### 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].

References:

1.  Wolfgang Rindler at UTD
http://www.utdallas.edu/physics/rindler-wolfgang/

2.  Other notes on WR
http://copaseticflow.blogspot.com/2013/06/rindlers-just-flat-out-pretty.html

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

4.  Excelent open access paper on gyroscopic precessions in special and general relativity
http://arxiv.org/abs/0708.2490v1

5.  Shapiro, et al.
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2643

6.  Corner reflectors
https://en.wikipedia.org/wiki/Corner_reflector

7.  Lunar Laser Ranging Review
http://physics.ucsd.edu/~tmurphy/papers/rop-llr.pdf

8.  Tom Murphy, current leader of the Lunar Laser Ranging project
http://physics.ucsd.edu/~tmurphy/

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr.

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

### Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…