### Gravitomagnetism and Antigravity for Experimentalists from Robert Forward and Bryce DeWitt

+Jonah Miller wrote about back of the envelope calculations today and it inspired me to finally write about the oft-quoted by fringe scientists work of Bryce DeWitt and the surprisingly less quoted work of Robert L. Forward.

The link between Forward's work and Jonah's article is that Forward wrote an excellent pair of articles entitled "General Relativity for the Experimentalist" for the Proceedings of the IRE[1], (the precursor to the IEEE), and "Guidelines to Antigravity" for the American Journal of Physics[2].  In these two articles, Forward encouraged scientists and engineers to do back of the napkin general relativity by using a method of linearizing Einstein's field equations in in the weak field flat space limit so that they could be treated in the same manner as Maxwell's EM equations.

So, where does Bryce DeWitt fit into the equation?  In 1966 he wrote an article about using superconductors to to detect gravitomagnetic fields, the same fields that Gravity Probe B eventually found.  He referenced an article from Schiff and Barnhill, (yes, the textbook author Schiff), that predicted the interior of a conducting cylinder in the presence of a gravitational field wouldn't have zero electric field.  It would actually have an electric field that produced an upward force on an electron exactly equal to the gravitational force that acts downward on the electron.  There's far more to this story, see the notes and interesting details section below.  I'll try to cover all these topics in upcoming posts because they're just fun.  Working from reasoning set out by Schiff and Barnhill DeWitt wrote down a set of equations that describe how gravitomagnetic fields would also be quantized inside a superconducting ring and provided a rough estimate of the size of the current that would be produced by the fields in the superconductor:

So, can we use a circular superconductor to detect a gravitomagnetic field?  This is where the napkin comes in.  Kappa in the equation above is the gravitational constant.  m is the mass of an electron, e is the charge of an electron and the other variables are described in the figure.  Using the modern-day napkin, a spreadsheet, we get for an estimate of the current that would be generated by a 200 gram disc spinning at about 2,900 rpm.

That's a current on the order of ten to the negative twenty amps.  We can detect currents in the nanoamp range, fairly easily, but to the best of my knowledge ten to the negative twenty is still a bit beyond our grasp.

Notes and Interesting Details
1.  Schiff was one of the architects of the Gravity Probe B experiment

2.  Schiff and Barnhills' work is often referenced in experiments and theories about whether a positron, (an anti-electron), will fall down or up.  If the positron does fall down, S & Bs' work implies that it will fall down a metallic cylinder twice as fast with twice the force of gravity while an electron will merely float in place.

3.  DeWitt's work referenced here discusses the fact that magnetic flux within a superconducting loop is quantized.

4.  William Fairbanks is noted for performing the experiment that verified the quantitization of magnetic flux.

5.  Fairbanks was also an architect of Gravity Probe B.

6.  Fairbanks also worked on detection of terrestrial fractional charge, a fancy term for unbound quarks.

7.  Fairbanks was indirectly involved in the Valentine's Day monopole.

6. DeWitt's work was sponsored by the same Air Force grant that sponsored some of Peter Higgs work on the Higgs boson.

References:
1.  Forward in the IRE
Forward R. (1961). General Relativity for the Experimentalist, Proceedings of the IRE, 49 (5) 892-904. DOI:

2.  Forward in the AJP
Forward R.L. (1963). Guidelines to Antigravity, American Journal of Physics, 31 (3) 166. DOI:

3.  DeWitt on gravitomagnetism and superconductors
http://dx.doi.org/10.1103%2FPhysRevLett.16.1092
DeWitt B. (1966). Superconductors and Gravitational Drag, Physical Review Letters, 16 (24) 1092-1093. DOI:

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

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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 differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…