The Alcubierre Drive's Tophat Shift Function In Motion

The Alcubierre drive works, (theoretically), by warping space 'downwards' in front of a spaceship and 'upwards' behind it, (picture 1).

The net effect is that the spaceship always feels like it's free-falling through space.  We have to be careful not to let the spaceship approach either the upward or downward walls of the curved space shown above.  In these regions, gravity due to the curvature of space is changing rapidly, and will cause uneven forces, known as tidal forces, on different portions of the spaceship that can cause it to break apart.  That's where Alcubierre's tophat function comes in.  The tophat is the input to the operation that describes how space is warped by the drive.  It's nice, flat top becomes the flat area in the middle of the warp shown above where the spaceship can safely rest.

In case you weren't there, here's the description of the tophhat function I provided in the last post on the Alcubierre drive[1], (stay tuned after the rehash for an explanation of the math behind the warp moving through space):

"The shift mentioned in Alcubierre's formulas[2] as beta 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".

Here's the sage demo of the tophat function's sigma and R parameters effect it:

You may have noticed that the all the graphs of the tophat function have been in terms of r, and not our usual spatial axis, x.  The value of r itself is another mathematical function defined by Alcubierre.  The value of r is the distance away from the center of the spaceship at a given point in space.  r is defined as:

$r(x_s) = \sqrt{(x-x_s)^2+y^2+z^2}$

For our example we're setting x and y to zero, so that we only move along the x direction.  With this simplification, we get

$r(x_s) = (x-x_s)$

OK, that leaves us with the question, what is $x_s$ ?  $x_s$ is the trajectory we want the ship to travel along with respect to time, something like this:

In other words, suppose we want to move from the location $x=0$ to the location $x=17$, then $x_s$ will take the values 1 to 17 as time progresses.  For our example below, we're going to assume time progresses in a uniform fashion and we advance one unit in x for each time tick.

Now, if we graph what's going on in terms of space, (the x variable), instead of terms of r, (the displacement from the spaceship), we get a different, and moving picture for the tophat function which is now:

$f_{tophat}\left(r\left(x_s\right)\right) = \dfrac{tanh(\sigma((x-x_s)+R))-tanh(\sigma((x-x_s)-R))}{2tanh(\sigma R)}$

Let's call this a chained function, (since it will fit in nicely in the next post where we'll use the chain rule for taking derivatives to get the curvature of space caused by the warp drive).  The position value returned by the trajectory for each instant of time is fed to the formula r which is fed to the tophat function.  And...

by stepping the value of our trajectory, $x_s$, from 1 to 17 as discussed, we get a moving tophat function that travels along the path we want to move through in space.  We're not all the way to the warp function yet, that'll be next time.

The animation above was created with sage.  As of yet, I haven't been able to get animations to work in an embedded sage cell, but for those who would like to play with the math, here's the sage code:

ftophat(s,r, R)=((tanh(s*(r+R)))-(tanh(s*(r-R))))/(2*tanh(s*R))
fhts = [plot(ftophat(23, x-xs, 1), (1,17), color=Color(0,0,0), ymin=-0.1,ymax=1.1,xmin=0,xmax=18) for xs in sxrange(2.1,17,.2)]
b = animate(fhts) b
b.show()

The code has also been placed in the github repository[3].

References:

1. Previous post on the Alcubierre drive
http://copaseticflow.blogspot.com/2014/06/the-alcubierre-warp-drive-tophat.html

2.  Alcubierre's paper:
http://copaseticflow.blogspot.com/2014/06/the-alcubierre-warp-drive-tophat.html

3.  github repository:
https://github.com/hcarter333/alcubierre

A year ago today:
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A month ago:
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Picture of the day: San Francisco Fog