### Special Relativity and Hyperbolic Trigonometric Functions

Just a few brief notes on circular and hyperbolic trigonometry today.  First, if there's anyone who'd like to offer, any clarifications, expansions, or other cool and interesting facts, please, you're more than welcome!  As usual, this is stuff that I learned in high school that didn't become blindingly clear and meaningful until it cropped up in grad school physics.  First, the equations for the circle and the hyperbola (picture 1)

Notice that they differ only by a single negative sign.  Now, for their graphs, (picture 2).

Each of these figures can also be expressed in parametric form as follows, (this is where the trigonometric and hyperbolic trig functions come in).  (picture 3)

Now for the notes and other thoughts.

Angular Arc Length and Angular Area
Until just a few weeks ago, I had always wondered why the inverse trig functions were called arcsine, and arccosine.  The answer makes perfect sense, it just hadn't occurred to me.  The arc prefix indicates that the function returns the arc length around the unit circle for the angle whose value of sine, (for example), is the input to the function.  A numeric example shows it pretty well.  Suppose we take a right angle, equal to pi/2 in radians, and calculate it's sine.  The value returned will be 1, and obviously if I plug that into arcsine, I'll get back the angle, pi/2, but....  On the unit circle, the angle is also equal to the arc length around the circle, (circumference from the x axis to the angle), so while it's true that arcsine returns the angle for the value of sine it's given, it's also returning the arc length around the circle for that angle, hence the name.

Hyperbolic trig functions share a similar property, their inverse function return the area between the x axis and an angle line drawn out to the hyperbola from the origin.  In the figure shown from Wikipedia[1], the red area is equal to half the angle returned by arcsinh for the hyperbolic angle measured by the line from the origin to the hyperbola.

Here's where we get into the stuff I don't know enough about yet, but which makes for interesting extra reading.

Geometric Algebra
David Hestenes has been writing about geometric algebra and how it can be applied to better understand physics since the mid '60s [2].  His articles are very, very interesting, and I highly recommend them and wish I had more time to read through them.  In each of his articles on geometric algebra, the fact that the square root of -1, i, can be interpreted geometrically as an area or volume always comes up.  I don't know if one has anything to do with the other, (and I don't have time to check right now which is part of why I'm recording these speculative notes here), but hyperbolic sine and cosine can be represented in the following form using complex numbers

So, we have sine and cosine sprinkled with imaginary numbers and we wind up with something that traces out area rather than arc length.

The Special Relativity Angle
You might have noticed that the infinitesimal line length mentioned in the gamma derivation post a few days back[3] looked kind of like the Pythagorean theorem, but with a few minus signs that didn't quite fit.

These minus signs essentially put calculations in special relativity into a hyperbolic, or Minkowski, geometry[4], and the end result is the use of hyperbolic trigonometric functions throughout.  More about this soon.

References:
1.  Wikipedia on hyperbolic trigonometric funcitons
http://en.wikipedia.org/wiki/Hyperbolic_sine

2.  Hestenes' Oersted medal lecture
http://dx.doi.org/10.1119%2F1.1522700
Hestenes D. (2003). Oersted Medal Lecture 2002: Reforming the mathematical language of physics, American Journal of Physics, 71 (2) 104. DOI:

open access
http://c2.com/cgi/wiki?HestenesOerstedMedalLecture

3.  Rindler's derivation of gamma
http://copaseticflow.blogspot.com/2013/06/rindlers-just-flat-out-pretty.html

4.  On Minkowski geometry

### 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…