 Hi everyone, I'm Lucy. So I'm really glad that there are actually two relativity talks in this session because relativity is all about space time, right? So that talk was about time and this one's going to be more about space. So here's a question that I asked myself a while ago. So if you took a photo of an object moving at some velocity near the speed of light. Would you mind taking the microphone? Yeah. Thank you. So here's a question that I asked myself not a year ago. So if you took a photo of an object moving at some velocity near the speed of light, what would that photo look like? So if you want, you can actually just do the math or you can get a computer to do it for you, which is what this is about. So before I get started on that and talk about what is relativity and so on, let's just talk about how computers render 2D images of three-dimensional objects. So before we even talk about computers, I'm just going to talk about pinhole cameras. So the way a pinhole camera works is that you have this box and it has this pinhole on the front and that's the aperture where light goes in and on the back you have some photographic film. And if the light rays coming from this object are at the right angle, they go into the aperture, they strike the photographic film and that's how you get this image. So it's sort of like a geometrical setup. By the way, if you're getting a sense of deja vu from looking at this tree, it's from the slides from yesterday. It's like a domain image. It's good. So ray casting is this algorithm in computer graphics that sort of simulates the same process. So what this is is that if you want to generate this 2D image, you can have the same geometrical setup. If you have this geometrical setup, what you can do is you can basically simulate the light rays that go from your simulated pinhole camera to the 3D object. And this is what you do if you have some tree or something in a video game and you want to render a particular two-dimensional view of it and that's how you get this pixel array of colors over there on the right. So let's take a slightly closer look at how this works. Okay, so for each image pixel you define a ray in three dimensions which starts at this pixel and passes through this pinhole. The way that you do that is you can just define these two points and two points to find a line. I put my aperture right at the origin. And the next step is that you can compute the point where this ray intersects the 3D object. This is basically a geometry problem. And this final step, so this is a really sort of simplified view of what ray tracing is, you know, I'm not going to talk about shadows or textures or shading. Basically what you do is you just set that pixel color equal to the color of the 3D object at that point and that's good enough. And from that we can like render this picture. Okay, so that's all fine if your tree isn't moving. What we want to talk about in general is like what happens if this object is moving. So specifically if your object is moving at some significant portion of the speed of light then it turns out that the finite speed of light actually really matters here. And to describe exactly what's going on we're going to need to do some physics. So this next section is going to be about special relativity. So special relativity is this physical theory that describes space time. And it's got these two postulates from which in principle you could derive like the rest of the content of relativity. I'm going to talk about these two postulates in the context of how they're going to be useful to us. So the first thing is that the laws of physics are the same in all non-accelerating reference frames. What this means for us is it's basically sort of a statement about symmetry. So let's say that you're on a train and it's moving and your friend is on the platform and it's not moving. So from your friend's point of view the train is moving forward and you and the train are moving forward but from your perspective and your reference frame you and the train are both actually still but your friend on the platform is moving backward. And what this says is that neither of you is more correct than the other physically speaking because both of your reference frames are equivalent and neither is privileged over the other. The second postulate says that the speed of light in a vacuum is always the same for all observers regardless of the motion of the light source. This seems pretty innocuous but this is the thing that's going to lead to some non-classical and possibly counterintuitive consequences for how different observers experience space and time. So I have a little thought experiment. I think for basically like historical reasons every thought experiment about special relativity has to involve at least one train so I hope you guys like trains. So you're on a train. In your reference frame the train isn't moving and let's say you like turn a flashlight on or something just for an instant at the exact midpoint of the train and so what's going to happen is that this spherical way front of light is going to expand outward symmetrically and then it's going to fill up the entire train and the light is going to reach the front and the end of the train at the same time. So now let's look at this from the perspective of your friend who's on the platform. From their reference frame the train is moving but it's the same physical setup so you turn the flashlight on again at the midpoint of the train but here's the thing that second postulate says that the motion of the train which is the light source doesn't actually matter from your friend's reference frame the light is still going to spread out in these concentric circles. Okay so the light now reaches the front of the train after it already reached the back. So what this means is that you and your friend actually disagree on whether the light reaches the train at an end point simultaneously. You think it does your friend says no like the light reaches the front after it's already reached the back but you're actually both right either of you is more right than the other. So more generally speaking different observers can actually disagree on various things about space and time. First of all as we've already seen they can disagree on whether two events are simultaneous and hopefully that will make the next two things seem more plausible even though you know we haven't fully gone into it. Different observers can also disagree on what the length of objects are. So the faster an object moving the shorter it gets. In your reference frame if the train is a certain length in your friend's reference frame from which the train is moving it's actually going to be shorter. The other thing is that time intervals can also be disagreed on so the faster a clock is moving it turns out more slowly it runs. One thing that's important to note here is that these are actually geometric properties of space time. This is nothing to do with cameras or human observers or anything like that. Let's make this a little more quantitative. So what it basically said amounts to the statement that for you and your friend on the train you basically have different coordinate systems and the way that you convert between these coordinate systems is that you do something called the Lorenz transformation. This is a linear transformation of the coordinates. If the train is moving in one spatial dimension which we'll call X and not this other time dimension which we'll call T and so this is like a 2 by 2 matrix that defines this coordinate transformation and it might be easier to see this if you plug in some numbers. So if the train is moving at 60% of the speed of light which means this parameter beta is 0.6 then this is what that matrix looks like essentially. And more generally we don't just live in one spatial coordinate we actually live in three. So a more fully general formula if you have three dimensions or something you know two observers moving co-moving in three spatial dimensions then you just have this bigger 4 by 4 matrix with T and XYZ. So now that we know about special relativity we can get back to this ray tracing problem. So this is just to review that same algorithm that we had before it just has these three steps. It turns out the modifications you need to make to make this relativistic are like not actually that difficult. So here I've just bolded all the parts that actually changed from the last slide. So for this first step it's basically the same except there's some points of clarification. One point is that when you're defining this ray to start out with that you're casting you define it in the cameras reference frame. And also these endpoints that define the ray you're going to have to make them into 4D because things are happening in time. But all of this is pretty straightforward so far. The second step here is the actually new thing. And what you have to do in this step is you have to transform it from the cameras reference frame to the moving objects reference frame. So all you have to do there is just apply this 4 by 4 matrix transformation. And that's basically it. The rest of the steps are the same. This third step which used to be the second step in our old algorithm is computing the point where the ray intersects the object. But since you already did the transformation both the ray and the 3D object are in the same reference frame. So you can just compute that intersection geometrically like you would before. And the fourth step is also the same. So let's actually see some pictures. So here's what you get if you just ray trace this cube. You're looking straight at one of the faces. The cube is not moving. And so let's say you know this cube is actually moving either to the left or to the right across this slide. Add some appreciable fraction of the speed of light. So we were just talking about length contraction. You know we expect this cube to get compressed in this direction. You're probably going to expect it to look something like this. And since this is an accurate physical representation of what the cube is doing when it's moving to the left or right but here's the actual like output of that ray tracer which I just took it a bunch of different time slices. This is a cube moving at 60% of the speed of light and it actually looks rotated. I haven't rotated it or anything but this is actually just what it looks like if you take these pictures at these time slices. So the question is why does it look rotated? It turns out that there's this visual phenomenon which is called Tarell rotation. And what's going on here is that because you have a camera that's it's actually taking all these light rays and light signals from the back of the object take longer to reach the camera compared to light signals from the front of the object. So objects appear to be rotated. If you look at this GIF then you can see that this bottom parameter here is what fraction of the speed of light you're moving at. And as you move faster the cube compresses more and more but also the apparent rotation angle becomes more and more extreme. And so the interesting thing here is that you can also do this with a sphere. And I've put in just a still shot of what you get if you do this ray tracer thing on a sphere. And it turns out that moving spheres always appear to have a circular outline. And this is because these two effects both the actual physical length contraction and this visual distortion of Tarell rotation actually cancel out always. And it turns out that there's actually some interesting math behind this. It's related to the theory of conformal mappings which are a type of mapping that always preserve angle and circles. And so the last thing I have for you is just some historical tidbits and why I find this interesting in the first place. Special relativity was discovered by Einstein in 1905 but Tarell rotation was not widely known for a while. It was only described by Tarell and Penrose in 1959. So that's a couple decades during which this effect wasn't widely known. And I think you can actually even find textbooks where this optical distortion wasn't talked about. Which is fine. This is not of extreme scientific importance but I think the interesting thing here is that now anybody with access to a computer can write a relativistic ray tracer and if you do that then this interesting physical effect just pops out at you. So basically in general you can actually use computers to visualize and understand and explore the physical world and that's what I find most interesting about this. And that's all I've got. You can look at the ray tracer code on GitHub and I've got some references if you want to look at that. Thanks.