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[Note: This transcript will not make much sense without the equations that I show in the video.]Today I want to answer a question that was sent to me by Ed Catmull who writes:
“Twice, I have read books on relativity by PhDs who said that we travel through time at the speed of light, but I can’t find those books, and I haven’t seen it written anywhere else. Can you let me know if this is right or if this is utter nonsense.”
I really like this question because it’s one of those things that blow your mind when you learn about them first, but by the time you have your PhD you’ve all but forgotten about them. So, the brief answer is: It’s right, we do travel through time at the speed of light. But, as always, there is some fine-print to what exactly this means.At first, it does not seem to make much sense to even talk about a speed in time. A speed is distance per time. So, if you travel in time, a speed would be time per time, and you would end up with the arguably correct but rather lame insight that we travel through time at one second per second.
This, however, is not where the statement that we travel through time at the speed of light comes from. It comes from good, old Albert Einstein. Yes, that guy again. Einstein based his theory of special relativity on an idea from Hermann Minkowski, which is that space and time belong together to a common entity called space-time. In space-time, you do not only have the usual three directions of space, you have a fourth direction, which is time. In the following, I want to show you a few equations, and for that I will, as usual, call the three directions of space, x, y, and z, and t stands for time.
Now, here’s the problem. You can add directions like North and West to get something like North-West. But you cannot add space and time because that’s like adding apples and oranges. Space and time have different units, so if you want to add them, you have to put a constant in front of one of them. It does not matter where you put that constant, but by convention we put it in front of the time-coordinate. The constant you have to put here so that you can add these directions must have units of space over time, so that’s a speed. Let’s call it “c”.
You all know that c is the speed of light, but, and this is really important, you do not need to know this if you formulate special relativity. You can put a dummy parameter there that could be any speed, and you will later find that it is the speed of massless particles. And since we experimentally know that the particles of light are to very good precision massless, that constant is then also the speed of light.
Now, of course there is a difference between time and space, so that can’t be all there is to space-time. You can move around in space either which way, but you cannot move around in time as you please. So what makes time different from space in Einstein’s space-time? What makes time different from space is the way you add them.
If you want to calculate a distance in space, you use Euclid’s formula. A distance, in three dimension, is the square-root of the of the sum of the squared distances in each direction of space. Here the x is a difference between two points in direction x, and y and z are likewise differences between two points in directions y and z.
But in space-time this works differently. A distance between two points in in space-time is usually called s, so that’s what we will call it too. A distance in space-time is now the square-root of minus the squares of the distances in each of the dimensions of space, plus c square times the squared distance in time.
Maybe let me mention that some old books on Special Relativity use a different notation, in which, instead of just putting a minus in the space-time distance, one uses a prefactor for the time-coordinate that is i times c. This has the exact same effect because the i square will give you a minus. The I turns out to be useless otherwise though, so this notation is not used today any more.
But why would you define a space-time distance like this, why not just all plusses? Well, for one, if you do it differently it doesn’t work. It would not correctly describe observation. That’s an answer, but not a very insightful one, so here is a better answer.
Einstein based special relativity on the idea that the speed of light is the same for all observers. You cannot do this in a Euclidean space where all the signs are plusses. But you can do it if one of the signs is different relative to the others. 
That’s because a space-time distance that is zero for one observer is zero for all observers. This is also the case in Euclidean space, but in Euclidean space, this just means zero in each of the directions of space.
But what does a zero distance mean in space-time? Well, let’s find out. For simplicity, let us look at only one dimension of space. So if the distance in space-time is zero, this means that the distance in space divided by the distance in time equals plus or minus c. And that’s the same for all observers. So this speed, c, is an invariant speed.But, well, we are not light, so we do not travel with the speed of light through space, and we do actually cover a distance in space-time. So let us look at this equation for the space-time distance again. Now let us divide this by the time difference. Now what you have on the left side is the space-time distance per time. And under the square root you have roughly something like the squares of the velocities in each of the directions of space. Plus c2.
And there you have it. Relative to yourself, you do not move through space, so these velocities are zero. You then only move into the time-like direction, and in this direction, you move with the speed of light. So, we indeed all travel through time with the speed of light.
I always try to show you equations because physics is all about equations. But to really understand what these equations mean, you have to use them yourself. A great place to do this is Brilliant, who have been sponsoring this video. Brilliant offers a large variety of interactive courses on topics in science and mathematics. They do for example have a course on Special Relativity, that will teach you all you need to know about space-time diagrams, Lorentz-transformations, and 4-vectors.
To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.