A Mathematical Intro to General Relativity, Part 1

Special relativity, which we introduced a while ago, is a really good theory. The combining of time and space into a single manifold, spacetime, turns out to be really convenient. It also predicts some important effects like time dilation. Quantum mechanics also works really well in a background of special relativity.

However, there’s one very serious problem with special relativity.

EinsteinExclamation

It doesn’t work with gravity.

EinsteinOof

In Newtonian gravity, the force of gravity depends on how far away, say, the Earth is, right “now.” The farther away you are, the weaker gravity is.

The problem is, in special relativity, there is no consistent idea of “now.” As we discussed before, it’s impossible for everyone to agree on which things are simultaneous. And so, it would be impossible, in special relativity, for gravity to “decide” how far away the Earth is.

To be fair, Einstein himself realized this. While electromagnetism and quantum mechanics fit amazingly well into the framework of special relativity, gravity doesn’t fit in very comfortably.

In order to reconcile gravity with special relativity, something more was needed.

Fortunately, Einstein had a good idea.

EinsteinIdea

Consider a person in a space station.

InSpace

If there weren’t windows to look out, she couldn’t tell the difference between whether the space station was in free-fall around the Earth…1

SpaceStation

…or whether the space station was far away from any gravitational source.

SpaceFarAway

In both cases, it feels like you’re not going anywhere. Sitting there doing nothing when there isn’t gravity feels the same as traveling incredibly fast while orbiting the Earth.

That’s a bit weird, when you think about it. But it’s also the key observation for general relativity. When you free-fall, whether in lots of gravity or none, you don’t feel like you are turning.

To tie this back into something we recently talked about, remember geodesics?

Geodesics are “straight” lines in a curved manifold.2 In that post, we explained how, even though they may look curved in our visualization of the manifold, to someone living in the manifold, they are straight lines. When you follow a geodesic, you don’t feel like you are turning.

greatCircleSphere

The key idea for general relativity is to equate these concepts.

General relativity says that “straight” should mean “geodesic” which should mean “free-falling.” Einstein’s goal was to somehow figure out how to write down a metric for spacetime, so that geodesics are the paths that someone would follow if they were freely-falling.

Thus, the basic idea of general relativity is that freely-falling people (and matter and energy) follow geodesics in the spacetime, but also, that matter and energy bend the spacetime itself which determines how everything will travel.

Gravity is not represented by a force, but rather by geometry.

That idea isn’t too hard to write down, but how would you go about actually finding those equations for the spacetime metric that would satisfy those dual principles?

There’s a reason it took even Einstein eight years to figure out the right equations.

EinsteinWorking

But he did eventually figure it out.

The Einstein equations are R_{ij} - \frac12 R g_{ij} = T_{ij}. Nice and short.

huh

Of course, it’d be nice if you knew what any of that meant. Let’s work on that.

The first is a notational thing. The indices (subscripts), like in R_{ij}, mean that that quantity is a matrix. The i and j mean we’re looking at the matrix entry in the i-th row and j-th column. The matrices here are four by four, with each row/column representing the quantity in a particular coordinate direction.

matrices

In other words, while the Einstein equations look like a single equation, they’re really sixteen equations, one for each entry in the matrices.

Next, what does each term represent?

For two-dimensional surfaces, we earlier talked about Gaussian curvature. While for two-dimensions, that’s the only kind of curvature there is, it’s perhaps unsurprising that in higher dimensions, there are more types of curvature.

The matrix R_{ij} and the function R are some of these new types of curvature.

scalar

R is the scalar curvature. It’s essentially the same as Gaussian curvature,3 in that positive curvature makes a surface compress in on itself, like on a sphere. That means a circle has shorter perimeter than you would expect for its radius, and contains smaller area.

Ricci

The matrix R_{ij} is the Ricci curvature (said “REE-chee”). The geometric intuition is definitely harder for this curvature, but the general interpretation is similar. A positive entry of the Ricci curvature makes a surface compress in on itself in a particular direction. So scalar and Ricci curvatures act pretty similarly, it’s just scalar is the overall stretching, while Ricci gives the stretching in each direction.

Scalar and Ricci curvatures aren’t all the possible kinds of curvature in higher dimensions, but I’ll leave more discussion of that to a footnote.4

metric

The g_{ij} is actually something we already know, the metric. Before, we’ve written metrics something like ds^2 = \sin^2(x) dx^2 + 2x dxdx + dy^2. We can rewrite this information as a matrix, as \begin{bmatrix} \sin^2(x) & x\\ x& 1\end{bmatrix}. (The 2x changes to x since there are two of the terms in the matrix, but only one in the ds^2 form.) We would then write the entry in the i-th row and j-th column as g_{ij}.

stress-Energy

The last term, T_{ij} is the stress-energy tensor. It contains all the information about where matter and energy are, and their current speeds.

So, there you have it. Einstein’s equations.

soProud

So, how does gravity work?

Since “negative mass” isn’t really a thing, you expect the stress-energy tensor to be “positive.”5

Looking at the left hand side of Einstein’s equations, positive matter will mean positive curvature.

And positive curvature causes the spacetime to compress in on itself, or in other words, for the metric to get smaller.

But as we talked about when we introduced geodesics, if the metric is smaller, it makes the geodesics in the manifold want to bend towards where the metric is smaller. This is the effect we see for gravity.

So, the Earth is freely-falling around the sun, and so is traveling along a geodesic in spacetime. If there wasn’t a sun, the geodesic would just be a straight line. But the sun is big, and so all that (positive) matter compresses the spacetime around it, and so the Earth’s geodesic wants to bend toward the sun. And it just so happens that the amount of bending means that the Earth ends up orbiting the sun.

This is slightly different than the standard explanation that gravity is like a bowling ball on a mattress.

Spacetime lattice analogy

That mental picture is useful, but it’s not really what’s going on.

This picture implies that gravity is “down,” and mass just pulls spacetime “down,” leaving a pit for matter to spiral into. But gravity is not an omnipresent force, universally pulling everything down. Rather, mass warps spacetime, which makes “straight lines” curve around the mass.

In fact, we only ever feel gravity when we’re not following a geodesic. Take Einstein sitting on his chair, furiously working:

EinsteinWorking

Because he’s not freely falling into the Earth (thanks to the pressure of the chair), Einstein is actually accelerating away from the geodesic that gravity wants him to take. That acceleration is what we feel as the force of gravity.

This is similar to the centrifugal force you feel when you drive a car around a corner. Relative to the car, you aren’t moving, but relative to the road, you’re accelerating sideways. You feel the acceleration as a sideways force.

In the same way, when Einstein is sitting in his chair, relative to the desk, he’s not moving. But relative to the underlying spacetime geometry, he’s accelerating away from a geodesic. Just sitting there, he is not traveling straight.

That seems like a good place to stop for today. There’s still quite a lot to say to introduce general relativity, so we’ll continue next week. Relativity is quite the complicated mathematical problem!


  1. In case you haven’t heard: astronauts float not because there is no gravity, but because they are falling. They just happen to be traveling so fast sideways that they manage to miss ever actually hitting the Earth. 
  2. They also represented shortest paths. 
  3. In fact, for two-dimensional manifolds, it’s just twice the Gaussian curvature. 
  4.  The Riemann curvature tensor R^i{}_{jkl} contains all possible information about curvature (including the Ricci and scalar curvatures) and, in fact, contains all the information the metric does. However, it’s quite complicated.  The Riemann tensor can be represented by a four-dimensional matrix, since it has four indices, i, j, k and l. The Ricci curvature is calculated from the Riemann tensor by R_{jk} = R^i{}_{jik}, where the up and down i‘s mean you sum up the entries over each possible value of i, i.e., i =1,2,3,4 in the case we’ll care about. The scalar curvature is calculated from the Ricci curvature by R = R_{ij} g^{ij}, where again we sum over repeated indices, and g^{ij} is supposed to be the inverse matrix of g_{ij}. (This notation of adding up over repeated indices is called the Einstein summation convention, since there would be way too many summation signs running around otherwise.) The Riemann curvature tensor can be calculated from the metric, but the equations are… messy, to say the least. Fortunately, we shouldn’t ever need the precise equations in the blog. 
  5. I put positive in quotes since it’s not quite that each entry is necessarily positive, but the overall matrix is in some sense positive. For normal matter and energy that we see in everyday life, the matrix is positive in this way. 
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