A Mathematical Intro to General Relativity: Part 2

In the last post, we introduced general relativity.

The key and foundational idea is that gravity, instead of being a force, is represented by the geometry of the spacetime. Freely-falling mass or energy is supposed to travel along “straight” paths, or, in other words, along geodesics. How much the spacetime bends and curves in turn depends on where the mass and energy are.

In special relativity, which is just the special case of general relativity where there isn’t any gravity, everyone agreed that light traveled at 45 degree angles.1

Of course, special relativity is based in nice, simple $\mathbb{R}^4$. But in general relativity, the manifold we’re working on could be a lot worse…

And I do mean a lot worse. We’ll need to be more careful…

In order to talk about the paths light can take, we need something other than coordinates.

The key idea is to use the metric instead.

Recall the metric for special relativity, $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$. This was something like a modified Pythagorean theorem for measuring lengths of vectors (and thus measuring speeds).

For a normal (Riemannian) manifold, the metric works like you would think, and $ds^2$ gives the square of the length of a vector, i.e., the square of the speed.

But in special relativity, we have a negative term in the metric, and so it doesn’t quite make sense to talk about $ds^2$ as the square of the length or speed, as a square should always be positive. The metric $ds^2$ can now be positive, zero or negative, but it’s still an important quantity.

And those cases where the metric is zero are the key.

A path at 45 degrees has to move in the $t$ and in a space direction (like $x$) at the same rate. If you take that path’s tangent vector and put it into the metric for special relativity, we get that $ds^2 = 0$!

In general relativity, we choose these zero directions to be the directions that light can travels.

Along with the four-dimensional manifold as the background, general relativity has a metric $ds^2$ of the same type as in special relativity: a metric with one “negative” direction.2

If we have a vector somewhere in our spacetime, and the metric $ds^2$ is zero on that vector, then we say the vector is null.

A null path is a path through your spacetime whose tangent vector is always null. A null geodesic is a null path that happens to also be a geodesic.

Light travels along null geodesics.

A good way to visualize this is the light cone. If light is emitted at a point in the spacetime for one instant, the light travels out in a spherical fashion. This shape, if we draw it in three dimensions instead of four, looks like a cone:

Well, at least, that’s what it looks like in special relativity. In general relativity, it’s probably bent.

Recall that the idea of geodesics is that they are “straight.” In special relativity, the straight paths at 45 degrees that light travels on are the null geodesics.

In general relativity, remember that gravity bends geodesics! Positive mass compresses spacetime, which makes geodesics want to bend closer to the mass.

This means that gravity bends light!3

Now, the light-bending effect of gravity is small. For instance, light coming from far away and passing just past the surface of the sun (where the sun’s gravity is the strongest) bends by only 1/2000 of a degree.

That’s not much, but it is noticeable, and it’s a very important effect. In fact, the validation of this effect catapulted general relativity to worldwide fame.4

In astronomy, they sometimes go the other way: they measure the light deflection, and use that to measure how much mass there is bending the light.

A good example of this is Einstein’s cross.

The four blue stars look like four different stars. But actually, all of them are light from the same star.5 That light is bending around a galaxy from multiple directions, and so we see it four times!

Pretty cool, huh?

What kinds of paths do other objects (besides light) travel on?

In special relativity, recall that people or rocketships (or really anything other than light) travel on paths that are more “vertical” (i.e., point more in the $t$ direction) than the path light travels.

In terms of the light cone (of either relativity), this means the paths of non-light objects are inside the cone. In terms of the metric, the metric $ds^2$ applied to the tangent vector is negative, since “up” is the time direction, and the time direction is the negative direction.

If the metric $ds^2$ applied to a vector is negative, we say the vector is timelike. Similarly, if $ds^2$ is instead positive, the vector is spacelike. Along with null vectors, these types of vectors tell us what kind of direction each vector is pointing in: a space direction, a time direction, or a light direction.

Like with null paths and geodesics, a timelike path (or geodesic) is a path (or geodesic) with a timelike tangent vector. In other words, it’s a path that is traveling into the future.

Anything traveling slower than the speed of light travels on a timelike path, into the future.

If you remember, anything freely falling is supposed to travel on a geodesic. For instance, the Earth is traveling on a timelike geodesic. If we draw this path in the spacetime, it spirals up into the future.

Note that earlier I didn’t say that “everything travels on a timelike path.” General relativity doesn’t actually say that, any more than special relativity does, though it’s often presented that way.

There’s nothing stopping a particle going faster than the speed of light.

In science fiction, if you ever hear the word “tachyons,” those are theoretical (probably not real) particles that go faster than the speed of light.6 In other words, they travel on spacelike paths.

What you can’t do is accelerate from a slow speed to faster than the speed of light.

To accelerate, you need to use energy. However, as your velocity approaches the speed of light, it takes more and more energy to get the same amount of acceleration. In order to accelerate up to the speed of light, you need an infinite amount of energy. That turns out to be a bit of a roadblock.

But, tachyons are fine since you don’t accelerate them to faster than the speed of light. They are always going faster.

The last thing we’ll talk about in this post is perhaps a bit dry, but really important for understanding relativity, so put your thinking caps on.

It’s called coordinate invariance.

At first glance, everything we’ve talked about seems to depend on coordinates. For instance, when we talk about the vector $(1, 1, 0, 0)$ using $(t, x, y, z)$ coordinates, it seems our vector depends on the coordinates.

But, actually, if we were a bit more… careful… in our definitions, vectors are a thing, independent of coordinates!

The idea is, roughly, to think of a vector as an arrow on your manifold.That arrow is there, and is the same arrow, no matter which coordinate system you use to describe it.

For instance, consider spherical coordinates. In spherical coordinates for $\mathbb{R}^3$, the three coordinates are $(r, \phi, \theta)$, where $r$ is how far from the origin (of space, not spacetime) you are (i.e., the radius of the sphere you’re sitting on), and $\phi$ and $\theta$ are the longitude and latitude on that sphere.

The vector $(1, 1, 0, 0)$ in $(t, x, y, z)$ coordinates is $(1, 1, \pi/2, 0)$ in spherical spacetime coordinates $(t, r, \phi, \theta)$.

Changing coordinates changes the representation of a vector, for sure, but the vector remains the same vector.

A similar idea is true to the metric and the curvature.

If you look at the metric from special relativity (which, remember, is just a special case of general relativity), the metric using the standard $(t, x, y, z)$ coordinates is $ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$, which looks like it is dependent upon the coordinates we’re using.

But it actually isn’t.

If we looked at that metric in spherical spacetime coordinates $(t, r, \phi, \theta)$, the metric in terms of those coordinates is $ds^2 = -dt^2 + dr^2 + r^2 d\phi^2 + r^2 \sin^2(\phi) d\theta^2$.

This looks completely different, but it is the same metric–it measures the same vectors to be the same length.

Being able to freely change between coordinate systems is useful because a different coordinate system (and thus a different visualization) can bring out different properties of a spacetime that were obfuscated7 by the original coordinate system.

We’ll use this next time when we talk about black holes! Everyone loves black holes.

You can take your thinking caps off now.

1. Well, 45 degree angles when using geometrized units, i.e., saying that the speed of light is 1.
2. This kind of metric is called a Lorentzian metric, to differentiate it with the more usual Riemannian metric, which has only positive directions.
3. Actually, Newtonian gravity already predicted that light should bend. Einstein also derived the effect from the equivalence principle, but in either of those calculations you get half the effect that the full theory of general relativity predicts.
4. In 1919, just four years after Einstein’s theory was first published, Sir Arthur Eddington measured the deflection of starlight by the sun during a solar eclipse. While the measurements weren’t very precise, this experimental confirmation of Einstein’s new theory was widely reported
5. Well, really it’s a quasar. It’s hard to make out individual normal stars from 8 billion light years away.
6. In physics literature, tachyon can sometimes instead refer to an “imaginary mass field,” but I’m a mathematician, so I only see the term in sci-fi.
7. Obfuscated means hidden, or made unclear. Kind of like my sentence, thanks to my word choice.

4 thoughts on “A Mathematical Intro to General Relativity: Part 2”

1. Harsh Gupta says:

Kind sir,

Thank you for this wonderful blog that explains math and physics in a very intuitive way. Even though I have not understood a 100% of the matter covered, I feel like I have finally begun to grasp Special and General Relativity thanks to your enlightening posts.

I wish I had a wonderful teacher like you during school/college. Please keep continuing the great work ad-infinitum.

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1. Dr. Dilts says:

Thanks for the kind words.

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2. The bar tender says, “We don’t serve tachyons here.”
A tachyon walks into a bar.

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