Wednesday, April 2, 2014

Rindler the Fokker-DeSitter precession and Lunar Laser Ranging

I'm reading an excellent article[3] by one of my all-time favorite authors, Wolfgang Rindler[1][2].  In the article, Rindler and Perlick show how to use a generalized form of the line element to derive the circular geodesics and the associated gyroscopic precessions of a number of different metrics including everyone's standby, the Schwarzschild metric.

Using the Schwarzschild metric and his newly defined method for calculating circular geodesics, Rindler first derives the Thomas precession which was originally a special relativistic result having to do with the precession of the spin of an electron around an atomic nucleus.  He then goes on to show how the Fokker-De Sitter precession of a gyroscope orbiting a massive body, (like the sun), can be calculated.

See reference [7]
I'd never heard of the Fokker-De Sitter precession before, so I read on.  It turns out that this precession contains a component due to the geometry of the Schwarzschild metric as well as a Thomas precession-like component.  The math worked out by Rindler is of course, concise, pretty easy to understand and elegant.  Here's the really cool part: Rindler mentioned that the precession which could be used to test the predictions of general relativity and which was factoring into plans for the then nascent Gravity Probe B, had already been experimentally verified using the Earth-Moon system as a giant natural gyroscope orbiting the sun!

The experiment had been performed by I. I. Shapiro, et al. using lunar laser ranging data gathered from the corner reflectors left on the moon by the Apollo mission.  You can see a picture of a corner reflector, (picture 1), above and read more about them on Wikipedia.  What I most remember about them from freshman physics is that they reflect back light exactly in the direction it came from, making them great for bouncing laser's off the moon.  Here's the cool part, (to me anyway), if you look in one, you see black because you're viewing a refection of your pupil.  At any rate, Shapiro and company did a careful analysis of the data gathered between the Apollo missions and 1988 revealed that the predicted precession did take place to an accuracy of 2%[5]!

For more on the ongoing lunar laser ranging mission, check out the team leader, Tom Murphy[8].  He wrote an excellent review of the topic[7].


1.  Wolfgang Rindler at UTD

2.  Other notes on WR

3.  Rindler's article, (sadly not open access)

4.  Excelent open access paper on gyroscopic precessions in special and general relativity

5.  Shapiro, et al.

6.  Corner reflectors

7.  Lunar Laser Ranging Review

8.  Tom Murphy, current leader of the Lunar Laser Ranging project

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