Applied Mathematics
Vol.3 No.12(2012), Article ID:25465,7 pages DOI:10.4236/am.2012.312264

One Common Solution to the Singularity and Perihelion Problems

Branko Sarić1,2

1Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia 2College of Technical Engineering Professional Studies, Čačak, Serbia


Received September 18, 2012; revised October 18, 2012; accepted October 25, 2012

Keywords: Celestial Mechanics; Planets; Rings


With a view to surmounting the singularity problem on the one hand, as well as the moving perihelion problem of the planets on the other, as two acutely vexed questions within Newton’s gravity concept, the goal of this paper is a modification of Newton’s gravity concept itself.

1. Introduction

It would be difficult to exaggerate the influence of Newton’s theory of gravitation on the subsequent development of physics. As well as explaining Kepler’s laws of planetary motion, Newton’s theory was central to the successful mathematization of physics using the newly-invented calculus and it served as a paradigm for the later theories of electrostatics and magnetostatics. However, new insights into Milky Way satellite galaxies raise awkward questions for cosmologists: Do we have to modify Newton’s theory of gravitation as it fails to explain so many observations? In other words, although Newton’s theory does, in fact, describe the everyday effects of gravity on Earth, things we can see and measure, it is conceivable that we have completely failed to comprehend the actual physics underlying the Newton’s force of gravity. In addition, Newton’s theory does not fully explain the precession of the perihelion of the orbits of the Planets, especially of planet Mercury. Namely, it has been experimentally stated that the perihelion of Mercury’s orbits moves into the plane of its planetary motion around the Sun. In other words, all planetary motions of Sun’s planetary system depart from elliptical orbits obtained from Newton’s gravity theory, [1]. By the strict Schwarzshild-Droste’s solution to the static gravitational field with spherical symmetry, in the general Einstein’s relativity theory, the perihelion problem has been approximately solved, [1]. On the other hand, Einstein’s theory has some difficulties hard to be overcome such as the problem of singularity, that occurs in Newton’s theory too (all relevant physical variables, such as velocity, force, kinetic and potential energy, don’t exist at point of singularity). Accordingly, in order to solve simultaneously these two acutely vexed questions within Newton’s gravity theory, we present, in this research paper, an approximative modification of Newton’s gravity concept itself. The outline of this article is as follows: In the Preliminaries, the space-time continuum (the integral space), as an ambient space, is completely defined. In Section 1 we establish a causal connection between the expression for the kinetic energy of a material point and the Minkowski metric in the four-dimensional space-time continuum. In addition, in two separate subsections of this section we derive Newton’s equations of motion and the relativistic Hamilton-Jacobi equation for a free particle. Since the dynamic (Newton’s) equations of motion are formally derived from geodesic equations in Section 2, this section together with Appendix at the end of the paper provide a possibility of further work on the modification of Newton’s gravity theory in Section 3. In this last section we show that a comprehensive analysis of particle motion under the modified Newton’s gravity force leads to the perihelion motions of a Planet’s elliptical orbit.

2. Preliminaries

By a material point, introduced for the purpose of an useful idealization, one means a geometrical point, which is spatially no dimensional on the one hand and exactly fixed mass on the other. Closely related to the notion of a geometrical point is the set of values of some arbitrary variables denoting the contravariant co-ordinates of the real -dimensional configurative space. The geometrical point, defined by a set of zero values, is the zero co-ordinate point. If one of arbitrary variables is the time variable, then the space aforementioned becomes the space-time continuum (shortly called the integral space), [2]. As it was noted in [3] the value of is called moment or instant.

The set of all geometrical points of the spatial subspace of the integral space, to which the mass can be joined in some strictly monotonous sequence of permitted instants of the time, makes an odograph usually referring to the a trajectory (motion path) of. The time variable is taken for a unique independent variable, so that all remaining spatial variables are functional variables. Across all the future text Greek indices take values, and Latin ones. In the space-time continuum the aforementioned trajectory of blossoms into an integral curve. The vectors

and defined with respect to the origin are position vectors of in the space-time continuum and in the spatial subspace of the integral space, respectively. The concept of a vector in vector hyper-dimensional spaces should be conditionally comprehended in the sense of its geometrical presentation in a form of segments. Hence it bears a name linear tensor, [4]. Covariant vectors, where

denotes, form a covariant vector basis

of the integral space. The vectors, such that at any point of the space, where the second order system (Kronecker’s delta-symbol, [5]) is the identity matrix, form a dual basis of

. The differential of the position vector

of is defined by, where the so called Einstein’s convention is applied to a summation with respect to the repetitive indexes (uppers and lowers), herein as well as in the further text of the paper.

3. The Action Metric in the Integral Space

Since the integral space is a metric affine space, whose linearly independent basis (fundamental) co-ordinate vectors reduced to the origin form an -hedral basis, it follows that if is a line element of the metric affine space of the spatial continuum, then the expression for the kinetic energy of can be stated in more appropriate form:


considering the fact that the basic mechanical (kinematics and dynamics) parameters of are its velocity, quantity of motion and kinetic energy. A term of, where is nominally equal to the light velocity in vacuum, can be added to both sides of the previous equation, as follows


For let be such that. Then, (2) becomes


This means that if, where is a line element of the metric affine space of the space-time continuum, then the four-dimensional integral space has the Minkowski metric, [1,4]. So, in this case the Minkowski metric (3) represents the kinetic energy of in the integral space. Hence, the Minkowski metric is the kinetic metric of the integral space, [6].

If the Pfaff form is absolute differential, that means that there exists a scalar valued function such that, then and


where, and is the total mechanical energy of.

Now, we can start with the action in the Lagrange sense along a motion path of in the integral space [4, 6],


Since it follows from (4) and (5) that


Let us introduce an action line element, thoroughly explained in [6], in such a way that


Accordingly, the action metric is as follows



are the metric tensors of and, respectively.

3.1. Newton’s Equations of Motion

By the well-known Maupertius-Lagrange’s principle [4], the path motion of is just the path along which the action is stationary, more precisely along which the following two mutually equivalent conditions


where is the variational operator, are satisfied. By (7), the previous conditions are reduced to


The second condition in (9) leads to the Euler Lagrange equations


where and

which yield Newton’s equations of motion


3.2. The Relativistic Hamilton-Jacobi Equation for a Free Particle

Analyze (5) again, but now let be a function ofthat means that. As


we introduce the functional, nominally equal to, such that


as well as the functional satisfying the condition


which together with (15) yields


since. Hence, is Lagrangian of. Further, since for, see (14), it follows from (15) that and


The previous equation is the Hamilton-Jacobi one, so that is the principal Hamilton’s functional of. Clearly, the Hamiltonian of is equal to, more precisely to the integral of motion, considering the fact that the kinetic energy of is a homogenous square function of. Now, by (14) and (15), we have, so that


This together with (17) leads to the second form of the Hamilton-Jacobi equation


In addition,


which together with (8) yields


where. These two equations are obviously analogous to the relativistic Hamilton-Jacobi equation for a free particle, see [7,8].

4. The Binet Differential Equation

As is well-known from the tensorial analysis, see [6], all curves of the integral space, for which the condition (10) is satisfied, are geodesics, and the absolute Bianchi (covariant) derivative of the unit tangent vector along geodesics is equal to zero (the vector projection of onto the tangent hyper-plane of the integral space is equal to zero). Thus, the geodesic equations are as follows


where denotes, and

are the second kind Christoffel symbols with respect to the action metric space. Let. Since, see (8), it follows that


where are the second kind Christoffel symbols with respect to the Euclidean metric space. A new form of the geodesic equations (22), for a constrained material point, is as follows


which yields





So, (25) represents the Euler-Lagrange differential equations of the extremal curve in the explicit form, and at the same time Newton’s second law of motion under the action of a potential force in the contravariant form:


Accordingly, one may conclude that the dynamic (Newton’s) Equations (26) of motion are formally derived from the geometric Equations (22).

In the case of the free motion of, when, both the kinetic and action metric form of the integral space are pseudo-euclidean, while integral curves are straight-lines (see Appendix), as it was thoroughly explained in the monograph by [6]. On the other hand, the well-known Binet differential equation for central force motion of


is obtained by differentiating (58) (see Appendix).

5. Modified Newton’s Gravity Concept

For the conservative Newton’s gravity force the expression is as follows, where is the gravitational radius, so that (27) is reduced to


where and.

Since, in the limit as, Newton’s gravitational potential tends to infinity, it is logical to assume that is the first-order MacLaurin series approximation of the exponential function, so that and


Accordingly, the modified Binet differential equation for the modified central Newton’s gravity force


is as follows


5.1. A motion Under the Action of

Start with Newton’s second law of motion


Multiply (32) on the right by the sector velocity vector as follows


Since it follows from (33) that


where the vector satisfying the relation is no longer an element of Milankovic’s constant vector elements, more precisely is no longer Laplace’s integration vector constant, see [9]. If we now multiply by, we get


Since, it follows from (35) that


where is an angle between and. This equation describes the motion of under the action of the modified Newton’s gravity force. Conditionally speaking, there is no formal difference between (36) and its analog in the ordinary Newton’s gravity theory. The key difference lies in the fact that is no longer constant vector.

For let and, where is the unit vector orthogonal to. Then, from (34) we get


which together with (36) yields


where and


In addition, the dot product leads to




If is the angle between and, it follows from (36) and (40) that




Note that, whenever. If we now multiply (38) by we get


which together with (41) and (42) yields


whenever. If, then


Since, see (38), it follows from (36) that


which together with (45) finally yields


This result we can also get explicitly from (46). Namely, if is the polar angle, then. Therefore, it follows from (46) that


Since we have


that is just the same as (47). Hence,


So, as, where and are the semimajor axis and the eccentricity of the orbit, the following angle value


is a very good approximation for the perihelion regression per one revolution of the Planets.

5.2. The Modified Perturbing Force

If, in addition to the modified Newton’s gravity force, we include the modified perturbing force [10]

where and are the radial vector between the Planet and the perturbing planet (whose orbit is assumed to be circular and coplanar with Mercury’s orbit) and the gravitational radius for the perturbing planet, respectively, then

where is the angle between and, and is the unit vector perpendicular to. Thus, the second equation of (35) becomes

where and

This vector is the modified Laplace’s integration vector (or more precisely, the modified Laplace-RungeLenz vector). Their original versions come from the ordinary Newton’s gravity theory. If we denote

by, then we have

6. Conclusion

The mathematical model of a material point motion in the three-dimensional spatial subspace of the four-dimensional space-time continuum and in the field of the action of a conservative active force is analogous to Newton’s mathematical model of the classical mechanics. In addition, the metric of the integral space, which represents the kinetic energy of a material point from the viewpoint of that space, is the Minkowski metric from Einstein’s relativity theory. Accordingly, it can be said that in the paper a new connection has been established, in contrast to an approximative one, between the classical Newton’s mathematical model and the relativistic Einstein’s mathematical model.

On the other hand the approximately modified Newton’s gravity concept is not, from any point of view, in collision with old Newton’s one. At the same time it solves the acutely vexed questions within old Newton’s gravity concept (the singularity and perihelion problems). Furthermore, analyzing the analytical expression for the modified Newton’s gravity force, we can separate the four indicative domains of its field of the action (see Figure 1). The first one is a domain of the weak action on finitely small distances. The second one is a domain

Figure 1. Modified Newton’s gravity force.

of the strong action in a neighborhood of the gravitational radius

. The third one is a domain of the action on finitely large distances relative to the gravitational radius and with the relatively small velocities relative to the light velocity, and the fourth on finitely large distances relative to the gravitational radius and with velocities that are comparable to the light velocity. Previously separated domains of the field of the action of the modified Newton’s gravity force it would be desirable to compare to the fields of the action of the four so far non-unified fundamental forces (weak and strong nuclear interactions, gravity and Lorenz’s electromagnetism). Clearly, all of these facts aforementioned could be subject of further analyses. Note at the end that a correction to Newton’s gravity law in the form of the functional dependence irresistibly reminding of the modified Newton’s gravity force, and obviously wrongly called the fifth force, has been revealed by a reexamination of the old attraction data and careful new force measurements presented in [11].


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Appendix: The Freee Motion of in the Integral Space

Let us start with the Euler-Lagrange equations




as the condition for the action (12) to be stationary. The geodesic Equations (13) are explicitly obtained from it in a known way. If spatial co-ordinates are spherical ones, then the components of depend only on and, so that it follows from (52) that




that leads to




Let the polar extension and the polar angle be intensities of and an angle between the position vector and the polar axis passing through the origin and the perihelial point, respectively. Then, since, where is the so-called sector velocity vector, it follows from the condition (57) that the motion is the plane one and. As then we obtain finally from (5), (10) and (57) that


that just leads to the Binet differential equation for free motion in plane polar co-ordinates


The solution, where is the perihelial distance, to this differential equation, defines a straightline in plane polar co-ordinates.