Golden Metric Tensor, Geodesic Equation, Coefficient of Affine Connection, Newton’S Planetary Equation, Einstein’S Planetary Equation

International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 DOI: 10.5923/j.ijtmp.20170702.02

The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

Y. Nura1,*, S. X. K. Howusu2, L. W. Lumbi3, I. Nuhu4, A. Hayatu1

1Department of Physics, University of Maiduguri, Maiduguri, Nigeria 2Theoretical Physics Program, National Mathematical Centre Abuja, Nigeria 3Department of Physics Nassarawa State University, Keffi, Keffi, Nigeria 4M.Sc Program, Department of Physics, University of Jos, Jos, Nigeria

Abstract In this paper we introduced super general geodesic equation and golden metric tensors. We derived linear velocity vector and linear acceleration vector using golden metric tensor in spherical polar coordinates. Coefficients of affine connection were evaluated based upon golden metric tensor. Based on the evaluated linear velocity vector and acceleration vector, we obtained super general geodesic planetary equation in terms of r, θ, and x0 to obtain Riemannian acceleration due to gravity in terms of r, θ, and x0 known as gravitational scalar potential that played an important role in dealing with planetary phenomenon. Also in this paper we derived generalized planetaryϕ equations based upon Riemannian geometry and the golden metricϕ tensor. The planetary equation obtain in this paper contained Newton’s planetary equation, Einstein’s planetary equation and the added or contribution term to the existing planetary equations known as post Newton’s planetary equation and post Einstein’s planetary equation. Whose solution of these equations will be consider in the next edition of this paper. Keywords Golden metric tensor, Geodesic equation, Coefficient of affine connection, Newton’s planetary equation, Einstein’s planetary equation

defining a geometrical quantity called the Riemann 1. Introduction curvature tensor, which describes exactly how the space (or space time) is curved at each point. In general relativity, the The properties of geodesics differ from those of straight metric and the Riemann curvature tensor are quantities lines. For example, on a plane, parallel lines never meet, but define at each point in space time. It is based on the above this is not so far geodesics on the surface of the earth. For argument explanation. Howusu introduced, by postulation, example, lines of longitude are parallel at the equator, but a second natural and satisfactorily generalization or intersect at the poles. Analogously, the world lines of test extension of the Schwarzschild’s metric tensor from the particles in free fall are space time geodesics, the straightest gravitational fields of all static homogeneous spherical possible lines in space time. But still there are crucial distribution of mass to the gravitational fields of all differences between them and the truly straight lines that spherical distributions of mass-named as the golden metric can be traced out in the gravity-free space time of special tensor for all gravitational fields in nature [3]. relativity [1]. Einstein’s equations are the centre piece of general relativity. They provide a precise formulation of the relationship between space time geometry and the 2. Golden Metric Tensor properties of matter, using the language of mathematics. More concretely, they are formulated using the concepts of In this paper we introduced golden metric tensor for all Riemannian geometry, in which the geometric properties of gravitational fields in nature as follows [4]. The covariant a space (or a space time) are described by a quantity called form of all golden metric tensor for all gravitational fields a metric. The metric encodes the information needed to in nature as 1 compute the fundamental geometric notions of distance and 2 0 g11 = 1 + 2 f(r, , , x ) (2.1) angle in a curved space (or space time) [2]. The metric c − 1 function and its rate of change from point to point can be �2 2 θ ϕ �0 g22 = r 1 + 2 f(r, , , x ) (2.2) c − 1 * Corresponding author: 2 � 2 θ2 ϕ � 0 g33 = r sin 1 + 2 f(r, , , x ) (2.3) [email protected] (Y. Nura) c − Published online at http://journal.sapub.org/ijtmp 2 g = 1 +θ �f(r, , , x0θ)ϕ � (2.4) Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved 00 c2

− � θ ϕ � 26 Y. Nura et al.: The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

guv = 0, otherwise (2.5) Where f is the gravitational scalar potential of the space time, the golden metric tensor and also contains the following physical effects:  Gravitational space contraction  Gravitational time dilation  Gravitational polar angle contraction  Gravitational azimuthal angle contraction While the contravariant form of golden metric tensor given as: 2 g11 = 1 + f(r, θ, , x0) (2.6) c2 22 1 2 0 g = � 1 + f(r, θϕ, , x�) (2.7) r2 c2 1 2 g33 = � 1 + f(ϕr, θ, �, x0) (2.8) r2sin 2θ c2 1 00 � 2 0ϕ � g = 1 + 2 f(r, θ, , x ) (2.9) c − uv g = 0−, otherwise� ϕ � (2.10)

3. Formulation of the General Dynamical Laws of Gravitation It is well known that all of Newton’s dynamical laws of gravitation are founded on the experimental physical facts available in his day. The instantaneous active mass mA passive mass mP and the inertial mass mI of a particle of non-zero rest mass m0 are given by:

mA = mP = mI = m0 (3.1) In all proper inertial reference frames and proper times. This statement may be called Newton’s principle of mass. According to the classic scientific method introduced by G. Galileo (the father of mechanics) and Newton the natural laws of mechanics are determined by experimental physical facts. Therefore today’s experimental revisions of the definitions of inertial, passive and active masses of a particle on non-zero rest mass induce a corresponding revision of Newton’s dynamical laws of gravitation which are now formulated (Howusu, 1991). Super General Planetary Equation 1 1 2 1 u 2 2 2 2 (mI)H = (mP)H = 1 2 1 + 2 f . 1 + 2 f m0 (3.2) c c − − c as given by equation (3.1) above � − � � � � � gH = Riemannian acceleration due to gravity

uH = Riemannian velocity vector But force is defined as the rate of change of momentum

1 1 1 1 2 1 2 2 1 d u 2 2 2 u 2 2 2 2 1 2 1 + 2 1 + 2 f m0uH = 1 2 1 + 2 f . 1 + 2 f m0gH (3.3) d c c − − c c c − − c

2 τ �� − � � � � � � � − � � � � � Where 1 + f is Riemannian factor. c2 Equivalently, � � d (m ) . u = (m ) g (3.4) d I H H P H H d (mτ ) a + [(m ) ] u = (m ) g (3.5) �I H H d� I H H P H H 1 d aH + � [τ(mI)H] u�H = gH (3.6) (mI)H d

aH general� τacceleration� vector. Equation (3.6) above referred to as super general geodesics equation of motion. ≡ Velocity Tensor x = {x1, x2, x3, x0 } (3.7) μ ̇ ̇ ̇ ̇ ̇ International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 27

x1 = r, x2 = , x3 = , x0 = ct, x = r, , , ct (3.8) θ ϕ 𝑖𝑖 Velocity vector μ ̇ � ̇ θ̇ ϕ̇ ̇� 1 2 3 0 uH = g11x , g22x , g33x , g00x (3.9) Based upon golden metric tensor �� ̇ � ̇ � ̇ � ̇ � 1 2 2 ur = g11r = 1 + 2 f r (3.10) c − 1 � ̇ � 2� 2̇ u = g22 = r 1 + 2 f (3.11) c − θ 1 � θ̇ � � 2 θ̇ 2 uϕ = g33ϕ = rsin 1 + 2 f (3.12) c − 1 � ̇0 θ �2 2 � φ̇ u0 = g00x = 1 + 2 f ict (3.13) c − Where we have make use of Golden metric tensor in equations� ̇ (2.1)�-(2.4). � ̇

4. Theoretical Analysis Acceleration tensors define by geodesics equation as: a = x + Γ x x (4.1) μ μ μ μ α β Γ αβ Where αβ is defined as coefficient of affine connection gives̈ as: ̇ ̇ 1 Γ = g g + g g (4.2) 2 , , , μ με Putting μ = 1 αβ � αε β εβ α − αβ ε� 1 1 2 1 2 1 2 a = r + Γ11(r) + Γ22 + Γ33 (4.3) Putting = 2 ̈ ̇ �θ̇ � �ϕ̇ � 2 2 2 2 2 μ a = + 2Γ12 r + Γ33 (4.4) Putting = 3 θ̈ � ̇θ̇ � �ϕ̇ � a3 = 2 + 2Γ3 r + 2Γ3 (4.5) μ 13 23 Putting = 0 ϕ̈ � ̇ϕ̇ � �θ̇ ϕ̇ � a0 = ct + 2cΓ0 (tr) (4.6) μ 01 Using ̈ ̇ ̇ x0 = ct By employing equation (4.2) above, in our paper (I) coefficients of affine connection are evaluated given in equation [(30) – (58)] [9] 1 1 2 Γ = 1 + f f (4.7) 00 c2 c2 ,1 1 1 1� 1 � 2 Γ01 = Γ10 = 2 1 + 2 f f,0 (4.8) c c − 1 1 1 − 1� � Γ11 = 2 1 + 2 f f,1 (4.9) c c − 1 1 −1 � 1 � 2 Γ12 = Γ21 = 2 1 + 2 f f,2 (4.10) c c − 1 1 1 − 1 � 2 � Γ13 = Γ31 = 2 1 + 2 f f,3 (4.11) c c − 2 1 1 r − 2� � Γ22 = 2 1 + 2 f f,1 r (4.12) c c − r2sin 2 2 1 Γ1 =− � 1 + � f f− rsin2 (4.13) 33 c2 c2 ,1 θ − and � � − θ

28 Y. Nura et al.: The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

1 2 Γ2 = 1 + f f (4.14) 00 c2r2 c2 ,2 1 2 2 � 1 � 2 Γ02 = Γ20 = 2 1 + 2 f f,0 (4.15) c c − 1 2 1 �1 � Γ11 = 2 2 1 + 2 f f,2 (4.16) c r c − 1 2 2 � 1 1 � 2 Γ12 = Γ21 = 2 1 + 2 f f,2 (4.17) r c c − 1 2 1 −2 � � Γ22 = 2 1 + 2 f f,2 (4.18) c c − 1 2 −2 � 2 � 2 Γ23 = Γ32 = 2 1 + 2 f f,3 (4.19) c c − sin 2 2 1 Γ2 = 1−+ � f f � sin cos (4.20) 33 c2 c2 ,2 θ − and � � − θ θ 1 2 Γ3 = 1 + f f (4.21) 00 c2r2sin 2 c2 ,3 1 3 3 1θ � 2 � Γ03 = Γ30 = 2 1 + 2 f f,3 (4.22) c c − 3 1 2 Γ = �1 + f �f (4.23) 11 c2r2sin 2 c2 ,3 1 3 3 1θ � 1 �2 Γ13 = Γ31 = 2 1 + 2 f f,1 (4.24) r c c − 1 3 1 − 2� � Γ22 = 2 2 1 + 2 f f,3 (4.25) c sin c − 1 3 3 θ � 1� 2 Γ23 = Γ32 = cot 2 1 + 2 f f,2 (4.26) c c − 1 3 1 2θ − � � Γ33 = 2 1 + 2 f f,3 (4.27) c c − and − � � 1 0 1 2 Γ00 = 2 1 + 2 f f,0 (4.28) c c − 1 0 0� 1 � 2 Γ01 = Γ10 = 2 1 + 2 f f,1 (4.29) c c − 1 0 0 1 � 2 � Γ02 = Γ20 = 2 1 + 2 f f,2 (4.30) c c − 1 0 0 1 � 2 � Γ03 = Γ30 = 2 1 + 2 f f,3 (4.31) c c − 3 0 1 � 2 � Γ01 = 2 1 + 2 f f,0 (4.32) c c − 2 3 0 − r � 2 � Γ22 = 2 1 + 2 f f,0 (4.33) c c − r2sin 2 2 3 Γ0 = − � 1 +� f f (4.34) 33 c2 c2 ,0 θ − Γ =−0; otherwise� � (4.35) μ Hence the acceleration vector αβ 1 2 3 0 aH g11a , g22a , g33a , g00a (4.36) By putting the results of coefficients of affine connection, covariant form of golden metric tensors into equation (4.15), �� we obtained acceleration vector equations as: 1 1 2 2 1 2 2 2 2 2 2 2 ar = 1 + 2 r 2 1 + 2 f f,1(r) – r 1 + 2 f rsin 1 + 2 f (4.37) c − c c − c c 1 � � � ̈ − � �2 2 ̇ 2 � � �θ̇ � − 2 θ � � �ϕ̇ � � a = r 1 + 2 f + r sin cos (4.38) c − r θ � � �θ̈ � ̇θ̇ � − θ θ�ϕ̇ � �

International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 29

1 2 2 2 a = rsin 1 + 2 f + r + 2 cot (4.39) c − r 1 ϕ θ � 2 � �ϕ̈ 2� ̇θ̇ � 2 θ �θ̇ ϕ̇ �� a0 = 1 + 2 f ct + 1 + 2 f,1 (4.40) c c c − Hence equations (4.37), (4.38), (4.39) and (4.40)� , are linear� 𝑖𝑖̇ � ̈acceleration� � vector� equations based upon golden metric tensor.

5. Riemannian Acceleration Due to Gravity Riemannian acceleration due to gravity in terms of r, , and x0 is given by [10] 1 d aH + [(mI)H] uH = gH (5.1) (mI)Hθ ϕd

Where aH = Riemannian acceleration vector [6] � τ � uH = Riemannian velocity vector gH = Riemannian acceleration due to gravity (mI)H = Riemannian inertial mass Using 1 2 2 u = g r = 1 + f r r 11 c2 − and � ̇ � � ̇ 1 1 2 2 ar = a 1 + 2 f (5.2) c − From equation (4.37) that is � � 1 1 2 2 1 2 2 2 2 2 a = 1 + r 1 + f f, (r)2 r 1 + f rsin2 1 + f r c2 − c2 c2 − 1 c2 c2 Hence Riemannian� radial acceleration� � ̈ − �due to gravity� is giveṅ − by� � �θ̇ � − θ � � �ϕ̇ � � 1 d ar + [(mI)r] ur = gr (5.3) (mI)r d Putting (4.37) into equation (5.3) we have � τ � 1 1 1 2 2 1 2 2 2 2 2 2 2 1 d 2 2 1 + 2 r 2 1 + 2 f f,1 (r) r 1 + 2 f rsin 1 + 2 f + [(mI)r] 1 + 2 r = gr c − c c − c c (mI)r d c − (5.4) � � � ̈ − � � ̇ − � � �θ̇ � − θ � � �ϕ̇ � � � τ � � � ̇ Riemannian acceleration due to gravity is given by 1 d a + [(mI) ] u = g (5.5) θ − (mI) d Putting equation (4.38) into equation (5.5) weθ get θ � τ θ � θ θ 1 1 2 2 2 2 1 d 2 2 1 + 2 f r + r sin cos + [(mI) ] 1 + 2 f = g (5.6) c − r (mI) d c − Riemannian acceleration� � due to�θ ̈ gravity� ̇θ ̇ is� −givenθ by θ�ϕ̇ � � θ � τ θ � � � θ̇ θ 1 d a + (mI) u = g (5.7) ϕ − (mI) d Putting equation (4.39) into equation (5.7) weϕ get ϕ � τ � ϕ�� ϕ ϕ 1 1 2 2 2 1 d 2 2 1 + 2 f rsin + r + 2 cot + (mI) rsin 1 + 2 f = g (5.8) c − r (mI) d c − Riemannian x�0 acceleration� θ due�ϕ̈ to gravity� ̇θ̇ � is givenθ �byθ̇ ϕ ̇ �� ϕ � τ � ϕ�� θ � � ϕ̇ ϕ 1 d a0 + [(mI)0] u0 = g0 (5.9) − (mI)0 d Putting equation (4.40) into equation (5.9) we have � τ �

30 Y. Nura et al.: The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

1 1 2 2 2 1 d 2 2 1 + 2 f ct + 1 + 2 f,1 + [(mI)0] 1 + 2 f ct = g0 (5.10) c c c − (mI)0 d c − 0 Hence equation (5.4), (5.6),� (5.8)� 𝑖𝑖anḋ � ̈(5.10)� are super� general� planetary� τ equations� � for the� components𝑖𝑖 of r, , and x . Using equation (3.2) above we get 1 1 1 1 θ ϕ 2 1 2 1 u 2 2 2 2 u 2 2 2 2 (mI)H = (mI)P = 1 2 1 + 2 f 1 + 2 f m0(mI)H = 1 2 1 + 2 f 1 + 2 f m0 (5.11) c c − c c c − c Putting equations (5.11) into� the− equation� (5.4),� � (5.6),� (5.8)� and (5.10) we �get,− � � � � � 1 1 2 2 1 2 2 2 2 2 2 1 + r 1 + f f, (r)2 r 1 + f r 1 + f sin2 1 + f c2 − c2 c2 − 1 c2 c2 c2 ̇ ̇ � � �1 ̈ − � � ̇ − � �1�θ� − � 1 � θ � � �ϕ� � 2 1 2 2 1 d u 2 2 2 2 + 1 + 1 1 1 + f 1 + r g (5.12) c2 d c2 c2 c2 r − u2 2 1 2 − − 1 1+ f c2 c2 � � � − � τ � − � � � � � � ̇� ≡ Equation (5.12) above give� − rise� to super� � general radial gravitational intensity (or acceleration due to gravity) based upon the golden metric tensor. Hence equation (5.6) becomes

1 1 1 1 2 1 2 2 2 2 2 1 d u 2 2 2 2 1 + r + r sin cos + 1 + 1 1 1 + f 1 + r g c2 r c2 d c2 c2 c2 − − u2 2 1 2 − − 1 1+ f c2 c2 θ � � �θ̈ � ̇θ̇ � − θ θ �ϕ̇ �� − � τ � − � � � � � � θ̇ � ≡ � − � � � (5.13) Equation (5.12) above give rise to general gravitational intensity (or acceleration due to gravity) based upon the golden metric tensor. Equation (5.8) becomes θ − 1 2 2 2 1 + r sin + r + 2 cot c2 − r

� � 1 θ �ϕ̈ � ̇ϕ̇ � θ �θ̇ ϕ̇ �� 1 1 2 1 2 2 1 d u 2 2 2 2 + 1 + 1 1 1 + f 1 + r sin g (5.14) c2 d c2 c2 c2 − u2 2 1 2 − − 1 1+ f c2 c2 ϕ � � � − � τ � − � � � � � � θ ϕ̇ � ≡ Equation (5.16) above give rise� −to � �gravitational� intensity (or acceleration due to gravity) based upon the golden metric tensor. Equation (5.10) reduced to ϕ −

1 1 1 1 2 1 2 2 2 2 2 1 d u 2 2 2 2 1 + f ct + 1 + f, + 1 + 1 1 1 + f 1 + ct g (5.15) c2 c c2 1 c2 d c2 c2 c2 0 − − u2 2 1 2 − − 1 1+ f c2 c2 � � 𝑖𝑖̇ � ̈ � � � � � � − � τ � − � � � � � � 𝑖𝑖̇ � ≡ Equation (5.15) above give rise to x0 gravitational� − �intensity� � (or acceleration due to gravity) based upon the golden metric tensor. Hence equation (5.12), (5.13), (5.14) and− (5.15) are referred to as general gravitational intensity (or acceleration due to gravity) based upon the golden metric tensor in spherical polar coordinates.

6. General Gravitational Intensity or Acceleration due to Gravity Based upon the Golden Metric Tensor in Spherical Polar Coordinates

0 2 1 gr = (x ) Γ00 (6.1) g = (x0)2Γ2 (6.2) − ̇ 00 0 2 3 g θ = (x ) Γ (6.3) − ̇ 00 0 2 0 ϕ ( ) g0 = − ẋ Γ00 (6.4)

− ̇

International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 31

1 2 3 0 where Γ00, Γ00, Γ00, Γ00 are coefficient of affine connection evaluated using the formula 1 Γ = g g + g + g (6.5) 2 , , , μ με Equations (6.1), (6.2), (6.3) and (6.4) reduce toαβ � αε β εβ α αβ ε� 0 2 1 1 g = (x ) 1 + f f, (6.6) r c2 c2 1 1 1 g = − (ẋ 0)2 � 1 + �f f, (6.7) c2r2 c2 2 1 1 gθ =− (ẋ 0)2 � 1 +� f f, (6.8) c2r2sin 2 c2 3 1 ϕ − 0̇ 2 1 θ1� � g0 = (x ) 2 1 + 2 f f,0 (6.9) c c − where − ̇ � � x0 = ct (xo)2 = c2t2 (6.10) ̇ ̇ equating equation (5.15) with equation (6.6) we get ̇ ̇ 1 1 2 2 1 2 2 2 2 2 2 1 + r 1 + f f, (r)2 r 1 + f r 1 + f sin2 1 + f c2 − c2 c2 − 1 c2 c2 c2

1 1 ̇ 1 1 ̇ � � � ̈ − � � 1 ̇ − � � �θ� −1 � � θ � � �ϕ� � 2 2 u2 2 2 d u2 2 2 2 2 + 1 + 1 1 + f 1 1 + f 1 + r c2 − c2 c2 − d c2 c2 − c2 −

� 1 � � 2 − � � � � � − � � � � � � ̇ = t2 1 + f f, τ (6.11) c2 c2 1 Equating equation− ̇ (5.16)� with� equation (6.7) we get 1 2 2 2 1 + r + r sin cos c2 − r 1 1 1 1 � � �θ̈ � ̇θ̇ � − θ 1 θ �ϕ̇ �� 1 2 2 u2 2 2 d u2 2 2 2 2 + 1 + 1 1 + f 1 1 + f 1 + r c2 − c2 c2 − d c2 c2 − c2 − ̇ � 1 � �2� − � � � � � − � � � � � � ̇θ� = t2 1 + f f, τ (6.12) r2 c2 2 equating equation− ̇ (5.17)� with� equation (6.8) we get 1 2 2 2 1 + r sin + r + 2 cot c2 − r 1 1 1 1 � � θ �ϕ̈ � ̇ϕ̇ � 1 θ �θ̇ ϕ̇ �� 1 2 2 u2 2 2 d u2 2 2 2 2 + 1 + 1 1 + f 1 1 + f 1 + r sin c2 − c2 c2 − d c2 c2 − c2 − ̇ � 1� �� −2 � � � � � − � � � � � � θ ϕ� = t2 1 + f f, τ (6.13) r2sin 2 c2 3 equating equation− ̇ (5.18)θ �with equation� (6.9) we get 2 2 2 1 1 + f ct + 1 + f, c2 c c2 − 1 1 ̈ 1 1 1 � � 𝑖𝑖̇ � � � �1 1 2 2 u2 2 2 d u2 2 2 2 2 + 1 + 1 1 + f 1 1 + f 1 + ct c2 − c2 c2 − d c2 c2 − c2 −

� � �� 1− � � � � � − � � � � � � 𝑖𝑖̇ � 2 2 τ = t 1 + 2 f f,0 (6.14) c − − ̇ � �

32 Y. Nura et al.: The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

7. Application to Planetary Theory

Consider a planet of rest mass m0 moving in the gravitational field exterior to the sun which is assumed to be a homogeneous spherical body of radius R and rest mass M0, shown in the diagram below

M0 m0

r R

In the first place let us assume that the gravitational field is that of Newton: GM K f(r, , , x0) = 0 = (7.1) r r Where K = GM0 θ ϕ − − Where G is the universal gravitational constant. K f, = f = (7.2) 1 r r2 ∂ ,2∂= 0 (7.3) , = 0 (7.4) 𝑓𝑓 3 , = 0 (7.5) 𝑓𝑓 0 u2 = r2 + r2 2 + r2sin2 2 (7.6) 𝑓𝑓 2 1 1 u 2 1 2 2 2 2 2 2 2K 2 1 + 2 f = 2 ṙ + r θ̇ + r sin θϕ̇ × 1 2 (7.7) c c − c c r − 1 1 2 1 � � � ̇ u θ̇ 2 2θϕ̇ � 2� −2 � t = 1 2 1 + 2 f 1 + 2 f (7.8) c c − c − Then super general geodesic equation of motion ̇ become:� − hence� equation� � (6.11)� becomes� K K 2K2 2K 2 2K 2 d 1 1 K 2K2 1 + r + (r)2 r 1 r 1 sin2 + tr = t2 (7.9) c2r c2r2 c4r3 c2r c2r d t c2 r2 c2r3 or� � � ̈ − � � ̇ − � − � �θ̇ � − � − � θ�ϕ̇ � � � ̇ ̇ � τ � ̇ �� − ̇ � − � K K 2K 2 2K 2 1 K 2K2 1 + r (r)2 r 1 r 1 sin2 [rt 1] = t2 (7.10) c2r c2r2 c2r c2r 2 r2 c2r3 − Equation �(6.12) becomes� � ̈ − � � ̇ − � − � �θ̇ � − � − � θ�ϕ̇ � � − 𝑐𝑐 ̇ ̇ − ̇ � − � 1 1 1 2K 2 2 2K 2 2K 2 d 1 2 1 2K 1 2 r + r sin cos + 1 2 t 1 2 r = t 2 1 2 f,2 (7.11) c r − r c r − c r d t r c r or � − � �θ̈ � ̇θ̇ � − θ θ �ϕ̇ �� − � � ̇ � − � � τ � ̇ �� ̇θ̇ � − ̇ � − � 2K 2 r 1 1 r + r sin cos 0 (7.12) c2r r t c2 ̇θ̇ Equation (6.13) becomes � − � �θ̈ � ̇θ̇ � − θ θ �ϕ̇ �� − � ̇ � ≡ 1 2K 2 2 1 r sin + r + 2 cot c2r − r 1 ̈ 1 ̇ ̇ ̇ � − 2�K 2 θ �ϕ2K 2� ̇dϕ�1 θ �θϕ1�� 2 1 2K + 1 2 t 1 2 r sin x 2 = t 2 2 1 2 f,2 (7.13) c r − c r d t c r sin c r 2K ̇ 2τ ̇ ̇ ̇ 1 1 θ � − 1 +� � �r sin− � +� �r� + 2θcotϕ� � − r�sin− =�0 (7.14) c2r r c2 t 0 Note that planetary equation� do not� containθ � ϕẗ that� iṡϕ ̇ time� parameterθ �θ̇ ϕ ̇ or�� −in the� ̇caseϕ̇ � x θparameter, also the same equation that planetary equation contains no polar angle , but retained only azimuthal angle , radial portion of the equation also retained. Based on this argument equation (6.14) and equation (7.11) vanishes only equation (7.10) and equation (7.14) remain for further evaluations. Equation (7.10) reduceθ to ϕ K K 2K 2 2K 2 1 u2 K K u2K 1 + r (r)2 r 1 r 1 sin2 1 + (r) = + (7.15) c2r c2r2 c2r c2r c2 2c2 c2r r2 c2r2 � � � ̈ − � � ̇ − � − � �θ̇ � − � − � θ�ϕ̇ � � − �� − � ̇ � −

International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 33

Equation (7.14) becomes K 1 u2 K 1 + r sin + 2 sin r + 2r cot sin 1 + r sin = 0 (7.16) c2r c2 2c2 c2r For motion along� the equatorial� � θ planeϕ̈ θ � ̇ϕ̇ � θ θ �θ̇ ϕ̇ �� − � � − � θϕ̇ � a = = 0 = 0 sin = 1 2 2 2 π2 2 θ K Kθ̇ 2Kθ̈ 1 3K r = + + (r)2 + + r2 c2r3 c2r2 c2 r3 c2r4 K2u2 2Ku2 1 𝑙𝑙 3𝑙𝑙K 2u2 r = ̈ Ku−2 + + (r)2 +̇ (r) + 2u−3 (7.17) c2 c2 c2 c2 𝑙𝑙 Note that ̈ − ̇ ̇ 𝑙𝑙 − 1 u = r For motion along the equatorial plane a = = 0 = 0 sin = 1 2 2 Equation (7.16) become π θ θ̇ θ̈ 2 1 + r = (7.18) r c2 Solution of equation (7.18) above gives ϕ̈ � ̇ϕ̇ � ϕ̇ 2r + 0 r = First approximation ̇ r2 ϕ̈ ϕ̇ ≈ Where𝑙𝑙 is a constant of the motion. Hence the radial equation becomes. ϕ̇ 𝑙𝑙 8. Solution of Radial Equation of Motion Solution of radial equation of motion, equation (1.17) above becomes 2 let = 2 = r2 r4 𝑙𝑙 𝑙𝑙 K K2 2K 1 2 3K 2 ϕ̇ ⇒ ϕ̇ r = + + (r)2 + + (8.1) r2 c2r3 c2r2 c2 r3 c2r4 𝑙𝑙 𝑙𝑙 1 let u( ) = ̈ − ̇ − r( ) ϕ dr dr d dr dr du 1 du ϕ r = = = = = d d d d du d u2 d ϕ 1 du̇ ̇ 1 du ̇ du ̇ r = ∙ ϕ = u2 ϕ =ϕ �− � (8.2) τ rϕ2 τu2 d ϕ u2 ϕd d ϕ 𝑙𝑙 Hence ̇ �− � ϕ 𝑙𝑙 �− � ϕ −𝑙𝑙 ϕ d du du2 r = u2 = 2u2 (8.3) d d d 2 Putting r and r into equation (8.1) we geẗ −𝑙𝑙 ϕ �𝑙𝑙 ϕ� −𝑙𝑙 ϕ du2 K2u2 2Ku2 1 3K 2u2 2u2 = Ku2 + + (r)2 + (r) + 2u2 (8.4) ̇ ̈ d 2 c2 c2 c2 c2 𝑙𝑙 Finally obtain −𝑙𝑙 ϕ − ̇ ̇ 𝑙𝑙 − d2u K 3Ku2 K2u 2K du 2 1 du + u = + + + (8.5) d 2 2 c2 c2 2 c2 d u2c2 d By neglecting the last two terms or where⇒ ϕ contribution𝑙𝑙 for the− mean𝑙𝑙 time� ϕ� 𝑙𝑙 � ϕ� d2u K + u = Newton s planetary equation (8.6) d 2 2 ′ d2u K 3Ku2 +ϕu = +𝑙𝑙 ⇒ Einstein s palnetary equation (8.7) d 2 2 c2 ′ The new equation obtained in this work givesϕ two 𝑙𝑙sets of equation⇒

34 Y. Nura et al.: The Generalized Planetary Equations Based Upon Riemannian Geometry and the Golden Metric Tensor

d2u k2 K + 1 + u = (8.8) d 2 c2 2 2 which give rise to post Netwton s planetary equationϕ and� 𝑙𝑙 � 𝑙𝑙 ′ d2u k2 K 3Ku2 + 1 + u = + (8.9) ⇒ d 2 c2 2 2 c2 which give rise to post Einstient s planetaryϕ equation� . 𝑙𝑙 � 𝑙𝑙 ′ ⇒ u2 2 2 9. Conclusions and Results 1 1 f … 1 f … c2 c2 c2 In this paper we have succeeded in deriving equations ≈ � − u2� − 2�� − � [(3.10)-(3.13)] which referred to as velocity vectors 1 2 … 1 2 f equations based upon the golden metric tensors. We further c c obtained equations [(4.16)-(4.19)] which referred to as ≈ �� − u2 � �2 − 2u2�� t2 1 f + f … linear acceleration vector equations. Also in this paper c2 c2 c4 concerted effort were made in order to obtained equation ̇ ≈ � −u2 2−K � (5.15), (5.16), (5.17) and (5.18) are referred to as general 1 + … (2) c2 c2r gravitational intensity (or acceleration due to gravity) based upon the golden metric tensor in spherical polar Next ≈ � − � coordinates. 1 = (t) 1 The results obtained in this paper are now available for t − 1 1 both physics and mathematician alike to apply them in ̇ 1 ̇ u2 2 2 2 2 solving planetary problems based upon Riemannian = 1 1 + f 1 + f geometry. More effort and time are being devoted in order c2 c2 − c2 − to offer solution to equations (5.15-(5.18) obtained in this � − u2� 2 � � � 2 � 1 + 1 f … 1 + f … paper. Also in this paper we offered solution to equation 2c2 c2 c2 (7.18) which gives rise to azimuthal angle equation along 2 the equatorial plane. Solution of equation (7.17) obtained ≈ � u � − ��2� � 1 + … 1 + f … also in this paper which leads as to equations (8.8) and (8.9). 2c2 c2 Equation (8.8) is referred to as post Newton’s planetary ≈ �� u2�� 1 �� equation while, equation (8.9) referred to as post Einstein’s (t) 1 1 + + f + 2c2 c2 planetary equations. These equations played an important − u2 K role in planetary phenomena if handle properly. This paved ̇ 1 +≈ + ⋯ (3) 2c2 c2r the way for applied mathematician and physics to investigate the implications of those equations. We hope to ≈ offer solution of those equations using a powerful method Appendix B of successive approximation in the next edition of this 1 paper. 2K 2 2K 1 1 + c2r − c2r 1 � − 2K� 2 ≈ � 2K� Appendix A 1 1 + c2r − c2r Expansion of t, t2 and t 1 to order of c 2. � − � ≈ � � We get − − ̇ ̇ ̇ 1 1 1 u2 2 2 2 2 t = 1 1 + f 1 + f c2 c2 − c2 − REFERENCES ̇ � −u2 � 2 �1 � � 1� [1] Gnedin, Nickolay Y. (2005). Digitizing the universe, Nature 1 2 1 + 2 f − + 1 2 f … 435 (7042): 572-573. Bibcode 2005 Nature 435…572 G, doi: c c c 10. 1038/435572a PMD 15931201. ≈ � −1 � u2 � ⋯ � � − � 1 f (1) [2] Hogan, Craig J. A Cosmic Primer, Springer, ISBN c2 2c2 0-387-98385. But ≈ − − [3] Howusu, S.X.K (1991). On the gravitation of moving bodies. u2 2 1 2 1 t2 = 1 1 + f 1 + f Physics Essay 4(1): 81-93. c2 c2 − c2 − [4] Howusu, S.X.K (2011). Riemannian Revolution in ̇ � − � � � � � Mathematics and Physics (iv).

International Journal of Theoretical and Mathematical Physics 2017, 7(2): 25-35 35

[5] Howusu S.X.K (2010). Exact analytical solutions of [9] Nura, Y. Howusu, S.X.K and Nuhu, I. (2016). General Einstein’s Geometrical Gravitational field equations (Jos Linear Acceleration Vector Based on the Golden Metric university press). Tensor in Spherical Polar Coordinates (Paper I), International Journal of Current Research in Life Science: [6] Howusu S.X.K Vector and Tensor Analysis (Jos university Vol. 05. No. 12, Pp 627 – 633. press, Jos, 2003) 55-92. [10] Nura, Y. Howusu, S.X.K and Nuhu, I. (2016). General [7] Anderson J.L (1967a). Solution of the Einstein equations. In Gravitational Intensity (Acceleration due to Gravity) Based principles of Relativity physics, New York Academic. upon the Golden Metric Tensor in Spherical Polar Coordinates (Paper II), International Journal of Innovation [8] Braginski, V. B and Panov, V.I (1971). Verification of the Science and Research: Vol. 05. No. 12, Pp 930 – 936. equivalence of inertial and gravitational masses. Soviet physics JETP34: 464-464.