 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 you know 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 rings frame? Well in the pink rings 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 rings 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 how y axis is the same as the height Bob measures along his so in other words yb is equal to ya. Okay so now if we want to represent the general coordinates of some event we represent it like this. So these are what we call space-time coordinates the first coordinate is the x the second is the y the third is the time and lastly the subscript a means this is in Alice's reference frame so this would mean one meter along Alice's x-axis three meters up Alice's y-axis and half a second after the time Alice calls time zero so here we have Bob and Bob chucks a ball forward three meters per second relative to him and just as he does that he starts timing on a stopwatch so that stopwatch measures his time axis now the ball is heading towards a wall 13 and a half meters ahead of Bob so in Bob's frame what are the space-time coordinates of the event a of the ball leaving his hand be of where the ball is after two seconds and see when the ball hits the wall assume that there's no gravity the ball just travels straight ahead without falling now we also have Alice standing 20 meters in front of Bob so now she's not moving relative to Bob Alice starts her own stopwatch two seconds after Bob's again what a A B and C but this time in Alice's frame