Hello everyone, welcome to Conic sections and light cones where we explain the concept of special relativity geometrically. Let's start with some basics of fundamentals. Let's start with the space time. You can think of it as a four-dimensional mathematical space in which we take three dimensions of space and one dimension of time. For the purpose of the illustration, because we do not have access to four dimensions, obviously, we're going to stick with two dimensions of space as seen in this beautiful blue plane and we take the z direction, the height direction, as being the time. So if you're going above the plane, you're progressing in the positive direction, you're going to the future as the arrow indicates. And if you're going downwards in the negative direction of the z, be low to plane, then you're going towards the past. Now, of course, it's all relative. That's from the perspective of an observer being on the plane at the time level, being at the present. If you're on the plane, you're at the present and if you propagate into the future, you're going upwards in parallel planes to the present plane, of course. This comes from the so-called Minkowski space. However, there's one constraint and that is the speed of light. You cannot travel in space faster than the speed of light. So that will limit how far you can move in space, given your progression in time. And that will be illustrated by a line elevated from the surface with the slope of one with respect to the surface. And in fact, it's not one line. It's a family of lines, which rotates around this common center of the observer in space time. Now, here we draw a line perpendicular to the plane only in the time direction or the time dimension and we'll intersect that line with another line that is 45 degrees away from the perpendicular line to the plane. So 45 degrees from the plane. And then we rotate that tilted line around the point of intersection, which illustrates the families of line of restrictions that we were talking about before. This happens to shape a cone. This cone is called a light cone. If we take the point of intersection, the observer's point, and the source of light starts emitting from that point, the light can only travel on the surface of this light cone in space time. The space inside the cone is called the time-like space, which is part of the space time or the universe that our observer at the center or at the tip of the cone can observe into the future or into the past if you're looking at the downward cone. Or as the fall outside of the cone are called space-like and cannot be observed by an observer at the center of our double cone or tip of each cone. If we take ourselves to be this observer at this point in space time and continue the path of the downward cone onto the beginning of the time, everything inside the downward cone, it will be the so-called observable universe. If we intersect and space plane to the cone into the future of the observer, the points of intersection will shape a circle, which has zero eccentricity. The circle and the area it contains are all the simultaneous events that can happen in the future of the observer, which are also possible to be observed by the observer himself. We can also have a narrower cone within the light cone, which in the upward direction will indicate a set of possible events in space time that an observer traveling with the speed proportional to the speed of light by the factor indicated as the inverse of the generator lines slope, which respect to the space plane, can observe if the observer is immobile and is stationary in the space plane, all those set of events will necessarily be contained on the line perpendicular to the space plane. Our cone will reduce to a line.