Landscapes and atmospheric refraction

Sometimes I'm bored and I'm getting involved in discussions with various kinds of pseudoscientists. Such discussions are often a waste of time, but it's possible occasionally to get something out of them - after all, if you want to explain to someone why there are wrong, you need to have a good understanding of the topic yourself. If your knowledge is not enough to counter the opponent's arguments, you need to expand it, and so you are learning. It was the case for me this time.

It all began with two flat-earthers appearing on a certain forum. The exchange started with standard arguments like timezones, seasons, eclipses, the rotation of the sky... what have you. As usual in such cases, those arguments were met with silence or really far-fetched alternative explanations. I'll omit the details, interested people can find standard flat-earth arguments on the web.

Well, you can't sway a person that is completely confident in their beliefs with arguments, so the discussion has become somewhat futile. Both sides stuck to their positions and mulling over the same issues time and time again has started. That is, until one of the flat-earthers started presenting photos which, according to them, proved that the Earth "can't be a ball with a 6371-6378 km radius", with descriptions that can be expressed shortly as "explain THAT!". Alright.

Challenge accepted!


Events and space-time

The first entry in the series will be quite basic, but I think that some problems will nevertheless be quite interesting. We'll be talking about what is the space-time, events, and we will show where the theory of relativity comes from. So, let's go :)

The notion of space-time is briefly mentioned at school, but usually the profound consequences of combining space and time into a single entity aren't explained too much. To understand this, one must first go a bit deeper into the details of this idea.


Hyperbolic functions - what sorcery is this?

If you are like me, your first contact with the hyperbolic functions was as "this strange, useless something on the calculator". There were just some weird buttons labeled "sinh" and "cosh". The school finally explained what "sin" and "cos" are, but there was no mention of those variants with the final "h". What is this about? The names suggest some similarity to the trigonometric functions, let's see what happens:

 \begin{array}{ll} \cos (1) = 0.54030230586 & \cosh (1) = 1.54308063482 \\ \cos (10) = -0.83907152907 & \cosh (10) = 11013.2329201 \end{array}

(You will get these results if you have the calculator set to radians - if you use degrees, then the cosine results will be different; it has no influence on the hyperbolic functions and we'll see later why that is.)

Right, these 11 thousand for cosh(10) look very similar to the trigonometric functions. This "h" apparently changes quite a bit, but what exactly...?

If you encountered complex numbers during your later education, you could stumble upon such definitions:

 \begin{array}{ll} \cos x = \frac{e^{ix} + e^{-ix}}{2} & \cosh x = \frac{e^x + e^{-x}}{2} \\ \sin x = \frac{e^{ix} - e^{-ix}}{2i} & \sinh x = \frac{e^x - e^{-x}}{2} \end{array}

Some similarity is visible here, but... Why such a form? What does this have to do with hyperbolas? If you don't know it yet, you will know after reading this article.


The shape of a black hole's event horizon

Yesterday, while browsing the internet, I stumbled upon a thread which looked like a typical question asked by someone interested in science, and turned out to be a really interesting problem.

The question that has been asked concerned the shape of a black hole. A few people replied that the event horizon (the boundary - or the "surface" in a way - of a black hole) has the shape of a ball (which should be actually described as a sphere, since the horizon is a 2-dimensional surface, and not a 3-dimensional shape). Someone suggested that it's not exactly true, because black holes usually spin, which flattens them. I entered the thread then and said that even when a black hole is spinning, its horizon is still spherical - it's described by an equation like r = const. But is that really so...?


Part 3 - the metric


We already mentioned the notion of the magnitude of a vector, but we said nothing about what it actually is. On a plane it's easy - when we move by v_x in the x axis and by v_y in the y axis, the distance between the starting and the ending point is \sqrt{v_x^2 + v_y^2} (which can be seen by drawing a right triangle and using the Pythagorean theorem - see the picture). It doesn't have to be always like that, though, and here is where the metric comes into play.

The metric is a way of generalizing the Pythagorean theorem. The coordinates don't always correspond to distances along perpendicular axes, and it is even not always possible to introduce such coordinates (but let's not get ahead of ourselves). We want then to have a way of calculating the distance between points \Delta x^\mu apart, where x^\mu are some unspecified coordinates.


Part 2 - coordinates, vectors and the summation convention

The basic object in GR is the spacetime. As a mathematical object, formally it is a differential manifold, but for our purposes it is enough to consider it as a set of points called events, which can be described by coordinates. In GR, the spacetime is 4-dimensional, which means that we need 4 coordinates - one temporal and three spatial ones.

The coordinates can be denoted by pretty much anything (like x, y, z, t), but since we will refer to all four of them at multiple occasions, it will be convenient to denote them by numbers. It is pretty standard to denote time by 0, and the spatial coordinates by 1, 2 and 3. The coordinate number \mu will be written like this: x^\mu (attention: in this case it is not a power!). \mu here is called an index (here: an upper one). By convention, if we mean one of the 4 coordinates, we use a greek letter as the index; if only the spatial ones are to be considered, we use a letter from the latin alphabet.


Part 1 - partial derivatives

As I mentioned in the introduction, I assume that the reader knows what a derivative of a function is. It is a good foundation, but to get our hands wet in relativity, we need to expand that concept a bit. Let's then get to know the partial derivative. What is it?

Let's remember the ordinary derivatives first. We denote a derivative of a function f(x) as f'(x) or \frac{df}{dx}. It means, basically, how fast the value of the function changes while we change the argument x. For example, when f(x) = x^2, then \frac{df}{dx} = 2x.

But what if the function depends on more than one variable? Like if we have a function f(x,y) = x^2 + y^2 that assigns to each point of the plane the square of its distance from the origin. How do we even define the derivative of such a function?


The mathematics of black holes

The science of gravity is being taught at schools as a part of the standard curriculum. Teachers say that bodies attract each other, they give the Newton's equation and that's about all. If it is at all mentioned that this is only a huge simplification of our current knowledge of this field, it is done so only on the side. The existence of something like General Relativity is only being hinted at - but it is not without reason. Full understanding of the mathematics behind that theory requires years of studying physics and is far beyond the reach of a typical high school student.

As a curious person, I was always interested in what this theory looks like and what is so hard about it. It was actually one of the main reasons that pushed me towards studying physics. I was eager to know what exactly is the curvature of spacetime and how to describe it. If it interests you, too - you are in the right place.

Thinking about this recently, I arrived at the conclusion that it should be possible to describe the basics of General Relativity using only high school mathematics. Therefore I aim to create here a series of articles in which I will explain the mathematical concepts behind GR and show how to use them to describe gravity - everything with the assumption that the most advanced concept known to the reader is that of a derivative of a function. If I succeed, we will see how the curvature of spacetime manifests as the attraction between bodies, how black holes affect time and why it is impossible to escape when you are inside.

So, feel free to read the series :) Remarks and suggestions are welcome as always.