It's only funny if you're in the audience...

Hi,

Today, I would like to talk about math.

Now before you go getting that glazed expression, start hyperventilating, and decide to become a Luddite after all if this is the sort of thing you have to face while online... this is a very simple idea that deals with math that's taught reasonably early in grade school: the line in two dimensions.

If you've ever had to deal with lines, you may or may not remember that they are normally described as having a slope (which is how many units on a Cartesian plane the lines goes up [positive slope] or down [negative slope] for every unit you move to the right on that Cartesian plane). To fully describe a line, you also need to specify some point on the line (usually a Cartesian point — an ordered pair, for example an x-coordinate and y-coordinate; or an x or y intercept point [where either x or y is 0]). If you're feeling adventurous, the standard form of a line equation we are taught is: y = mx + b (where m is the slope, and b is the y-intercept value). A line can also be specified (in any number of dimensions) by two points (of an appropriate number of dimensions).

Here's the problem: the basic Cartesian representation of the line is fundamentally broken. Specifically, for a vertical line, the slope is infinite, and there is no y-intercept (that is, the vertical line runs parallel to the y-axis and never intercepts it; unless the line runs along the y-axis, in which case it intercepts it at an infinite number of points). One of the most basic notions in grade-school math breaks down horrifically when confronted with one of the most basic of geometric notions: a vertical line.

The two-dimensional solution is a simple one: use θ (it specifies an angle from 0 to 360 degrees) and a point (you can't use polar coordinates because it is always centered on the point (0,0) and that will only allow for the description of slope). As a note, the point can't be something like the y-intercept or x-intercept because not all lines will intercept both of them (i.e. vertical and horizontal lines). I would argue that if we taught kids to do lines this way instead, it would provide a much more useful tool. Firstly, a line always falls on a two-dimensional plane (no matter how many dimensions it exists in, you can always plot it on a plane using θ and a single point... a two-dimensional point if it's on the plane itself or a point in however many dimensions you're working with if the line is parallel to a reference plane). In grade school, you'd just teach how to work with them in two dimensions. In high school, it would be extended to three dimensions (most people work with three dimensional geometry at some point in their lives, even if it's just laying out a garden or putting something together). In late high school or early university, the whole thing can be generalized to n-dimensions (even in finance, plotting a line through multi-dimensional data can lead to valuable insights... it doesn't have to be physics or something to make use of it). It's only when you have to deal with "lines" in non-linear spaces (for example, general relativity) that you need to go beyond the basics of what would be learned in grade school with θ and a single point description of a line.

It may sound complicatatious (that's my "word" for the day), but if taught from early on, it would be no more complex (that's a math pun) than anything we've learned to do ourselves.

And by that I, of course, mean the secrets of the universe. I have been told my entire life — well as much of it as people were talking to me about this sort of thing or I could get my hands on books that were comprehensible to me at the time — that the special theory of relativity (1905) came about because the failure of the Michelson-Morley experiment (1881) to detect the “æther” led Einstein to reason that another explanation was needed. In reality, Einstein, to the best of anyone's knowledge, had never even heard of the pivotal experiment, but rather was led to his theory because of Maxwell's equations (1864) which said that light was actually an electromagnetic wave (and which, in one swoop, united the previously separate fields of electricity, magnetism, and optics). The problem with the equations was that if you applied classical Galilean transformations to them you got nonsensical answers. Specifically, if you you measured the speed of light in a “reference frame” moving at a constant velocity (say a person in a car) relative to, and sent from, another reference frame that is considered “at rest” (say a person standing by the road with a flashlight that the car is moving towards), the classical transformations said that the person in the car would measure the speed of light as the speed it left the flashlight (“c”) plus the speed of the car (say “v”). Maxwell's equations said that you should measure the speed of light as “c” in both reference frames, and this clearly did not agree with classical predictions, so something was wrong somewhere. Einstein's success (or genius as people suggest) was in accepting what Maxwell's equations were saying, as well as the Galilei-Newton basic Principle of Relativity that the laws of nature must be the same for all observers, to arrive at two “postulates” that he considered fundamental (note, inertial frames are just two systems moving at constant velocity with respect to one another, like the person in the car relative to the one by the road):

1. The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.
2. The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

What is amazing is that accepting those two very simple statements (and the ability to apply some fairly basic arithmetic) leads pretty much directly to the special theory of relativity. It gets a little more complex when applied to Maxwell's equations (well mathematically, it would take an average person about 4 to 6 months of mathematical training to be able to do), but Einstein realized that the Lorentz transformation factor γ=1/(1 − v²/c²)^1/2 would solve the problem, but required that the concepts of absolute time and space be torn down and replaced with the notion that time and space depended on one's relative motion to what one was measuring in another reference frame... again, his success was in accepting what the equations told him rather than trusting his very limited senses (we didn't do a lot of travelling near the speed of light in 1905... or at least as far as we had observed and understood). A quick aside on the Lorentz transformation: note that if v=0 (or close to 0 relative to the speed of light, which is very large), then the factor γ is equal to 1, which is what we observe every day around us when we throw balls or drive cars. It's not until v approaches c that γ starts to get large (the bottom of the fraction will get smaller and smaller, never reaching zero, and 1 divided by a small number gives a big number).

In his own words, “What led me more or less directly to the special theory of relativity was the conviction that the electromagnetic force acting on a body in motion in a magnetic field was nothing else but an electric field”. Whereas I have called his other reasoning his success, this leap of understanding is a true flash of inspiration that can be called genius without any reservation on my part. I don't know that it is possible for me to express how powerful an impact this had on me reading it for the first time last week. Like any great work of art (which I consider it to be), either it speaks to you or it doesn't, and if it does, what it says is utterly individual. The textbook I was reading (2nd year physics) indicated that the math was well beyond what I likely have been exposed to at this point; however, the concepts don't need the math to understand. An electric field is generated when two things have electric charges: if the charge sign (+/−) is the same, they repel; if they are opposite, they attract. Magnetic fields are generated when electric charges move: they generate a flux that permeates space and is perpendicular (90°) to the direction of motion of the charge. So imagine a charged object (say with a positive charge for simplicity's sake) that is moving outside of and along the length of a copper wire. The wire is conducting electricity and thus, because there are moving electric charges in the wire, it is generating a magnetic field. Again, for simplicity's sake, assume the object is moving in the same direction and at the same speed as the negative charges in the wire. Viewed from the reference frame of the wire, the object is moving through its magnetic field and because of its motion will experience a force that will push the object away from the wire — it feels no electric force because the wire is electrically neutral... even though there are moving negative charges generating a magnetic field, the number of negative charges is the same as the number of positive charges and the electric fields cancel each other out (it's just a bunch of copper atoms). Note: this is how electric motors work, it's a pretty important principle in modern life.

Now suppose we jump onto the object and that becomes our new reference frame (we've hopped in the car) and we do the experiment again. Well, once more for simplicity's sake, remember that the object is moving at the same speed as the negative charges moving in the wire and so, from the perspective of the object, it is not moving with respect to the magnetic field generated by the wire and so feels no force from it... but it still “sees” a force that is pushing it away from the wire. The magnetic field that was acting on the object from the perspective of the wire is not acting on it from the perspective of the object itself; but no matter the reference frame, a force is pushing the object away from the wire (observers in both frames would see the object and the wire moving apart because of the force, presuming no other force was acting on it). The answer lies in the Lorentz transformation between the two inertial frames! To the object, the negative charges in the wire appear stationary; however, the positive charges in the wire (that are stationary from the reference frame of the wire) appear to be moving to an observer on the object (at the same speed, but opposite direction, as the negative charges are moving from the reference frame of the wire). But earlier, I said that the positive and negative charges balanced in the wire (and so there was no electric force on the object), but that was from the perspective of the reference frame of the wire. From an observer on the object, the positive charges in the wire are moving and, from the reference frame of the object, will experience a compression of space in the direction of their motion per the Lorentz transformation and will appear to be closer to each other than do the negative charges that are not moving relative to the object. Because, to the object, there are more positive charges per length of wire than negative changes, the wire will appear to have a positive charge and will repel the object due to an electric field!

Again, words can't express how beautiful I find this result. With the publication of his special theory of relativity, Einstein negated the need for there to be an æther for electromagnetic waves to travel through, changed the way we view and experience space and time, and drew a firm line between “classical” physics and “modern” physics (oh, and his simultaneous publications of a paper explaining Brownian motion as an effect of the motion of molecules, and another paper fixing the field of thermodynamics by introducing the quantum nature of light as the source of the photoelectric effect didn't hurt either). His general theory of relativity, fyi, extends the two “postulates” to non-inertial (i.e. accelerating) frames and models gravity as a curvature of spacetime as a result (and was more perspiration than inspiration at that point, but amazing that he accomplished it in his lifetime given he could easily have left the task to others and sat on his laurels).