 So here's Bob's reference frame and Alice's reference frame is moving with respect to this with speed v. Now let's say an event happens. So we've seen that we can record the coordinates of this event with respect to either reference frame and using the Galilean transformations, we can transform from one reference frame to the other. So there are some problems with the Galilean transformations, however. The first is that they don't take into account the fundamental effects that we've seen, like length contraction, time dilation, and relativity of simultaneity. The second is that they don't treat time properly in that Alice and Bob should each have their own time. The time Bob records an event taking is different to the time Alice records, while in the Galilean transformations there's only one time for everyone. Lastly, the Galilean transformations don't agree with the second postulate. So if I was running at half the speed of light and then I turned on a torch pointing forwards, emitting photons at the speed of light, Galilean transformations would suggest that someone standing and watching me would see those photons moving at one and a half times the speed of light. However, the second postulate says that everyone agrees on the speed of light, regardless of their initial reference frame. So what we're going to do now is derive the Lorentz transformations. So these are transformations between inertial reference frames. Only these are relativistic. They take into account all of the effects that we've discussed, but they also reduce to the Galilean transformations when your velocity is much less than the speed of light. As they should, because in everyday life we don't see these relativistic effects and everything works according to Galilean relativity. So there are three spatial coordinates, x, y, and z. But because it's easier to draw, we're just going to count x and y. So we're going to be assuming, like we have all this time, that Alice's and Bob's axes look like what we've got above. The y-axis are parallel, the x-axis lie across each other, and Alice moves along Bob's x-axis with a constant velocity v. So no acceleration, no turning. So we can start off by seeing, how do we transform the y-coordinate between frames? So we've seen that with the x-coordinate, we can have things like length contraction. A train in Alice's frame will have a different length to its length in Bob's frame. However, there is no length contraction along the y-axis. To see this, let's suppose that there was length contraction in the y and z directions. So when an object moves really fast in the x direction, its y and z axes will contract. Now in this universe, we have two rings. One green, one pink, both with the same size, and they're flying towards each other at equal speed. Now in the reference frame of the green ring, it is stationary while the pink ring is moving towards it. So if there's some sort of length contraction, the green ring will see the pink ring being shrunk. So if we imagine that there's some paper stretched in the middle of both of these rings, the pink ring will punch a hole in the green ring's paper. But now what about in the pink ring's frame? Well, in the pink ring's frame, it is stationary while the green ring is flying towards it, which means the green ring must be length contracted. But in this case, the green ring would punch a hole in the pink ring's paper. Now this is a contradiction, because depending on our frame, we think one ring remains intact while the other ring is punctured. So because of this, there can be no length contraction in the directions orthogonal to the motion. And similarly, there can't be any sort of length expansion. So this tells us that the height Alice measures along her y-axis is the same as the height Bob measures along his. So in other words, yB is equal to yA.