General equation for Space-Time geodesics and orbit equation in relativistic gravity

  • Orbit equation and orbital precession

General Relativity explains gravity as Space-Time curvature and orbits of planets as geodesics of curved Space-Time. However, this concept is extremely hard to understand and geodesics hard to compute. If we can find an analytical orbit equation for planets like Newtonian orbit equation, relativistic gravity will become intuitive and straightforward so that most people can understand.

From gravitational force and acceleration, I have derived the analytical orbit equation for relativistic gravity which is equation (1). Below I will explain the derivation of this equation. Albert Einstein had correctly predicted the orbital precession of planet Mercury which had definitively validated General Relativity. Equation (2) is the angle of orbital precession that this orbit equation gives, which is identical to the one Albert Einstein had given [1][2].

If this orbit equation gave the same result than Space-Time geodesics, then everyone can compute the orbit of any object in gravitational field which obeys General Relativity using personal computer rather than big or super computer. Also, everyone can see how gravity leads to Space-Time curvature without the need of knowing Einstein tensor.

The derivation of the orbit equation is rather tedious and lengthy. So, for clarity of the reasoning and explanation, I have collected all the mathematical equations in the last section “Derivation of equations”, in which full details are provided to help readers for checking the validity of my mathematics.

  • Relativistic dynamics
    a) Velocity in local frame

Take an attracting body of mass M around which orbits a small body of mass m, see Figure 1. We work with a polar coordinate system of which the body M sits at the origin. The position of the body m with respect to M is specified by the radial position vector r, of which the magnitude is r and the polarangle is q.

Let the frame of reference “frame_m” be an inertial fame that instantaneously moves with m. Frame_m is the proper frame of m where the velocity of m is 0. So, Newton’s laws apply in this frame. Let am be the acceleration vector of m in frame_m and the inertial force of m is m·am, see equation (3). The gravitational force on m is given by equation (4). Equating (4) with (3), we get equation (5), the proper acceleration of m caused by gravitational force in frame_m.

Let “frame_l” be the local frame of reference in which M is stationary. In frame_l m is under the effect of gravity of M, the velocityvector of m is vl and the acceleration of m is a l. As frame_m moves with m, it moves at the velocity vl in frame_l.

The acceleration of m in frame_m and frame_l are respectivelyam and al. To transform al into am we use the transformation of acceleration between relatively moving frames which is the equation (18) in «Relativistic kinematics and gravity»[3][4], in which we replace a1 with al, a2 with am and u with vl. Then the transformation between am and a l is equation (6).

Equating (5) with (6) we get equation (7), both sides of which are then dotted by the vector vl·dtl, see equation (8). On the left hand side of (8), we find dr the variation of the radial distance r, see (9). On the right hand side, we find the variation vector of velocity dvl, see (10), which, dotted by the velocity vector vl, gives dvl2/2 in (11).

Plugging (9) and (11) into (8), we get (12), both sides of which are differential expressions, see (13) and (15). Then, plugging (13) and (15) into (12) gives (16) which is a differential equation. (16) is integrated to give (17), with K being the integration constant. Then, we rearrange (17) to express vl2/c2 in (18), which relates the local orbital velocity vl to the gravitational field of M.

The value of eK is determined at a known point 0 at which the velocity is v0 and the radial distance is r0, see (19).

General equation for Space-Time geodesics and orbit equation in relativistic gravity