Paper, astrolabe, ruler, compass: a short introduction to the math behind general relativity

I’m going attempt to give an accessible introduction to general relativity for non-mathematicians without glossing over the mathematical objects one must get a feel for to be able to follow research in the area. Let me know how successful I am!

The visual summary of the text below is here:gr visual summary

General Relativity

At its heart, general relativity (GR) is a theory that posits that what we experience as gravity is a result of of the shape of the space in which we live–spacetime. The fundamental equations of GR, the Einstein Field Equations, can be written as:

quantities quantifying the "shape of the space"=quantities quantifying matter in that space

meaning we can easily switch between the two descriptions. If we have all the information about the matter, we can draw the shape of the space, and if we have the shape of the space, we know all we need to about the matter.

How does gravity result from all of this? Well particles under no other forces are as lazy as possible–they move along straight lines. Only if the space they are moving in is curved, then there are no straight lines in the traditional sense (try drawing a straight line on an orange for example), so they simply move as straight as they possibly can on the curved surface.

The “Shape of a Space”

What is meant by the “shape of a space” in my description is actually a technical term. Let’s go back to the example of an orange and shrink ourselves down to be explorers on the surface of the orange, or conversely blow the orange up to be the size of a planet and go for a nice relaxing walk on it. We might take it as an unknown planet to be explored! If we are tasked with describing the surface in detail, we might want to walk over every point first and after a few days write our best description about what the orange as a whole might look like. If we want to make a more precise map, we’ll need some equipment, just like surveyors of the Earth need tools like compasses to determine the direction of a line, chains to measure distances, transits and  theodolites to measure angles, levels to determine elevation, etc.

Only, since we will be working in the language of mathematics our tools are more up to date: analogous to precise computers which can calculate to a degree of accuracy much greater than the traditional surveyor is able to with his or her clunky tools. Here I will introduce the tools used by mathematicians for surveying a space, show you how to use them, and then link you to further information about why and how they actually work, internally (this is all in the domain of math referred to as differential geometry).

So first, the tools! First and foremost we have mathematical objects called the tangent space and the cotangent space, the metric, and the Riemann curvature tensor. Those sound rather fancy, so rather we can think of these visually as a stiff piece of paper, an astrolabe, a ruler, and compass–with special properties–respectively, and they’ll rather dramatically form more than everything we need to survey the entire space.

Your goal with the piece of paper is to touch it to the surface of the world at exactly one point. The purpose of the astrolabe is that you can input any line you draw on the paper going through that point and the astrolabe in turn spits out a number (your boss has built the astrolabe and you can look at the plans to see exactly why and how the astrolabe does this). The ruler allows you to measure the distances between two points, and the compass has the peculiar property that it doesn’t measure N-S-E-W but rather, if you go in a small circle, taking your measurements with the paper, astrolabe, and ruler all the while, while trying to keep a line on your paper as straight as possible, the compass quantifies the difference between the starting and the finishing positions of the line. In general the compass carries out its measurement with the help of your measurements from the paper and the astrolabe.

So let’s get to surveying. Your boss gives you the rather simple task of stopping by every point in the world, walking in a small circle around that point, and taking measurements with you piece of paper, astrolabe, ruler and compass. If we weren’t mathematicians we’d eventually get tired and quit (visiting every point sounds rather tough!) but since we are we can just as well. Once you report back to the boss, at any point or at any two points she knows the shape of the surface, so in turn by the fundamental insight of general relativity she knows how gravity behaves on it.

But for now all we need to know, is that if we go around with our piece of paper, astrolabe, ruler and compass, we’ll be able to measure everything we need to know about the surface to quantify its shape in enough detail to see exactly how gravity will behave.

If you want to learn more, check out my introduction to general relativity from my Ph.D. thesis which has links to more resources.

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