Schwarzchild solution and orbit precession

[oldman]

In the Schwarzschild geometry of a static spacetime, elliptical test-particle orbits precess at a rate that (famously) agrees with observations of the inner solar system. Yet the model system considered is isolated, spherically symmetric with only the radial coordinate non-Euclidean.

I can't figure out what the axes of the ellipse precess "relative to" in this highly symmetric situation. Neither the "fixed stars" nor the CMB provide any explicit reference frame for the model. Does the analyst somehow provide an implicit reference?

Indeed I fail to see what physically causes the GR precession in such a symmetric model situation. How does the feature of orbital precession get built into the model?

 

[pervect]

In the Schwarzschild geometry of a static spacetime, elliptical test-particle orbits precess at a rate that (famously) agrees with observations of the inner solar system. Yet the model system considered is isolated, spherically symmetric with only the radial coordinate non-Euclidean.

I can't figure out what the axes of the ellipse precess "relative to" in this highly symmetric situation. Neither the "fixed stars" nor the CMB provide any explicit reference frame for the model. Does the analyst somehow provide an implicit reference?

Indeed I fail to see what physically causes the GR precession in such a symmetric model situation. How does the feature of orbital precession get built into the model?

Look at the effective potential formulation for orbiting bodies in GR. r and theta both vary periodically as a function of time, but the period in the variation of r is different than the period in the variation of theta in the GR formulation.

In the Newtonian case, the period of r exactly matches the period of theta, and the orbit is closed. Not so in the GR case - the fact that the perodicity of r is different from the periodicity of theta implies that the orbits are not closed, but precess.

There is a detailed reference to this in Goldstein, "Classical mechanics". Much of the discussion in Goldstein covers the effective potential approach, and in general relates to how one solves the differential equations for an orbiting body for general force laws.

There is even a discussion specifically of the precession of Mercury's perihelion in Goldstein, but the differential equations themselves is not derived there, the necessary equations are rather imported from MTW's "Gravitation". This is the same source used for the above webpage. Note that you will also find most of the same material in "Exploring Black Holes" (which is probably an easier read than MTW, which is graduate level) if you want a textbook reference. The very quick summary is that if you replace Newtonian t by GR tau, the differential equations for the orbit are formally very similar, except for an added 1/r^3 term in the GR equations. This is referred to in MTW, for example, as "the pit in the potential". It is this "pit in the potential" which causes the periodicity of r to change relative to the perodicity of theta.

Thus, it is the direction of mercury's orbit itself which determines the direction of the precession. If you reverse the orbital direction, you reverse the precession.

 

[oldman]

Look at the effective potential formulation for orbiting bodies in GR... Thus, it is the direction of mercury's orbit itself which determines the direction of the precession. If you reverse the orbital direction, you reverse the precession.

Yes, now I see it clearly. Thanks for the detailed help -- it's much appreciated.

As for the first part of my post:

In the Schwarzschild geometry ...somehow provide an implicit reference?

I now wish I hadn't made such a silly comment.

I've realized that describing the path of, say, Mercury as an orbit that is a precessing ellipse, while apt, is just using familiar words to describe a geodesic that is in fact not a closed path at all. I was thinking of an orbit as a path that is identically traced over and over again.

 

[Jorrie]

I can't figure out what the axes of the ellipse precess "relative to" in this highly symmetric situation. Neither the "fixed stars" nor the CMB provide any explicit reference frame for the model. Does the analyst somehow provide an implicit reference?

In addition to pervect's answer, I think that the Schwarzschild metric provides a local angular reference. After all, we set up a non-rotating coordinate system with r, phi and theta as parameters. This makes it possible to have a sign for dphi/dt and dtheta/dt and hence a sign for the orbital precession.

 

[oldman]

In addition to pervect's answer, I think that the Schwarzschild metric provides a local angular reference. After all, we set up a non-rotating coordinate system with r, phi and theta as parameters. This makes it possible to have a sign for dphi/dt and dtheta/dt and hence a sign for the orbital precession.

I now realize that the ultimate reason for precession is the particle's non-radial velocity vector. This introduces asymmetry into an otherwise spherically symmetric situation, making the GR geodesic an open path, in contrast to a Newtonian orbit.

Thanks, Jorrie.