24 Steps To The Precession Of The Perihelion Of Mercury
By Doug Sweetser | April 24th 2012 12:13 AM | 25 comments | Print | E-mail | Track Comments

Trying to be a semi-pro amateur physicist (yes I accept special relativity is right!). I _had_ my own effort to unify gravity with other forces in...

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I tried to figure out the precession of the perihelion of Mercury calculation out three or four times from my collection technical books on gravity.  There was never enough detail for me to follow their work.  The authors can rightly figure that anyone reading this part of their textbook is exceptionally good at physics compared to the general populace and will be able to fill in any missing details.

For those part-timers who wish to move beyond the "Brief History of Time" level of physics, this is an obvious thing to try and figure out.  Because gravity does not work instantaneously, there is a wee bit more wobble in the orbit of Mercury.  This blog hopes to provide all the detail needed.

The book that got me over the technical hurdle was Sean Carroll's "Lecture Notes on General Relativity" from a course he taught at MIT in 1996 (chapter 7).  The lecture notes are not casual reading.  It took me 3-4 months to chew through them.  I made my own crib sheet of the contents.  Do I feel like I have an MIT graduate level understanding of general relativity?  Absolutely not, I would get silence by a standard problem set.  I do now have an appreciation for many more of the technical issues of the subject.

To teach someone in my station this kind of calculation, I developed a format.  Each step is numbered.  The first step must always define the starting point.  Nothing is left as an exercise. Each step should be described briefly - one sentence is ideal - but flexibility is allowed.  The 24 steps that follow are almost exactly what is on my lunchbox I used daily.  The difference is that my lunchbox references a retracted proposal of mine in the first four steps, so I modified the text accordingly.

What is the big picture view before diving into the detail?  The starting point is the Schwarzschild metric solution of general relativity.  Physics is the art of knowing what is conserved and what can be simplified.  The first seven steps set out to simplify and find the two key conserved quantities, the energy and angular momentum of the system.

The next collection of steps (8-13) are common place for people who do celestial mechanics, to switch the 1/R variable to U.  Those steps still don't feel familiar.  It is a rewrite that does make the resulting equation look simpler.  This is past the halfway mark.

Then one solves the equation where only Newton's theory pushes the perihelion around.  We need to understand that dominant solution first which is good because it is simpler.  Unfortunately, it is not that simple.  Planetary orbits are not circles but are ellipses.  Real physicists are good enough to deal with something other than circles :-)  It involves the eccentricity of the ellipse.

The solution to the Newtonian equation provides an answer to what U is.  The correction term depends on U squared.  Plug that back in to the differential equation.  One can pluck out the one of three possible terms that is going to "row with Newton" and therefore always contribute its little bit to the precession.  The on resonance term comes with its own collection of factors which are used in the final two steps to calculate the 42.9" per century contribution to the precession of the perihelion of Mercury due to the finite speed of gravity.

That is the big picture.  Here now are the 24 steps.

1. Start with a solution to the general relativity field equations for a spherically symmetric, non-rotating, uncharged mass. The solution is the Schwarzschild metric written in spherical Schwarzschild coordinates.

$d \tau^2 = \left( 1 - 2 \frac{G M}{c^2 R} \right) d t^2 - \frac{1}{\left( 1 - 2 \frac{G M}{c^2 R} \right)} ~ \frac{d R^2}{c^2} - \frac{R^2}{c^2} ~ \left( d \theta^2 + \sin^2 \theta ~ d \phi^2 \right)$

2. Simplify by working in the plane of rotation, so  $\inline d \theta = 0$ and $\inline \sin^2 \theta = 1$:

$d \tau^2 = \left( 1 - 2 \frac{G M}{c^2 R} \right) d t^2 - \frac{1}{\left( 1 - 2 \frac{G M}{c^2 R} \right)} \frac{d R^2}{c^2} - \frac{R^2}{c^2} d \phi^2$

3. Multiply through by $\inline \left( 1 - 2 \frac{G M}{c^2 R} \right)$:

$\left( 1 - 2 \frac{G M}{c^2 R} \right) d \tau^2 = \left( 1 - 2 \frac{G M}{c^2 R} \right)^2 d t^2 - \frac{d R^2}{c^2} - \frac{R^2}{c^2} \left( 1 - 2 \frac{G M}{c^2 R} \right) d \phi^2$

4. Divide by $\inline d \tau^2$:

$\left( 1 - 2 \frac{G M}{c^2 R} \right) = \left( 1 - 2 \frac{G M}{c^2 R} \right)^2 \left( \frac{d t}{d \tau}\right)^2 - \left(\frac{d R}{c d \tau} \right)^2 - \frac{R^2}{c^2} \left( 1 - 2 \frac{G M}{c^2 R} \right) \left(\frac{d \phi}{d \tau} \right)^2$

5. Define the Killing vectors for this metric.* Notice that the expression in 4 is not a function of either time t or angle phi. This means there is a conserved quantity associated with a change in time (energy E) and a change in angle (angular momentum L). Come back at a later time, and the expression stays the same. Spin around a few degrees, and the metric stays the same. The conservation law which uses a Killing vector has this form:

$\rm{ (conserved \; thing) = (Killing \;vector)\cdot(\rm{velocity \; vector})}$

Here are the two Killing vectors for 4:

$\\ \frac{E}{m c^2} = K_t V_t = (1 - 2 \frac{G M}{c^2 R}, 0, 0, 0) \cdot ( \frac{d t}{d \tau}, 0, 0, 0) = (1 - 2 \frac{G M}{c^2 R}) \frac{d t}{d \tau} \\ \\ \frac{L}{m c} = K_{\phi} V_{\phi} = (0, 0, 0, \frac{R}{c}) \cdot (0, 0, 0, R \frac{d \phi}{d \tau}) = \frac{R^2}{c} \frac{d \phi}{d \tau}$

*The proper way to do this critical step is to write out the equations of motion, which turns out to be four equations that have an affine parameter.  An analysis of the constants of the equations of motion leads to this result.  Much more work is required to really get this step right.

6. Calculate the squares of the energy and angular momentum:

$\\ \left( \frac{E}{m c^2}\right)^2 = \left(1 - 2 \frac{G M}{c^2 R}\right)^2 \left( \frac{d t}{d \tau}\right)^2 \\ \\ \\\left( \frac{L}{m c}\right)^2 = \frac{R^4}{c^2} \left( \frac{d \phi}{d \tau} \right)^2$

7. Plug 6 into 4:

$\left(1 - 2 \frac{G M}{c^2 R}\right) = \left( \frac{E}{m c^2}\right)^2 - \frac{1}{c^2} \left( \frac{d R}{d \tau}\right)^2 - \left(1 - 2 \frac{G M}{c^2 R}\right) \left( \frac{L}{m c R}\right)^2$

What has happened? We have introduced two constant quantities, the energy E and the angular momentum L. We still have two factors of $\left(1 - 2 \frac{G M}{c^2 R}\right)$.

8. Prepare for a change of variable, to $\inline U = \frac{1}{R}$:

$\frac{d R}{d \tau} = \frac{d R}{d \phi} ~ \frac{d \phi}{d \tau} = \left(- \frac{1}{U^2} \frac{d U}{d \phi} \right) \left( \frac{L U^2}{m} \right) = - \frac{L}{m} ~\frac{d U}{d \phi}$

9. Plug 8 into 7:

$\left(1 - 2 \frac{G M}{c^2} U \right) = \left( \frac{E}{m c^2} \right)^2 - \frac{L^2}{m^2 c^2} \left( \frac{d U}{d \phi} \right)^2 - \left(1 - 2 \frac{G M}{c^2} U \right) \left( \frac{L U}{m c} \right)^2$

10. Bring all the terms to one side:

$0 = \left( \frac{E}{m c^2}\right)^2 - \frac{L^2}{m^2 c^2} \left( \frac{d U}{d \phi}\right)^2 - 1 - \frac{L^2 U^2}{m^2 c^2} + 2 ~\frac{G M}{c^2} ~U + 2 ~\frac{G M L^2 U^3}{m^2 c^2}$

11. Take the derivative of 10 with respect to phi:

$0 = - 2~ \frac{L^2}{m^2 c^2} ~ \frac{d U}{d \phi} ~ \frac{d^2 U}{d \phi^2} - 2~ \frac{L^2 U}{m^2 c^2} ~ \frac{d U}{d \phi} + 2 ~\frac{G M}{c^2} ~ \frac{d U}{d \phi} + 6 ~\frac{G M L^2 U^2}{m^2 c^2} ~ \frac{d U}{d \phi}$

12. Divide all the terms by $- 2~ \frac{L^2}{m^2 c^2} ~ \frac{d U}{d \phi}$:

$0 = \frac{d^2 U}{d \phi^2} + U - \frac{G M m^2}{L^2} - 3 ~\frac{G M}{c^2} ~ U^2$

The first three terms are classical Newtonian gravitational physics (implied by the lack of a factor of c). The fourth is the correction required by general relativity.

13. Write out the Newtonian equation:

$0 = \frac{d^2 U}{d \phi^2} + U - \frac{G M m^2}{L^2}$

14. Solve the Newtonian equation. This is a slight variation on an equation with the cosine solution. We must account for the eccentricity of the circle and the constant factor. Guess a solution:

$\\ U = \frac{G M m^2}{L^2} (1 + \epsilon \cos (\phi - \phi_0))\\ \\ \frac{d U}{d \phi} = - \frac{G M m^2}{L^2} ~\epsilon \sin (\phi - \phi_0) \\ \\ \frac{d^2 U}{d \phi^2} = - \frac{G M m^2}{L^2} ~\epsilon \cos (\phi - \phi_0)\\ \\ \\ \frac{d^2 U}{d \phi^2} + U - \frac{G M m^2}{L^2} \\= - \frac{G M m^2}{L^2} ~ \epsilon \cos (\phi - \phi_0) + \frac{G M m^2}{L^2} ~(1 + \epsilon \cos (\phi - \phi_0)) - \frac{G M m^2}{L^2} = 0$

OK!

15. At the perihelion, $\inline \cos (\phi - \phi_0) = 1$ and $\inline R = 1 / U = a (1 - \epsilon)$. Plug into U found in 14:

$\frac{1}{a (1 - \epsilon^2)} = \frac{G M m^2}{L^2}$

16. Use the Newtonian solution found in 14 as a start for the correction
term, U2:

$U^2 = \frac{G^2 M^2 m^4}{L^4} ~(1 + 2~ \epsilon \cos (\phi - \phi_0) + \epsilon^2 \cos^2 (\phi - \phi_0))$

17. Plug this into 12:

$0 = \frac{d^2 U}{d \phi^2} + U - \frac{G M m^2}{L^2} - 3~ \frac{G M}{c^2} \frac{G^2 M^2 m^4}{L^4} (1 + 2 \epsilon \cos (\phi - \phi_0) + \epsilon^2 \cos^2 (\phi - \phi_0))$

18. Keep only the second U2 correction term. The factor of $\inline \frac{G^3 M^3}{c^2}$ will make the U2 correction tiny. Only if a term is on resonance'' - in effect pushing the swing at the same time as the main solution for U - can a term eventually make a contribution to the precession:

$\inline 0 = \frac{d^2 U}{d \phi^2} + U - \frac{G M m^2}{L^2} - 6 ~\frac{G M}{c^2} ~\frac{G^2 M^2 m^4}{L^4} ~\epsilon \cos (\phi - \phi_0)$

19. Guess a solution. It must be composed of the previous solution in 14, plus a way to drop the additional cosine term:

\begin{align*} U &= \frac{G M m^2}{L^2} (1 + \epsilon \cos (\phi - \phi_0) + 3 ~\frac{G^2 M^2 m^2}{c^2 L^2}~ \epsilon~\phi~ \sin (\phi - \phi_0))\\ \\ \frac{d U}{d \phi} &= - \frac{G M m^2}{L^2} (\epsilon \sin (\phi - \phi_0) + 3 \frac{G^2 M^2 m^2}{c^2 L^2} \epsilon~\phi \cos (\phi - \phi_0) + 3 \frac{G^2 M^2 m^2}{c^2 L^2} ~\epsilon\sin (\phi - \phi_0) )\\ \\ \frac{d^2 U}{d \phi^2} &= - \frac{G M m^2}{L^2} (\epsilon \cos (\phi - \phi_0) - 3 \frac{G^2 M^2 m^2}{c^2 L^2} ~ \epsilon \sin (\phi - \phi_0) + 6 \frac{G^2 M^2 m^2}{c^2 L^2} ~\epsilon \cos (\phi - \phi_0)) \end{align*}

The final term in the second derivative of U will cancel with the -6 cosine term in 3U2. OK!

20. Bring the correction term into the cosine. Because $\inline \frac{G^2 M^2 m^2}{c^2 L^2}$ is so small, only the first term of $\inline \epsilon \phi \sin (\phi - \phi_0)$ will make a contribution to cosine. Rewrite U in 19:

$U = \frac{G M m^2}{L^2} \left(1 + \epsilon \cos \left(\phi - \phi_0 - 3 \frac{G^2 M^2 m^2}{c^2 L^2} \phi\right) \right)$

21. Calculate the ratio of the advance of phi in one rotation between the Newtonian solution, and the general relativity correction:

$\Delta \phi = 2 \pi \left(\frac{1}{1 - 3 ~\frac{G^2 M^2 m^2}{c^2 L^2}} \right)\approx 2 \pi \left(1 + 3 ~\frac{G^2 M^2 m^2}{c^2 L^2}\right)$

22. Plug the relation in 15, $\inline \frac{1}{a (1 - \epsilon^2)} = \frac{G M m^2}{L^2}$, into 21.

$\phi_{\tmop{advance}} = 6 \pi \frac{G M}{a (1 - \epsilon^2) c^2}$

23. Collect the relevant numbers:

$\\ G = 6.67 \times 10^{- 11} \mathrm{m^3 / \tmop{kg} s^2}\\ \\ M = 1.99 \times 10^{30} \tmop{kg}\\ \\ c = 3.00 \times 10^8 m / s\\ \\ a = 5.79 \times 10^{10} \mathup{m}\\ \\ \epsilon = 0.206$

$\left( \frac{1 ~\rm{revolution}}{88.0 ~\rm{days}}\right) \left( \frac{365 ~\rm{days}}{\rm{year}}\right) \left( \frac{100 ~\rm{years}}{\rm{century}}\right) = 415 ~\frac{\rm{rev} .}{\rm{century}}$

$\left( \frac{180^{\circ}}{\pi~ \rm{radians}}\right) \left( \frac{60'}{^{\circ}} \right) \left( \frac{60''}{'} \right) = 2.0610^5~ \frac{''}{\rm{radians}}$

24. Plug values in 23 into 22:

$\\ 6 \pi \rm{rad} . \frac{6.67 \times 10^{- 11} \mathrm{\frac{m^3}{\rm{kg} s^2}} ~1.99 \times 10^{30} \rm{kg}}{5.79 \times 10^{10} \mathup{m} (1 - 0.206^2) (3.00 \times 10^8 \frac{m}{s})^2} ~ 415 \frac{\rm{rev} .}{\rm{century}} ~2.06 \times 10^5 \frac{''}{\rm{rad} .} \\= 42.9 \frac{''}{\rm{century}}$

Bingo, bingo.

This number is tiny.  Jupiter changes the precession an order of magnitude more than this. [Clarification: In box 40.3 of "Gravitation" by Misner, Thorne and Wheeler, they point out that the other planets Newtonian gravity also changes the precession of the perihelion of Mercury by 531" per century.] How folks did that calculation back in the days before computers is impressive.

Does Sean Carroll do the calculation this way?  In chapter 7, he _derives_ the Schwarzschild solution.  There is also far more discussion of Killing vectors that is well worth the read.

Doug

Snarky puzzle: The only metric that is spherically symmetric and solves the Einstein field equations is the Schwarzschild metric, or so sayith Jørg Tofte Jebsen aka Birkhoff's theorem. [Correction: The theorem applies for a spherically symmetric, non-rotating, and uncharged source for the vacuum field equations.]

Study an alternative, that will have both the right answer infinitely far away (the flat Minkowski metric), could pass weak field tests like the precession of the perihelion of Mercury, but is not a solution to the Einstein [vacuum] field equations.  I like to call this the exponential metric, but it is known as the Rosen metric in the literature:

$d \tau^2 = e^{- 2 \frac{G M}{c^2 R}} d t^2 - e^{2 \frac{G M}{c^2 R}} ~ \frac{d R^2}{c^2} - \frac{R^2}{c^2} ~ \left( d \theta^2 + \sin^2 \theta ~ d \phi^2 \right)$

Get to step 4 of the above derivation.

Google+ hangout, Saturdays, 11am (3 people last time, a new record :-)

Next Monday/Tuesday: It's a bug, no a feature, of general relativity

"The only metric that is spherically symmetric and solves the Einstein field equations is the Schwarzschild metric"

Consider a hypothetical stationary spherically symmetric star (not a blackhole). Are you saying such an object cannot exist accordint to Einstein? (Hint: You are dropping some necessary conditions.)

Another counter-example (to your statement, not Birkhoff):
http://en.wikipedia.org/wiki/Reissner-Nordstr%C3%B6m_metric

To be honest, I don't fully understand the limits of Birkhoff's theorem myself. For I thought one gave the boundary conditions of asymptotic flatness to derive it, whereas wikipedia makes it sounds like this is a derived fact from the theorem. But then wouldn't a lambda vacuum solution violate this? Or really anything with non-zero cosmological constant?

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"This number is tiny. Jupiter changes the precession an order of magnitude more than this."

I'm not sure what you mean here.
Do you mean the gravitational tug on Mercury by Jupiter affects the orbit of mercury more than the GR effect, and so needs to be accounted for? Or are you saying the precession in Jupiter's orbit has a larger effect (why did they use mercury? was the period of Jupiter's orbit just too long for people to have measured it well?).

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"Study an alternative, that will have both the right answer infinitely far away (the flat Minkowski metric), could pass weak field tests like the precession of the perihelion of Mercury, but is _not_ a solution to the Einstein field equations."

I think that any smooth metric you can write down can be a solution of Einstein's field equations. I think you meant to say that it is not a vacuum solution to Einstein's field equations.

Since your goal seems to be to find a linear theory of gravity, I have to ask: Do you think a theory that could give a metric with exponential terms for an isolated mass would be a linear theory? Why or why not?

The main blog has been adjusted to correct the points raised.  The Birkhoff theorem is for the vacuum solutions of a spherically symmetric, non-rotating and uncharged source.  One thing Carroll points out is the Schwarzschild metric is stationary and static.  The stationary part means there is a timelike Killing vector.  It is also static because there are no cross terms, things like dt dR.  The source can be an imploding star so long as it maintains spherical symmetric, indicating the source need not be static.

Regular Newtonian-at-the-speed-of-light gravity from Jupiter advances the precession of the perihelion of Mercury, as do the other planets for smaller amounts.  They added up all of these effects, and were unable to explain what was seen.  This led some to propose a missing interior planet.

I think that any smooth metric you can write down can be a solution of Einstein's field equations. I think you meant to say that it is not a vacuum solution to Einstein's field equations.
I gather one can pick out a stress-energy tensor to fit the need.  Yup, I meant the vacuum solution.
Do you think a theory that could give a metric with exponential terms for an isolated mass would be a linear theory? Why or why not?
I should wimp out on this question since I don't have a theory (and I prefer the word proposal since so many of my efforts get knocked around).  Either the equations of motion or the field equations would be the giveaway.  Electric fields are not sources of electric charge.  The same kind of statement would be true of a proposal for gravity that was linear: gravity fields don't gravitate.

I vaguely recall some people claiming that there is experimental support of the idea that gravity fields gravitate (ring any bells readers?).

"Electric fields are not sources of electric charge. The same kind of statement would be true of a proposal for gravity that was linear: gravity fields don't gravitate."

In an attempt to encourage discussion and thinking:

In static situations, the electric field is only sourced by electric charge.
But according to Maxwell's equations, the electric field can be sourced by a changing magnetic field. And the magnetic field can be sourced by a changing electric field. Does this mean electromagnetism can't be linear in dynamic cases?

Einstein's equations are usually written with a clean separation of metric terms on one side, and matter terms on the other side (the stress energy tensor). Gravitational waves do not contribute to the stress-energy tensor. So what precisely do you mean by "gravity fields", and in what sense are you claiming they have a gravity 'charge' / stress-energy themselves?

The Ricci curvature scalar is linearly related to the trace of the stress energy tensor. Does this make GR linear?

As a disclaimer some of these questions (or variations at least) were historically debated. Feynman used a pseudonym at a conference to submit a related answer (rumored to be because he was disgusted at the state of the field arguing over the questions). So I don't expect anyone to come up with the answers on their own, but definitely think about it and read up to learn more. CuriousReader, please don't answer these questions before Doug has time to think and begin discussion. And everyone please give Doug the room to think openly on these discussions without fear of people pouncing on him.

"I should wimp out on this question since I don't have a theory (and I prefer the word proposal since so many of my efforts get knocked around)."

Why do you feel you need a specific gravity theory to analyze this?
Define what qualities you want in a theory and then analyze if they are compatible.
It could save you a lot of time, and also build skills at asking useful questions.

"I vaguely recall some people claiming that there is experimental support of the idea that gravity fields gravitate (ring any bells readers?)."

I think that question is equivalent to the "strong equivalence theorem", which has been tested.
http://en.wikipedia.org/wiki/Tests_of_general_relativity#The_equivalence...

Beyond that, tests of "strong field" gravity which match GR could be argued to support non-linear, as GR is non-linear. If a linear theory was made to fit the weak field tests of GR, it would then necessarily fail the strong field tests.

Let me take one step back to my sloppiness with the word "vacuum", since that might also trip up others.  It is slightly odd that one talks about sources and the vacuum solution.  It was that conflict that prevented me from using the word "vacuum" as often as I should have in the blog.

A vacuum solution applies to a volume of spacetime that does not include the source.  If there is no source too, then there is nothing to talk about, spacetime is flat.  So if there is a source, but we want to understand the fields as the fields exist outside of the source, then we can study the vacuum solutions.  People have studied the field equations inside of a source, but that is not my focus.

If one has an electric charge over on the right, and one chooses to study a volume of spacetime on the left, the relevant equations are the vacuum solutions to the Maxwell equations.  The "empty" spacetime on the left has an electric field.  It may or may not have a magnetic field.  If one doubles the electric charge on the right, then the electric and the magnetic fields on the left is exactly doubled (assuming no changes in the distances involved).  To answer the question, I do think electromagnetism can be linear in a dynamic case.  If this were not the case, then a change in an inertial reference frame would have the field be linear to one observer and not another, a very bad thing.

EM quickly becomes non-linear.  I believe that is fundamental in QED.  One starts with a perfectly solvable problem, say a simple harmonic oscillator with a charged particle, then adds more and more perturbations trying to get a better, more accurate solution.  It is the interaction between the charged particle and the field that leads to non-linearity. [note: it feels like I may be mixing up linearity and exactly solvable.]

What is really clean in general relativity is the metric side.  What remains foggy to me is the stress-energy tensor.  But like EM, I am discussing the vacuum solutions where the stress-energy tensor is zero.  There is a spherically symmetric, non-rotating, electrically neutral mass on the right, and I hope to characterize the gravitational field on the left.  If I double the amount of mass on the right, then due to the non-linear nature of gravity, the gravitational field on the left will not be exactly double.  To be perfectly honest, I don't feel confident about saying the gravitational field will be more or less.

Why do you feel you need a specific gravity theory to analyze this? Define what qualities you want in a theory and then analyze if they are compatible. It could save you a lot of time, and also build skills at asking useful questions.

The snarky puzzle just has a metric.  One could test it against the Schwarzschild metric, but such a test is beyond the state of the arts for light bending around the Sun.  I would need two separate things.  The first is a Mathematica notebook that goes through a list of about ten properties for the fields and forces starting from a well-defined Lagrangian.  The second is a reason for all the math machinery to be found in the notebook, a good old fashion reason why gravity works as universally as it does.

In my experience, it has always taken a year or more to figure out a new proposal one I comprehend a technical objection.

This is an interesting claim:

If a linear theory was made to fit the weak field tests of GR, it would then necessarily fail the strong field tests.

Strong field tests are distinct from the strong equivalence principle, correct?  The strong field tests would look at additional coefficients of the Taylor series of a metric solution.  The strong equivalence principle definition I look to is this one:

The strong equivalence principle suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity) and does not have any extra fields associated with it.

I could only discuss that issue with field equations in hand (oh, and a math trick too since to vary a Lorentz invariant Lagrangian by the metric leads to a rank two tensor which means it binds to a second rank stress-energy tensor which means the gravity is non-linear in the GR sense).

Interesting points.

"A vacuum solution applies to a volume of spacetime that does not include the source. ...
To answer the question, I do think electromagnetism can be linear in a dynamic case. If this were not the case, then a change in an inertial reference frame would have the field be linear to one observer and not another, a very bad thing."

You seemed to have side stepped what I was trying to get at. Before you said "Electric fields are not sources of electric charge" and used this to argue that EM was linear. But I pointed out that in the dynamic case, changing electric fields can source a magnetic field and vice versa. So if your definition of vacuum is no sources, then is a dynamic electromagnetic field a source and therefore not a vacuum solution of EM?

Or re-using your approach above, would one observer call everything outside a static charge a vacuum, while another that sees that charge move insist that it isn't a vacuum accordintg to your definition?

If I gave you Maxwell's equations, and asked if they are linear, how would you test that? (What does "linear" actually mean here? Is it a mathematical property, or do I also need to specify some properties of the electromagnetic fields "outside" of what is specified in those equations? such as specifying if they have a charge or something?)

"If I double the amount of mass on the right, then due to the non-linear nature of gravity, the gravitational field on the left will not be exactly double."

I'm still not sure how to interpret this, as you skipped defining what you mean by "gravitational field". In a region where the stress-energy tensor is zero, R will be unaffected by the doubling of the mass. R_uv will also be unaffected. Are you considering the full Reiman curvature tensor the "gravitational field"?

If you want a theory which has linear equations for the "gravitational field" it would be worthwhile to think about defining what that field is precisely. I'd don't think there is "one" particular answer, and the phrase "gravitational field" is used fairly vaguely when people discuss GR, and some people even encourage not using the phrase. I don't see any harm in the phrase, but defining what you mean would be good.

"Strong field tests are distinct from the strong equivalence principle, correct?"

Yes.

"The strong equivalence principle definition I look to is this one: ... the metric alone determines the effect of gravity ..."

That is a fine definition. It is useful to get the rough idea as well. The idea is that regardless of the "interaction" (even gravity) the equivalence princple will hold. Imagine for instance that it didn't hold for gravity. In the Newtonian limit, we can define a clear gravitational binding energy between two masses. This reduces the energy of the system. If the "force" of gravity didn't care about this binding energy, then it would take a different path than a single object with the energy of that system. Or, as you put it, the path must be determined by more than just geometry of spacetime. Make sense?

"I would need two separate things. The first is a Mathematica notebook that goes through a list of about ten properties for the fields and forces starting from a well-defined Lagrangian."

You don't need to get lost in the weeds of mathematical details to answer questions.

For example, if you wanted to require your theory obeyed the weak (or strong) equivalence principle, can your theory just couple to a scalar "propertry" of a particle (like the electric charge)?
Sometimes trudging through the math is a good learning exercise, but it is useful to train yourself to ask questions that can help you avoid the math.

I'm particularly interesting in seeing you think out what exactly you mean by a linear gravitational field theory. If you are careful in your definitions, I think it is very possible that the things you require will either be incompatible (you could prove! to yourself that no such theory is possible), or it will be so restrictive that you could derive the form of the theory, and look at the consequences of that instead of doing the opposite (you seem to want to guess ideas and then check if it meets your requirements, which can never lead to a definitive answer if the answer happens to be no). It is worth it to try to design your questions so that you can do more than just mark one very specific idea as not working.

I just don't think I have to the time to address these question as reasonable as they are.  The ice here is thin for me, and I would like to see others response.  The comment that bothered me the most was this one:
For example, if you wanted to require your theory obeyed the weak (or strong) equivalence principle, can your theory just couple to a scalar "propertry" of a particle (like the electric charge)?
Sometimes trudging through the math is a good learning exercise, but it is useful to train yourself to ask questions that can help you avoid the math.
It seems to me the most such an exercise could do was clarify the assumptions that go into reaching this reasonable conclusion.  Should one make a proposal with critical differences in the math, then the proof would not apply.  That happens in research.

"I just don't think I have to the time..."

That is disappointing. While I consider the discusison interesting in its own right, the process of answering these questions was hopefully to _save_ you time in the long run.

" I would like to see others response"

The answers are useful, but the answers don't mean as much as the learning process of trying to figure them out. In your case, it is especially important since you seem to just dismiss all the answers others give you as potentially 'not applicable'. If you aren't willing to listen to others, _and_ are not willing to work it out yourself, where does that leave you?

"It seems to me the most such an exercise could do was clarify the assumptions that go into reaching this reasonable conclusion. Should one make a proposal with critical differences in the math, then the proof would not apply."

You won't know until you try. Dismissing the process before you start is not a good idea. Remember, with the definition of the strong equivalence principle you gave, it is incredibly restrictive: gravity is described by the metric, and no auxilarly fields. By studying this one question, you will probably be able to wipe out a huge swath of "theory space". If your complaint is that you might be able to eliminate all but a couple exceptions, that is not a complaint. That is even more reason to think this out. It would mean the work focussed your search to a few exceptions, and you could proceed forward with those. Regardless, the restriction is quite strong, and should be used as a tool, not as a "check" you run in mathematica. Invert your thinking to use restrictions as probing questions and you will learn a lot more a lot faster.

Mathematically, what is the "gravitational field" which you wish to be linearly related to some "source"? Is it the Ricci curvature scalar, the Ricci tensor R_ab, the Reimann curvature tensor R_abcd, some auxilliary scalar field, some vector field, some tensor field, maybe just the metric itself, some combination of these, etc. ?

---

Note that a particular combination of the objects mentioned above is called the Einstein Tensor G_uv, and is linearly related to the stress energy tensor in GR with no cosmological constant. With a cosmological constant, the g_uv Lambda term could be subsumed into the defintion, giving a new tensor G' and the equation of GR as just: G'_uv = T_uv. These are full fledged tensors, with on one side geometric content, and on the other side matter, energy, momentum, etc. If you called that the "gravitational field", then would that not be a "linear" coupling to the "source"? This field is boring in that it does not propogate. It is also apparent to me that this is not the theory you are looking for; I'm merely using it to hopefully encourage you to think out exactly what you mean by a linear theory and a "gravitational field". I truly hope your answer isn't "I'll know it when I see it", in which case you should really drop all other work to first define what you are even searching for. (As a historical note: Einstein defined a lot of what he was looking for in a gravity theory before he got the equations of GR work out.) Currently you seem to not be very clear on what exactly you are looking for, nor have a strategy in working towards this elusive goal.

As a chess player, I'm sure you're aware playing haphazardly without a strategy is really nothing but a recipe for losing. You need a strategy, and while other have tried to help, I'm trying (hopefully) to do it in a nicer manner. Ultimately though, you need to make the choice yourself.

I think we have far more overlap than the discussing might suggest.  There is also a clear point of departure.  Let me do one friendly modification of a line in your comment:

the definition of the strong equivalence principle you gave, it is incredibly restrictive: gravity [must be] described by the metric, and no [auxiliary] fields.

I do feel like a study of that definition will necessarily lead to the Riemannian curvature tensor as the simplest structure which can describe second order changes in a metric tensor.  From there it is a short drive to the Ricci curvature scalar that is the Lagrangian of GR for a vacuum.  I cannot see any wiggle room in that.

Here is my seamingly small variation:

the definition of the strong equivalence principle you gave, it is incredibly restrictive: gravity [may be] described by the metric, and no [auxiliary] fields.

The person creating the description of the system in question now has a choice: use a dynamic metric with no auxiliary fields, or use a mix of a dynamic metric and an auxiliary potential field, or use a static metric and an auxiliary potential field.  One can easily construct an ad hoc and not elegant theory consistent with weak and strong gravitational field tests by using a scalar potential theory for the g00 term, and a dynamic metric for the spatial term:

$\\ \phi = e^{- 2 \frac{G M}{c^2 R} } \\ \\ g_{00} = 1\\ g_{RR} = - \frac{1}{1 - 2 \frac{G M}{c^2 R}} \\ g_{\theta\theta} = -R^2 \\ g_{\phi\phi} = -R^2 \sin^2(\theta)$

That is what changing must to may makes possible.  This combo of a scalar potential theory and a dynamic metric will be consistent with all our weak and strong field experiments to date.  To get along with the strong equivalence principle, phi has to go all boring and let g00 pull the sled again.

The Ricci scalar is the only player in the GR Lagrangian for a vacuum.  It is like playing chess with a black king versus a white queen and king.  The white queen makes one further rule: when it is white's turn, only she can move.  That would be a game that never ends.  The queen needs to work with her scrawny king of a husband and then a checkmate will proceed quickly.

The connection must work with the potential.  As such, that eliminates working with the Reimann curvature tensor Rabcd, its contraction the Ricci tensor Rab, or its contraction the Ricci scalar R.  Those are the tools of the "must be" crowd because there is no place for a potential.

I cannot say the gravity field is just a 4-potential.  Where is the geometry in that?  Plus any solution has to be elegant to take on the house of Einstein's gravity.  That is why I am not in a rush.

Why are you avoiding his questions? First you blame time. Then you go and spend time to avoid the questions by trying to explain an excuse for avoiding the questions. But in doing so your "seamingly small variation" is anything but, and is irrelevant. You took a question which is essentially "how does requiring X restrict the possible theories?", and said it isn't really worth studying because you can come up with math that violates that restriction and could instead ask "what if we _don't_ require X?". You aren't pointing out anything useful at all here, except for maybe that you are completely missing the point.

The point is to learn to ask useful questions. Learn to use your requirements themselves as useful probes, to narrow the possibilities, instead of hoping you can randomly walk through all of theory space eliminating every idea one by one. If you don't want to require the strong equivalence principle, fine. I'm not sure you ever even said that you did, but again that was clearly not the point.

"I do feel like a study of that definition will necessarily lead to the Riemannian curvature tensor as the simplest structure which can describe second order changes in a metric tensor. From there it is a short drive to the Ricci curvature scalar that is the Lagrangian of GR for a vacuum. I cannot see any wiggle room in that."

You added some restrictions to yourself along the way, so of course there is "wiggle room". Try approaching the question again.

Here's my thoughts based just on things I saw you skip. You could use other curvature scalars, or functions of the Ricci curvature like in f(R) theory, etc. Or to make it more interesting to you, I'll add the restriction that the field theory is linear.
Consider for example:
http://en.wikipedia.org/wiki/Linearized_gravity
It is derived as an approximation to GR, but there is nothing preventing you from studying the resulting field theory as a theory of its own. Given a source and boundary conditions, one can solve for the geometry of spacetime. You have "on one side geometric content, and on the other side matter, energy, momentum, etc." As gravity is still described by geometry with no auxiliary fields, this appears to satisfy the strong equivalence principle definition you gave.

Please humor him, or humor David and his leading questions. Don't be so dismissive, give it a try, and you may be surprised how much you learn.

CuriousReader, please don't give Doug answers. He needs to learn to think critically for himself. And please leave him the room to think these out freely.

Doug, you clearly spent time on that response, which unfortunately was dismissive to the process instead of actually putting effort towards answering the questions. Please instead choose to spend time on thinking and learning. If you are rushed for time, try just answering the question I laid out above regarding what you mean by "gravitational field". You can even restrict yourself to GR if you'd like to narrow the scope.

Wow I have never seen that worked out by hand. Way to go.
I figured this out for a course in computational physics using Matlab.  Like you I had access to Sean Caroll's work, but in the form of his book "Spacetime and Geometry: An Introduction to General Relativity"... and I had been taught a formal course in the subject by Dr. Arthur Licht at UIC in Chicago.   We had to BEG for year to get them to offer General Relativity as a course.   It's one of those things thats it the catalog of every school but few really teach it.

Basically it had two big steps.

1.) I found a program that simulated the orbit of a object around a star.

2.) I recalled the potential energy due to the Schwarzchild metric.

3.)Substitute that for the Potential Energy in the program.

4.)Fiddle with the units so that the computer does not blow up or melt down when I run the program.

5.)Simmulate the precession using the adaptive Runge-Kutta method.

%% Perhelion - Program to compute the orbit of Mercury.
% This program is derived from the "orbit.m" used for assignment 3 in class
%I have modified it minimally.  The most noteable modifications have been
%to the aceleration and potential which are now based on General
%Relativity, and the norm of r is not determined using the Schwarzchild
%metric.  This is valid because the problem here is formulated in the weak
%field approximation.
clear all;  help orbit;  % Clear memory and print header

%% * Set initial position and velocity of the comet.
r0 = 0.466697; % radial distance between mercury and the sun at aphelion
v0 = 6.07; %velocity of mercury at aphelion.
r = [r0 0];  v = [0 v0];
state = [ r(1) r(2) v(1) v(2) ];   % Used by R-K routines

%% * Set physical parameters (mass, G*M, L, and epsilon)
GM = 4*pi^2;      % Grav. const. * Mass of Sun (au^3/yr^2)
mass = 0.000000166;        % Mass of mercury in proportion to GM
time = 0;
L = 3.1054e-19;%The rotational Killing vector times d(phi)/d(tau) the angular momentum of the orbit.
epsilon = 1; %see the accompanying paper.
%% * Schwarzchild metric and related Relativistic Hardware.
%I will only use those parts wich correspond to r and phi
%% * Loop over desired number of steps using specified
%  numerical method.
nStep = input('Enter number of steps: ');
tau = input('Enter time step (yr): ');
%NumericalMethod = menu('Choose a numerical method:', ...
for iStep=1:nStep

%* Record position and energy for plotting.
rplot(iStep) = norm(r);           % Record position for polar plot
thplot(iStep) = atan2(r(2),r(1));
tplot(iStep) = time;
kinetic(iStep) = .5*mass*norm(v)^2;   % Record energies
potential(iStep) = 0.5 - GM/norm(r) + L*L/(2*norm(r)^2) - GM*L*L/norm(r)^3;

%* Calculate new position and velocity using desired method.
%if( NumericalMethod == 1 )
%accel = - GM*r/norm(r)^3 + L*L*r/(norm(r)^4) - 3*GM*L*L*r/norm(r)^5;
%r = r + tau*v;             % Euler step
%v = v + tau*accel;
%time = time + tau;
%elseif( NumericalMethod == 2 )
%accel = - GM*r/norm(r)^3 + L*L*r/(norm(r)^4) - 3*GM*L*L*r/norm(r)^5;
%v = v + tau*accel;
%r = r + tau*v;             % Euler-Cromer step
%time = time + tau;
%elseif( NumericalMethod == 3 )
%state = rk4(state,time,tau,@gravrk,GM);
%r = [state(1) state(2)];   % 4th order Runge-Kutta
%v = [state(3) state(4)];
%time = time + tau;
%else
r = [state(1) state(2)];   % Adaptive Runge-Kutta
v = [state(3) state(4)];
%end

%% * Graph the trajectory of the comet.
figure(1); clf;  % Clear figure 1 window and bring forward
polar(thplot,rplot,'+');  % Use polar plot for graphing orbit
xlabel('Distance (AU)');  grid;   % Pause for 1 second before drawing next plot

%% * Graph the energy of the comet versus time.
%figure(2); clf;   % Clear figure 2 window and bring forward
%totalE = kinetic + potential;   % Total energy
%plot(tplot,kinetic,'+',tplot,potential,'--',tplot,totalE,'-')
%legend('Kinetic','Potential','Total');
%xlabel('Time (yr)'); ylabel('Energy (M AU^2/yr^2)');
F(iStep) = getframe;
end
movie(F,20)
One would also need the function rka.m  to make this work.
function [xSmall, t, tau] = rka(x,t,tau,err,derivsRK,param)
% Inputs
%   x          Current value of the dependent variable
%   t          Independent variable (usually time)
%   tau        Step size (usually time step)
%   err        Desired fractional local truncation error
%   derivsRK   Right hand side of the ODE; derivsRK is the
%              name of the function which returns dx/dt
%              Calling format derivsRK(x,t,param).
%   param      Extra parameters passed to derivsRK
% Outputs
%   xSmall     New value of the dependent variable
%   t          New value of the independent variable
%   tau        Suggested step size for next call to rka

%* Set initial variables
tSave = t;  xSave = x;    % Save initial values
safe1 = .9;  safe2 = 4.;  % Safety factors

%* Loop over maximum number of attempts to satisfy error bound
maxTry = 100;
for iTry=1:maxTry

%* Take the two small time steps
half_tau = 0.5 * tau;
xTemp = rk4(xSave,tSave,half_tau,derivsRK,param);
t = tSave + half_tau;
xSmall = rk4(xTemp,t,half_tau,derivsRK,param);

%* Take the single big time step
t = tSave + tau;
xBig = rk4(xSave,tSave,tau,derivsRK,param);

%* Compute the estimated truncation error
scale = err * (abs(xSmall) + abs(xBig))/2.;
xDiff = xSmall - xBig;
errorRatio = max( abs(xDiff)./(scale + eps) );

%* Estimate new tau value (including safety factors)
tau_old = tau;
tau = safe1*tau_old*errorRatio^(-0.20);
tau = max(tau,tau_old/safe2);
tau = min(tau,safe2*tau_old);

%* If error is acceptable, return computed values
if (errorRatio < 1)  return;  end
end
%* Issue error message if error bound never satisfied
:)

Science advances as much by mistakes as by plans.

Clearly there is code that is concerned with truncation errors, so the question is whether it works.  I suppose you tested your work by having it run for 100 years.  How close to 42.9" per century did it get?

I have no idea how one would modify this bit of code to take into account the other planets.

It's hard to tell.   I was just happy when my PC did not over heat.
Science advances as much by mistakes as by plans.
Since I KNOW computers can mislead, I always want to have a few checks.  Here is one thought.  Do the century calculation.  Make sure the program shows lots of digits, and save the value.  Then, if I am reading your code right, the 3 critical lines are:
potential(iStep) = 0.5 - GM/norm(r) + L*L/(2*norm(r)^2) - GM*L*L/norm(r)^3;
...
%accel = - GM*r/norm(r)^3 + L*L*r/(norm(r)^4) - 3*GM*L*L*r/norm(r)^5;
...
%accel = - GM*r/norm(r)^3 + L*L*r/(norm(r)^4) - 3*GM*L*L*r/norm(r)^5;

If the corrections are the last terms, chop 'em off, rinse and repeat.  Compare those two numbers.  I think it should be the phi_advance term in step 22 which is 5.17 x10-7.  That is the difference I would expect to see between the two calculations.
Ah I would have never thought of that.  Other than simulating a century of two of the orbit I could not think of a way to do that.
Thanks.
Science advances as much by mistakes as by plans.
The calculation is much simpler and and physically more intuitive if you use the phase-plance approach discussed here:

Bruce Dean, "Phase-plane analysis of perihelion precession and Schwarzschild orbital dynamics," American Journal of Physics -- January 1999 -- Volume 67, Issue 1, pp. 78

Sounds interesting.  If you or your affiliated institution has a subscription, it is available on line.  I am not willing to pay the \$30, so will need to wait until I drive into MIT. I have put the citation (QC.A513, 1999, v67:1, p78) on a sticky note for my wallet.
I'm not following the steps here.

At some point you start interpreting this as an equation of motion.

You get the right answer, but you are sweeping something under the rug. Can anyone see it?

The first and only time I reference the equations of motion is in step 5 in an effort to justify the two Killing vectors, the Killing fields, and the conserved quantity that results.  That is the most muddled step which was acknowledged in the blog.  Energy and angular momentum are conserved quantities.  Those are the expressions for energy and angular momentum everyone uses.  Step 6 squares the energy and angular momentum, step 7 substitutes that in.

I feel more like I am tripping on the rug in step 5.  I don't have enough technical experience with Killing vector fields.
In mathematics, a Killing vector field (often just Killing field) , named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object.
Sounds like a deep but relevant description of the issue.  By step 7, the Schwarzschild metric is still the same animal, but is written in terms of energy E and angular momentum L.
"The first and only time I reference the equations of motion is in step 5 ..."

Think about this: How can you refer to the orbital precession at all if you don't treat the equation as the equation of motion for a planet at the end?

You start with a line element that is true for all paths in spacetime. How do you know which path the planet will take?

You don't learn anything by just copying someone else's math. Learn the logic behind the steps, so that you could solve a similar problem by yourself in the future.

Also, in #5:
Why don't you use the metric when evaluating your dot product?
Based on what you wrote, I'm guessing the two killing vectors you are using are actually just (1,0,0,0) and (0,0,0,1).

Also I think you accidentally wrote many of the components of the velocity four-vector as zeroes.

in #11:
What is the meaning of taking the derivative with respect to phi? (it seems to remove the possibility of a circular orbit?)

This is the only context where I have ever dealt directly with Killing vectors.  I would not be surprised if my notation was in error.  I am pretty sure I saw something like this 6 years ago when I created the lunchbox.

While I was writing out the blog, I wondered why that step was required (and saw the same step in done in other derivations).  Look what happens going from step 10 to 11: the equation no longer depends on the constant energy E.  All the remaining terms have dU/dphi, so that too drops out.  What remains is a pretty simple second order differential equation between U (1/R), phi, the masses and L.
Note that R in the Schwarzschild coordinates is not a radial distance. But you treat it like a radial distance when comparing to the Newtonian solution. Since you get the correct answer, I guess that doesn't matter, but why?

When I wrote the blog, I was wondering if I should bring up this issue.  Measurements in space are done in isotrophic coordinates, not Schwarzschild coordinates.  Here is the isotrophic coordinate solution, found on page 840 of Misner, Thorne, and Wheeler as part of an exercise at the end of a chapter:
$d \tau^2 = \left( \frac{1 - \frac{G M}{2 c^2 R}}{1 + \frac{G M}{2 c^2 R}}\right)^2 d t^2 - \left( 1 + \frac{G M}{c^2 R}\right)^4 \left( d R^2 + R^2 d \theta^2 + R^2 \sin^2 \theta ~d \phi^2\right)/c^2$

Is that butt ugly or what :-)  For this calculation, one only uses the first order terms of GM/c2R.  That turns out to be the same as when the metric is written in Schwarzschild coordinates.  The Taylor series expansion that gets tested in all weak field tests of gravity is this one (p. 1097 of MTW):

$d \tau^2 = \left( 1 - 2 \frac{G M}{c^2 R} + 2 \left( \frac{G M}{c^2 R}\right)^2\right) d t^2 - \left( 1 + 2 \frac{G M}{c^2 R}\right) \left( d R^2 + R^2 d \theta^2 + R^2 \sin^2 \theta ~d \phi^2\right)/c^2$

The precession of the perihelion of Mercury is a test of the changes is space caused by gravity.

[There was a discussion in a previous blog about the Newtonian case.  I believe we got into a long discussion about two different types of parameterization, one known as beta-delta, the other as alpha-zeta. I confess much confusion in that part of the comments.  I have seen at least one statement from David that for this is a correct way to write the purely Newtonian theory of gravity.  What I neglected to do was to put the qualifiers on the metric below (low mass density, low speeds, non-rotating, uncharged). Parameterized Post-Newtonian formalism only makes sense if one can fine Newton's theory which happens when all ten parameters using either approach are equal to zero, the zeroeth order case.]
It is good to know Newton's law in its metric form:

$d \tau^2 = \left( 1 - 2 \frac{G M}{c^2 R} \right) d t^2 - \left( d R^2 + R^2 d \theta^2 + R^2 \sin^2 \theta ~d \phi^2\right)/c^2$

In this form, Newton's old gravity law in new clothes has no chance of getting the bending of light calculation correct for the space bending part.  The old theory does get a half right answer.
"It is good to know Newton's law in its metric form:"

That is not equivalent to Newton's gravity. Think about it and see if you can come up with some thought experiments that could distinquish a gravity theory specified by that equation, and the actual Newtonian mechanics and gravity.

(Weirdly, I think this was already discussed at length previously, so I was surprised to see you bring it up again.)