This is an 'engineering-view' of relativistic orbits, written in the terms and mathematics that most engineers are comfortable with, yet it is scientifically rigorous. It is an attempt to demystify the rather complex behavior of relativistic orbits.

The relativistic orbit of a particle with mass m at distance r from an uncharged, non-rotating black hole with mass M can be obtained from a set of 'flat space' equations of pseudo-forces^[1], a-la Newton's gravity:

Here g_tt = 1-2GM/(rc²) , the gravitational redshift factor squared, valid for r > 2GM/c². Force F_r is the radial gravitational pseudo-force on a particle moving with radial speed v_r and transverse speed v_t. It is easy to spot that it is the Newtonian force, modified by some relativistic terms. Force F_t is an apparent transverse gravitational pseudo-force on the particle, as is evident from its trajectory - it does not exist in Newtonian gravity. Numerical integration of equations (1) and (2) gives the relativistic orbit.

The origins^[2] of these equations will be discussed later, but in brief: the normal Newtonian gravitational force is modified by the curved spacetime through which the orbiting particle moves. It causes additional apparent pseudo-forces on the particle, as viewed in the Schwarzschild coordinate system. Below we will discuss the pseudo-forces term by term and show their effects on the orbit of a particle around a black hole.

For a pure Newton orbit, we simply set c => infinity, hence g_tt = 1, F_r = -GMm/r², F_t = 0 and we have the familiar Kepler ellipse that repeats over and over. The black circle at the focus of the ellipse represents the Schwarzschild (event horizon) radius R_H=2GM/(rc²) of the point mass, drawn to scale. The small blue circle represents the particle at time t. The periapsis of the orbit is at 5R_H and the (transverse) velocity at that point is v_t = 0.3744c, chosen for a convenient orbit size and some other esthetical considerations. All the orbits shown here have these initial conditions and are drawn to the same scale.

If we set c to its normal value, but ignore the influence of velocity, i.e., we only consider the effect of gravitational time dilation (or redshift) g_tt = 1-2GM/(rc²), the radial force F_r becomes:

The Newtonian force has been multiplied through for clarity. That positive 2(GMm)²/(r³c²) term reduces the magnitude of the negative Newtonian force and modifies the orbit as shown below.

Apart from a larger ellipticity due to the lesser gravity, there is also a retrograde precession of the orbital ellipse. The closer to the event horizon, the more the Newtonian force is reduced, meaning that the orbit curves less when close to the black hole. This causes the orbit to precess in a retrograde direction.

If we now also factor in the effect or radial velocity v_r, while ignoring transverse velocity, we have the radial force as:

The radial velocity term reduces the magnitude of the negative Newtonian force further and gives the orbit an increased ellipticity. The radial velocity does not change the orbital precession appreciably, so the retrograde periapsis shift remains. At the periapsis, the radial velocity is zero, so there the third term vanishes. Near the periapsis it is small, explaining the small effects on ellipticity and precession.

That third term can be viewed as a sort of 'anti-gravity' effect that is proportional to the inverse square of the distance from the hole and to the radial speed of the particle squared. It is however a pseudo force that is due to the curvature of space and the relativistic effects of velocity and sadly, it cannot be used to propel a spaceship.

Next we also factor in the effect or transverse velocity v_t, but we keep the transverse force F_t = 0 for now. The radial force equation is now complete:

Note the differences between the radial and transverse velocity terms - apart from the factors +3 and -2 respectively, there is a g_tt below the line in the radial term, caused by the fact that radial movement encounters changing spatial curvature, while pure transverse movement stays at a constant curvature.

The fact that the last (negative) term works with the normal negative Newtonian force, drastically reduces the ellipticity of the orbit and surprisingly, it causes the orbital precession to become prograde. This is due to the large value of v_t near periapsis, causing a stronger inward force and hence the orbit curves more than in the previous cases. The prograde precession for the particular initial conditions is about 25 degrees per orbit, periapsis to periapsis.

One may think that this is it - there can after all only be radial forces in the symmetrical gravitational field of a 'point source'. Not so in the curved spacetime around a black hole.

Only when we also consider the transverse force, eq. (2) above, do we have the full relativistic orbit. It has an increased ellipticity and an increased pro-grade precession, caused mainly by the transverse pseudo-force. The particular orbit chosen precesses by exactly 90 degrees per orbit, so that after four revolutions it repeats itself, creating a simple four-leaf pattern.

Note from eq. (2) that the transverse force is dependent on the product of the radial and transverse velocities. If the product v_rv_t is positive, i.e., during the outwards portions of the orbit, the transverse pseudo-force works with the orbital movement. During the inwards portions of the orbit, v_rv_t is negative and the transverse pseudo-force works against the orbital movement. This balances the energy equations nicely, but it must obviously cause deviations from Newtonian orbits.

It is clear that the transverse pseudo-force causes the bulk of the periapsis precession of relativistic orbits. It is however not a real force, but again an artifact of the Schwarzschild coordinates chosen. While the particle moves in the curved spacetime, its coordinate positions over time correspond to all the pseudo forces discussed above.

It is hoped that this post and possible discussions that may follow will make relativistic orbits a little less mysterious!^[3]

Jorrie

[1] All Newtonian gravitational forces on a point particle are pseudo-forces, because a freely moving particle does not 'feel' accelerations. It is only when the particle's movement is measured in some static coordinate system that one can deduce that there is a Newtonian forces acting on it. In general relativity, such motion is called geodesic movement and is force-free.

[2] The mathematical details (and a lot of explaining) of the relativistic orbital accelerations are contained in Appendix C of Relativity-4-Engineers.

[3] Also look at the beautiful patterns of the highly relativistic orbit in the previous post in this CR4 Blog again. Note that that 'four-leaf-clover' performs a few close-by, nearly circular orbits around the black hole before zooming out to the apoapsis and back again. The same equations and principles do however apply.

-J