Selling Science with Glossy Pictures and FTL Travel Baby!!!

The Washington Post published pictures of NASA's concept of what a Alcubierre drive spaceship might look like.  A few pundits immediately pointed out that perhaps hyping what amounts to a set of mathematical equations with a spacecraft design might not have been the classiest move on NASA's part.  NASA collaborator Mark Rademaker[1] maintains it was done with the intent of convincing people that STEM is cool, you know, 'for the kids'.  Here's Mark's big, glossy, futuristic design

It's cool and soooo pretty!

Discussion of how STEM should be sold aside, there are now conversations circulating the internet regarding whether or not the ship would violate causality by flying faster than the speed of light.  The answer is, that this might be an issue if the ship actually violated the speed of light by traveling 4.3 light years in 14 days.  As it is though, it doesn't.  Read on:

Travelling Faster than the Speed of Light Without Really Trying, (but not really)
The description in the Washington post article triggers a pretty common misconception:

"If an object reaches a distance x light years away in under x years, then it must be travelling faster than the speed of light."

What the article failed to mention is that the 14 days quoted is in the reference frame of the ship.  The equation for the distance travelled with respect to time in the frame of the ship, (known as proper time), is

$distance = \dfrac{c^2}{a}cosh\left(\dfrac{at}{c}\right)-\dfrac{c^2}{a}$,

where $a$ is the acceleration of the ship and $c$ is the speed of light.

Using this formula, it can be shown that at an acceleration of 188g, (188 times the acceleration due to gravity), the ship could reach alpha centauri in 14 days of ship time.  You might point out that 188 g's would surely smush everyone against the back wall of the ship, but the beauty of the theoretical drive described is that you carry your own gravity well along with you and therefore, you're always in freefall and don't feel the acceleration.

Here's the problem though.  The time that will have elapsed here on Earth will be much, much greater than the 14 days that elapsed on the ship.  The expression for the time elapsed on Earth is

$Earth\ time\ elapsed = \dfrac{c}{a}cosh\left(\dfrac{at}{c}\right)$,

which can be used to show that when the ship reaches alpha centauri, 817 years will have passed here on Earth.

The calculations shown here are nothing new, by the way.  Rindler applied them to the problem of relativistic space travel for the first time in 1960 in a Physical Review article titled "Hyperbolic Motion in Curved Space Time""[2].

References

2.  Rindler, W., "Hyperbolic Motion in Curved Space Time", Phys. Rev. 119 2082-2089 (1960).

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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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 differe…

Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…