What Is It in Einstein's Theory of General Relativity That Makes the Orbit of Mercury Precess?
David Eckstein: The Precession of the Perihelion of Mercury: "The difference... in the case of Mercury...
...was so big that it demanded an explanation.... Einstein was overjoyed when he calculated near the end of 1915 that his new theory predicted an addition of just 43 arc-seconds per century to the precession of the perihelion of Mercury! He derived the following formula:
where Δφ is the extra rotation per orbit in radians; RS is the Schwarzschild radius of the sun; a is the length of the semi-major axis of the orbit; and ε is the eccentricity of the ellipse.
The effect decreases with increasing distance from the sun and is also greater with highly elliptical orbits than with circular orbits. Therefore Mercury was an ideal candidate...
I know that in the Schwartzchild metric the effective potential looks like:
and this extra term produces, in addition to the inverse-square attractive force of the Newtonian potential, an inverse-quartic force also proportional to the square of angular momentum. But what is the intuition behind where this force comes from as we transition from Newtonian to Einsteinium physics? And why does a weak inverse-quartic force cause precession?
I really ought to know the answer to these, at some level. It saddens me that I do not...
: General Relativity Made Concise: "General Relativity Made Concise John Baez writes:
Einstein's Equation: We promised to state Einstein's [general relativity] equation in plain English, but have not done so yet. Here it is:
Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the x direction at that point, plus the pressure in the y direction, plus the pressure in the z direction. In the final section of this article, we will prove that this sentence is equivalent to Einstein's equation. The reader who already knows general relativity may be somewhat skeptical of this claim. After all, Einstein's equation in its usual tensorial form is really a bunch of equations.... It is hard to believe that the single equation (2) captures all that information. It does, though, as long as we include one bit of fine print: in order to get the full content of the Einstein equation from equation (2), we must consider small balls with all possible initial velocities -- i.e., balls that begin at rest in all possible local inertial reference frames... That is remarkably concise. Of course, the amount of auxiliary stuff you need to surround that equation with in order to calculate things with it is enormous...
And I wish somebody, sometime would tell me why Einstein's theory differs from Newton's in its prediction of the precession of the perihelion of Mercury..."
