 So in the last section we derived the Lorentz transformations to move between inertial reference frames. Now what we're going to do now is we're going to tidy these up a little bit. So gamma is our Lorentz factor. Now what I'm going to do is define beta. So beta is just v over c. Now let's write these equations in terms of beta. So factorizing xb, pulling out that gamma, we get this, substituting in b. So what about tb? Now what I'm going to do is I'm going to multiply both sides by c, the speed of light. And then a Lorentz transformation looks like this, which is very symmetric in x and t. This new form is the one I'm going to be using from now on, because the symmetry between space and time makes it a bit easier to remember. Feel free to use whichever you're most comfortable with, because it's easy to switch between them just by substituting in beta. Symmetries are really important in maths and physics and can give us deep and fundamental insights. Here the fact that space and time are being treated so similarly tells us that even though they seem very different in our everyday experience, space and time are connected in a very fundamental way. And it is for this reason that we often talk about space-time rather than space or time separately. The other thing you may be wondering is the presence of this c next to tb. Why are we considering ct rather than just t by itself? Now there are many ways of thinking about this. One way is that if you look at the units, ct has units length just like x. So if we're looking for a symmetric form of the equations, we'd need to be considering things which have the same units. Another way to look at it is that we're just rescaling our unit scale. So the speed of light is 299,792,458 meters per second, where a meter was originally defined as 110 millionth of the distance from the north pole to the equator passing through Paris. And a second was a 60th of an hour where an hour relates to just the amount of time our planet takes to spin around the sun. So none of these units are particularly fundamental to the universe. They're just things we happen to come up with. So if an alien civilization or an American were to come along they might not use the same units that we are using. So if you were to ask what are the natural units of the universe, what you'd do is you'd look for fundamental constants of nature. The speed of light as we've seen from the second postulate is fundamental. Everybody regardless of their reference frame will agree on what the speed of light is. And if so if you are trying to use the fundamental units of the universe it would make sense to choose units where the speed of light is one. In that case ct can be thought of as just rescaling t. So we're moving from our unit system to the more fundamental one. So this is called Planck units where you set a number of the fundamental constants of the universe to one. Now these aren't very practical for everyday usage, right? I mean you wouldn't want to look on the highway and see speed limit 0.0000001. But in fundamental physics computations they often make things a lot nicer by getting rid of a lot of constants. For example if we rewrite these equations by setting z to one then gamma is just one over root one minus v squared. B is just equal to v and the equations become which is very nice and very simple and the symmetry between space and time is really evident.