^{1}

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The planetary bodies are more of a spheroid than they are a sphere thereby making it necessary to describe motions in a spheroidal coordinate system. Using the oblate spheroidal coordinate system, a more approximate and realistic description of motion in these bodies can be realized. In this paper, we derive the Riemannian acceleration for motion in oblate spheroidal coordinate system using the golden metric tensor in oblate spheroidal coordinates. The Riemannian acceleration in the oblate spheroidal coordinate system reduces to the pure Newtonian acceleration in the limit of c
^{0} and contains post-Newtonian correction terms of all orders of c
^{-2}. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.

Most planetary bodies have been assumed to be spherical and consequently, many treatments of motion involving these bodies have been taken into consideration the spherical approximation of the bodies [

It is worth noting that the description of the planetary bodies mentioned so far have been based on the theory of orthogonal curvilinear coordinates [

In this formulation we have chosen the spheroidal coordinate system based on the approximate representation of the planetary bodies as oblate spheroids. The surface generated by the rotation of an ellipse about its minor axis

is an oblate spheroid. The oblate spheroidal coordinate system

where a is the ellipse’s focal distance and this distance is one-half the ellipse’s foci such that

Therefore, the differential length of a line element

where

Following from Equations (1)-(3), we can define the space time position tensor in oblate spheroidal coordinate system as the set of four labelled quantity

Equation (9) can be written explicitly in terms of the coordinate axes as

and in Einstein’s coordinates

A fundamental quantity in Riemannian coordinate geometry is the metric tensors. Therefore the metric tensor in the oblate spheroidal coordinate system is necessary for the formulation of the Riemannian acceleration in the spheroidal coordinate system. Thus the golden metric tensor in the oblate spheroidal coordinates,

where

is the gravitational scalar potential. From Equations (11)-(15) we can obtain the corresponding contravariant metric tensors,

With the metric tensors in Equations (11)-(15) and Equations (17)-(21), we can proceed to define another quantity which depends on the metric tensors. This quantity is the Christoffel’s symbols of the second kind or the coefficient of affine connections. The coefficient of affine connection or Christoffels’s symbol of the second kind is the set of labelled quantities,

Thus, using Equation (22), and with the metric tensors given by Equations (11)-(15) and Equations (17)-(21), we can obtain all the nonzero terms of the coefficient of affine connections. Hence, after some mathematical simplification, the non-zero terms of Equation (22) is obtained as follows:

and

and

and

and

Therefore, the equations given by (23)-(51) denote all the coefficients of affine connection or Christoffel’s symbols of the second kind, where for example,

The first rank tensor,

is called the Riemannian space-time or 4-dimensional “linear acceleration” tensor;

where

The Equations (54)-(57) are the physically measurable four dimensional acceleration components along the corresponding coordinate axes

Now, substituting Equations (23)-(51) into Equation (52) and after some mathematical simplification, we can then write Equations (54)-(57) explicitly.

Consequently, Equation (54) for

Equation (55) for

Equation (56) for

and Equation (57) for

where

In this paper, we have derived the Riemannian acceleration for the oblate spheroidal coordinate system

Newtonian linear acceleration in the limit of

N. E. J.Omaghali,S. X. K.Howusu, (2016) Riemannian Acceleration in Oblate Spheroidal Coordinate System. Journal of Applied Mathematics and Physics,04,279-285. doi: 10.4236/jamp.2016.42035