I transferred yesterday's Mathematica file with the Alcubierre warp drive line element and space curvature calculations to the +Sage Mathematical Software System today, (the files been added to the public repository). 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, as well as a bit about my first experience with using Sage.
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.... 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
If you're using Windows and you want to install Sage to run locally, you're gong to swear a little bit at the gods o' Linux. They've kind of obviously written the documentation pages. However, like me, after you calm down a bit, you'll realize there are only really two steps. First, download the gigantic Sage binary from http://sagemath.org/download-windows.html . It's an ova file, (if you don't know what that is don't worry yet). Next, download VirtualBox and just take all the installation defaults. Select the box that says something like 'Run VirtualBox' as the last installation step. A nice shiny,new GUI will appear asking you which machine you want to run, (this won't make any sense at fit either necessarily). Now, click on the .ova download in your browser. If you're as lucky as Iwas, it will come up in the VirtualBox GUI and just start Sage in your brand-spanking new Linux virtual machine. For a detailed explanation see: http://wiki.sagemath.org/SageAppliance .
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
3. Alcubierre Drive Open Science Repository on GitHub