 Welcome to video 10 in this series on the general theory of relativity. In our three remaining videos, we're going to look at the predictions of the theory that we've been developing, and also at tests that have verified many of those predictions. In the present video, we'll explore the ideal case of a uniform gravitational field, or equivalently of an observer undergoing uniform acceleration. In the next video, we'll deal with spherical bodies in black holes, and in the last video, we'll tackle basic Big Bang cosmology. The gravitational fields of real objects, such as Earth, are not uniform. But the uniform case is fairly simple to analyze, and it provides a reasonable description near an object's surface. Moreover, it will allow us to accurately describe the relativistic effects that would be seen in the plausible scenario of a uniform-accelerating space traveler. As we examine various predictions, we're going to come face-to-face with one of the most, at least to me, unsettling and brain-frying aspects of the theory. Recall from video four the happiest thought of Einstein's life, which occurred in 1907. The equivalence between falling and floating, likewise between gravity and acceleration, was his key physical insight. So why did it take Einstein so many more years to work out the complete theory? One reason, as we've seen in several of these videos, was the complexity of the math. But according to Einstein, the main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate, metrical meaning. Let's think about the way people have been trained to use coordinates to do physics. Say we want to describe the trajectory of a cannonball. We imagine some spatial coordinates superimposed on the physical world. We give them names like x and y, and we assume that these have, as Einstein said, immediate, metrical meaning. That is, their numerical values represent measurable physical distances. We also assume a similarly meaningful time coordinate, t, can be defined. Then the laws of motion allow us to track the x and y positions through time, such that changes in the space and time coordinates have predetermined physical significance. We might say that this point of view imposes a, quote, prior geometry on the physical world. In contrast, general relativity allows us to assign coordinates, including time, in an essentially arbitrary fashion. However, not only do these not need to have any assumed metrical meaning, we actually cannot initially assign them precise meaning. Instead, the trajectory of the cannonball and the physical meaning of the coordinates are parts of a single problem that has to be solved as a whole. Only after we have a solution in the form of the metric tensor can we describe the coordinates of the cannonball trajectory and be certain of the physical meaning of those coordinates. We might say that general relativity allows no prior geometry to be assumed. The very geometrical structure of the world is itself part of the physics. This is indeed a bizarre way to look at the world. In his 1916 paper, Einstein describes this problem as follows. In the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial coordinates can be directly measured by the unit measuring rod, or differences in the time coordinate by a standard clock. The method hitherto employed for laying coordinates into the spacetime continuum in a definite manner thus breaks down. Now, of course, the classical prior geometry physics of Galileo and Newton work very well in our day-to-day experience. So we expect that this no prior geometry weirdness will only become apparent in extreme limits where classical physics breaks down and general relativity begins to reign supreme. In video three, we described how Einstein concluded that light falls in a gravitational field. Hence, light rays are bent. This requires the speed of light to change with distance from a massive body. This seems a bit perturbing. After all, isn't the speed of light a universal constant? We'll settle this question shortly. Let's look at an analogy that at least cruelly gives us some perspective on our problem. Suppose you're in Cairo, driving south at 100 kilometers per hour. You ring up your friend in Cape Town and she says, hey, she too is driving south at 100 kilometers per hour. You've got one of those 11G phones that can directly access satellite imagery. So you ping a satellite that's directly above Cairo, and because you've also got a car tracking app, you observe the paths of your two cars. You are indeed going 100 kilometers per hour, but your friend seems to be going quite a bit slower. You're about to mention this when she tells you that she's been washing from a satellite that's directly above Cape Town, and although she's clearly going 100 kilometers per hour, you seem to be going quite a bit slower. Not only that, but you've been talking about your vacation in somewhere called Europe and there's no such place visible in her satellite image. Of course, these issues are due to the viewing of a curved surface from different perspectives. You can't see Europe because it's outside of her horizon, and you can because it's inside your horizon, and you each appear to the other to be going slower due to perspective foreshortening. The solution seems pretty obvious. Get rid of the perspective problem by drawing Earth's curved surface on a flat map. You try this and someone complains that you've distorted some areas. Well, you try again and now you've distorted some angles, and no matter how you try, there's always something clearly distorted about your map. So why don't we just come up with the one and only one corrupt map projection and be done with it? Well, it can't be done. Every possible map projection will distort some aspect of Earth's surface. But we have an intuitive understanding of this problem because we have direct experience of two-dimensional curved surfaces viewed from three-dimensional flat space. But now our problem is that relativity tells us that four-dimensional space time is curved. Presumably, if there was a fifth dimension, we could step into five-dimensional flat space and view our four-dimensional curved space time, and it all make perfect sense. Unfortunately, we're stuck inside curved space time. And because of this, we're forced to accept the reality summed up in this quote from the book Exploring Black Holes. In general relativity, every coordinate system is partial and limited, correctly representing one or another feature of curved space time and misrepresenting other features. Figures and diagrams that display these coordinate systems embody the combination of clarity and distortion. No illustration, graph, or animation we can ever produce to represent space time can ever be entirely free of some form of distortion. The minute we pull out a piece of paper and start drawing, we are, in this sense, doomed. And that's a pretty humbling realization. So we've got to just choose a projection and live with it. Well, here we go. After all those warnings about coordinate systems, we're going to start by imagining a coordinate system. We'll use common rectangular coordinates with x, y, and z axes and a time coordinate t. If there's no gravity, then this is the system of special relativity. And each coordinate has a precise meaning in terms of time or distance. From video eight, we know the yardstick of space time in this case. The proper time increment ds that describes the time between two events recorded by a clock that freely falls between them. ds squared equals dt squared minus dx squared minus dy squared minus dz squared. Now we go through the looking glass and assume there's a gravitational field with acceleration A pointing down the x-axis. We take the yz plane to represent the ground or equivalently the floor of an accelerating spaceship. In an appendix video with link in the description box, we verify the following solution to our problem. The special relativity metric is modified by multiplying the time increment and the spatial increment in the direction of acceleration, x in this case, by the factor e to the 2ax. We note that if a equals zero, then you have e to the zero which equals one and this reduces as it must to the special relativity case. Now we consider how clocks operate in a gravitational field. Consider two clocks at essentially the same position. One is at rest, that is it feels its own weight as it sits on a shelf, say. The other falls freely and therefore measures proper time. As time progresses, they follow different trajectories and will in general record different times. But for the first instant, the following clock is not moving, it has acceleration but no velocity. Therefore the two clocks coincide and initially measure the same proper time increment ds. With dx, dy, and dz all zero. ds squared will in general be some coefficient, let's call it gtt times dt squared. Or, taking square roots, ds equals the square root of gtt times dt. Now if gtt changes with position, we have the result that a clock in a gravitational field appears to run at different rates at different positions. This is called gravitational time dilation. In our current problem, the gtt factor is e to the two ax. The square root of that is e to the ax. So the increment of a clock tick, ds, is equal to e to the ax times the increment of coordinate time dt. We can also write dt equals e to the minus ax times ds. For x equals zero, that is on the ground, ds and dt are equal. So our metric is telling us that our time coordinate t corresponds to the reading of a clock sitting on the ground. And we can call this ground time. Now consider two clocks. One's on the ground at x equals zero and another sitting on a shelf at height x equals h. On the ground, the time coordinate tick, dt, is equal to the clock tick, ds. But at height h, the time coordinate tick is equal to e to the minus ah times the clock tick, ds. Since e to the minus ah is less than one, dt is less than ds. Even though to a local observer, each clock ticks with the same increment, ds, the time coordinate ticks, ground time increments, are different. As seen from the ground, a clock at height h takes less time to tick, hence it appears to run faster by a factor of e to the ah. Let's see what this predicts we'll observe on earth. Earth's gravitational acceleration is about 9.8 meters per second squared. That is, if you drop a rock from rest after one second, it's going to be traveling at 9.8 meters per second. If it could accelerate like that for an entire year, 365.25 days times 24 hours times 60 minutes times 60 seconds, then classically it would end up going about 309 million meters per second, just a bit above the speed of light. So if we measure time in years and distance in light years, about 9,460 trillion meters, the speed of light will be one light year per year, one. And earth's acceleration, a, will conveniently be about one light year per year squared. So we'll take a equals one in what follows. Also, here's a useful little math fact. If x is very small, e to the x is essentially equal to one plus x. We'll use this also. So let's consider Mount Everest. Its height is about one trillionth of a light year. So e to the h is about one plus one trillionth. And a clock on Everest should run faster than a sea level clock by about one part in one trillion. One trillionth of a day is 86.4 nanoseconds. Now current atomic clocks have uncertainties of much less than a nanosecond per day. So we could easily measure gravitational time dilation by watching atomic clocks on Everest and at sea level for a day or so. Satellites are much higher above sea level than Everest. So as we mentioned in previous videos, the atomic clocks of the global positioning system show a very measurable gravitational time dilation effect, which must be compensated for in order for the GPS system to work at all. So this incredible prediction of general relativity has been spectacularly verified. Moreover, gravitational time dilation is even measurable on much smaller scales. In 1959, Pound and Repka performed an experiment between two floors of a laboratory, 22.6 meters apart. They took a loudspeaker and attached a sample of radioactive iron. This emitted gamma rays, which traveled from the top floor to the ground floor where they passed through another iron sample. The gamma rays are absorbed by the second sample, only when the frequency is essentially the same as the emitted frequency. In this manner, the two iron samples act at its very precise clocks. As we've seen, the top floor clock will appear to run faster. Hence, the gamma rays observed at the ground floor will experience a gravitational blue shift to a higher frequency. By causing the loudspeaker to move the upper iron sample away from the ground, a Doppler red shift to a lower frequency could be introduced. When these two frequency shifts precisely canceled out, the ground sample was observed to strongly absorb the gamma rays. And since the speed of the sample on a loudspeaker could be measured, this allowed an experimental measurement of the gravitational time dilation. The results verified general relativity to within 10%.