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

metryka1The series' table of contents

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

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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

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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?