Spherical field in rotating space in 5D

A geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory. A proposed phenomenological model predicts 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 planets.


I. INTRODUCTION
A five-dimensional model of the space-time was proposed by Nordstrom [1] and Kaluza [2] for unity of gravitation and electromagnetism. Klein [3] suggested a compactification mechanism, 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 more high-dimensional theories of supersymmetry, supergravity and string theory. They admit scenario, in which particle has a rest mass in 5D [4,5]. Exact solutions of Kaluza-Klein and low limit of bosonic string theories in 5D-6D [6] with toroidal compactification are equivalent. Analogous conclusion is made in [7] with comparison space-time-mass theory based on geometric properties of 5D space without compactification and braneworld model. Predictions of five-dimensional model of extended space and its experimental tests are considered in [8,9]. Cosmological model with motion of matter in fifth dimension also is examined [10]. Astrophysical applications of braneworld theories, including Arkani-Hamed-Dimopoulos-Dvali and Einstein-Maxwell models with large extra dimensions, are analyzed in [11]. EM model in 6D has become further development in [12], where linear 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 rotations of particles in (4+n)D space periodically returning in 4D is phenomenologically predicted in ADD model [14], where radius of rotation defines the size of additional dimension. Phenomena described by one-time physics in 3+1 dimensions appear as various "shadows" similar phenomena that occur in 4+2 dimensions with one extra 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 cylindric frame. It is studied also (3+2)D space-time with time-like additional dimension, where the motion is hyperbolic. A motion of the particle in certain domain 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 exhibits itself in 4D as action of central force. In Kaluza-Klein model this force divides into components, the part of which associates with to electromagnetic field.
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 efforts 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 from route of radial motion with respect to Sun. A radial Rindlerlike acceleration [23,24] in itself also can't explain the Pioneer effect because it is absent in planets motion [22].

II. GEODESICS IN ROTATING SPACE
Five-dimensional space-time having 4D spherical symmetry is considered in coordinate frame X i s = (τ, a, θ, ϕ, χ), where a, θ, ϕ, χ are spherical space coordinates and τ = ct, where c is light velocity constant and t is time. Rotating space-time with space-like fifth dimension [25] is described by metric where function f (χ) is continuously increasing and it is taken g = df /dχ. For the domain under review it is assumed where K is constant. Transition to five-dimensional cylindrical coordinates X i c = (τ, r, θ, ϕ, y) for 0 ≤ χ ≤ π is performed by transformation r = a sin f (χ), y = a cos f (χ). (2.3) Geodesic equations in 5D for particle having rest mass are where U i are components of five-velocity vector and Γ i kl are 5D Christoffel symbols of second kind. In spherical coordinates for metric (2.1) these equations, with take form (2.10) For particle with rest mass the solutions of these equations must correspond to given by metric condition From the second equation of the system we obtain where µ is ±1. We name corresponding solution a type I for µ = −1 and a type II for µ = 1. When particles move along geodesics, which are arcs of circle: equations of motion have following solutions: and where σ is 1, −1.
Change of passage of time is defined as relation between intervals of proper time T = dS and coordinate time τ . For geodesic of type I for solution (2.14) chosen σ = 1 we have Geodesic equations will be Components of five-velocity vector corresponding to coordinates r and y are found by differentiation of transformation (2.3) and will be V 1 = sin χU 1 + a cos χU 4 , V 4 = cos χU 1 − a sin χU 4 . (3.8) Condition given by metric (3.2) for the time-like path is In coordinate frame X c non-zero components of five-velocity vector corresponding to solutions of geodesic equations in coordinates X s (2.13)-(2.15) of types I and II is rewritten as For y = 0 (χ = π/2) they correspond with stationary in 4D particle. For motion in the neighborhood of point (τ 0 , r 0 , π/2, 0, 0) with conditions V 2 0 = V 3 0 = 0 geodesic equations are reduced to Condition (3.9) takes form For circular motion (3.10)-(3.13) Eq. (3.4) yields radial accelerations

IV. METRICS WITH TIME-LIKE FIFTH COORDINATE
A space-time having hyperbolic motion with coordinatesX i g = (τ ,ǎ,θ,φ,χ) is described by metric whereχ is assumed to be time-like andǨ is constant. This metric can be obtained from (2.1) for (2.2), (2.5) by substitution K = −iǨ, χ = π 2 − iχ and addition of (ˇ) in the notation of other coordinates. The geodesics equations for a particle motion along time-like path are For particle with rest mass the solutions of these equations must correspond to given by metric condition Second equation of the system yieldš For solution of type I time dilation is absent: and for type II increase of proper time passage is given by Transition to cylindrical coordinatesX i c = (τ ,ř,θ,φ,y) is realized by transformatioň r =ǎ coshχ,y =ǎ sinhχ. (4.13) Corresponding components of five-velocity vector arě (4.14) Inverse coordinate transformation is written aš Substituting this in (4.1) gives The same line element can be obtained by replacement K = −iǨ, y = iy and addition of (ˇ) in the notation of other coordinates in (3.2). Geodesics equations for motion of particle having rest mass is written as   Condition given by metric (4.16) for the time-like path is A non-zero components of five-velocity vectors corresponding hyperbolic solutions (4.9), (4.10) arě For motion in the neighborhood of point (τ 0 ,ř 0 , π/2, 0, 0) with conditionsV 2 0 =V 3 0 = 0 local solution is found from reduced Eqs. (4.17)-(4.21), which turned out to  and potentials where it is denoted a ε = r 2 − εy 2 . If 4D metric satisfies cylindrical conditions ∂gij ∂y = 0 electromagnetic field is defined by F ij = ∂ i A j − ∂ j A i . Ratio of electric charge to mass in 4D is written as with scalar function In more general case with 4D metrical coefficients being dependent on y the relationship between Q and q/m is not identical [4], but value Q = 0 also corresponds to q = 0. For considering space-time after substituting components of five-velocity vector (3.10) and (4.23) we obtain for the geodesics of the type I: This value is interpreted as neutral charge of a test particle. For the type II scalar function is The light trajectory is assumed to be isotropic curve both in 4D and in 5D: ds=dS=0. From

VI. 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: (2.14), (2.15) and (3.10)-(3.13) or (4.9), (4.10) and (4.23)- (4.26). It is suggested also that in cylindrical frame matter moves along fifth coordinate in single direction, which is opposite to antimatter motion.
In case of space-like fifth dimension function f from metric (2.1) is chosen so that its meaning is continuously increasing in intervals I + n = [2πn, π + 2πn] for integer n. Since value r < 0 is inadmissible in cylindric frame we must assume that function f has discontinuity on the endpoints of I n , which prescribes singularity. It can be avoid if model of binary world consisting of universe -anti-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 I − n = [−π + 2πn, 2πn] put into accordance with packet of 4D anti-universes. With condition (2.5) intervals I + n , I − n contain values of additional coordinate χ. 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 CPTsymmetry of the universe and anti-universe. Thus a birth of the pair particle-antiparticle is assumed to occur in points y = − + a 0 , r = 0, after which they move through opposite packets of branes and annihilate, when y = + − a 1 , r = 0.

A. Basic properties of Pioneer effect model
Recently much attention was attracted to the Pioneer effect, which consists in additional acceleration of spacecrafts Pioneer 10/11 [20,21,31,32] a p = (8.74 ± 1.33) × 10 −8 cm s −2 directed to the inner part of the solar system. We will analyze how much studying models of rotating space conform to this data. Motion of the spacecrafts and the planets is considered in the frame of the Sun [33].
For this analysis we must use geodesics of the first type because, as was shown in Sec. 5, for geodesics with constant radial coordinate they correspond to the neutrally charged particles. Their proper time coincides with coordinate time for trajectories, which are the arcs of circle (2.16) or hyperbola (4.11). Also motion of light is assumed to correspond with equation (5.8), 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 where G i is gravity force vector without part related to space rotation and left terms correspond to Eqs. (3.3)-(3.7) or (4.17)-(4.21). Denoting acceleration W i = dV i dS we divide it into W i = W i g + W i F , where W i F conforms in case G 2 = 0, θ = π/2 in the neighborhood of point (τ 0 , r 0 , π/2, 0, 0) to equations Accelerations W i g correspond to Eqs. (3.14)-(3.16) or (4.27)-(4.29). By using analogy with motion of particle in central gravity field in 4D, we take U 1 I and dU 1 I /dS for χ = π/2 in spherical coordinates orǓ 1 I and dǓ 1 I /dS forχ = 0 in hyperbolic coordinates as radial velocity and acceleration observed in 4D surface of five-dimensional space-time.

C. Additional acceleration of planets
In this section we will test proposed model by finding additional acceleration for planets of the solar system and comparing them with observations data. Further we will use following denotations: γ is gravity constant, M is the Sun's mass, Ω is its semimajor axis, e is eccentricity, n = γM/Ω 3 is unperturbed Keplerian mean motion, P = 2π/n is orbital period and ξ is eccentric anomaly.
Parameters of motion are given by equations r = Ω(1 − e cos ξ), nt = ξ − e sin ξ.  For closed to hyperbolic motion (4.9) non-vanishing five-velocities are written in form  With condition |V 1 | << 1, 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 I,II).

VII. CONCLUSION
Solutions of geodesics equations is found for rotating space in 5D with angular velocity, being inversely proportional to the square root of radius. They describe motion of particle having rest mass in a circle with space-like fifth coordinate in spherical (2.13-2.15) or cylindric (3.10)-(3.13) frames and hyperbolic motion with space-like fifth coordinate (4.9)-(4.10), (4.23)-(4.26). Time dilation is absence for solutions of the first type (2.16), (4.11), and in Kaluza-Klein model they corresponds to neutrally charged particle (5.6).
Proposed toy-model of particles motion 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-antiparticle in mirror worlds.
Analogy with motion in central gravity field in 4D is employed 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 cylindric frame for geodesics (6.7)-(6.9), (6.29)-(6.31) deviating from having constant radius. With space-like fifth coordinate a body in 4D space-time with appropriate radial velocity will have centripetal acceleration (6.21) being proportional to square of radius and directed towards 5D geodesic axis. That roughly conforms to underlying properties of the Pioneer-effect, namely, constant additional acceleration of apparatus