Hey, guys. In this video, we're going to start talking about Lorentz transformations. When we first introduced relativity, we talked about the addition of velocities or the addition of speeds, which was the way in Galilean relativity to take a measurement in one reference frame and move it to an inertial reference frame. Now in special relativity, we can't do something as simple as addition of speeds, but we can still move measurements from one inertial frame to another using more complicated Lorentz transformations. Alright. Let's get to it.

So like I said, in Galilean relativity, the process of moving one measurement into another is relatively simple. The position in one reference frame is easily relatable to the position in another reference frame based on how quickly that reference frame is moving relative to the first. For example, if you make a measurement of x in s, and s' is moving at a velocity or speed u relative to s, then x' is going to be x plus or minus u times t. It depends on the relative direction. But it's basically just the initial position in s plus how far the frame moves in your time measurement t. Now, it's much more complicated in special relativity because this thing, time, sort of messes it up for everybody because you can't just say that the time measured in one frame is equal to the time measured in another frame, you have to consider time dilation. It makes the calculations a lot more complicated.

Lorentz transformations are what are going to allow us to relate these measurements in different frames, taking into account phenomena like time dilation and length contraction. Galilean transformations like this one will always work if the speed is low enough, but they are not going to work as you get closer and closer to the speed of light. This only works if you are considering inertial frames. It does not work if you're talking about accelerated reference frames, like non-inertial frames.

To use Lorentz transformations, we need two things. First of all, we need two inertial frames. Here, we're saying s is the rest frame, just like we've been using in all of our other videos, and s' is the moving frame. They are inertial so the velocity, u, is constant. We are choosing that their origins are aligned at the initial time t0, so coordinate systems are going to look like this at t' equals t equals 0. We are choosing to align their origins.

Now, what we need to do is we need to choose a particular orientation of our frames such that the boost, which is what we call this relative velocity, is along some axis. Commonly, we are going to choose the x direction. If you look at the images here, the axis along which the boost lies is the x-axis. There is not going to be any relative velocity for the y axes or the z axes, so like we saw when talking about length contraction, there's not going to be any length contraction along those axes. Length contraction will only apply along the x direction because that's the direction of the boost.

So now we can actually write down our transformations, our coordinate transformations. Because there's no boost in the y or z direction, that means there's no length contraction along those directions, so y' equals y, z' equals z. The equations for time dilation and length contraction become: t ' = γ ( t - u x c 2 ) and x ' = γ ( x - u t )

Alright, guys. That wraps up this discussion on Lorentz transformations. We're going to follow this up with some practice problems. I'll see you guys in another video. Thanks so much for watching.