In the general theory of relativity, the orbits are given by geodesics. A geodesic is a path that extremizes the path length. The path length is defined as the integral of over the path, where is the metric. For the case of Painleve coordinates on the Schwarzschild metric, is given by:
. Let’s let our path start at time t=0 and end at time t=1. For the path to be a geodesic, we must extremize the following integral (I’ll quickly sneak in a minus sign to make the path be timelike instead of spacelike):
To make life easier for us, we will make the assumption that the orbital motion is in the plane so there’s no dependence. That turns the angular part of the square root into . Furthermore, since the simulation is going to use Cartesian, (x,y) coordinates, we might as well replace with , and with , their Cartesian equivalents. And put M=1, we can always fix it later by dimensional analysis.
And one last thing. As long as we’re going to piss off people who think that there is only one way to compute orbits in general relativity, we might as well make our notation easier by assigning and . This is in complete violation of the usual general relativistic notation that the dot refer to differentiation with respect to the “affine parameter” s. Our purposes is to put the gravitation of Painleve coordinates into Newtonian form, and so it is natural for us to use Newtonian notation, where the dot means the derivative with respect to coordinate time. So as a result of the conversion to Cartesian coordinates, and using dots to reduce the size of the equation, our problem is to extremize S given by:
Note on the Physics
The principle that orbits are defined by geodesics amounts to the requirement that orbits be stationary with respect to the experienced proper time of the orbiting object. If one were to think of the object as a quantum object, or a wave function, this is very natural. The classical path corresponding to the wave function needs to be one where the wave function won’t interfere with itself and end up with zero net probability. That happens if small changes to the path (i.e. the spread out wave function) experience no net change in the passage of proper time. This sort of correspondence principle appears repeatedly between quantum mechanics and classical mechanics.
As a sociological aside, I should mention that I’ve learned that one of the aspects of general relativity know-it-alls is a belief that Cartesian coordinates cannot be used in GR. It’s kind of surprising that they’re too stupid to search for “cartesian coordinates” and “general relativity” in arXiv before informing me that there is no such thing. If you’re such a person, read this paper, or read these U. Colorado class notes.
As another sociological aside, in writing this up on Physics Forums, I learned that it is apparently possible to attend a class in general relativity without understanding enough about the subject to realize that the method used here is a valid (but rarely used) method for obtaining orbital equations. It’s like these people are so arrogant that if they see someone doing something that is new to them, they immediately jump to whatever conclusions are necessary to support the contention that they are right and the person doing things the unusual way is wrong. If you’re such a person, then kindly read exercise 24.5 in this Cal Tech GR web book. The above integral is the integral from 24.5 for Painleve/Cartesian coordinates. In the remainder of this post, we will vary this integral and find the orbital equations — in Cartesian Newtonian form.
So much for the easy part.
Now comes the hard part, the computation of the orbital equations. We need to minimize / extremize the above integral. Notice that the integral is written as a definite integral over time t, and the stuff inside the integral includes and nothing else.
The method of extremizing this sort of mathematical object was discovered by Leonhard Euler and Joseph Louis Lagrange in the 1750s. It is also called the “calculus of variations.” Yet another sociological feature of GR know-it-alls is that when you say the word “Lagrange”, no matter if it’s part of “Euler-Lagrange” they immediately conclude that you don’t know how to properly spell “Lagrangian” and are screwing up the calculation. No, there are no energy principles hiding here. We’re just going to apply the Calculus of Variations to a definite integral.
The first step in applying the Euler-Lagrange method is to write the integral in the form . We’ve done that above with L being the big square root sign and the stuff under it. Next we compute all the partial derivatives of L with respect to . The Euler-Lagrange equations for the extremization of S is then given by the partial differential equations:
The fact that L is written with a square root makes the application of the Euler-Lagrange method far more computationally difficult. Terms end up with powers of sqrt(L) all over the place. This is in contrast to the simplicity of the calculation without the big square root. The usual methods of computing orbits avoid the ugly square root by working in proper time instead of coordinate time. The resulting differential equation uses Christoffel symbols, but they are four differential equations in proper time s, not coordinate time t. As such, they would have to be converted into t form in order to write them as a Newtonian force.
In addition to a very complicated set of Euler-Lagrange equations due to the square roots, one finds in addition that the resulting equations are not split between x and y. In the usual application of Euler-Lagrange to a Lagrangian, shows up as a mass term, for example, . The partial derivative of this with respect to is quite simple, i.e. . When this is the only way that appears, the derivatives with respect to t turn this term into . Thus the two Euler-Lagrange equations each have only one second derivative, one giving , the other giving . This is conveniently in the form of a force, F=ma, already split into x and y components.
Contrast this simplicity witht he complications that arise when one applies the same method to Painleve coordinates. The d/dt applies to a mixture of and and this becomes a mixture of and . The good news, of course, is that they’re linear. So to separate them out, you solve two linear equations in two unknowns. The bad news is that this activity complicates the equations that much more.
The above explanation should be enough to explain the equations given on this Physics Forums post. But the equations as written here are not in fully reduced form. Because one had to solve two linear equations in two unknowns, one ends up with a solution that has a ratio. It turns out that you can cancel out almost all of that ratio and reduce the equations of motion for Painleve in two dimensions down to:
In the above equations, I’ve added a column on the right to give the negative powers of the radius for the x equation (the y equation of course is similar). Some of these powers don’t match the power of r in the term because the term has a factor of x or y in the numerator that cancels part of the power of the radius. In such cases, the power listed is the “worst case” or smallest power of the radius.
Let’s take a closer look at the result. The y differential equation is comparable to the x. The reader is invited to use symmetry to turn the gravitational force here back into a 3-dimensional force instead of 2-dimensional, but we won’t do it here. In the following, I’ve marked the Newtonian gravity term in red:
Most of the terms have factors of or . For masses moving at speeds << c, these terms will be close to zero and the force can be approximated by the “static force” which consists of two terms, marked in blue above.
It’s interesting that in Painleve coordinates, the static force is quite simple, with only two terms. This is in contrast to the Schwarzschild case discussed briefly below. In a later post, we will take up the task of attributing the static force terms to a particle force.
It’s a little surprising that the lowest order terms are of order rather than the of the Newtonian theory. These terms will be detectable only for very fast moving test masses as they are multiplied by terms that are 3rd order in velocity. But for any given non zero velocity, these terms will dominate at sufficiently large radius r.
MOND or “Modified Newtonian Dynamics” is a theory of gravity intended to match the unusual rotation curves seen in galaxies and various other places. In MOND, the acceleration of gravity is adjusted when the acceleration is very small. One chooses a smooth function that, as the Newtonian acceleration “a” approaches zero, replaces that acceleration with where is a constant.
The effect of the MOND modification is to make gravity act as a 1/r force at large radii. The dominant terms in Painleve coordinate forces are not quite that strong. But they are stronger than one would expect in a theory that is supposed to be Newtonian at large radii, so it leaves me wondering if there is a correction to Painleve coordinates would look like a constant, 1/r.
Schwarzschild Equations of Motion
One can also execute the above procedure for Schwarzschild coordinates. The result is not so simple as Painleve coordinates, and since it is Painleve coordinates that are picked out as special by GR done with geometric algebra, we will not discuss them much (the calculations are here) except to give the results:
The Schwarzschild coordinates include a singularity at r=2. At that singularity, particles falling into the black hole get stuck and cease movement. So I expected to get an equation of motion with a factor of (r-2) in the numerator. Instead, the equations of motion have an (r-2) in the denominator and a few more copies of the same scattered through the numerator.
To see how particles get stuck on the event horizon in Schwarzschild coordinates one has to also take into account the fact that the velocity of the particle is going to zero. If one looks only at the static elements of the force (that is, if one puts ), one finds that one does, in fact, end up with a factor of (r-2) in the numerator.
Schwarzschild and Painleve Force Compared
To see how the Schwarzschild force falls off with radius, we look for the highest power of r, x, or y in the numerator. That turns out to be 4, and since there is a 6th power of r in the numerator, it turns out that in Schwarzschild coordinates, the force falls off the same as the Newtonian force, that is as .
The reason Painleve falls off as instead of is due to how one converts between Painleve and Schwarzschild events. The recipe looks like . This is a change to the coordinates that falls off as , hence the difference in how the two coordinate systems fall off at large radii.