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.

Table of contents

  1. Partial derivatives
  2. Coordinates, vectors and the summation convention
  3. The metric