- 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 *a*_{m} be the acceleration vector of m in frame_m
and the inertial force of m is m*·**a*_{m}, 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 *v*_{l}
and the acceleration of m is *a*_{ }_{l}. As frame_m moves with m, it moves at the velocity* v*_{l} in frame_l.

The acceleration of m in frame_m and frame*_l* are respectively*a*_{m} and *a*_{l}. To transform *a*_{l} into *a*_{m} 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 *a*_{1} with *a*_{l}, *a*_{2} with *a*_{m} and *u* with *v*_{l}.
Then the transformation between *a*_{m} and *a*_{ }_{l} is equation (6).

Equating (5) with (6) we get equation (7), both sides of which are then dotted by the vector *v*_{l}·d*t*_{l}, 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 *d***v**_{l}, see (10), which, dotted by the velocity vector *v*_{l}, gives d*v*_{l}^{2}/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 *v*_{l}^{2}/c^{2}
in (18), which relates the local orbital velocity* v*_{l} to the gravitational field of
M.

The value of e^{K} is determined at a known point 0 at which the velocity
is *v*_{0} and the radial distance is *r*_{0}, see (19).

…

**General
equation for Space-Time geodesics and orbit equation in relativistic gravity** https://www.academia.edu/44540764/Analytical_orbit_equation_for_relativistic_gravity_without_using_Space_Time_geodesics

https://pengkuanonphysics.blogspot.com/2020/11/analytical-orbit-equation-for.html