Monday, December 17, 2012

Gem(?) of the Week - the Minkowski metric

The Pythagorean theorem is one of those gems that I won't go into a lot of detail about, because it has been so well-covered before by others. Let's just say that it is a property of Euclidean space, in particular a consequence of the parallel postulate. I just mention it here because it is an example of a topic I am going to go over in great detail.

In 2D Euclidean space, we can figure out the distance of any point from any reference point by setting up a rectangular coordinate system with the reference point at the origin. If there were just two points we cared about, we could align the axes such that the X axis went through the other point, and then we could just read the distance off that axis. That's not really exploiting 2D space, so we will think about one reference point but lots of other points all over the plane. We can use the Pythagorean theorem to measure the distance from the origin to any point in the plane:


With this coordinate frame you can figure the distance between any two points, even if one is not at the origin, as follows:


By using the standard delta-notation from engineering, we can simplify this back to 

\[s^2=\Delta x^2+\Delta y^2\]

Now what is true for 2D Euclidean space is also true for 3D. The Pythagorean formula works, you just have to extend it to cover the third dimension
\[s^2=\Delta x^2+\Delta y^2+\Delta z^2\]

Those crazy topologists have generalized the Pythagorean theorem to fit their weird bent rubber spaces. They say that any space, along with a function which takes two points as an argument and returns a number, is called a metric space, where that function is called the metric. The metric must obey certain axioms, most important of which is commutativity - the distance from point A to B is the same as the distance from B to A.
Some of the topologists twisted imaginings don't really admit such a thing as a straight line. They get around this by saying that if you look at any small enough piece of the space, it is close enough to flat that we can define a metric there. Some spaces are so bent as to not even permit this, but those spaces which do, are called manifolds. In a manifold we talk about points which are close together, and represent this in our metric with differential notation


Now on a manifold we can specify a series of points to draw a path through, find the distance between each, add them all up, and get the length of the path. In calculus-speak, we have every point on the path, and we integrate along the curve to get the distance. However for flat space, there is such a concept as a straight line, and it is the shortest distance between two points. If you take the delta-form above and integrate the differential form, you end up with the same thing. Because of this, we will just show things in differential form from now on.

The film Dimensions is all about extending the same concept to four- and higher- dimensional space. All the Euclidean axioms apply, and the fourth dimension is exactly like the other three, so it works into the metric the same way:


One narrator talks about how 4D space is the prettiest, because it contains such things as the 24-cell. Also he says that it may be because real physical space is 4-dimensional also once you consider time. Blah blah Einstein aggressively ignore history blah blah blah. Spacetime is 4-dimensional, but here's the weird part.

Time is not the same as Space.

"Wait a minute" I hear you all saying - "Obviously time isn't the same thing as space. Duh." But, the whole reason we call time a dimension, and the same reason we don't call temperature a dimension, is that there are coordinate transformations that mix in time with space. I'm going off of memory, but this argument came to me through a little book called "Relativity and Common Sense". For instance, imagine a plane where every point is painted a different color. So, at each point we can measure three things, its x and y coordinate, and its color. If we rotate the coordinate frame, we mix together the x and y coordinates

x'=& &x \cos \theta&+&y\sin \theta \\
y'=&-&x \sin \theta&+&y\cos\theta\ \end{eqnarray*}\]

But, there is no rotation, no coordinate transformation which preserves our understanding of what coordinate transformation means, which can mix color and spatial coordinates. Color is not a dimension in this sense.

Well, Time is.

The transformation isn't just a rotation, but the Lorenz transformation from coordinates measured by one observer to coordinates measured by a relatively moving observer, depends on the relative speeds of the observers. And here's the weird thing - time is a dimension, but it is not just like the other three dimensions. In fact, the metric for the spacetime observing the Lorenz transformation is called the Minkowski metric, and it looks like this:


See the minus in the time term? It says that the longer the time between two events, the shorter the distance, all other coordinates being equal.

It gets weirder than that. If the time difference is long enough, it drags the whole right side negative. The squared distance between two events is negative. The distance between two events is imaginary. In special relativity, we say that when \(ds\) is real and positive, it is called proper distance, and it is the distance between two events as seen by some observer who sees them happening at the same time. Further there is no way for a single observer to be present at both events without exceeding the speed limit in the space, which introduces its own problems. When \(ds\) is imaginary, the (real) coefficient is called proper time, and represents the time interval between two events as seen by an observer who sees them happening in the same place. A single observer can be present at both events without exceeding the speed limit.

See what I mean by weird? In Minkowski space, there is a speed limit imposed as part of the fundamental geometry of space. If we though of time as just a dimension like space, this is equivalent to saying that it is impossible for a line to exceed a certain slope (change in space dimension per unit change in time dimension). No such limit exists in Euclidean space. Spacetime is 4D, but not Euclidean. It is not the same 4D space discussed in Dimensions. The Pythagorean theorem is false, and therefore the parallel postulate is false. Minkowski space is the only flat (metric works across long distances) space I know of which is non-Euclidean.

Now the question is, are the regular polytopes the same in Minkowski space? Does it make sense to talk about polytopes? Is a polytope regular from one point of view but not from another? These and other questions can be answered by the Minkowski metric, but I don't know the answers. I was only recently even able to form the questions.

Let's finish this off with a visualization:

Sunday, December 16, 2012

Putting my hardware where my mouth is...

I had been keeping this quiet, but it is the natural culmination of all that I have written on this blog.

At my day job, I work with space projects, but I have only been in the same room as flight hardware once, and that was purely as a tourist, to get my picture taken with it. I have written code that has gone into space, but never touched the hardware that carried it.

In October 2013, that will change. I am building space hardware. In a sense, that has already changed, since I have touched the hardware, but it's not space hardware yet. I have arranged for a version of the Rocketometer to fly on a rocket all the way into space.

The people I work for at my day job run an instrument in space that needs to be calibrated every so often. Every year or so, we fly a sounding rocket with a copy of our instrument, pop it up above the atmosphere for a few minutes, then let it fall back down into the atmosphere and descend on a parachute. It goes into space (well over 100km) but not into orbit.

On the campaign building up the rocket for the last flight this past summer, I was tinkering with my rocketometer (actually the 11DoF) and got to talking to my scientists about it. I told them about my daydream to actually get this thing on the rocket, and they said it was a great idea. Naturally it was far too late to get on board that last rocket, but with this next one I have plenty of time, especially considering that the hardware is finished, and I potentially could fly it now. The baseline mission is just to collect the data as fast as possible, and not bother with any on-board processing. That code was demonstrated with the speed test I did a few weeks ago.

My test plan and to-do list then looks like this:

  1. Adapt the old code to the new hardware. A couple of the I/O lines were reassigned to simplify the board design.
  2. Write an offline Kalman filter to process the data. IDL will work fine for that. Mostly I just need a set of equations of motion that allow the compass, gyro, and accelerometer to calibrate each other.
  3. Calibrate the sensors. I have an old record player with no needle which will be perfect for this. I may be able to use some stuff in the labs at work to help with this.
  4. Do a test flight in a model rocket. These generate a similar scale of forces and rotation rates, just for much shorter durations, seconds rather than minutes. The Rocketometer was designed to be carried in any rocket with a payload section 1" or larger in diameter.
  5. Get USB Bootloader++ working. This is low priority, as I can program the part over serial as I have been doing for a while.
  6. Consider on-board processing of the data.
I will be using the Rocketometer2148 with an MPU6050 6DoF sensor, an ADXL377 analog high-g accelerometer, an AD7991 12 bit ADC to read it, an HMC5883 compass, and a BMP180 pressure/temperature sensor.

With a couple of changes to main.cpp and gpio.cpp to tell it where the sensors and lights are on this board, the thing works! It may also be working at my goal rate of 1000Hz, attributable to a faster SD card, reading the compass only 1 of 10 times that the 6DoF is read, and not reading the HighAcc.