to underlying properties of the Pioneer-effect, such as constant additional accelerat ion of apparatus towards the Sun on distance from 20 to 50 a.e., its increase from 5 to 20 a.e., observed absence of one in motion of planets. However, presence of analogous acceleration in mot ion of bodies with similar radial velosity, for examp le, asteroid Icarus, will be significant fo r confirmat ion of this model. In case of space-time with t ime-like fifth coordinate it isn't found geodesics corresponding to this effect.