Math is Pretty

One hundred students line up to walk by one hundred closed lockers. The first student walks by opening every locker that is closed. The second student then walks by and opens every second locker that is closed and closes every second locker that is open. The third student does the same for every third locker and the nth student does the same for every nth locker.

After all the students have walked by the lockers, how many lockers are open.


Based on the responses in our group, this is either a very easy or a somewhat difficult puzzle. It was the latter for me. It took over thirty minutes for me to find the reason and provide a proof. I personally believe that type of frustration is good and I’m working on increasing the amount of time I spend trying to solve problems before giving up and looking for the answer.

Over the next few months, I’m learning the theories of Special and General Relativity with a group. Part of this learning experience is teaching each other the content of the different sections of the book we’re studying (The Geometry of Spacetime: An Introduction to Special and General Relativity). The first section that I had to teach was on Galilean relativity. The following are my cliff notes on the subject.

Introduction

Imagine two engineers:, one on a train moving 20 m / s, and another stationary on the ground. The engineer on the train throws a ball towards the back of the train at 20 m / s. What does the engineer on the ground see? What does the engineer on the train see? Will those observations be in agreement or translatable to each other.

A world in which the answer was no would be a purely subjective one. Physics could not exist because no objective facts could ever be correlated. A world in which observations can be correlated is a world in which a principle of relativity exists – i.e., some subset of objective facts or events have the same form in all frames of reference.

A principle of relativity works not by proscribing the same observations in different frames of reference but mathematical formulas that can translate between them. The engineer on the train will not observe the same trajectory of the ball as the engineer on the ground. Instead, they will be able to use a principle of relativity to translate their observations of the event so that they are in agreement.

Galilean relativity is a principle of relativity that was first proposed before special and general relativity. It stipulates that all observers moving in constant motion – without speeding or slowing down – will be able to translate their observations of the same events by factoring in the relative velocities of the observers.

In this introduction to Galilean relativity, we will walk though a concrete example of Galilean relativity by translating the observations of the first engineer when the ball is thrown in the air to the observations of the second engineer. We will then work through the first example in this section of the book, taking care to point out potential stumbling blocks for mathematicians. We will close by briefly providing an overview of the principles of relativity at work in Newtonian mechanics and special relativity.

Engineers and Trains

Let’s return now to our engineers and walk through a simple example of translating their different observations of the ball..

Relative motion is the key to translating events in Galilean relativity. The equations for such translations are:

τ = t

ξ = vt + z

Intuitively, this equation states that the observations of the two engineers will be translatable once their relative velocities are factored out.

In this example, the relative velocity of the engineer on the train is + 20 m/s with respect to the engineer on the ground. Or, alternatively because there are no privileged reference frames, the engineer on the ground is moving – 20 m/s with respect to the engineer on the train.

The engineer on the train throws the ball at – 20 m/s away from him. As expected, it travels in a parabola away from the engineer.

Assuming the engineers start at the same position, after three seconds the ball will be 60 meters away from the engineer on the train.

For the engineer on the ground, we can determine the position at any moment by using the above equation. Intuitively, the forward velocity of the train of 20 m/s and the backward velocity of the train at 20 m/s will cancel out and the ball will appear as if it is falling straight down.

We can predict this plugging in the formula. At one second, the engineer on the train will observe the ball at -20m. The engineer on the ground will see the ball at +20m/s (1s) -20m = 0m. At two seconds, the engineer on the train will see the ball at -40m and the engineer on the ground will see it at 20m/s (2s) – 40m = 0m. And so on.

Exercise 1: Potential Stumbling blocks

The Question: Suppose the events E₁ and E₂ have the coordinates (1,0) and (2,0) in R. What is the spatial distance between them, according to R? What is the spatial distance according to G?

One of the great things about this books is that the questions are largely written for mathematicians. In general, the treatment is fairly standard, but there is one potential stumbling block in notation and another in perspective that can sometimes cause confusion. Both of these, to the author’s credit, are carefully explained in the book.

• Notation: The first coordinate in an event is the time and the author has a preference for representing events in 1 dimension of space and 1 dimension of time. Critically, this means that (1,0) is not a particle at x=1 and y=1 but the position of a particle at time t = 1 second and position z = 0.

• Perspective: From the perspective of a mathematician looking at a graph – or the graph reference frame – the particle is not moving. The question might even appear nonsensical. It’s critical to remember that this is just an illusion. The graph perspective of velocity 0 is just another reference frame. And it is not the reference frame that we are being asked about.

With these caveats in mind, the problem is simple. One second passes between the events. During that time, the first reference frame G moves with a velocity v away from the coordinate z=0 and the second reference frame R moves with a velocity -v away from the coordinate z=0. G will measure the distance between the events as being +v and R will measure the distance between the events as being -v.

This blog has been under a heavy spam load for some time. In the midst of deleting some of spam comments (no I do not want to visit your porn site even if you like my post very much), I accidentally deleted all of the real comments from readers (some of which had never been posted).

My sincere apologies to my readers — for now I’m disabling comments until the problem of spam can be addressed more elegantly.