 In classical physics so far, you're familiar with this idea of Newtonian momentum. So if an object has mass m, and it's moving with velocity v, we say it has momentum p equals mv. Now this is useful because momentum is conserved by interactions. And so we can use this to solve all sorts of physical problems. For example, by setting momentum before equals momentum afterwards, and then solving for the velocity. Now let's see how this formula holds up in a relativistic setting. So here we have Bob looking at two balls, a green and a pink one, flying towards each other with exactly the same velocity. So we can break this up into horizontal and vertical components. So for the green ball, these are v and w, where they're measured in Bob's frame. And for the pink ball, the components are s and r, also measured in Bob's frame. Now because these balls have the same velocity, we have vb equals rb, and wb equals sp. So after the collision, the balls bounce off each other. Now because the initial situation was symmetric, both of these balls are moving with exactly the same speed, after the collision, they'll still be moving with the same speed, because there's no reason for any one ball to have picked up more speed than the other. Let's look at Newtonian momentum conservation in the y direction. So before the collision, the green ball is coming downwards with speed wb, and the pink ball is coming upwards with speed sp. After the collision, the green ball is going upwards with speed wb, and the pink ball is going downwards with speed sp. Now because the balls have the same velocity, we have vb equals rb, and wb equals sp. And so the momentum before the collision is the same as the momentum after the collision. So momentum is conserved. But from what we know about relativity, there is no special inertial reference frame. So we don't just want momentum to be conserved in one frame, we want it to be conserved in all frames. So let's pick a frame. Suppose Alice starts off where the green ball starts off. Here are her coordinates, and she moves to the right with speed vb. So in other words, she's matching the horizontal velocity of the green ball. Let's see how things look in Alice's frame. So Alice sees the green ball initially going downwards with some speed wa, measured in her frame. And then after the collision, it just comes straight back upwards, again with speed wa. Meanwhile, the pink ball comes flying towards Alice. So its horizontal and vertical components are r and s, but now measured in Alice's frame. And then after the collision, it bounces off and flies back. Let's again look at Newtonian momentum conservation in the y direction. For the collision, the green ball is moving downwards with speed wa, and the pink ball is coming upwards with speed sa. And then afterwards, the green ball is moving upwards, and the pink ball is moving downwards. So is this equation satisfied? Simplifying this, we get this condition that we need. So in other words, for Alice to see momentum to be conserved, she needs sa to be equal to wa. However, this is not the case. Let's look at this intuitively. Suppose I'm on a train moving in this direction. You're on the platform watching me, and just as I go past, I give you a funny sort of a wave, an up and down wave, like that. Now let's suppose that the train's motion in this direction is very, very fast relative to you. What will you see? Well, from what we know about the fundamental effects, you'll see me undergoing time dilation by a factor gamma, and my wave to you will appear in slow motion. So what's this got to do with the particles? Well, imagine my hand as one of those balls. It's got horizontal motion because of the train, and it's got vertical motion from my wave. And if you go into a reference frame where the horizontal motion is very quick, because of time dilation, you will start to see the vertical motion slow down. So in Bob's frame, the red and the green ball have the same up and down velocity. But now Alice is matching the green ball's horizontal speed. From her point of view, it's not moving at all in the horizontal direction, and so she'll see it going up and down at full speed. The pink ball, however, is moving towards her. And so time dilation means that Alice will measure its vertical velocity to be slower. Thus, SA is smaller than WA, and so Newtonian momentum is not conserved in the y direction. It turns out that if we look at this problem and see quantitatively how the velocities transform between frames, we can get something that is conserved in both Alice's and Bob's frame, and this is p equals gamma mv. So v is the velocity vector of the particle, m is its mass, and gamma is the gamma factor for the particle's total velocity. v equals vx squared plus vy squared plus vz squared. For the pink particle, its initial momentum is m times SA on 1 minus its total velocity is SA squared plus rA squared. If we look at the green particle, its initial momentum in the y direction is m times minus WA on 1 minus WA squared. Now, just because p equals gamma mv is conserved in these two reference frames, it doesn't mean that it's conserved in every possible reference frame. Now, that is the duty of experiment. So from the theory, we have a reason to think this is a reasonable candidate for momentum. And so what we do now is we go and perform thousands of different experiments looking at this from many different reference frames in many different circumstances and see, is this conserved? And what we found is that, yes, in every experiment that has been performed in a relativistic regime, p equals gamma mv has been found to be a conserved quantity. And so because of that, we call this relativistic momentum.