### The Alcubierre Warp Drive Tophat Function and Open Science with Sage

I transferred yesterday's Mathematica file with the Alcubierre warp drive[2] line element and space curvature calculations to the +Sage Mathematical Software System today, (the files been added to the public repository[3]).  If you haven't used Sage before, it's a Python based software package that's similar in functionality to Mathematica.  Oh, and it' free.  I also worked a little more on understanding the theory, but frankly, I made far more progress with the software than the theory.  What follows will be a little more of the Alcubierre theory, plus, a cool Sage interactive demo of one of the Alcubierre functions[1], as well as a bit about my first experience with using Sage.

Theory
The theory is fun, but it's moving slowly.  Here's the chalk board from this morning's discussion

Alcubierre setup the derivation using something called the 3+1 formalism which means we consider space to be flat, (in this case), slices that are labelled with the advancing value of time.  The Greek letters alpha and beta above, are called the lapse, and the shift respectively.  The lapse is related to how the spacetime moves from time slice to time slice, and the shift is related to how space changes as time advances.

Understanding the Alcubierre Drive Equations: the shift and the tophat function
When you're cruising across the sky at faster than the speed of light, you don't want to drag too much other stuff with you in your space warp.  It's not energy efficient and it just looks silly!

The shift mentioned in the section above is what describes how the ship advances through space and it is a function of the velocity of the moving spaceship.  Alcubierre wanted to derive a set of equations that would advance the spaceship through space without advancing anything outside the size of the spaceship.  To make sure that the shift moves just the ship and not too much of anything else that happens to be near the ship, Alcubierre tailored the shift function to be very localized in space.  Consequently, the shift is the velocity of the ship times a very tophat looking sort of function denoted as f.  The tophat ensures that the shift variable is only non-zero and proportional to velocity near the spaceship.  In the following equation,  r is the distance away from the center of the spaceship, and R is a parameter that should be a little bit large than the ship.  Sigma is a parameter that determines how localized the velocity of the spaceship is.  You'll see that as sigma increases, the function becomes a nice little square pulse centered on 0 and about the width of the spaceship.

Now here's the really cool thing about Sage. Instead of me just telling you about the function and how it behaves with respect to sigma to become sharper and with respect to R to give you a larger warp for your spaceship, you can actually adjust sigma and R and see for yourself.  Voila...[1].  Make sure to hit the button  labeled, 'Analyze the Alcubierre tophat function', to start the demo.

My Experience with Sage
Sage rocks!  I was able to do everything I'd done with Mathematica, only better!  I was also able to publish the demo above with only a little fuss and muss.  I'll give you, I haven't learned how to properly size the demo cells yet, but I'll get there I bet.  For an example of how to publish Sage demos to the web, just check out the source of this web page.  Also, if you want to just play with Sage, please notice that you can type any Sage you like into the code box above.  If things become hopelessly intractable, just refresh the page to get back to the original starting point.

Oh, and if you asked if it can make the cool warp graph, well, of course it can!!!  Jr.'s into the color pink this month,so here's you're Alcubierre space warp in pink courtesy of Sage!

The Sage Install Experience

References:
1.  permalink to blog with the cool demo! (You're here, this is for folks reading outside the blog)

2.  Alcubierre's original open access paper
http://arxiv.org/abs/gr-qc/0009013v1

3.  Alcubierre Drive Open Science Repository on GitHub
https://github.com/hcarter333/alcubierre

Anonymous said…
Love it, this short but sweet post helped me see why alcubierre chose a top hat function as he did.

### 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…

### 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…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…