aa r X i v : . [ phy s i c s . g e n - ph ] J a n Spherical field in rotating space in 5D
W. B. BelayevCenter for Relativity and Astrophysics,185 Box, 194358, Sanct-Petersburg, Russiae-mail: [email protected]
A geodesic motion in rotating 5D space is studied in framework of Kaluza-Klein theory. Aproposed phenomenological model predicts basic properties of the Pioneer-effect, namely, a) constantadditional 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 gravitationand electromagnetism. Klein [3] suggested a compactification mechanism, owing to which internal space of the Plancksize forms additional dimension. In his theory a motion of particle having rest mass in 4D can be described byequations 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 ofsupersymmetry, 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 compactificationare equivalent. Analogous conclusion is made in [7] with comparison space-time-mass theory based on geometricproperties of 5D space without compactification and braneworld model. Predictions of five-dimensional model ofextended space and its experimental tests are considered in [8, 9]. Cosmological model with motion of matter infifth 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 6Dhas 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 possibilityof 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 in3+1 dimensions appear as various ”shadows” similar phenomena that occur in 4+2 dimensions with one extra spaceand one extra time dimensions (more generally, d+2) [15–17].In present paper it is considered some geometrical construction in (4+1)D space-time with space-like fifth dimensionand 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 withsufficient accuracy by geodesic equations. Their solutions lead to conclusion that rotation in 5D space-time exhibitsitself in 4D as action of central force. In Kaluza-Klein model this force divides into components, the part of whichassociates with to electromagnetic field.In astrophysical applications proposed model of space-time is of interest with respect to Pioneer effect. In somepapers [18, 19], see also review of efforts to explain anomaly [20], presence of signal frequency bias is associatedwith dependence of fundamental physical parameters from time. Though it should allow for data [21], which witnessindependence of direction of additional acceleration from route of radial motion with respect to Sun. A radial Rindler-like 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 is = ( 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. Rotatingspace-time with space-like fifth dimension [25] is described by metric dS = [1 − a B ( a ) ] dτ − da − a [2 B ( a ) g ( χ ) dτ dχ + sin f ( χ )( dθ + sin θdϕ ) + g ( χ ) dχ ] , (2.1)where function f ( χ ) is continuously increasing and it is taken g = df /dχ . For the domain under review it is assumed B ( a ) = Ka − / , (2.2)where K is constant.Transition to five-dimensional cylindrical coordinates X ic = ( τ, r, θ, ϕ, y ) for 0 ≤ χ ≤ π is performed by transforma-tion r = a sin f ( χ ) , y = a cos f ( χ ) . (2.3)Geodesic equations in 5D for particle having rest mass are dU i dS + Γ ikl U k U l = 0 , (2.4)where U i are components of five-velocity vector and Γ ikl are 5D Christoffel symbols of second kind. In sphericalcoordinates for metric (2.1) these equations, with f ( χ ) = χ, (2.5)take form d τdS + K dτdS dadS + Ka / dadS dχdS − Ka / χ ) (cid:18) dθdS (cid:19) − Ka / χ ) sin ϕ (cid:18) dϕdS (cid:19) = 0 , (2.6) d adS − K (cid:18) dτdS (cid:19) − Ka / dτdS dχdS − a sin χ (cid:18) dθdS (cid:19) − a sin θ sin χ (cid:18) dϕdS (cid:19) − a (cid:18) dχdS (cid:19) = 0 , (2.7) d θdS + 2 a dadS dθdS + 2 cot χ dθdS dχdS − sin(2 θ )2 (cid:18) dϕdS (cid:19) = 0 , (2.8) d ϕdS + 2 a dadS dϕdS + 2 cot θ dθdS dϕdS + 2 cot χ dϕdS dχdS = 0 , (2.9) d χdS + K (3 − K a )2 a / dτdS dadS + 4 − K a a dadS dχdS − (1 − K a ) sin(2 χ )2 (cid:18) dθdS (cid:19) −− (1 − K a ) sin θ sin(2 χ )2 (cid:18) dϕdS (cid:19) = 0 . (2.10)For particle with rest mass the solutions of these equations must correspond to given by metric condition1 = (1 − K a ) U − Ka / U U − U − sin χa ( U + sin θU ) − a U . (2.11)From the second equation of the system we obtain U = KU a / ( − µ (cid:20) aK U (cid:18) dU dS − a sin χ ( U + sin θU ) (cid:19)(cid:21) / ) , (2.12)where µ is ±
1. We name corresponding solution a type I for µ = − µ = 1.When particles move along geodesics, which are arcs of circle: U = U = U = 0 , (2.13)equations of motion have following solutions: U I = σ, U I = − σKa / (2.14)and U II = 2 σ √ − K a , U II = − σKa / √ − K a , (2.15)where σ is 1 , − T = R dS and coordinate time τ .For geodesic of type I for solution (2.14) chosen σ = 1 we have dτ = dT, (2.16)i.e. time dilation is absence. With motion of particle along circular geodesic of type II (2.15) we obtain dT = 12 p − K adτ . (2.17) III. REPRESENTATION IN CYLINDRIC FRAME
After substitutions of coordinate transformation being inverse to (2.3), namely, a = p r + y , f ( χ ) = arccot yr (3.1)metric (2.1) with (2.2) is rewritten as dS = (1 − K p r + y ) dτ − K ( r + y ) − / dτ ( ydr − rdy ) − dr − r ( dθ + sin θdϕ ) − dy . (3.2)Geodesic equations will be d τdS + K r p r + y dτdS drdS + K y p r + y dτdS dydS + Kry r + y ) / (cid:18) drdS (cid:19) − K ( r − y )2( r + y ) / drdS dydS −− Kry ( r + y ) / (cid:18) dθdS (cid:19) − Kry ( r + y ) / sin ϕ (cid:18) dϕdS (cid:19) − Kry r + y ) / (cid:18) dydS (cid:19) = 0 , (3.3) d rdS − K r p r + y (cid:18) dτdS (cid:19) − K ry r + y ) / dτdS drdS − K y − K p r + y r + y ) / dτdS dydS −− K ry r + y ) / (cid:18) drdS (cid:19) + K y ( r − y )2( r + y ) / drdS dydS + r ( K y − p r + y ) p r + y (cid:18) dθdS (cid:19) ++ r ( K y − p r + y ) p r + y sin θ (cid:18) dϕdS (cid:19) + K ry r + y ) / (cid:18) dydS (cid:19) = 0 , (3.4) d θdS + 2 r drdS dθdS − sin(2 θ )2 (cid:18) dϕdS (cid:19) = 0 , (3.5) d ϕdS + 2 r drdS dϕdS + 2 cot θ dθdS dϕdS = 0 , (3.6) d ydS − K y p r + y (cid:18) dτdS (cid:19) + K r − K p r + y r + y ) / dτdS drdS + K ry r + y ) / dτdS dydS ++ K yr r + y ) / (cid:18) drdS (cid:19) − K r ( r − y )2( r + y ) / drdS dydS − K r y p r + y (cid:18) dθdS (cid:19) − K r y p r + y sin θ (cid:18) dϕdS (cid:19) −− K r y r + y ) / (cid:18) dydS (cid:19) = 0 . (3.7)Components of five-velocity vector corresponding to coordinates r and y are found by differentiation of transfor-mation (2.3) and will be V = sin χU + a cos χU , V = cos χU − a sin χU . (3.8)Condition given by metric (3.2) for the time-like path is1 = (1 − K p r + y ) V − K ( r + y ) − / V ( yV − rV ) − V − r ( V + sin θV ) − V . (3.9)In coordinate frame X c non-zero components of five-velocity vector corresponding to solutions of geodesic equationsin coordinates X s (2.13)-(2.15) of types I and II is rewritten as V I = σ, V I = − σKy ( r + y ) / , V I = σKr ( r + y ) / , (3.10) V II = 2 σ (4 − K p r + y ) / , (3.11) V II = − σKy ( r + y ) / (4 − K p r + y ) / , (3.12) V II = σKr ( r + y ) / (4 − K p r + y ) / . (3.13)For y = 0 ( χ = π/
2) they correspond with stationary in 4D particle.For motion in the neighborhood of point ( τ , r , π/ , ,
0) with conditions V = V = 0 geodesic equations arereduced to d τdS + K dτdS drdS − K r / drdS dydS = 0 , (3.14) d rdS − K (cid:18) dτdS (cid:19) + 3 K r / dτdS dydS = 0 , (3.15) d ydS + K r − K r / dτdS drdS − K drdS dydS = 0 . (3.16)Condition (3.9) takes form 1 = (1 − K r ) V − Kr / V V − V − V . (3.17)For circular motion (3.10)-(3.13) Eq. (3.4) yields radial accelerations dV I dS = − K , (3.18) dV II dS = − K − K p r + y . (3.19) IV. METRICS WITH TIME-LIKE FIFTH COORDINATE
A space-time having hyperbolic motion with coordinates ˇ X ig = (ˇ τ , ˇ a, ˇ θ, ˇ ϕ, ˇ χ ) is described by metric dS = (1 + ˇ K ˇ a ) d ˇ τ − d ˇ a + ˇ a [2 ˇ K ˇ a − / d ˇ τ d ˇ χ − cosh ˇ χ ( d ˇ θ + sin ˇ θd ˇ ϕ ) + d ˇ χ ] , (4.1)where ˇ χ is assumed to be time-like and ˇ K is constant. This metric can be obtained from (2.1) for (2.2), (2.5) bysubstitution K = − i ˇ K, χ = π − i ˇ χ and addition of (ˇ) in the notation of other coordinates.The geodesics equations for a particle motion along time-like path are d ˇ τdS − ˇ K d ˇ τdS d ˇ adS − ˇ K ˇ a / d ˇ adS d ˇ χdS − ˇ K ˇ a / χ ) (cid:18) d ˇ θdS (cid:19) − ˇ K ˇ a / χ ) sin ˇ ϕ (cid:18) d ˇ ϕdS (cid:19) = 0 , (4.2) d ˇ adS + ˇ K (cid:18) d ˇ τdS (cid:19) + 3 ˇ K ˇ a / d ˇ τdS d ˇ χdS − ˇ a cosh ˇ χ (cid:18) d ˇ θdS (cid:19) − ˇ a sin ˇ θ cosh ˇ χ (cid:18) d ˇ ϕdS (cid:19) + ˇ a (cid:18) d ˇ χdS (cid:19) = 0 , (4.3) d ˇ θdS + 2ˇ a d ˇ adS d ˇ θdS + 2 tanh ˇ χ d ˇ θdS d ˇ χdS − sin(2ˇ θ )2 (cid:18) d ˇ ϕdS (cid:19) = 0 , (4.4) d ˇ ϕdS + 2ˇ a d ˇ adS d ˇ ϕdS + 2 cot ˇ θ d ˇ θdS d ˇ χdS + 2 tanh ˇ χ d ˇ ϕdS d ˇ χdS = 0 , (4.5) d ˇ χdS + ˇ K (3 + ˇ K ˇ a )2ˇ a / d ˇ τdS d ˇ adS + 4 + ˇ K ˇ a a d ˇ adS d ˇ χdS + (1 + ˇ K ˇ a ) sinh(2 ˇ χ )2 (cid:18) d ˇ θdS (cid:19) ++ (1 + ˇ K ˇ a ) sin ˇ θ sinh(2 ˇ χ )2 (cid:18) d ˇ ϕdS (cid:19) = 0 . (4.6)For particle with rest mass the solutions of these equations must correspond to given by metric condition1 = (1 + ˇ K ˇ a ) ˇ U + 2 ˇ K ˇ a / ˇ U ˇ U − ˇ U − cosh ˇ χ ˇ a ( ˇ U + sin ˇ θ ˇ U ) + ˇ a ˇ U . (4.7)Second equation of the system yieldsˇ U = ˇ K ˇ U a / ( − µ (cid:20) − a ˇ K ˇ U (cid:18) d ˇ U dS − ˇ a cosh ˇ χ ( ˇ U + sin ˇ θ ˇ U ) (cid:19)(cid:21) / ) . (4.8)With hyperbolic motion for ˇ U = ˇ U = ˇ U = 0 corresponding five-velocity vectors have non-zero componentsˇ U I = σ, ˇ U I = − σ ˇ K ˇ a / , (4.9)ˇ U II = 2 σ p K ˇ a , ˇ U II = − σ ˇ K ˇ a / p K ˇ a . (4.10)For solution of type I time dilation is absent: d ˇ T = d ˇ τ , (4.11)and for type II increase of proper time passage is given by d ˇ T = 12 p K ˇ ad ˇ τ . (4.12)Transition to cylindrical coordinates ˇ X ic = (ˇ τ , ˇ r, ˇ θ, ˇ ϕ, ˇ y ) is realized by transformationˇ r = ˇ a cosh ˇ χ, ˇ y = ˇ a sinh ˇ χ. (4.13)Corresponding components of five-velocity vector areˇ V = cosh ˇ χ ˇ U + ˇ a sinh ˇ χ ˇ U , ˇ V = sinh ˇ χ ˇ U + ˇ a cosh ˇ χ ˇ U . (4.14)Inverse coordinate transformation is written asˇ a = p ˇ r − ˇ y , ˇ χ = arcoth ˇ y ˇ r . (4.15)Substituting this in (4.1) gives dS = (1 + ˇ K p ˇ r − ˇ y ) d ˇ τ − K (ˇ r − ˇ y ) − / d ˇ τ (ˇ yd ˇ r − ˇ rd ˇ y ) − d ˇ r − ˇ r ( d ˇ θ + sin ˇ θd ˇ ϕ ) + d ˇ y . (4.16)The same line element can be obtained by replacement K = − i ˇ K, y = iˇ y and addition of (ˇ) in the notation of othercoordinates in (3.2).Geodesics equations for motion of particle having rest mass is written as d ˇ τdS − ˇ K ˇ r p ˇ r − ˇ y d ˇ τdS d ˇ rdS + ˇ K ˇ y p ˇ r − ˇ y d ˇ τdS d ˇ ydS + ˇ K ˇ r ˇˇ y r − ˇ y ) / (cid:18) d ˇ rdS (cid:19) − ˇ K (ˇ r + ˇ y )2(ˇ r − ˇ y ) / d ˇ rdS d ˇ ydS −− ˇ K ˇ r ˇ yc (ˇ r − ˇ y ) / (cid:18) d ˇ θdS (cid:19) − ˇ K ˇ r ˇ y (ˇ r − ˇ y ) / sin ˇ ϕ (cid:18) d ˇ ϕdS (cid:19) + ˇ K ˇ r ˇ y r − ˇ y ) / (cid:18) d ˇ ydS (cid:19) = 0 , (4.17) d ˇ rdS + ˇ K ˇ r p ˇ r − ˇ y (cid:18) d ˇ τdS (cid:19) + ˇ K ˇ r ˇ y r − ˇ y ) / d ˇ τdS d ˇ rdS − ˇ K ˇ y − K p ˇ r − ˇ y r − ˇ y ) / d ˇ τdS d ˇ ydS −− ˇ K ˇ r ˇ y r − ˇ y ) / (cid:18) d ˇ rdS (cid:19) + ˇ K ˇ y (ˇ r + ˇ y )2(ˇ r − ˇ y ) / d ˇ rdS d ˇ ydS + ˇ r ( ˇ K ˇ y − p ˇ r − ˇ y ) p ˇ r − ˇ y (cid:18) d ˇ θdS (cid:19) ++ ˇ r ( ˇ K ˇ y − p ˇ r − ˇ y ) p ˇ r − ˇ y sin ˇ θ (cid:18) d ˇ ϕdS (cid:19) − ˇ K ˇ r ˇ y r − ˇ y ) / (cid:18) d ˇ ydS (cid:19) = 0 , (4.18) d ˇ θdS + 2ˇ r d ˇ rdS d ˇ θdS − sin(2ˇ θ )2 (cid:18) d ˇ ϕdS (cid:19) = 0 , (4.19) d ˇ ϕdS + 2ˇ r d ˇ rdS d ˇ ϕdS + 2 cot ˇ θ d ˇ θdS d ˇ ϕdS = 0 , (4.20) d ˇ ydS + ˇ K ˇ y p ˇ r − ˇ y (cid:18) d ˇ τdS (cid:19) + ˇ K ˇ r + 3 ˇ K p ˇ r − ˇ y r − ˇ y ) / d ˇ τdS d ˇ rdS − ˇ K ˇ r ˇ y r − ˇ y ) / d ˇ τdS d ˇ ydS −− ˇ K ˇ y ˇ r r − ˇ y ) / (cid:18) d ˇ rdS (cid:19) + ˇ K ˇ r (ˇ r + ˇ y )2(ˇ r − ˇ y ) / d ˇ rdS d ˇ ydS + ˇ K ˇ r ˇ y p ˇ r − ˇ y (cid:18) d ˇ θdS (cid:19) + ˇ K ˇ r ˇ y p ˇ r − ˇ y sin ˇ θ (cid:18) d ˇ ϕdS (cid:19) −− ˇ K ˇ r ˇ y r − ˇ y ) / (cid:18) d ˇ ydS (cid:19) = 0 . (4.21)Condition given by metric (4.16) for the time-like path is1 = (1 + ˇ K p ˇ r − ˇ y ) ˇ V − K (ˇ r − ˇ y ) − / ˇ V (ˇ y ˇ V − ˇ r ˇ V ) − ˇ V − ˇ r ( ˇ V + sin ˇ θ ˇ V ) + ˇ V . (4.22)A non-zero components of five-velocity vectors corresponding hyperbolic solutions (4.9), (4.10) areˇ V I = σ, ˇ V I = − σ ˇ K ˇ y (ˇ r − ˇ y ) / , ˇ V I = − σ ˇ K ˇ r (ˇ r − ˇ y ) / , (4.23)ˇ V II = 2 σ (4 + ˇ K p ˇ r − ˇ y ) / , (4.24)ˇ V II = − σ ˇ K ˇ y (ˇ r − ˇ y ) / (4 + ˇ K p ˇ r − ˇ y ) / , (4.25)ˇ V II = − σ ˇ K ˇ r (ˇ r − ˇ y ) / (4 + ˇ K p ˇ r − ˇ y ) / . (4.26)For motion in the neighborhood of point (ˇ τ , ˇ r , π/ , ,
0) with conditions ˇ V = ˇ V = 0 local solution is found fromreduced Eqs. (4.17)-(4.21), which turned out to d ˇ τdS − ˇ K d ˇ τdS d ˇ rdS − ˇ K r / d ˇ rdS d ˇ ydS = 0 , (4.27) d ˇ rdS + ˇ K (cid:18) d ˇ τdS (cid:19) + 3 ˇ K r / d ˇ τdS d ˇ ydS = 0 , (4.28) d ˇ ydS + ˇ K ˇ r + 3 ˇ K r / d ˇ τdS d ˇ rdS + ˇ K d ˇ rdS d ˇ ydS = 0 . (4.29)Condition (4.22) takes form 1 = (1 + ˇ K ˇ r ) ˇ V + 2 ˇ K ˇ r / ˇ V ˇ V − ˇ V + ˇ V . (4.30)For hyperbolic motion Eqs. (4.23)-(4.26) radial accelerations are d ˇ V I dS = ˇ K , (4.31) d ˇ V II dS = ˇ K K p ˇ r − ˇ y . (4.32) V. KALUZA-KLEIN MODEL
In Kaluza-Klein theory the line element is brought in form d S = g ij dx i dx j + ε Φ ( A i dx i + εdy ) , (5.1)where x i and g ij is coordinates and metrical tensor of 4D space-time, Φ and A are scalar and vector potential. Metricalcoefficients and potentials are functions of x i and y . Constant ε equals 1 for time-like fifth coordinate y and -1, whenit is space-like.In this form metrics (3.2) and (4.16) are represented by line-element of 4D space-time ds = (cid:18) − K y a ε (cid:19) dτ − Ka − / ε ydτ dr − dr − r ( dθ + sin θdϕ ) (5.2)and potentials A = Kra − / ε , A = A = A = 0 , Φ = 1 , (5.3)where it is denoted a ε = p r − εy .If 4D metric satisfies cylindrical conditions ∂g ij ∂y = 0 electromagnetic field is defined by F ij = ∂ i A j − ∂ j A i . Ratio ofelectric charge to mass in 4D is written as qm = Q p − Q , (5.4)with scalar function Q = ε Φ (cid:18) dydS + A i dx i dS (cid:19) . (5.5)In more general case with 4D metrical coefficients being dependent on y the relationship between Q and q/m is notidentical [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 thegeodesics of the type I: Q I = 0 . (5.6)This value is interpreted as neutral charge of a test particle. For the type II scalar function is Q II = εσKra / ε √ εK a ε . (5.7)The light trajectory is assumed to be isotropic curve both in 4D and in 5D: ds=dS=0. From (5.1)-(5.3) we obtainsolutions dydτ = 0 , (5.8)and dydτ = − ε Kra / ε . (5.9) VI. ASTROPHYSICAL APPLICATIONS
Considering phenomenology of particles motion in 5D we assume that stationary in 3D space particles, having restmass, 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 alongfifth 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 continuouslyincreasing in intervals I + n = [2 πn, π + 2 πn ] for integer n . Since value r < f has discontinuity on the endpoints of I n , which prescribes singularity. It can be avoidif model of binary world consisting of universe - anti-universe pair [26–28] is considered under the assumption thatit possesses a large number of copies [29, 30], in which a physical laws are identical. In bulk a space-time half I − n = [ − π + 2 πn, πn ] 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 shouldbe interpreted as motion of particle and anti-particle through opposite packets of branes, which conforms to CPT-symmetry of the universe and anti-universe. Thus a birth of the pair particle-antiparticle is assumed to occur in points y = − + a , r = 0, after which they move through opposite packets of branes and annihilate, when y = + − a , r = 0. A. Basic properties of Pioneer effect model
Recently much attention was attracted to the Pioneer effect, which consists in additional acceleration of spacecraftsPioneer 10/11 [20, 21, 31, 32] a p = (8 . ± . × − cm s − directed to the inner part of the solar system. We willanalyze how much studying models of rotating space conform to this data. Motion of the spacecrafts and the planetsis 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 constantradial coordinate they correspond to the neutrally charged particles. Their proper time coincides with coordinate timefor trajectories, which are the arcs of circle (2.16) or hyperbola (4.11). Also motion of light is assumed to correspondwith 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 dV i dS + Γ ikl V k V l = G i , (6.1)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 ig + W iF , where W iF conforms in case G = 0, θ = π/ τ , r , π/ , ,
0) to equations W F = G , (6.2) W F = G + r (cid:18) dϕdS (cid:19) , (6.3) W F = 0 , (6.4) W F = G − r drdS dϕdS , (6.5) W F = G . (6.6)Accelerations W ig 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 I and dU I /dS for χ = π/ U I and d ˇ U I /dS for ˇ χ = 0 in hyperbolic coordinates as radial velocity and accelerationobserved in 4D surface of five-dimensional space-time. B. Model with space-like fifth coordinate
In spherical coordinates in the neighborhood of point ( τ , r , π/ , , π/
2) Eqs. (3.14)-(3.16) correspond to system(2.6)-(2.10) reduced to d τdS + K dτdS dadS + Kr / dadS dχdS = 0 , (6.7) d adS − K (cid:18) dτdS (cid:19) − r / dτdS dχdS − r (cid:18) dχdS (cid:19) = 0 , (6.8) d χdS + K (3 − K r )2 r / dτdS dadS + 4 − K r r dadS dχdS = 0 . (6.9)For closed to circular motion (2.13), (2.14) non-vanishing five-velocities are written in form U I = σ + α , (6.10) U I = α , (6.11) U I = − σKa / + α , (6.12)where α i are functions of coordinates. Substitution of U iI in (6.7)-(6.9) yields dα dS = − K α α − Kr / α α , (6.13) dα dS = − σK α − σKr / α + K α + 3 Kr / α α + r α , (6.14) dα dS = − K − K r r / α α − − K r r α α . (6.15) TABLE I: Distance from Sun to Pioneer 11 r p , in AU, its velocity ˙ r p , in km s − and unmodeled acceleration a p , in 10 − cms − (See plans and figure in [20]), predicted magnitude of acceleration | ¨ r p | , in 10 − cm s − . r p r p . ± . ± . a p . ± . . ± . | ¨ r p | . ± .
75 5 . ± . Equation (2.11) gives 0 = 2 σα + (1 − K r ) α − Kr / α α − α − r α . (6.16)We consider case | α i | << α = 0 on the surface χ = π/
2. First and second equations of system(6.13)-(6.15) reduce to dα dS = − K α α , (6.17) dα dS = − σK α , (6.18)that gives σα = α H, (6.19)where H is constant. Substituting this expression into Eq. (6.18) and choosing H = 0 we obtain dα dS = − K α . (6.20)The average spacecraft’s velocity on interval 20-50 a.e. is about ˙ r p = 15 ± − (See diagram in [21]) and forapproximation S = τ corresponds to α = 5 × − . Therefore this equation turns out to be¨ r p = − K ˙ r p K = (3 . ± × − cm − / . For this value and made choice of S, H
Eq. (6.19) conforms to (6.16)without small higher-order terms.Additional acceleration of Pioneer 11 on distance less than 20 a.e. and its predicted magnitude are contained inTable I.
C. Additional acceleration of planets
In this section we will test proposed model by finding additional acceleration for planets of the solar system andcomparing them with observations data. Further we will use following denotations: γ is gravity constant, M is theSun’s mass, Ω is its semimajor axis, e is eccentricity, n = p γM/ Ω is unperturbed Keplerian mean motion, P = 2 π/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. (6.22)Differentiation of these relations with respect to t yields˙ r = Ω e sin ξ ˙ ξ, ˙ ξ = n − e cos ξ (6.23)and radial velocity is rewritten as ˙ r = n Ω e sin ξ − e cos ξ . (6.24)0 TABLE II: Semimajor axes Ω in AU, eccentricities e , orbital periods P in years, mean squared radial velocities h ˙ r i in 10 cms − , coordinate r , predicted magnitude of additional radial accelerations | A p | in 10 − cm s − and determined from observationsanomalous radial accelerations A obs in 10 − cm s − for the planets [22] and asteroid Icarus [34].Yupiter Saturn Uranus Neptune Pluto IcarusΩ 5.2 9.5 19.19 30.06 39.48 1.077 e P h ˙ r i | A p | . ± . ± . ± . . ± .
022 27 ±
16 (14 . ± . × A obs ±
700 ( − . ± . × (0 . ± . × - - < . × A mean squared radial velocity during half-period h ˙ r i = P P/ Z ˙ r dt / (6.25)after following from (6.23) substitution dt = 1 n (1 − e cos ξ ) dξ (6.26)will be h ˙ r i = πe Ω P π Z sin ξ − e cos ξ dξ / . (6.27)Values h ˙ r i for the planets, corresponding them Pioneer-like acceleration A p = − K h ˙ r i A obs , obtained from observations are in Table II. Predicted additional radialacceleration for Yupiter, Saturn, Uranus is within the observation error and for asteroid Icarus it is close to upperlimit of A obs . D. Model with time-like fifth coordinate
In hyperbolic coordinates in the neighborhood of point (ˇ τ , ˇ r , π/ , ,
0) Eqs. (4.27)-(4.29) correspond to system(4.2)-(4.6) reduced to d ˇ τdS − ˇ K d ˇ τdS d ˇ adS − ˇ K ˇ r / d ˇ adS d ˇ χdS = 0 , (6.29) d ˇ adS + ˇ K (cid:18) d ˇ τdS (cid:19) + 3ˇ r / d ˇ τdS d ˇ χdS + ˇ r (cid:18) d ˇ χdS (cid:19) = 0 , (6.30) d ˇ χdS + ˇ K (3 + ˇ K ˇ r )2ˇ r / d ˇ τdS d ˇ adS + 4 + ˇ K ˇ r r d ˇ adS d ˇ χdS = 0 . (6.31)For closed to hyperbolic motion (4.9) non-vanishing five-velocities are written in formˇ U I = σ + ˇ α , (6.32)ˇ U I = ˇ α , (6.33)ˇ U I = − σ ˇ K ˇ a / + ˇ α , (6.34)1where ˇ α i are functions of coordinates. Substitution of ˇ U iI in (6.29)-(6.31) yields d ˇ α dS = ˇ K α ˇ α + ˇ K ˇ r / α ˇ α , (6.35) d ˇ α dS = σ ˇ K α + σ ˇ K ˇ r / α − ˇ K α − K ˇ r / α ˇ α − ˇ r ˇ α , (6.36) d ˇ α dS = − ˇ K K ˇ r r / ˇ α ˇ α − K ˇ r r ˇ α ˇ α . (6.37)Equation (4.7) gives 0 = 2 σ ˇ α + (1 + ˇ K ˇ r )ˇ α + 2 ˇ K ˇ r / ˇ α ˇ α − ˇ α + ˇ r ˇ α . (6.38)We consider case | ˇ α i | << α = 0 on the surface ˇ χ = 0. Equations (6.35), (6.36) reduce to d ˇ α dS = ˇ K α ˇ α , (6.39) d ˇ α dS = σ ˇ K α , (6.40)that gives σ ˇ α = ˇ α H, (6.41)where ˇ H is constant. Substituting this expression into Eq. (6.40) we obtain d ˇ α dS = ˇ K ˇ α H ˇ K . (6.42)This result doesn’t conform to the Pioneer effect, so far as in accordance with this expression with increase ofmagnitude of radial velocity corresponding growth of acceleration will be positive.For cylindrical coordinates in case ˇ V = 0 in point (ˇ τ , ˇ r , π/ , ,
0) from Eq. (4.28) we obtain d ˇ rdS = − ˇ K (cid:18) d ˇ τdS (cid:19) . (6.43)With condition | ˇ V | <<
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 thatcontradicts data of observations (Tables I,II).
VII. CONCLUSION
Solutions of geodesics equations is found for rotating space in 5D with angular velocity, being inversely proportionalto the square root of radius. They describe motion of particle having rest mass in a circle with space-like fifth coordinatein spherical (2.13-2.15) or cylindric (3.10)-(3.13) frames and hyperbolic motion with time-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 modelthey 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 notionthat closed geodesic of elementary particle having a rest mass corresponds to motion of the pair particle-antiparticlein mirror worlds.Analogy with motion in central gravity field in 4D is employed for determination of velocity and accelerationobserved in 4D sheet for particles moving in 5D bulk. We obtain approximate solution in the neighborhood ofsurface with zero fifth coordinate in cylindric frame for geodesics (6.7)-(6.9), (6.29)-(6.31) deviating from havingconstant radius. With space-like fifth coordinate a body in 4D space-time with appropriate radial velocity will havecentripetal acceleration (6.21) being proportional to square of radius and directed towards 5D geodesic axis. Thatroughly conforms to underlying properties of the Pioneer-effect, namely, constant additional acceleration of apparatus2towards the Sun on distance from 20 to 50 a.e., its increase from 5 to 20 a.e., observed absence of one in motion ofplanets. [1] G. Nordstrom, Uber die Moglichkeit, das Electromagnetishe Feld und das Gravitation Feld zu Vereinigen, Phyz. Zeitschr.1 (1914) 504.[2] T. Kaluza, Zum Unitatsproblem der Physik, Sitz. Preuss. Akad. Wiss. Phys. Math. K1 (1921) 966.[3] O. Klein, Quantentheorie und funfdimensionale Relativitatstheorie, Z. Phys. 37 (1926) 895.[4] J. M. Overduin and P.S. Wesson, Kaluza-Klein Gravity, Phys. Rept. 283, 303 (1997), gr-qc/9805018.[5] J. Ponce de Leon, Does force from an extra dimension contradict physics in 4D? Phys. Lett. B523, 311 (2001),gr-qc/0110063.[6] A. Herrera-Aguilar and O. V. Kechkin, Bosonic string - Kaluza Klein theory exact solutions using 5D-6D dualities,Mod.Phys.Lett. A16 (2001) 29-40, gr-qc/0101007.[7] J. Ponce de Leon, Equivalence Between Space-Time-Matter and Brane-World Theories, Mod.Phys.Lett. A16 (2001) 2291-2304, gr-qc/0111011.[8] D.Yu.Tsipenyuk, V.A. Andreev, 5-dimensional extended space model, Presented at 13th General Conference of the Euro-pean Physical Society: Beyond Einstein - Physics of the 21st Century (EPS-13), Bern, Switzerland, 11-15 Jul 2005.[9] D.Yu.Tsipenyuk, Field transformation in the extended space model: Prediction and experimental test., Gravitation andCosmology Vol.7(2001),No.4(28),pp.336-338.[10] W.B. Belayev, Cosmological model with movement in fifth dimension, Space-time and Substance 7:63,2001, gr-qc/0110099.[11] K. Koyama, The cosmological constant and dark energy in braneworlds, Gen. Rel. Grav. 40421 (2008), arXiv:0706.1557.[12] T. Kobayashi and Y. Takamizu, Hybrid compactifications and brane gravity in six dimensionsClass.Quant.Grav.25:015007,2007, arXiv:0707.0894.[13] F. Arroja, T. Kobayashi, K. Koyama and T. Shiromizu, Low energy effective theory on a regularized brane in 6D gaugedchiral supergravity, JCAP0712:006,2007, arXiv:0710.2539.[14] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, The Hierarchy Problem and New Dimensions at a Millimeter,Phys.Lett.B429:263-272,1998, hep-ph/9803315.[15] I. Bars, S.-H. Chen, G. Quelin, Dual Field Theories In (d-1)+1 Emergent Spacetimes From A Unifying Field Theory Ind+2 Spacetime, Phys.Rev.D76:065016,2007, arXiv:0705.2834[16] I. Bars, S.-H. Chen, Geometry and Symmetry Structures in 2T Gravity, Phys.Rev.D79:085021,2009, arXiv:0811.2510[17] I. Bars, Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime, Int.J.Mod.Phys.A25:5235-5252,2010,arXiv:1004.0688[18] K. Trencevski, Time dependent gravitational potential in the universe and some consequences. Gen.Rel.Grav.37:507-519,2005.[19] Belayev, W.B., Cosmological model in 5D, stationarity, yes or no, gr-qc/9903016.[20] Turyshev S.G. and Toth V. T., The Pioneer Anomaly, Living Rev.Rel.13:4,2010, arXiv:1001.3686.[21] Anderson J.D., Laing P.A., Lau E.L., Liu A.S., Nieto M.M. and Turyshev S.G., Study of anomalous acceleration of Pioneer10 and 11, Phys. Rev. D65, 082004 (2002); gr-qc/0104064.[22] L. Iorio, Can the Pioneer anomaly be of gravitational origin? A phenomenological answer, Found. Phys. 37 (2007) 897-918,gr-qc/0610050.[23] D. Grumiller, Model for gravity at large distances, Phys.Rev.Lett.105:211303,2010, arXiv:1011.3625.[24] L. Iorio, Solar system constraints on a Rindler-type extra-acceleration from modified gravity at large distances,arXiv:1012.0226.[25] W.B. Belayev, D.Yu. Tsipenyuk, Gravi-electromagnetism in five dimensions and moving bodies in galaxy area, Space-timeand Substance 22:49,2004, gr-qc/0409056.[26] A. Linde, Inflation, Quantum Cosmology and the Anthropic Principle, in ”Science and Ultimate Reality: From Quantumto Cosmos”, honoring John Wheeler’s 90th birthday. J. D. Barrow, P.C.W. Davies, C.L. Harper eds. Cambridge UniversityPress (2003), hep-th/0211048.[27] S.L. Dubovsky, S.M. Sibiryakov, Domain walls in noncommutative gauge theories, folded D branes, and communicationwith mirror world, Nucl.Phys.B691:91-110,2004, hep-th/0401046.[28] M. Sarrazin, F. Petit, Equivalence between domain-walls and ”noncommutative” two-sheeted spacetimes: Model-independent matter swapping between branes, Phys.Rev.D81:035014,2010, arXiv:0903.2498.[29] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, N. Kaloper, Manyfold Universe, JHEP 0012 (2000) 010, hep-ph/9911386.[30] G. Dvali, I. Sawicki, A. Vikman, Dark Matter via Many Copies of the Standard Model, JCAP 0908:009,2009,arXiv:0903.0660.[31] Anderson J.D., Laing P.A., Lau E.L., Liu A.S., Nieto M.M. and Turyshev S.G., Indication from Pioneer 10/11, Galileo, andUlysses data, of an apparent anomalous, weak, long-range acceleration, Phys. Rev. Lett. 81, 2858 (1998); gr-qc/980808;also http://ummaspl.narod.ru/papers.htm.[32] Anderson J.D. and Nieto M.M., Search for a Solution of the Pioneer Anomaly, Contemp. Phys. 48, No. 1. 41-54 (2007),arXiv:0709.3866.[33] Turyshev S.G., Nieto M.M., Anderson J.D., A Route to Understanding of the Pioneer Anomaly, The XXII Texas Sympo-3