### Rindler's Just Flat Out Pretty Derivation of the Special Relativistic Time Dilation Factor Gamma

Rindler demonstrates[2] the best way ever to derive the relativistic Lorentz contraction/time dilation factor, gamma, (of twin paradox fame), from four velocity!  I've been working through Rindler's paper on hyperbolic motion as a result of constant acceleration[1] lately.

Starting from the constancy of the speed of light, Einstein's theory of special relativity posited, and Minkowski refined the idea that the universe is actually four dimensional with space and time sitting on an equal footing.  Starting from here, we can write an expression for the distance squared along an infinitesimal line element in four dimensions, (think Pythagorean theorem)...

Brian Greene popularized the idea of four velocity[3] in one of his books and although it isn't mentioned as much as some of the other aspects of special rel, it's a simple idea.  Everything is moving at the speed of light.  Something might have more of it's velocity pointed into either the space dimensions, or the time dimension, but at the end of the day, its four dimensional speed is equal to the speed of light, c.  So, let's write down the infinitesimal distance along a four dimensional line divided by an infinitesimal amount of time as a velocity

which can be rearranged to give

From here on out, it's all Rindler.  Factor the increment of time out of the above expression

At this point, the velocity, (distance over time), squared is sitting in the second term, so we can write

Leaving us with

Which can be rearranged once again to give

Which is the formula for time dilation, (the time in the moving frame is equal to gamma times the time in the rest frame), so

and we're done!

References:
1.  http://dx.doi.org/10.1103%2FPhysRev.119.2082
Rindler W. (1960). Hyperbolic Motion in Curved Space Time, Physical Review, 119 (6) 2082-2089. DOI:

2.  Special Relativity by Wolfgang Rindler

3.  On four velocity

### 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…