Author: Watsisname
Original post on the forum: link

Background

Often in discussions of Mercury’s perihelion advance, the effect is shown greatly exaggerated.  Sometimes even the orbit is shown much more eccentric than it really is.  Before we dive in, let’s take a moment to get some perspective on the true shape of the orbit and appreciate the scale of the solar system.

 

The Solar System (June 12, 2018)

 
All orbits are ellipses, but most planetary orbits are close enough to circular that it can be hard to tell at a glance.  Mercury is the biggest exception with an eccentricity of 0.21.  (Or Pluto at 0.25).  What’s interesting, and what this post is all about, is that Mercury’s orbit is not stationary.  The ellipse slowly shifts around, it’s perihelion point advancing by about 574 arcseconds (or 0.159 degrees) per century.  Much of this (531 arcseconds) can be explained by the perturbations from the other planets.  However, the remaining 43 arcseconds per century are “anomalous”, unaccounted for by Newtonian mechanics.  

The puzzle of this additional observed precession was first noted by LeVerrier in 1859.  Following the successful prediction of the existence and location of Neptune from similar perturbations on Uranus’ orbit, LeVerrier proposed that Mercury’s orbital precession could be caused by another undiscovered planet inside of Mercury’s orbit, which he named Vulcan.  Astronomers searched for this missing inner planet during solar eclipses.  Of course, Vulcan was never found… 

The mystery was not solved until Einstein began developing his general theory of relativity.  When he applied his general relativistic equations to the problem in 1915, he found that they exactly predicted the additional 43 arcseconds per century.  He considered this one of his greatest achievements.

Imagine my joy at the recognition of the feasibility of general covariance and at the result that the equations correctly yield the perihelion motion of Mercury. I was beside myself for several days in joyous excitement.

Physics

So, why does it happen? Is there an extra energy involved?  (Spoiler: Yes, in a way!)

For an elliptical orbit, we can think of the orbital motion as being made of two parts: an oscillation in azimuth (angle around the Sun), and an oscillation radially (in and out).  In Newtonian mechanics, the periods of these two oscillations are exactly equal.  That is, each time the planet completes one cycle around the Sun, it also exactly completes one cycle radially in and out.  Therefore in Newtonian mechanics there is no precession of the orbit (besides that which is caused by the influences of the other planets).

In general relativity this is no longer true.  The periods of the two oscillations are different!  To see why, it is easiest to examine the shape of the “effective potential”, which you can think of as being just like the potential energy well around a spherical mass, except also taking into account the angular momentum of the orbiting body.  

If you are unfamiliar with the concept of a potential energy well, imagine a rolling landscape of hills and valleys.  If you place a ball at the bottom of a valley, it will just sit there.  It’s in a stable equilibrium.  On the other hand, the top of a hill is an unstable equilibrium.  If you displace the ball slightly from the top of the hill and let it go, it will continue to roll down, exchanging gravitational potential energy for kinetic energy as it drops in height.  Finally, if you have some bowl-shaped valley and drop the ball from rest at some initial height, then it will roll down to the bottom, and (ignoring friction) keep going, rolling back up the other side until it reaches its former height, and then roll back again, oscillating back and forth indefinitely.  

Because the gravitational potential energy near Earth’s surface is simply proportional to your altitude, if you make a plot of the potential energy of a ball in this landscape as a function of its position, it will be exactly the shape of the landscape.  This form of modelling of the potential is extremely useful.  In general, we can use “potential energy wells” as a way to understand motions of objects subjected to different kinds of attractive and repulsive forces, from balls rolling down hills to the vibrations of atoms bound together in molecules.

An elliptical orbit is an oscillation in an effective potential well, as well.  There is a hill to climb as you move outward due to the gravitational attraction of the star, and there is also a hill to climb as you move inward!  This is because the sideways motion is associated with angular momentum, and angular momentum acts like a repulsive centrifugal force.  By conservation of angular momentum, the sideways velocity increases as your orbit swings inward, increasing the centrifugal repulsion and making it harder to approach the center.  If the angular momentum is large enough, then the inward fall can be halted, and you swing back out again, making an orbit!  

So if we can compute the effective potential well for a planet orbiting the Sun, we can figure out some things about its motion.  The anomalous precession of an orbit can also be understood this way, by comparing the motion from the Newtonian potential with what happens in the general relativistic version of the potential.  Let’s try it!

Math

The goal here is to try to understand how things work conceptually, and I will try to emphasize those concepts (especially physical concepts) as we go along.  The hard truth is that much of it will still require going through the math in order to access it, but I’ll do my best to turn that math into something visualizable and comprehensible.

In Newtonian mechanics, the gravitational potential energy of a mass m a distance r around a spherical mass M is

This describes a simple curve which plummets downward indefinitely as you move to smaller radii.  As it should.  If you drop something near a massive object and give it no sideways motion (no angular momentum), then it will fall straight into it.

If we instead give the object some angular momentum L, then the potential is modified:

where μ is the reduced mass, which for a planet orbiting the Sun (m << M), reduces to approximately m.

Let’s see what this looks like for Mercury:

Again the way to think about this is that the potential describes a landscape that a ball will frictionlessly roll across.  To visualize that, I added the large red dot to represent Mercury’s position as a function of time.  The acceleration is determined by the slope of the potential, and I plotted the motion with 2 days per frame.  With 44 frames in all, this traces out one complete period of Mercury’s orbit in the radial motion.  I also added a horizontal line to show the total energy of Mercury (its gravitational potential energy + kinetic), which is constant along the orbit.  Where this line is above the effective potential defines the range of distances from the Sun that Mercury’s orbit will cover.  The vertical lines represent the extremes (Mercury’s perihelion and aphelion distances).

Why is the Newtonian effective potential shaped this way?  
Because of Mercury’s amount of angular momentum, the shape of the effective potential that it sees near the Sun is a valley with hills on either side.  The hill at large radii is due to the Sun’s gravitational attraction, while the hill at small radii is due to centrifugal repulsion.  Even though the gravitational force grows stronger at closer distances, for an orbit the centrifugal force gets stronger more quickly, due to the conservation of angular momentum which increases your sideways speed.

A useful concept to keep in mind here is that the period of the radial oscillation depends on the curvature of the potential well.  If the well opens up more sharply, then the average acceleration is greater, and the period is shorter. 

And here’s one other useful trick.  If the amplitude of the oscillation is not too large (does not reach too far away from the minimum of the well), then we can approximate the well as a parabola around the minimum.  Then for a parabolic well the motion is described very simply as a simple harmonic oscillator.  For Mercury this approximation isn’t particularly good (the well is noticeably asymmetric over the region Mercury covers), but it’s not terrible either.  

For a simple harmonic oscillator, the frequency is given by

which for the radial motion leads to

The frequency for the angular oscillation on the other hand is given by

Which leads to the exact same expression:

So we see in Newtonian mechanics the periods are exactly equal and there is no anomalous precession.  Now let’s see how this gets modified when we move to General Relativity.

The Effective Potential in General Relativity

Einstein’s general theory of relativity describes gravitation as distortion of the geometry of space-time.  I’m sure you’ve heard of the rubber-space-time sheet analogy, and perhaps seen the interactive displays at science museums where you can roll coins or marbles down a funnel.  Something like this (I love this guy’s presentation by the way).

These are classic and excellent tools for teaching how mass distorts the shape of space-time, and how the shape of space-time gives the orders for how other masses will move.  Now if you watched it carefully you might have noticed something interesting.  Not only can you get elliptical orbits in these demonstrations, but highly elliptical orbits that fall deep down into the well also precess!
We could just stop right here and say “this demo explains Mercury’s precession”!  But that would be not quite right.  The reason precessing orbits appear in these funnels is because their shapes do not match the Newtonian potential, and in general if you change the shape of the potential you can make all kinds of weird trajectories occur.  But these funnels also do not correctly reproduce the general relativistic potential.  So the motions we see on them do not correspond to real celestial motions, even if they appear qualitatively similar.
A motivation of your post was to go beyond the common but not completely correct explanations to get closer to “what’s really going on”.   So let’s get the motions “the right way”.  We will use the general relativistic effective potential for an orbiting body:
Again I don’t want to get lost in math, but it’s worthwhile just to look briefly at what the math is saying here.  Notice this still has the exact same two terms from the Newtonian effective potential: an attraction that goes as -1/r, and a repulsion that goes as +1/r2.  But a new term is added: another attractive term that goes as -1/r3. This means that at very small radii, the -1/r3 term dominates, and gravitation becomes attractive again, dominating even over the centrifugal effect of your orbital velocity.  
Next we will apply this to Mercury.

Mercury’s Orbit in General Relativity

Here’s where everything comes together.  Now we can gain some insight by plotting Mercury in the general relativistic potential.  We have one small hurdle though.  The Sun’s gravitational field is pretty weak by general relativistic standards.  If I plot the general relativistic potential on top of the Newtonian potential, you will not be able to see the difference between them.  
I guess I could just plot their difference on a log-scale… but I have a better idea.  I’ll instead “make general relativity stronger”, by reducing the speed of light to 1/1000th of its actual value.  Here’s what happens:
Observations:
  • The minimum in the effective potential well is deeper and displaced slightly inward.
  • The oscillation spreads over a wider range of radii for a given energy than before.
  • The well opens up a little more steeply.
Remember that for oscillations that are not too large about the minimum of a well, the frequency of the oscillation is related to the curvature.  Because this well opens up more steeply, we should expect Mercury’s radial oscillation to be a bit faster than before.  To check, I iterated through the radial and angular equations of motion and plotted the results, for the Newtonian case and for general relativity with c slowed by a factor of 1000.  The radial motion is the black curves while the angular motion is in blue.  Vertical blue lines represent the completion of one 360° circulation about the Sun, while each peak in the black curves represent one complete oscillation radially (from aphelion to aphelion).

Indeed, with general relativity “turned on”, the radial oscillation is faster than before.  But so is the angular oscillation, even more so!  The two oscillation periods are unequal, and Mercury completes one 360° revolution in less time than it takes to complete one oscillation radially.  Now at last we directly see the precession!  Here it is as an orbital plot:

Why is the time to complete one angular orbit reduced so much, and more so than the radial one?

By conservation of angular momentum, the sideways velocity increases at smaller radii.  Here we have a well which has changed shape and moved inward.  With General relativity turned on, Mercury plunges inward a little closer to the Sun.  There it not only moves faster sideways, but it also has a smaller circle to complete.  These effects conspire to allow the orbit to cycle around faster than it otherwise would in Newtonian gravity, and grow out of phase with the radial motion. 

 
It all ultimately arises from the change in the effective potential, and to call back to the question of whether energy is involved, an effective potential defines the change in potential and kinetic energy as an object moves through some landscape (curved space-time in this case).  So yes, this orbital precession can be thought of as an effect of how general relativity modifies the exchange of potential energy, by changing the geometry of space-time.
I should also say that this anomalous precesion does not only happen for Mercury.  It happens to all orbits!  But it is strongest for Mercury, since it is closest to the Sun.  These are values (arcseconds per century) for all the inner planets:
  • Mercury:  42.98
  • Venus:  8.62
  • Earth: 3.85
  • Mars: 1.35
That, I think, completes the story of Mercury’s Perihelion Advance.  We’ve seen how to model motions by using effective potentials, and applied them to Mercury to visualize the changes that arise from general relativity.  By “turning up general relativity”, we can see the precession in even a few orbits and understand why it happens.
However, there is still more that we can cover on this topic.  The Sun’s gravitational field is weak, so we’ve only explored the weak-field effects introduced by general relativity to orbital motions.  Really amazing things happen if we move into stronger fields!  For anyone interested, in the next section I move away from the solar system, and explore what happens to orbits near black holes.

Going Further: The Bizarre Orbits in Strongly Curved Space-Time

Recall back to the expression for the general relativistic effective potential, and the attractive -1/r3 term it introduced.  At very small radii, this term will dominate, even over the centrifugal effect caused by angular momentum.  This is exactly why there are black holes.  Get too close to a sufficiently massive and compact object, and the gravitation totally overwhelms.  Even light cannot move fast enough to withstand it.  

The Sun is very far from being a black hole.  To become one, it would have to be squeezed down into a space smaller than 6km across.  Whereas Mercury orbits at around 60 million kilometers away.  So Mercury doesn’t really explore this region of strong general relativistic effects.  But, as I was working on making the plots for this post, I suddenly realized “I’ve just made something that simulates orbits in general relativity.”  

Let’s change some parameters, and look at some orbits that happen very close to a black hole.  Don’t worry, there is no more math.  Only some neat figures and explanations.

The effective potential near a black hole:

Here I’ve plotted the potential for a particle with some angular momentum, and starting at 10 event horizon radii from the black hole.  The mass of black hole I used is 4×106 solar masses which is similar to the supermassive black hole at the center of our galaxy.  The energy of this particle is just enough so that its orbit can drop down close to the black hole without falling in.  Notice the hill on the left side of the graph.  The particle is starting at a point just below that peak’s height, so that when it “rolls down the well” it will stop and turn just below the peak.

(Aside:  The units ‘r/M’ for the horizontal axis means that I’m plotting distance in terms of units GM/c2.  In these units, r/M=2 represents the event horizon.  r/M = 3 defines the photon sphere where light can orbit around the hole, and r/M=6 defines the “ISCO” or the innermost stable circular orbit that can exist around the hole.)
Because the particle will slow down a lot near that peak, you might imagine it could spend a bit of time circulating there at that radius before moving back out.  In fact, it does.  


This trajectory is delicately balanced on a knife’s edge.  If it strayed just a little bit further in, it would plunge down the other side, into the black hole.  Like so:


So this is one remarkable feature of motions near a black hole.   You can get an orbit that drops down and then circulates around several times very close to the black hole, before either zooming back outward to safety, or fatally falling down in.  And the difference between the two is precariously thin.  

Another remarkable thing can happen here.  You actually can get orbits that retrace themselves — but they will be very different from ellipses.  With the right angular momentum and energy, the orbits become beautiful flowery patterns. There is absolutely nothing like these trajectories in Newtonian gravity.
There is a wonderful “Periodic Table” of black hole orbits, including ones for rotating (Kerr) black holes.  Sadly, as beautiful as these are, they probably do not occur in nature.  Not just because meeting the initial conditions for them would be unlikely, but also because for real objects in these orbits, gravitational radiation (gravitational waves) would be significant and cause them to decay and change shape fairly quickly.  However, it is faintly possible that we could sometime observe something at least briefly resembling one of these orbits, in the gravitational waves emitted by a binary black hole merger.  And that would be pretty amazing to see.

Don’t miss discussion of the original post on the forum: it has a lot of interesting questions and answers.