 Since space and time are so central to relativity, let's look at the scientific definitions of time and length. The second is defined as 9 plus billion periods of the microwave radiation produced by a particular transition of the cesium atom. That is what the second is, by definition, nothing more and nothing less. The meter is fixed by defining the speed of light to be 299 plus million meters per second. A light second is then this same number of meters. In our units, this means that the speed of light is, by definition, 1. To make a length measurement, shoot a light beam between two points and note the elapsed time on your cesium clock. The points need to be close enough so that any curvature of space time can be neglected. The elapsed time in seconds is the distance between the points in light seconds. In part A we develop the idea of gravitational time dilation. The time coordinate T corresponds to the tics of an atomic clock on the ground. At a height h in the gravitational field, the tics of an atomic clock is seen from the ground, take less time, and the clock therefore appears to run faster by a factor of e to the ah. Now we turn to the issue of the speed of light. We just said that, by definition, the speed of light is 1, but now we're going to say the speed of light varies with position and direction. We'll soon see how these two apparently conflicting claims are actually compatible. The x coordinate denotes position above the ground and the y coordinate position on the ground, as does the z coordinate which points out of the screen in which we don't show. Here's our metric for proper time. The proper time increment for a light pulse is always 0, so if we imagine light moving along the x axis, that is going up or down, the dy and dz increments are 0 and we're left with dx equals plus or minus dt. The plus corresponds to going up and the minus to going down. The change of position over time, the velocity of light in the up-down direction, is plus or minus 1. Okay, that accords with the defined speed of light. Now imagine we move in the y direction, so dx and dz are 0. We then end up with dy equals plus or minus e to the ax dt. The plus and minus corresponds to right or left travel. So the velocity of light parallel to the ground varies with height above the ground. However, on the ground, x equals 0 and e to the 0 is 1, so the velocity of light on the ground is indeed 1 in all directions as per the definition. But the speed of light parallel to the ground seems to change as we move up or down. As we already know from video 3, a varying speed of light goes hand in hand with gravitational bending of light rays. Here we show numerical calculations of light beams for the constant gravity case. Light appears to travel farther in the same time the higher we are above the ground and the light beams accordingly bend downward. The light is falling in the gravitational field. So is this just some mathematical artifact? The famous Eddington Eclipse experiment of 1919 showed that no, this is a real phenomenon. An eclipse allowed telescopes to view stars near the sun. The positions of those stars were compared to their known positions in the night sky. Stars near the sun appeared to move slightly away from the sun in the sky. In the illustration, the red star denotes the true position, and the blue star, the apparent position of a particular star. The observed deviations were in accord with the gravitational bending predicted by a general relativity. This result probably more than anything else catapulted Einstein into his public role as the archetypal genius. Now, before we move on, let's heed the warning from Part A that we cannot assume our coordinates have any prior geometrical meaning. Instead, we need to see what the theory tells us they mean. Our x-coordinate denotes position above the ground, and our y-coordinate denotes ground position in the left-right direction. There would also be a z-coordinate denoting ground position in and out of the screen. An atomic clock on the ground measures the time coordinate t. We mark off y equals zero and put a light source there, and then we mark off y equals one and put a mirror there. On the ground, our theory says the speed of light is one in the y-direction, so the time it will take for light to go around trip from our source to the mirror and back will be one plus one equals two. Based on the definition of length, we are justified in saying that the point y equals one is one second away from y equals zero. Likewise, we could do the same with the point y equals two and so on. So the ground observer feels justified in interpreting the y-coordinate on the ground as an accurate measure of distance. Now we do the same in the x-direction. In principle, we could fix a mirror at x equals one, and since the speed of light in the x-direction is one at all heights, the round trip time will be one plus one equals two seconds, similarly for x equals two. So now the ground observer feels justified in also interpreting the x-coordinate as an accurate measure of distance. Now we move the light source to y equals one on the ground and point it upward. By the same process, we can again fix x equals one and two and so on. Continuing on, we can directly measure the distance to any point on the ground and any point in space perpendicular to the ground. These measurements will correspond to the x, y, and z-coordinates, and our coordinates will seem to correspond to a familiar flat graph paper type coordinate system. Now notice, for these measurements, we only use light beams for which the theory says the speed of light is one. We never used beams that, according to the theory, would have shown a different speed of light. Now let's see how things appear locally. Say you're in a laboratory at rest at some height above the ground. You shoot a beam of light in an arbitrary direction and measure its speed. Your friend is in a laboratory that was catapulted upwards, such that it came to rest coincident with your laboratory precisely when you made your measurement. Since your friend's laboratory is in free fall, it represents an inertial reference frame in which special relativity holds. And therefore, both of you will measure a speed of light of one. By this argument, all local observers will measure the speed of light to be one in all directions. So it's only when a light path is interpreted by a remote observer, such as someone on the ground in this case, that light may appear to travel at a different speed. This brings back to mind the car analogy we had in part A. Each driver measures a local speed of 100 kilometers per hour but perceives the other far away driver as moving more slowly. So in general relativity, we get to have it both ways in a sense. The speed of light is a universal constant for all local observers, but to distant observers it can appear to vary over large regions of space-time, so as to give us curved light paths. Now this doesn't mean that the remotely observed variations in the speed of light are an illusion. They aren't, or if they are, then they're an illusion with observable consequences. In addition to the bending of light, quite direct evidence for a varying speed of light comes in the form of the so-called Shapiro delay. This was first demonstrated in the 1960s by direct measurement of the slowing of radio waves near the sun. Radio waves transmitted from radio telescopes, such as the Arecibo Observatory shown here, were shot towards the planet Mercury, from which they reflected and traveled back to Earth. At certain orbital positions, the radio waves traveled closer to the sun. The orbits of Mercury and Earth were precisely known, so the distance between them was also for all points in the orbit. By comparing the theoretical time delay, which is the round-trip distance divided by the speed of light, to the measured time delay, any significant variations in the speed of light could be observed. And indeed the data spanning almost two years, the points with air bars on this plot, showed a very clear extra time delay varying with orbital position precisely as predicted by general relativity. The solid curve. More recently, Shapiro delay has been observed in the Pulsar radiation of a neutron star orbiting a companion star. Since Pulsars emit radiation of a very stable frequency, when this radiation passes by an orbiting companion star in its way to Earth, the added Shapiro delay could be detected and used to calculate physical properties of the cellular system. Finally, we turn to what our constant acceleration solution to the equations of general relativity can tell us about the possibility of deep space travel. In video five, we investigated the twin paradox and saw that a person who travels from Earth to some destination and back at a velocity v will age more slowly than a person who remained behind on Earth. The elapsed time for the trip recorded by an Earth clock will be greater by the dilation factor beta. And beta can be made arbitrarily large by taking v arbitrarily close to the speed of light. But this is in a practical space travel scenario because the accelerations involved in instantaneously going from zero to near light speed or an instantaneous balance from an outgoing frame to an incoming frame would be infinite. However, our current uniform acceleration solution does provide, at least theory, a practical space travel scenario. As shown in an appendix video, this metric results in a vanishing curvature tensor, which means that by going into freefall, we must obtain a global inertial reference frame that corresponding to special relativity. We can use this reference frame, denoted by Greek letters, to represent the universe. In an appendix video, we verify that the following relations exist between our accelerated spaceship coordinates to Latin coordinates and universe coordinates. Now, if a spaceship accelerated at 9.8 meters per second per second, the occupants would experience the equivalent of Earth's gravitational field. It would be business as usual for the astronauts. As in special relativity, nothing changes for the two spatial coordinates that are not in the direction of motion. The interesting stuff happens between time and the coordinate in the direction of motion, t and x in accelerated coordinates, and psi and tau in universe coordinates. In addition to the exponential function e to the x, these transformations contain two functions of time, the so-called coche and cinch, the hyperbolic cosine and sine functions. If you have a calculator that does regular cosine and sine functions, there's usually a hype button that converts these to hyperbolic form. The exponential type growth of these functions are the key to the results we're going to see in a moment. If we assume our spaceship is at x equals 0, then as a function of spaceship time t, universe position psi is coche t minus 1, and universe time is cinch t. For t much bigger than 1, these are both approximately 1 half e to the t exponential growth. If we accelerate uniformly, our travel distance grows exponentially. The velocity of our spaceship grows as the ratio of the cinch and coche functions, which is the so-called hyperbolic tangent function. Here's the scenario. The Greek coordinates are fixed to the universe, and the lat coordinates are fixed to the spaceship. Our experience of constant acceleration is indistinguishable from a uniform gravitational field. The universe is an inertial frame, so from our perspective it appears to accelerate downward as it falls in the uniform gravitational field. Classically, with constant acceleration, our speed would increase linearly. After one year, we would be traveling the speed of light. But as already mentioned, general relativity says that our velocity will grow as the hyperbolic tangent of time, which is the blue curve. Even though our experience in the spaceship is of constant acceleration, our speed continually grows, but never reaches that of light. This is yet another confirmation that the speed of light is an absolute upper bound for the velocity of an object. Let's now analyze what would be involved in a trip to a distant object while undergoing constant acceleration equal to that of Earth's surface. With Earth as our reference, the distance psi we travel during spaceship time t is coche t minus 1. The lapsed Earth time is cinch t. Say our destination is a distance L light years away. The halfway point will divide our trip into four phases. In phase one, we start from rest at Earth and accelerate towards the destination. In phase two, we decelerate, that is, accelerate back towards Earth, ending at rest at our destination. Assuming we want to return to Earth, in phase three, we continue accelerating towards Earth. Upon reaching the halfway mark, we enter phase four, where we decelerate and finally come to rest back home at Earth. The four phases are symmetric with respect to distance traveled and elapsed time. So therefore, we can find the time for the first phase and multiply by two for the one-way time or by four for the round trip time. Therefore, to travel L light years, you find time t1 such that coche t1 minus 1 equals L over 2. And you've got your one-way and round trip times then. For example, say we wanted to travel to the Andromeda galaxy, a distance of about 2 and 1 half million light years. The first phase takes only a little bit less than 15 years, and we'd be at Andromeda in less than 30 years, back in Earth in less than 60 years. Of course, more than 5 million years would have elapsed on Earth, so who knows what we'd find. What if we wanted to travel 10 billion light years? The exponential growth of the coche function is such that the first phase would only take 23 years, and the one-way time would only be 46 years. Indeed, by merely traveling with constant Earth equivalent acceleration, it's theoretically possible to travel anywhere in the visible universe within a human lifetime. Of course, there are some massive practical obstacles. One is that the kinetic energy required is equal to the mass energy of the spaceship times the distance traveled, and this is an enormous amount of energy. For some perspective, the annual world energy consumption would only be enough to provide this constant acceleration for a 100 kilogram mass to a distance of about 50 light years. But if you could overcome that, quote, little hurdle, and a few more, the exponential distance traveled versus time means that in principle, your receding view would look something like that famous powers of 10 movie, link in the description box. There would be some serious visual distortion due to curved light paths and Doppler shifts. We won't get into those issues in this video, but we will investigate these concepts in the context of black holes in our next video.