 Let's shoot a light pulse from the left side of the plot directly at the black hole. Now another from the same point, but at an angle of 10 degrees, then 20 degrees, and 30 degrees. These all hit the horizon. But a pulse at 40 degrees, or at 50 degrees, continues on through space. Now imagine these pulses traveling in the reverse directions and ending up at your eye. The 0, 10, 20, and 30 degree pulses would never leave the black hole, so you would see nothing in those directions, pure blackness. But the 40 and 50 degree pulses could have been generated by some other objects, and would reach your eye. Say the 40 degree pulse came from this brown object. If the black hole were not there, you would see it at an angle of a bit less than 10 degrees. In the presence of the black hole, light from this object strikes your eye at 40 degrees, and because our brain's interpret light is moving in straight lines, you will therefore perceive the brown object as being in a very different direction in space. Let's see how this works with a simulated image of a star field. On the right, we'll place a black hole of increasing mass in the center and see how this distorts our view. The two central stars expand into rings while the other stars move outward and are distorted in shape. We also see some small star images inside the rings, and notice that the center is always dark. This is the black hole. The gravitational distortion of an object's image into a ring is another incredible prediction of general relativity, and has been observed on several occasions by the Hubble Space Telescope. These are, appropriately, called Einstein rings. Here's one of my favorite examples in some detail. Let's see what causes Einstein rings. Here we see an object directly behind the mass, relative to our eye. There are two symmetric light paths that will bring this object's light to our eye, so we'll see it on both sides of the sky. Because our eye, the mass, and the object all lie on a single line, there will be symmetric light paths at all angles about the brown line. The two solid red lines sweep out a cone, and we see a circle of light. This is Einstein's ring. If the rings come from objects directly behind the mass, where do the little images inside the rings come from? Back on our illustration, imagine an object off the brown line. There will still generally be two paths by which light can reach our eye from the object, the blue paths, but they will not be symmetric about the brown line. So we will see them at only one place in the sky. The solid red lines are where we see Einstein's ring, and the solid blue lines are where we see the two images of this object. One blue line is between the red lines, and the other is outside. Thus, we will see two images of every object, one outside the ring, and another, on the opposite side of the sky, inside the ring. In fact, because of boomerang light paths such as this one, your eye would, in principle, even see an Einstein ring image of itself in the sky. Let's again look at the trajectory of a light pulse falling into a black hole. Remember that the coordinates we plot here are those of a distant observer. Each dot denotes a one-second interval, and we see that the dots get closer together as the light approaches the black hole. The light appears to slow down, and actually seems to freeze at the source-shield singularity, the surface of the black hole, the so-called event horizon. Now the question is, does the speed of light really vary, or is this an illusion? We'll separately look at the speed of light in the altitude direction, the arc-coordinate, and the longitude direction, the phi-coordinate. All that we use coordinates in which the speed of light is perceived by a local observer in empty space is one. In part a, we saw that the speed of light in the longitude direction is this horse-shield factor, the square root of one minus two m over r, which is less than one. In the altitude direction, the speed of light is the square of this. But in video 10b, we saw that the speed of light is unity in all directions for all local observers. Therefore, for a local observer, light should move one unit of distance in one unit of time. So we write a displacement r d phi equals an increment ds of local time. For the distant observer, we have r d phi equals the source-shield factor times dt. What we already know from our discussion of gravitational time dilation, that ds equals the source-shield factor times dt. Therefore, the apparent variation of the speed of light in the longitude direction is simply a manifestation of gravitational time dilation. Light appears to move slower because time appears to move slower. Now we turn to the altitude or r direction. Here the distant observer perceives a speed of light that is the square of the source-shield factor. For a local observer, the speed is always unity, so we write a displacement dr equals a time increment ds. For the distant observer, dr equals the source-shield factor squared times dt. We can account for the time dilation by replacing one source-shield factor times dt by ds. But this still leaves a source-shield factor, and this result does not agree with what we assume for the local observer. We are forced to conclude that the local observer does not consider dr to be an increment of distance. Instead, we write an increment of distance as ds, and express the local observer's unity speed of light as ds equals ds. Comparing this to the distant observer's expression, we conclude that his perception of spatial displacement dr equals the source-shield factor times ds. The local observer's measure of displacement. That is, not only do the two observers disagree on time measurement, but they also disagree on distance measurement. Local distance measurement in the altitude or r direction is larger than perceived by the distant observer by one over the source-shield factor. As r approaches 2m, the source-shield singularity, a finite distance as perceived by the distant observer, would approach an infinite distance as perceived locally. This is an extreme distortion of space itself, independent of time dilation. These considerations lead us to the concept of proper distance. In video eight, we developed the idea of an invariant, quote, yardstick of space-time for special relativity, and then showed how to apply it in the general case. Lines in space-time correspond to events which occur at a specific time and place. Consider the three events shown here, and specifically the relation between the first event and the other two. Starting at event one, we can draw lines corresponding to the propagation of light in both directions in space and into the future and the past. We call this the light cone, and we think of it as dividing space-time into two regions, inside and outside the light cone. Event number three is inside the light cone. This means that the velocity required to travel from event one to event three is less than the speed of light. Therefore, it's possible, at least in principle, for a clock to move between the two events and measure the elapsed proper time, s. We call this a time-like interval. As we showed in video eight, s squared is the square of the difference of the time coordinates minus the square of the difference of the space coordinates. Event two, however, is outside the light cone. It's not possible for a clock to travel between these two events, and s squared would actually be a negative number. However, the negative of that is a positive number, which we call sigma squared. Sigma squared is the square of the difference of the space coordinates minus the square of the difference of the time coordinates. We call this a space-like interval. The arguments given in video eight can be applied to this interval to show that it too is an invariant on which all reference frames can agree. We call sigma the proper distance. We see that if the times t1 and t2 are the same, then sigma is simply a difference of space coordinates, a length. Back to the Schwarzschild metric. We've discussed the proper time ds. The time between two events is measured in a frame where they occur at the same place. We can also talk about the proper distance dsigma. The distance between two events is measured in a frame where they occur at the same time, and this is just the negative of the ds squared expression. The proper time interval, when dr and d phi are zero, when we stay in one place, is ds equals the Schwarzschild factor times dt. And as we've discussed, this expresses gravitational time dilation. The proper distance interval, when dt is zero, has two terms. If we limit motion to the altitude or r direction, then we find our previous result dsigma equals one over the Schwarzschild factor times dr. For motion in the longitude or phi direction, dsigma is simply rd phi. Because of the distortion of space, the altitude or r coordinate no longer has the flat space-time meaning of the distance to a point from the center of the mass. Instead, its physical significance can be described as a reduced circumference. For motion in the longitude or phi direction, proper distance does not depend on the mass m, so it retains its flat or euclidean form. But for motion in the altitude or r direction, proper distance does depend on the mass. Therefore it corresponds to a curved or non-euclidean spatial geometry. Arbitrary motion is a combination of these flat and non-flat components. Therefore a circle with coordinate r1 will have a circumference 2 pi r1, independent of the mass. So although r1 no longer has an unambiguous meaning in terms of the radius of a circle, it continues to represent the circumference of a particular circle. And we call it the reduced circumference of that circle. The same will be true for a second circle with coordinate r2. The circumferences of the circles will differ by 2 pi times the difference of the r coordinates, as they do in euclidean geometry. But the distance between the circles as measured by a local observer is no longer the difference of the r coordinates. Here's a representation of this phenomenon. Suppose I tell you I'm going to draw a series of circles equally spaced apart. At first you can believe me, but soon it's clear that these circles are getting closer together. Eventually the claim of equal spacing seems ridiculous. But I'm insistent. Just because they don't look equally spaced to you doesn't mean that they aren't. In fact when viewed from a different perspective you can clearly see that the distances between them are indeed equal. You are interpreting what you saw as if the circles lay on a flat surface, when in fact they lie on a curved surface. This curved surface is called an embedding diagram, and it's a way to visualize the distortion of space around a black hole. Now it's a curved two-dimensional surface embedded in a flat three-dimensional space. For an actual black hole we'd have to imagine a curved three-dimensional space embedded in a flat four-dimensional space. Let's go back to the scenario of a light pulse falling into a black hole. In addition to time dilation, the distortion of space contributes to the apparent decreasing velocity of light. We can imagine the light pulse moving at a constant speed down into the funnel-shaped embedding diagram. From our perspective above it appears to slow down. But viewed from another perspective it clearly moves at a constant velocity. Distant observers can see the effects of the changing curvature of the surface. However, local observers, looking at their own little flat patch on the surface, will see simple uniform motion at a constant velocity.