Angle of x' axis in Minkowski spacetime | Special relativity | Physics | Khan Academy

We've been doing some interesting things in the last few videos. We let go of our Newtonian assumptions that the passage of time is the same in all inertial frames of reference, that time is absolute, that one second in my frame of reference is the same as one second passing in your frame of reference. We even let go of the idea that space is absolute, that one meter in a certain direction in my frame of reference is going to be the same as one meter in your frame of reference if we're moving at relative velocities with respect to each other.

And what that allowed us to do? It allowed us to reconcile what is actually observed in the universe, and that's the idea that the speed of light is an absolute. Regardless of what inertial frame of reference we're in, regardless of the speed of the source of the light, we will always measure light traveling at three times, or roughly three times ten to the eighth meters per second. When we let go of our assumptions about absolute time and absolute space, and we did assume absolute velocity for light, it gave us a very interesting diagram because it took our x-prime axis from being on top of the x-axis and put it at an angle relative to the x-axis.

Now one question that you might have is, "Well, what is this angle?" The way I've drawn it, it looks like the angle between x prime and x is the same as the angle between t prime and t. As we did in the last videos, we turned our units in time, or the units along this axis, the ct axis. Instead of saying seconds, we're now going to measure it in meters. Once again, watch the last video if you're having an issue with that. But it looks like this angle and this angle are the same.

What I want to do in this video is feel good about the fact that they really are the same. So, one of the convenient things, when a couple of videos ago we went through this thought experiment of my friend's frame of reference—her emitting a photon of light and bouncing off of the spaceship that's three times ten to the eighth meters in front of her and then coming back to her—is we assumed that as we mark off the meters and as we mark off the seconds, the distance from the origin for three times ten to the eighth meters is the same as the distance from the origin as one second.

Now it makes even more sense because now we call this three times ten to the eighth meters and this times three times ten to the eighth meters as well. So the speed, when we depict the path of light, of a photon in either of these spacetime diagrams, is always going to be at either a positive 45-degree angle or a negative 45-degree angle depending on the direction that the light is traveling. In fact, that was what we used to establish that this x prime axis is not going to sit on top of the x-axis; it's going to be at an angle.

But now let's actually think about that angle. So if we say that this angle right over here is alpha, and if we were to continue this 45-degree line, we had to remember that any path of a photon is going to be at a 45-degree angle. So let me continue that. This is going to look like that, and then we could go back and continue it, right, like that. Well, we know that this is going to be 45 degrees, 45 degrees right over there.

This is going to be—whoops—this is going to be 45 degrees right over there. We know that this green triangle that I'm—oh, green isn't different enough; let me get a better color. This move—no, that's still green. We know that this purplish triangle that I'm setting up right here is an isosceles triangle. How do I know that this is an isosceles triangle? Well, this hash mark right over here on the ct prime axis, that is three times ten to the eighth meters, and this hash mark on the x prime axis is also three times ten to the eighth meters.

And so this side is equal to that side, and so we know that the base angles of an isosceles triangle are going to be congruent. So this angle is going to be congruent to that angle. Well, if those two angles are congruent, then this angle is going to be congruent to that angle because they are supplementary to those two base angles.

So notice, if we look at this triangle—let me do this in a new color—if we look at this triangle right over here and compare it to this triangle right over here, notice they both have a 45-degree angle, they both have this blue angle which is congruent, so the third angle has to be congruent. They always have to add up to 180 degrees. If you have two angles of two different triangles that are the same, then the third angle has to be the same.

So if this is alpha—actually, I'll do—let's see, I've already used—I’ll do four arcs there. If that angle is alpha, then this angle is going to be alpha as well. This is really interesting; it's a beautiful kind of symmetry, and it really comes out of the fact that the speed of light should always be measured at roughly three times ten to the eighth meters per second in any inertial frame of reference.

Notice, if we go back to my frame of reference, that photon that I emit from my flashlight, it will look like this. It will look like this; its path in this Minkowski spacetime from my point of view will look like that. And notice from my friend's point of view—the one that's traveling on a ship—for every amount, let's say, let's pick up a certain point in spacetime right over there, well, her x time coordinate is going to be right over there, and her ct prime coordinate is going to be right over there.

So that makes sense: in one, I guess we could say, light meter, the thing travels a meter, so it is still traveling at the speed of light from her point of view, even though she is moving at half the speed of light relative to me. And you could think about what would happen if she was moving even faster. Someone moving even faster, their ct prime axis would be at an even more severe angle, so it would look something like that.

And then what would their x prime angle look like? Well, then their x prime angle is going to be symmetric around that line that shows the path of light. So it would look like this. So this is someone, let's say, who's moving even faster relative to me; let's call that ct prime prime, and this would be their x prime prime.

You can imagine you get close as that second frame of reference gets closer and closer to the speed of light; from my frame of reference, their coordinate axes are just going to get more and more scrunched up around this 45-degree angle. So that's really neat. Now, another thing I want to think about—and I won't draw it in this video just because I think I've given you enough to digest—is there's a symmetry here. If my friend is moving at half the speed of light, it is moving at half the speed of light in the positive x direction according to me, then from her point of view, I'm moving at half the speed of light in the negative x prime direction.

So we should think about, "Well, what would—if I were to—if we were to look at this right over here, what would my frame of reference look like if I were to project it on top of that?"