 Before we continue, I want to tip my hat to the excellent introductory-level book Exploring Black Holes by Taylor and Wheeler. The late John Wheeler was the physicist responsible for coining the term black hole, and some of the topics and terminology in this video were motivated by this book. Imagine, rather whimsically, that we have a global observer who we'll call the bookkeeper. He has a clock that measures time T. He spies a distant tiny asteroid floating in deep space, and he sends a team of robots to build a physical coordinate system using this as its origin. The robots measure off a distance of one light second from the asteroid, and construct a solid ring with this as its radius, which they label with the coordinate r equals one. They measure out another light second, and build a second ring, r equals two, and so on with all the rings lying in a common plane. Since there is no significant gravitational field in this region of space, spacetime will be flat, and Euclidean geometry will hold. So we know that the circumference of a ring with radius r will be two pi r. Now attention turns to the second coordinate. A beam of a light is shot from the asteroid in the point where it intersects each ring as marked as coordinate phi equals zero. The beam is rotated by, say, 45 degrees, and coordinate phi equals 45 degrees as marked off, and so on. We now have a complete two-dimensional coordinate system. Anyone can figure out where they are by noting which ring and angle marking is nearest, and the bookkeeper can follow anyone's trajectory by noting what r and phi coordinates they pass by when his clock reads some time T. Now very whimsically, let's picture the robot construction team depositing more and more mass onto the asteroid until it eventually becomes a black hole. From previous discussions, we know that a ring with radial coordinate r will still have a circumference two pi r. However, the black hole will curve spacetime such that the locally measured distance between rings is no longer equal to the difference of their r coordinates. We visualize this as the rings being on a curved surface such that the distance between them as seen from above is less than the distance between them as measured on the surface. Now we launch a satellite and track it as it falls freely in this curved spacetime. The satellite has a clock that reads time s, the proper time for its trajectory. For each value of s, the satellite can record its current r and phi coordinates as well as bookkeeper time T. These are the values predicted by the equations of motion. At the same time, the bookkeeper can keep track of the satellite coordinates as a function of his time, T. We can also have local observers positioned at rest on the rings. They will measure proper distance sigma, where an increment d sigma is greater than an increment dr by a factor of one over square root one minus two m over r. And they will have their own clocks that measure local time. An increment dt local is less than an increment dt by a factor square root one minus two m over r. In an appendix video, we derive the equations of motion and explain how to solve them numerically. Now let's look at what they predict for a satellite released from rest near a black hole. The object starts at the upper right corner of the coordinate grid. We plot its position versus bookkeeper time T. The satellite initially accelerates, but then slows and comes to rest at the horizon. Now we plot position versus satellite time, s. The satellite accelerates continuously until it hits the horizon. At that point we stop the animation, not because the satellite stops, but because the equations of motion break down at the Schwarzschild singularity. Let's illustrate these two different views of the motion on a single plot. Here the vertical axis gives the r coordinate. We'll have the satellite fall from r equals five. The thick black line at the bottom indicates the horizon. The horizontal axis corresponds to time s for the satellite and t for the bookkeeper. The red dots show the satellite's perception of the motion while the blue dots correspond to the bookkeeper's view of things. While the satellite records itself as moving ever more rapidly toward the horizon, indicated by the red line always curving downward, the bookkeeper sees it first accelerate, then level off and finally slow down, indicated by the upward curving of the blue line near the end. Let's zoom in on the behavior near the horizon. There seems to be nothing special about the horizon from the satellite's perspective, but for the bookkeeper it represents an absolute limit of the satellite's motion. As the satellite gets closer, ever longer intervals of time pass for the bookkeeper. No matter how long he watches, the bookkeeper will never see the satellite reach the horizon. From these curves we can calculate apparent velocities. The vertical axis is the r coordinate, and the thick black line at the bottom represents the horizon. The horizontal axis shows velocity, with rest at left and increasing toward the right. The left blue curve is the satellite's velocity as perceived by the bookkeeper, dr, an increment of r coordinate, over dt, an increment of bookkeeper time. This starts at zero and increases as the satellite falls, up to a maximum of a bit more than point three, at r a bit more than two. It then rapidly decreases back to zero as the satellite gets closer to the horizon. The right red curve shows how fast the r coordinate changes from the satellite's perspective, dr over an increment of satellite time, ds. This also starts at zero, but continues to increase as the satellite falls. Now let's simulate the satellite falling from a point closer to the horizon, r equals 1.1. Once again the satellite sees itself as always accelerating, speeding right up to the horizon, while the bookkeeper sees it as accelerating and then decelerating, never reaching the horizon. And again, the apparent velocity plots reflect this. As we might expect since the satellite started its fall closer to the horizon, the maximum velocities are less than for the previous case. Not only does the bookkeeper see the satellite slowing down as it approaches the horizon, he sees all processes on the satellite slowing down too. Here we plot 100 clicks of the satellite's clock as it nears the horizon. Satellite time s is shown on the horizontal axis, and the increment of s between each tick is uniformly one millisecond. The total elapsed satellite time is 100 milliseconds, from s equals .585 to s equals .685 seconds. The corresponding bookkeeper time is plotted on the vertical axis. The total elapsed bookkeeper time is about 7 seconds, from a bit less than t equals 3 to a bit less than t equals 10. That's about 70 times the elapsed time experienced on the satellite. And this apparent slowing of time rapidly increases the closer the satellite gets to the horizon. The last three ticks of the clock shown on this plot are seen by the bookkeeper to take 389, 643, and 2308 milliseconds. This time stretching increases without limit, and the final clock tick seen by the bookkeeper would last forever. As a result, the frequency of any signal sent by the satellite, including light or other forms of radiation, will be seen by the bookkeeper to decrease to zero. That is to be infinitely redshifted. So, in a more accurate representation of the bookkeeper's view of the satellite's fall, the satellite initially accelerates, then slows down, and finally fades from sight as it approaches the horizon. Physics tells the bookkeeper that the satellite comes to rest at the horizon, but he never actually sees this. Now let's look at the velocity of the falling satellite as measured by local observers at rest on our coordinate rings. Their measurements of length and time differ from the bookkeepers. The bookkeeper sees the satellite velocity as the change in r-coordinate over a change in the t-coordinate. The local observer will measure the corresponding increment of proper distance, d-sigma, which is dr over square root 1 minus 2m over r. And she measures an increment of time, dt-local, which is dt times square root 1 minus 2m over r. Our velocity measurement is the ratio of these increments, d-sigma over dt-local, which is the bookkeeper velocity dr over dt divided by 1 minus 2m over r. As the satellite approaches the horizon, bookkeeper velocity approaches zero, but so does the factor 1 minus 2m over r. When we plot this for the fall from r equals 5, the red curve at right, we find that it starts at zero and continually increases. As the satellite nears the horizon, the locally measured velocity approaches one. In our units, that's the speed of light. For the fall from r equals 1.1, locally measured velocity once again starts at zero and then approaches the speed of light at the horizon. So approaching the event horizon, the bookkeeper sees a falling object coming to rest and fading to black. The local observer see a falling object approaching the speed of light. The same object is seen by different observers to both approach the smallest possible speed rest and the largest possible speed, that of light. This is the scenario that we somewhat crudely tried to visualize using the embedding diagram. Viewed from above the surface, corresponding to the bookkeeper's view, an object does indeed appear to slow down in its motion toward the black hole. But the same motion viewed by local observers displays no slowing at all. If things are so bizarre near the horizon, it's natural to ask what happens at the horizon itself. And if it's possible to pass through the horizon, what happens inside? Our problem is that we can't solve the equations of motion across the horizon because the form of our metric, from which everything is derived, is such that as the r-coordinate approaches its horizon value, the metric coefficient of dr squared blows up. This is the Schwarzschild Singularity. And yet our solutions show that the satellite speeds right up to the horizon. And local observers see it approaching the speed of light. So it certainly seems like the satellite would cross the horizon. Either that or something catastrophic happens there. To get some insight into this, let's look at the gravitational acceleration that would be measured by local observers at rest on the coordinate rings. The equations of motion for an object starting from rest look like acceleration in the r-coordinate equals minus m over r squared. Here are the double dots above the r-indicate acceleration. But the r-coordinate does not measure proper distance, so this expression is not the acceleration that would be measured by local observers. Instead, an increment of local measured distance equals an increment of r times the factor of 1 over square root 1 minus 2m over r. We have to add this factor to get the formula for acceleration that would be measured by local observers. But this expression blows up as we approach the horizon. Here's a plot of gravitational acceleration, which is also the gravitational force per mass versus the r-coordinate. As we descend toward the horizon, this grows arbitrarily large. In our system of units, acceleration is seconds per second squared. In other words, one unit of acceleration corresponds to going from zero to one light second per second in one second. That is, from rest to the speed of light in one second. The speed of light is about 300 million meters per second. One at Earth's surface is roughly 10 meters per second squared. So one unit of acceleration in our system corresponds to about 30 million g's, a truly astronomical quantity. Now we can understand how it is that no matter how close to the horizon an object falls from, it approaches the speed of light as seen by local observers. The closer to the horizon we are, the greater is the acceleration. And this grows arbitrarily large, arbitrarily close to the horizon. With an arbitrarily large acceleration, we can get an object to go from zero to the speed of light in an arbitrarily short period of time. Let's look at our metric again. Outside the horizon, r is greater than 2m, and the expression in parentheses is positive. Therefore, it's possible, at least in principle, to remain at rest, that is to have no change in the r-coordinate while having a change in the time coordinate. For dr equals 0, the metric reduces to ds squared equals a positive number times dt squared. Now, let's see what happens inside the horizon. There 2m is greater than r, so 1 minus 2m over r is negative. We show this explicitly by writing it as minus 2m over r minus 1. We can no longer have dr equals 0, because this leaves us with ds squared equals a negative number times dt squared. A positive number cannot equal a negative number. It follows that inside the horizon, no object can be at rest. If it falls through the horizon, moving toward r equals 0, it must continue to do so. Nothing can stop its motion and bring it to rest, not even in principle. So, at the horizon, gravitational force becomes infinite. And inside the horizon, you cannot remain at rest. You can only move toward the center r equals 0. Although the bookkeeper coordinate system can't describe motion across the horizon, coordinate systems which themselves are in freefall can. Details are given in the Taylor and Wheeler text. What can be said is that inside the event horizon, at least according to general relativity, no information can be transmitted to the outside, so no one outside can confirm a prediction of what happened inside. No stationary local observers can exist, and velocity measurements relative to a fixed frame cannot be made. We've seen that according to stationary local observers at fixed values of the arc coordinate, a falling object always approaches the speed of light at the horizon. Extrapolating this curve, we might expect that inside the horizon, the object is falling faster than the speed of light, but we aren't justified in extrapolating this curve inside the horizon because no stationary local observers can exist there, not even in principle, so the plotted quantity is not even defined. Likewise for the gravitational force measured by stationary local observers, what can be said is that no experiment performed anywhere, even inside the horizon, will ever measure an object moving locally faster than the speed of light. Everything that falls into the black hole ends up at r equals 0 in a finite proper time. The source of singularity is a singularity only in certain coordinate systems, such as bookkeeper coordinates. In other systems, such as freely falling coordinates, it's not a singularity, and motion across the horizon is readily described. However, the center at r equals 0 is a true singularity. There the curvature of space time becomes infinite.