Analytical equation for Space-Time geodesics and relativistic orbit equation

[pengkuan]

Orbit equation and orbital precession

General Relativity explains gravity as Space-Time curvature and orbits of space objects as Space-Time geodesics. This concept is extremely hard to understand and geodesics hard to compute. However, if we can express Space-Time geodesics in analytical form, that is, as orbit equation like Newtonian orbit equation for planets, relativistic gravity will be much simpler and most people can understand.

From gravitational and inertial accelerations, I have derived the orbit equation for relativistic gravity, see equation (57), of which I will explain the derivation below. I will refer this equation as the relativistic orbit equation. Albert Einstein had correctly predicted the orbital precession of planet Mercury which had definitively validated General Relativity. The orbital precession that this orbit equation gives is identical to the one Albert Einstein had given ^[1][2], see equation (65).

Because the relativistic orbit equation gives the same result than Space-Time geodesics, it is an analytical expression of Space-Time geodesics which is so simple that personal computer can use to compute geodesics rather than big or super computer. The direct derivation from gravitational and inertial accelerations is a new insight into General Relativity. 

 Relativistic dynamics

• Velocity in local frame

Figure 1 shows a body of mass m orbiting around a body of mass M which sits at the origin of the polar coordinate system. The position of m with respect to M is specified by the radial position vector r, of which the magnitude is r and the polar angle is q. 

Let frame_m be an inertial frame of reference that instantaneously moves with m. Newton’s laws apply in frame_m because m’s velocity is 0, so the inertial acceleration vector of m a_m equals the gravitational acceleration a_g, see (1).  

Let frame_l be the local frame of reference in which M is stationary, m move at the velocity v_l and the acceleration of m is a _l. As frame_m moves with m, its velocity is v_l in frame_l. We transform a_l into a_m using the transformation of accelerations between relatively moving frames, see equation (18) in «Relativistic kinematics and gravity»^[3][4]. This transformation is in (2) where we have replaced a₁ with a_l, a₂ with a_m and u with v_l. 

Equating (1) with (2) we get (3), both sides of which are dotted with the vector v_l·dt_l in (4) to give  on the left hand side, see (5), and  on the right hand side, see (6). Plugging (5) and (6) into (4) gives (7), which we integrate to give (8), with K being the integration constant. The value of e^K is given in (9). Rearranging (8) gives (10), which expresses the local orbital velocity v_l due to the gravitational field of M. 

• Velocity in the frame at infinitely far

The orbit of a planet is the curve drawn by its position with respect to a frame in which the gravitational effect of M is 0. This frame is infinitely far from M and is labeled frame_¥. In this frame the time is t_¥, the radial position vector is r and the velocity vector v_¥ equals the derivative of r with respect to t_¥, see (11). 

The velocity vector v_l in frame_l equals the derivative of the same r with respect to the time in frame_l, t_l. We transform v_l into the product of v_¥ and  in (12).  is the time dilation factor of the Schwarzschild metric around M, see (13). By plugging (13) into (12) we obtain the relation between v_l and v_¥ in (14). Substituting (14) for v_l in (10) gives (15), which expresses v_¥ with the gravitational field of M. In the case of planets,  is very small and the term  is simplified into the Taylor’s series in (16). 

Notice that the right hand side of (15) is the product of two parentheses, the first represents gravitational acceleration, see (10), the second represents the time dilation of Schwarzschild metric, see (13). 

Substituting (16) for  in (15) gives (17), from which we drop out the third order term  to get  in (18), where u_¥ is the velocity vector of m in frame_¥ correct to the 2nd order of   . 

• Differential equation for orbit

The general expression of velocity in polar coordinates is given in (19) and the velocity squared u_¥² in (20). For expressing u_¥² with h the constant of Kepler’s second law given in (21) ^[5][6], we transform  in (22), which is plugged with (21) into (20) to give (23), which expresses u_¥² in terms of    instead of r. 

We divide (23) with c² to express  in terms of , see (24). For transforming (18), we plug (24) and  into (18) to get (25) and (26), which is the differential equation for orbit in frame_¥. Solving (26) will give the relativistic orbit equation. 

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Figures and equations are in the pdf here:Analytical equation for Space-Time geodesics and relativistic orbit equation.pdf