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Current time:0:00Total duration:7:22

Angle of x' axis in Minkowski spacetime

Video transcript

- [Voiceover] We have 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 initial frame 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 are moving at relative velocities with respect to each other. And what that allows us to do, it allowed us to reconcile what is actually observed in the universe. And that's the idea that speed of light is an absolute. That regardless of what inertial frame of reference we're in regardless of the speed of the source of the light that we will always measure light travelling at three times or roughly three times 10 to the eighth meters per second. And 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 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. Or as we did in the last videos, we turned our units in time where the units along this axis, the ct-axis, instead of saying seconds, we're now going to measure it in meters, and once again watch the last video if you're having an issue with that. But it looks like this angle and this angle is 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 videos ago, we went through this thought experiment of in my friend's frame of reference, her admitting a photon of light bouncing off of the spaceship that's three times 10 to the eighth meters in front of her, and then coming back to her. As we assumed that as we mark off the meters, and as we mark off the seconds that the distance from the origin for three times 10 to the eighth meters is the same as the distance from the origin is one second. And now it makes even more sense 'cause now we call this three times 10 to the eighth meters and this times three times 10 to the eighth meters as well and so the speed... When we depict the path of light of a photon in either of these space time diagrams, it's always going to be at a either a positive 45 degree angle or a negative 45 degree angle depending on the direction that light is traveling in. In fact, that was what we used to establish that this x prime axis is going to be, it's not going to sit on top of the x-axis, it's going to be at an angle. But I don't 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 that we had to remember, any path of a photon is gonna be at 45 degree angle, so let me continue that. So, this is going to look like that and then we could go back and continue it right like that. Well, we know that this 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 and we know, we that this green triangle that I'm... It's not green, it's not different enough, let me get a better color. This??, no that's still green. We know that this, this purplish triangle that I'm setting up right here. This 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 10 to the eighth meters and this hash mark on the x prime axis is also three times 10 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 congruent. So, this angle is going to congruent to that angle. Well, if those two angles are congruent, then this angle is going to congruent to that angle. Because they are supplementary to those two base angles. And so notice, if we look at... 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 of the same, then the third angle has to be the same and so, if this is alpha, I shall 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. And, so, 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 three times 10 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, my flashlight it will look like this. It will look like this. It's path in this Minkowski space time from my point of view will look like that and notice, from my friend's point of view the one that's travelling on a ship for every amount let's say, let's pick a certain point in space time 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, in one I guess we could say light meter that thing travels a meter. So, it is still travelling 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 can think about what would happen if she was moving even faster. Someone moving even faster. Their ct prime axis would be yet at an even, at an even more severe angle so it might look something like that. And then, what would their x prime angle look like? Well then, their x prime angle is going to 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. As you can imagine, you get close as that second frame of reference gets closer and closer to speed of light from my frame of reference, their coordinate axis are just going to get more and more squanched up around, around this 45 degree angle, so that's really neat. Now, another thing I wanted to think about and now, I won't draw it in this video just 'cause 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 moving at half the speed of light in the positive extraction 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 you 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.