Relativity

Part 3 - the metric

metryka1

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.

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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.

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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?

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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.