 Hello, and welcome to General Relativity Part 2. In Part 1, we covered the equivalence principle, non-Euclidean geometry, and the Einstein field equations. Einstein had come up with a totally different way of looking at gravity. But was it a difference that made a difference? Or did his equations predict different physical phenomena than Newton's did? You may recall, if you've seen how small is it video book, that the theory for quantum mechanics and the standard model was developed to explain experimental evidence. But with General Relativity, there was no experimental evidence. The theory came first. So Einstein came up with three tests to demonstrate that his theory did indeed predict more accurately phenomenon like the orbit of Mercury, the bending of light, and gravitational redshift. We'll cover all three of these, and frame-dragging a more recent test around how rotating masses can twist and move space around it will end by building a black hole where all of these concepts come together. We'll use Gargantua, the black hole in the movie Interstellar. But first, Mercury. In 1916, the same year that Einstein published his General Relativity paper, Carl Schwarzschild published his exact solution for space around a large, non-rotating mass. His metric is now called Schwarzschild metric, and it works quite well for slowly rotating masses like the Earth and the Sun and the planets in our solar system. We'll use this metric for the first three tests. Let's take a look at what our space-time curvature looks like with this metric. If we draw the circumference of the Earth's orbit, we get a length that is 2 pi times our distance from the Sun. If we existed in flat Euclidean space, we would calculate the circumference of an orbit one kilometer closer to the Sun and see that the distance between the orbits is one kilometer. But because of our positive curvature, if we were to measure the circumference with a radius that is one kilometer shorter than the first, we'd find that it is less than 2 pi times the shorter radius. Which means that the distance between the circumferences would be greater than the one kilometer difference in the radii, but only a little. We can repeat this process all the way to the surface of the Sun. With each successive radius, the difference between the orbits would increasingly diverge from the Euclidean numbers. If we were to telescope this picture, you'd see the standard diagrams that are used to help explain general relativity. But diagrams like this are misleading in two ways. First, they represent an external curvature into another dimension, when in fact we are talking about intrinsic curvature. There is no evidence for the existence of a fourth spatial dimension. Second, it looks like you need a downward force on the object to get it to drop into the hole. That would be gravity, but that's what the lines were supposed to represent. So we'll avoid using this technique. For over a half of a century before Einstein's time, it was known that there was something odd about the orbit of Mercury. The elliptical path it carves around the Sun shifts with each orbit, leaving its perihelion, or closest point to the Sun, 56 arc seconds forward on each pass. Newtonian equations accounted for all but around a half an arc second per year. And of course, they couldn't take into consideration the effects of curved space, because the idea that space wasn't flat hadn't been considered yet. With Schwarzschild's metric, Einstein came up with the exact number to cover the mysterious half an arc second. He had passed the first test of his new theory. When light comes close to the Sun, the Sun's gravity bends it inward. This makes the star look like it's further away from the Sun in the sky than it really is. Both Einstein's and Newton's gravitational theories predicted this. But the theories predicted different values for the amount light would bend. Einstein suggested that a solar eclipse could be used to find the exact number. In 1919, a solar eclipse was slated to occur with the Sun silhouetted against the Hiades star cluster, the nearest open cluster to our solar system. Here's the Hiades star configuration with some of the brightest stars identified. The British astrophysicist Arthur Eddington took up positions off the coast of Africa and Brazil and simultaneously measured the cluster's light as it brushed past the Sun. The images were then superimposed on top of an image taken at night earlier in the year. When the eclipse and night images were compared, a gap was found. And when the gap was measured, it confirmed that Einstein's prediction was correct. This same light bending leads to the warping of light from distant galaxies as the light encounters supermassive galaxies on their path to us. This is called gravitational lensing. Here's a clip that shows how this lensing works on a grand scale. A distant galaxy would be seen here on Earth directly if there were no intervening massive cluster to bend the light. But with such a cluster, the light from the distant galaxy gets bent into rings and arcs to continue on to the Earth. This is Abel 1689, 2.2 billion light years away. It's one of the most massive galaxies clusters known. The gravity of its trillion stars plus dark matter acts like a 2 million light year wide lens in space. And here's another cluster, 5.7 billion light years away. It's the latest from Hubble on gravitational lensing released in late 2015. These foreground galaxy clusters are magnifying the light from the faint galaxies that lie far behind the clusters themselves. These faint lensed galaxies are around 12 billion light years away. It's the gravitational lensing that allows us to see that far back in time. Without the magnification, these galaxies would be invisible for us. One of the key implications for bending of light is its impact on what's physically possible in heavily curved spacetime. Here's a two-dimensional slice of the future light cone that we developed in the previous segment on special relativity. This purple line represents a path by anything with mass. It's called the whirl line and can be anywhere inside the light cone. In this representation, whirl lines have to remain between the two arms of the light cone because nothing can travel faster than the speed of light. The speed of light lines are the divider between events that are in your future, if it's your light cone, and events that are not. By in your future, I mean that you can be connected to them physically in some way. Now suppose there is a great mass energy density to the left of the cone. The light would be bent in its direction. We see that points that were impossible to reach before now fall inside the light cone and are reachable. And we see that points that were reachable inside the cone now fall outside the cone and are no longer reachable. This is light cone tipping. The closer we get to the source of the gravity, the greater the space-time curvature. And the larger the matter curving the space, the greater the curvature. We'll take another look at this when we get to black holes. One of the most dramatic consequences of general relativity is how space-time curvature affects the flow of time. We'll use the elevator thought experiment to illustrate how clocks can run at different rates in the box according to their distance from the source of the gravity. We'll see that a clock closer to the source of the gravitational field runs slower than a clock further away. To help see how this works, we'll take another look at the lightning strike for the person on the train and the person on the ground that we used in our segment on special relativity. Only this time, we'll map the events to our space-time graph. The whirl line for the person standing on the ground is shown in purple. We'll label the lightning strikes A and B and place the two events on the space-time graph with A to the left of the person on the ground and B to the right. The plane containing A and B contains all the points that are simultaneous for the person on the ground at the time of the two strikes. We call this the simultaneity plane. The light from both events travels at the speed of light, so their whirl line always moves at a 45 degree angle. They reach the person on the ground at the same time. This of course is what makes them simultaneous from the point of view of the person on the ground. Now let's repeat the lightning strike so that from the point of view of the person on the moving train, they strike at the same time. In order for the light to reach the person on the train at the same time, the strike behind him will need to hit first from the person on the ground's point of view because it will have to travel further to get to the moving person than the light from the strike that hits in front of him. So we see that the simultaneity plane with a moving person is necessarily tilted up on the right. Now we can map the movements of A and B in the accelerating elevator to the space time graph. The center is the source of the acceleration, or gravity. A is to the right of it, and B a bit further to the right, reflecting their distances from the source of the gravity. As the elevator accelerates, the whirl lines on the space time graph are not straight lines. They curve outwards because their velocity increases with every second. Here we have clocks that measure the proper time elapsed along each person's world line. They mark the time in their own reference frame. At the start, they are both at rest, so their simultaneity plane is horizontal and they each read each other's clocks reading zero. In this example, we see that after two seconds, we have a slightly tilted simultaneity plane. B sees that, at the same time, his clock ticks two, A's clock ticks one. A also sees his own clock reading one when B's clock reads two. Continuing to a higher velocity, with a steeper slope for the simultaneity plane, B sees A's clock reading two when his own clock reads four. A also sees his own clock reading two when B's clock reads four. A and B both agree that A's clock is ticking slower than B's clock. The equivalence principle tells us the same thing will happen near a massive body. Gravity slows down time. Newton's gravitation has no such implication. We see this with our global positioning systems. In our segment on special relativity, we saw that time dilation due to velocity differences have GPS satellites losing time every day, time that must be corrected for to get the right position on the surface of the Earth. They must also take into account gravitational time dilation due to their being further away from the Earth than clocks on the ground. Based on the Schwarzschild metric, calculations show that the satellite's clocks will gain over 45,000 nanoseconds a day due to this general relativity effect. The accuracy of our GPS system is strong evidence for the correctness of general relativity. In 1959, physicists Robert Pound and Glenn Rebka performed an experiment in the Jefferson physical lab at Harvard to demonstrate gravitational redshift. It was based on physicist Rudolf Mossbauer's effect discovered two years earlier that involves the emission and absorption of gamma rays from the excited states of iron nucleuses. Here we have an iron atom's nucleus in an excited state. When it falls to a lower energy level, a gamma ray photon carrying the energy is emitted. Once this photon enters a like atom, it will be absorbed, raising the energy level of the encountered atom's nucleus. The problem is that when the gamma ray is ejected, the nucleus recoils. Because of energy momentum conservation, the recoiling energy reduces the energy of the gamma ray. The gamma ray is no longer a match for the other nuclei, and it moves right through. There is no absorption. What Mossbauer discovered was that if he embeds the iron atoms in a crystal, the recoil is reduced dramatically, and absorption can be reestablished. Pound Rebka used this Mossbauer effect. They placed an emitter at the bottom of a tower in the laboratory and installed a detector 22.6 meters above it. No absorption was detected because gravitational time dilation changed the frequency of the emitted gamma rays so no energy match existed in the detector. The calculated shift was extremely small, but the Mossbauer effect is sensitive enough to measure it. They adjusted the detector's velocity down until absorption occurred. We get the amount the frequency changed using the well understood relativistic Doppler red shift equation, just like the Doppler shift in Starlight. These results came in within 1.6% of the value predicted by Einstein's field equations using Schwarzschild's metric. Although this experiment did not produce new results, it showed that gravitational time dilation, one of General Relativity's most significant findings, was consistent with all physical conservation laws. This gave the General Theory of Relativity three successes out of three tests. Gravitational time dilation is also the answer to the twin paradox that we covered in the previous segment on special relativity. The key interval is at the halfway point. As the spaceship approaches Vega, it decelerates to a stop and then re-accelerates back to the Earth. The traveling twin finds that she is in a gravitational field. Let's say her acceleration is 10 G's or 98 meters per second squared. At this rate, it would take her 35 days to decelerate to zero and another 35 days to re-accelerate back to 99% of the speed of light. Gravitational time dilation shows that as her clock ticks 70 days, her twin's clock on Earth will have ticked 18,134 days. That's 48 years. The twin on Earth agrees. So instead of both twins thinking the other should be younger, they both agree that the twin on the rocket to Vega and back is younger. No contradiction is involved and the paradox is resolved. Our last test is the most recent. It was designed to measure the twisting of space around a rotating mass. This twisting is called frame dragging, where space is literally dragged along with the rotating mass. The effect was derived in 1918 by physicist Joseph Lenz and Hans Thiering. It is known as the Lenz-Thiering effect. They predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess like a gyroscope. This does not happen with Newtonian gravity, where the gravitational field of a body depends only on its mass, not on its rotation. Up until now, we've been using the Schwarzschild metric, which does not show this effect. It wasn't until 1963 that a mathematician named Roy Kerr discovered the significantly more complicated metric for rotating bodies that made it possible to calculate the precession one can expect from a given mass and rotation of an object like the Earth. To test this effect, NASA developed a satellite called Gravity Probe B and put it into orbit 264 kilometers above the Earth in 2004, where it operated for a year. It used a set of super sensitive gyroscopes to measure precession due to frame dragging. It also included a non-gravitational drag identification gyro and compensation micro thrusters to maintain a non-gravitational drag-free environment. It compensated for solar radiation drag and atmospheric disturbances drag. By 2011, data analysis had confirmed that frame dragging did occur and measured it to within 15% of the amount predicted by the Kerr metric for Einstein's field equations. One of the most interesting consequences of general relativity is the structure and impact of a black hole. In the Milky Way segment of the How Far Away Is It? video book, we discussed how they are formed from collapsing massive stars, too big for neutron pressure to halt their collapse to a point called a singularity. The Schwarzschild metric showed that if a mass of a body should contract to a small enough radius, it could capture light itself. This radius is known as the Schwarzschild radius and forms a sphere known as the Event Horizon. One of the best illustrations of a black hole was created for the 2015 movie Interstellar with the help of theoretical physicist Kip Thorn. His black hole, called Gargantua, was given a mass of 100 million suns and a super high rotation rate of 99.8% of the speed of light. With this kind of rotation, we see that Gargantua is indeed a Kerr black hole. At 100 million solar masses, the Schwarzschild radius is around the distance from the sun to the earth. That's far enough away to make the tidal forces at the horizon quite unnoticeable. We'll use Gargantua to illustrate the properties of general relativity that we have discussed in this segment. So let's build this black hole from the ground up. We are viewing it from the equatorial plane and the object is rotating in on the left and out on the right. The center is dark out to the Schwarzschild radius. The Kerr metric shows that light can also be captured in stable orbits outside the Event Horizon. For a rapidly rotating black hole, the orbital volume around the black hole would be significant. This would produce a photon sphere shell, encasing the black hole with light from all stars in the universe accumulated over the entire age of the universe. It would be a sight to see. But given that the light is trapped in orbit, we can only see what leaks out. This thin ring around the black hole represents the cross section of this shell we'd see because of light that leaks out in our direction. It is flattened on the left because light rotating with the black hole's rotation can get closer to the horizon than light rotating against the black hole's rotation. Next we see a dense sprinkling of stars with a pattern of concentric shells. This is the pattern produced by the gravitational lensing. Further out we see the dislocation of star positions due to the bending of light by the gravity of the black hole. This black hole has the remnants of an accretion disc that is no longer feeding the black hole. If the disc were not gravitationally lensed, the black hole should have looked like this. But because of gravitational lensing, the massive amount of light rays emitted from the disc's top face travel up and over the black hole, and light rays emitted from the disc's bottom face travel down and under the black hole. This combination gives us the full image of how the black hole would actually look. In the movie, one hour on Miller's planet equals seven years on Earth. Some of this came from time dilation due to the planet's speed. It's traveling at 55% of the speed of light in order to maintain its orbit. But the bulk of the time comes from gravitational time dilation. And the fact that gargantua's rotational energy is so large. This intensifies time dilation considerably. If you watch the movie again, you might note that it is the curmetric on the professor's blackboard. We can use tipping light cones to show how all objects, unfortunate enough to cross the event horizon, are captured forever. Here's a light cone far from the black hole. The horizontal axis represents distance from the singularity. As we saw earlier, when spacetime is curved by the presence of mass energy, the light cone structure gets distorted. When the mass is a black hole, the tilting reaches 45 degrees at the event horizon. This means that all events beyond the horizon are no longer in the future light cone for any object that has gone past it. No possible world line gets you out. All remaining world lines lead to the singularity. Distance from the singularity decreases inside the black hole's horizon as surely as time increases outside the horizon. In Interstellar, Cooper flies his ship into the black hole while Brand watches from a higher orbit. We can use our spacetime diagram along with light cone bending again to illustrate what each of them would have seen. First we'll take a look at it from Cooper's point of view. As he heads directly into Gargantua, he sees periodic signals from Brand. She is far enough away from the horizon for her light signals to all travel in a parallel manner at 45 degrees along her light cone boundary. Cooper crosses the event horizon without even noticing it, as signals continue to arrive at regular intervals. Eventually, he will feel the tidal forces of the singularity. Things are quite different from Brand's point of view. As Cooper approaches the event horizon, his light cone tips towards the singularity. This means that his light signals back to Brand are taking longer with each kilometer traveled. The effect is hyperbolic, and the light signal he sends from the horizon itself will never get to Brand. She sees his clock slowing down to the point that it stops. She never sees him enter the black hole. What's more, because of gravitational redshift, the image of Cooper and his ship shift to the red. At the horizon, it has shifted into the infrared and can no longer be seen by Brand. For her, Cooper grinds to a halt and goes invisible. Quite different from what Cooper sees. The general theory of relativity is now 100 years old. In spite of the fact there have been a number of tests, questions remain. One of the theory's most interesting predictions is gravitational waves. But as yet, no gravitational waves have been found. If they are ever found, would there be an associated elementary boson particle, the graviton, like photons for the electromagnetic force? A great deal of active research is underway to find out. The fate of general relativity remains in the hands of experimental physicists.