 Welcome to this ongoing video series on the theory of relativity. In this video we want to take a look at the seminal 1916 paper by Albert Einstein, the foundation of the general theory of relativity. To get us started, here's a quote from Einstein. And this will be our only Einstein quote that's not from the 1916 paper. If you will not take the answer too seriously and consider it only as a kind of joke, then I can explain general relativity as follows. It was formerly believed that if all material things disappeared out of the universe, time and space would be left. According to the relativity theory, however, time and space disappear together with the things. Well let's see if we can figure out what he means by this. Now consider this video. A ball is thrown from a rotating platform and we see that it goes in a straight line at a constant velocity, a inertial motion. But the same process viewed by a camera fixed on the platform shows the ball following a curved trajectory. It's accelerating. In the classical Newtonian perspective, this is explained as follows. The ball travels in a straight line relative to the earth. The platform is rotating relative to earth and therefore it sees the ball following a curved trajectory. What it's really seeing is the effect of its own acceleration. There's no gravitational field causing the ball to accelerate. But what if everything was quote at rest and just as the ball was thrown, a space giant grabbed hold of the earth and spun it? From the Newtonian point of view, we'd say that the platform would stay at rest and the ball would go in a straight line and the platform would see it going in a straight line but the earth which is spinning would now see the ball following a curved trajectory. Now let's think relativistically. In two situations in which you have exactly the same relative rotation between the platform and the earth, the trajectory of the ball is dramatically different. What is the cause of this difference? The Newtonian answer is not that it's the earth or that it's this galaxy over here or that it's this galaxy over here or some combination of those things. Quote space is the cause. In the one case you're viewing from a frame quote fixed in space and in the other from a frame quote rotating in space. But Einstein responds, the law of causality has not the significance of a statement as to the world of experience. Except when observable facts ultimately appear as causes and effects. Space and time are not observable facts. They're concepts, coordinates really, that we use to describe observable facts. In special relativity we concluded that there is no velocity relative to a non-thing. So how can there be rotation relative to a non-thing? Nope, those objective concepts of space and time gotta go. Einstein now extends his principle of relativity into a principle of general covariance. The general laws of nature are to be expressed by equations which hold good for all systems of coordinates. That is our covariant with respect to any substitutions, whatever. This requirement of general covariance takes away from space and time the last remnant of physical objectivity. The observable phenomena we see must be caused in turn by observable phenomena and the relative positions and motions and properties of those phenomena. If we want to use this coordinate system to describe the world, that's fine. If we want to use this coordinate system to describe the world, that's fine too. In this latter case, we'll have to conclude that there's a gravitational field that causes the ball to curve. That gravitational field will, like the gravitational field that causes this ball to curve, have the property that all objects will experience the same gravitational acceleration. Because of this, Einstein then points out, it will be seen from these reflections that in pursuing the general theory of relativity, we should be led to a theory of gravitation, since we are able to produce a gravitational field merely by changing the system of coordinates. There is indeed a deep connection between gravity and geometry. Let's consider the following geometric illustration. Suppose the screen represents the ground, and we pick two points A and B and we drive stakes into the ground at those points. We then stretch a rope between A and B and pull it top. That will form the shortest path between those points, which would be a line in this case. Then using a yardstick, we could measure distance S along the path from A towards B. So I could tell you how to get to any point on that line. Start at A, travel eight yards along the rope. Those are all physical operations that are independent of any coordinate system. But to describe geometric problems in general, we need to introduce a coordinate system. So suppose we introduce this rectangular graph paper type coordinate system. This is a two-dimensional surface, so we'll need two coordinates. Let the vertical red lines represent a coordinate Y1 is equal to a constant, and the horizontal blue lines represent a coordinate Y2 is equal to a constant. As we travel along the path defined by the rope, at every point we can read off the Y1 and Y2 coordinates, and then produce a plot of Y1 and Y2 versus distance S. Every point along the path will then have two corresponding values of Y1 and Y2 value. If we consider the distance S to take the place of time, then by analogy with physics, we'd say, look, our position as a function of time varies linearly. This is straight line motion and a constant velocity. There's no acceleration, hence there's no gravitational field. This is natural inertial motion. For the exact same physical situation, let's introduce a different coordinate system. Let's call these coordinates X1 and X2. This is a curved coordinate system. X1 equal to a constant will correspond to the magenta circles. X2 equal to a constant will correspond to the green radial lines. For these coordinates, the coordinate plots would not be straight lines, but instead would be curves. Following the physics analogy, we'd say, look, through time this object is being accelerated. It's following a curved path. There must be a gravitational field present. Now, of course, we've already seen that there is a coordinate system in which the path would be described by straight lines. We could say the reason you're seeing curved lines is because you have a curved coordinate system, not because there's gravity present, but we've already seen in a previous video how, by simply going into free fall, you can transform away a, quote, real gravitational field. So this argument goes both ways. I could claim that the reason that you feel the gravitational force while standing on Earth is because you're actually accelerating upward relative to a freely falling reference frame. A great description of these ideas is given by Nobel laureate physicist Max Born in his book, Einstein's Theory of Relativity. It is meaningless to call gravitational fields that occur when a different system of references chosen fictitious in contrast with the real fields produced by near masses. A gravitational field is neither real nor fictitious in itself. It has no meaning at all independent of the choice of coordinates. In the one case, it is particularly the near masses that produce an effect. In the other, it is the distant masses of the cosmos. These concepts, together with the concepts developed in video four of this series, where we showed that the geometry of spacetime must, in some sense, correspond to that of a curved surface, have led us to the very threshold of the general theory of relativity. Incredibly, as we will see, essentially all the math that was needed for general relativity was already lying on the table in the field of differential geometry, waiting just for Einstein to pick it up and apply to physics. Accordingly, in our next video, that is the very topic we will take on.