 It's 1907 and Albert Einstein has just had what he would later describe as the happiest thought of his life. What do you suppose it was? Maybe a complicated mathematical expression? Nope, it was this. If a person falls freely, he will not feel his own weight. Yup, that's it. Because as we saw in video three of this series, that insight allowed him to recognize the physical equivalence between floating and falling frames of reference. At least if the frame is very small. Since inertial reference frames are described by special relativity, which we developed in videos one and two, this insight allowed him to apply special relativity in the presence of a gravitational field. But there's that very small caveat. What's up with that? Gravitational acceleration falls off as you get further from an attracting body. Yes, you can enclose two widely separated objects in their own individual falling frames, but neither of those frames can be extended to, quote, fit both objects and still look like a single inertial reference frame. Even if two objects are at the same height, if they fall from different directions, there is no single falling frame that will, quote, fit both. In general relativity, this fact is explained by saying, space time is curved. To understand why, let's say you operate flat land industries that provide surveying surfaces for the inhabitants of flat land. You have a standard recipe for drawing a triangle. Pick three points. Between each pair of points, stretch a rope. The taut rope will follow the shortest distance between the two points, and that will be a straight line. As a check, if you measure the three resulting angles in some of them, you should always get 180 degrees. Now play along with me and assume that if you're on the ground, you can only view a very small area at a given time. You don't have the bird's eye view that we're showing here. You complete a triangle job and measure angles of 57, 60, and 63 degrees, which indeed some do 180. Now you happen to know someone who has a new fangled flying machine who agrees to take airborne pictures for you. She reports back that indeed you drew a triangle, and from directly above it has the claimed angles. However, if she flies around and looks at the same object from different points of view, the angles change, although the object remains a triangle with straight sides, and the angles still sum to 180 degrees. You get a call for triangle services from folks in far away Kervland. You've never worked there before, but you go and you follow your triangle recipe, and you measure angles of 79, 82, and 95 degrees, which of course sum to 256 degrees. So you call in your flying friend to see what went wrong. She tells you that you connected the points with some kind of curves, not straight lines. She finds out that she can fly around and find a point of view in which any given side appears as a straight line, but she can't find a single point of view from which all three sides appear as lines at the same time. Of course what's happened is that you're trying to do geometry on a curved surface. On such a surface straight lines may not even exist, but we still have the idea of the shortest path between two points, and we call such a path a geodesic. Their shape can meander around as they try to avoid hilltops and the like, and the normal rules of Euclidean geometry break down. For example, a circle is a set of all points equidistant from a central point, and the circumference is supposed to be pi times the diameter. But on a curved surface, the circumference can be less than this. Look that from the appropriate point of view, we can see why. The curve that serves as the diameter is curved, and so has a greater length than what the diameter would be on a flat surface. We can also get circles with a circumference greater than pi times the diameter. Yes, this is a circle. Every point on this curve is at the same distance from a central point. Same distance in the sense of a geodesic. Since the surface curves both up and down, the circle has to undulate, causing the circumference to be larger than pi times d. Why does Euclidean geometry break down on a curved surface? It has nothing to do with how we designate position, that is our coordinate systems. On a flat surface, we can use different types of coordinate systems, but straight lines still exist. They're just described differently. The problem is something intrinsic to the surface. No matter how we designate position, in general, on a curved surface, we cannot form things like normal triangles and squares. Now for small enough patches on the curved surface, we can attach more or less flat pieces of graph paper, and Euclidean geometry will work on those patches. But if you try to attach a large square of graph paper, you'll be able to get one, two, even three of the edges pasted to the curved surface, but not the entire sheet. You end up having to cut the sheet and manipulate the pieces, and in doing so, you're going to alter the geometric properties of the graph paper. Alternately, you could imagine the graph paper made out of rubber, and then you have to stretch it in weird ways. In either case, no intact graph paper equals no Euclidean geometry. What Einstein came to realize is that inertial reference frames are like the flat graph paper, and special relativity is like Euclidean geometry. You can map small individual pieces of inertial frames, that is freely falling frames, to various regions of space-time, and special relativity will work in those pieces. But the fact that you can't map one big inertial reference frame to all of space-time means that space-time itself must be curved by massive objects. What remained was to work out the details. Now there were a lot of details, and true, those details involved a lot of math. For starters, physics involves not just space, but also time. So that's four dimensions of space-time. And what exactly is a four-dimensional space-time surface? What limits does nature put on its curvature, and so on? But keep in mind that math was just the crank that needed to be turned to produce specific results. The physical insight, which was the true genius of Einstein, was not primarily about math. In fact, a mathematician named David Hilbert presented the full equations of general relativity before Einstein did. So why are they named after Einstein and not Hilbert? As brilliant as Hilbert was, he was just turning the admittedly quite complicated mathematical crank needed to express Einstein's insight. As he himself said, every boy in the streets of Goodenken understands more about four-dimensional geometry than Einstein. And in spite of that, Einstein did the work and not the mathematicians.