 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, but first, Mercury. The Planet Mercury's orbit test, Einstein's General Relativity Theory, actually starts with the planet Uranus, discovered by William Herschel in 1781. By the early 1800s, it was understood that planetary orbits were elliptical, with small deviations called perturbations, and that each orbit's closest approach to the sun, called a perihelion, shifted slightly over each orbit. This is called precession. Using Newton's gravitational equations, all known perturbations and precessions were calculated and found to fit the observations for all the planets except one, Uranus. A full study over decades showed that Uranus's orbit did not fit Newton's equations. At times it was moving faster than predicted, and at other times it was moving slower. There were two schools of thought at that time. One held that Newton's theory did not hold up that far from the sun, indicating that a new theory was needed. The other proposed that there is another planet beyond Uranus that pulled on it. This is observed deviations could be explained as perturbations. If correct, this would keep Newton's theory intact. Astronomer Urbain Laveret went to work to try to discover this new planet. Early in 1846, he published calculations that came very close to where it actually was. On September 23rd of that same year, Johann Gaul, an astronomer at Berlin Observatory, and a student, Heinrich-Louis de Arrest, found the new planet looking where Laveret replaced it. This planet is now called Neptune. The reason this is relevant for Mercury is that the overall thinking at the time was similar. Newton's theory does not fully explain the observed precession of Mercury's perihelion. In 1859, astronomers, including Laveret, theorized that another planet inside the orbit of Mercury could account for the observations, much like how Neptune explained Uranus's orbital irregularities. The proposed planet between the Sun and Mercury was even given a name, Vulcan, but no such planet was ever observed. Another school of thought held that Newton's theory simply did not hold up that close to the Sun. Einstein was one of them, and his general theory of relativity, describing the impact of the curved space near the Sun, provides a full explanation for the observed precession without the need for an extra planet. Einstein himself thought that this result was the most critical test of his theory. Here's how it works. 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 a 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. As seen from Earth, the precession of Mercury's orbit is measured to be 0.56 arc seconds per orbit. An arc second is one-thirty-sixth-hundredths of a degree, taking into account all the perturbation effects from all the other planets, as well as a very slight deformation of the Sun due to its rotation, and the fact that the Earth is not an inertial frame of reference, Newton's equations predict a precession of 0.5557 arc seconds. That's 0.0043 arc seconds short. With Schwarzschild's metric, Einstein came out with 0.0043 due to the curvature of space near the Sun. This was the exact number to cover the difference. He had passed the first test of his new theory. It's the curved space around the Sun, defined by the Schwarzschild metric, that produces this small additional precession on each orbit. Here's what it looks like. 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 as we move on to the bending of light by the Sun. One of Newton's laws of motion states that an object in motion remains in motion at a constant speed in a straight line unless acted on by a force. In this view, gravity is a force that can act on light and divert it from its usual straight line motion. Einstein, on the other hand, had massive objects curving the space around them. An object in motion traveling through this curved space follows geodesics, the shortest path between two points, unless acted on by a force. It's important to remember that in general relativity, gravity is not a force. But light will bend. Both theories have light bending when traveling near a massive object. The larger the mass of the object, the larger the bending. And the closer to the center of the object, the larger the bending. But the two theories predict different amounts of bending for the same mass and distance measurements. Light passing near the surface of the Sun, Newton's theory predicts a deflection angle of 0.87 arc seconds. Einstein's theory predicts a deflection angle of 1.74 arc seconds. Twice Newton's prediction. Einstein pointed out that the best way to test his theory was to study apparent star locations during a total eclipse of the Sun. 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. 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. Here's an enhanced picture produced a hundred years later by the Heidelberg Digitized Astronomical Plates Project and released by the European Southern Observatory. They scanned one of Eddington's photographic glass plates and applied modern image processing techniques like noise reduction. This version identifies some of the stars used in Eddington's analysis. But the Sun's corona is strong. It interferes with all the measurements. It is estimated that errors as large as 20% are inherent in Eddington's and other visible starlight-bending experiments around the Sun. But other tests have produced much more accurate results. For example, the European Space Agency's Hipparchus Satellite, from 1989 to 93, designed to measure parallax distances to 100,000 stars, charted to positions of stars so accurately that no eclipse was needed to see the effect of the Sun's gravity. They produced numbers with only a 0.1% error. In 2003, using radio frequency light and measuring techniques that eliminated the error-producing impact to the Sun's corona, astronomers measured how much waves sent from the Earth to the Cassini satellite and back again were deflected by the Sun. Their error rates were around 0.03%. These and many other light-bending experiments have confirmed that Einstein's equations are correct. One of the key implications for bending of light is its impact on what's physically possible in heavily-cured space-time. 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. This is 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 spacetime 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. The third test of Einstein's relativity theory proposed by Einstein himself involved the shifting of light wavelengths to the red in the curved space of a gravitational field. To see how this works, we'll take a minute to review just what redshift is. Most people have had the experience of hearing a pitch of a car horn, train whistle, or ambulance siren drop off as the source moves past. As the sound source moves towards the observer, the sound waves are compressed, making the pitch of the sound higher. As the sound source moves away from the observer, the sound waves are stretched out, making the pitch of the sound lower. The same effect works for light. Here we have the visible spectrum from a star. Hydrogen in the star's atmosphere creates absorption lines with a unique pattern. Here's the pattern for a star at rest with respect to the observer. Light from an approaching star has its wavelengths shortened. We see that the lines shift to the blue. They are said to be blue-shifted. And light from a receding star has its wavelengths lengthened. We see the lines shift to the red. They are said to be red-shifted. The key to measuring the Doppler effect is to measure the change in position of the spectral lines. The further the shift, the faster the radial velocity. When the shift to the red is caused by gravity instead of receding velocity, the phenomenon is called gravitational redshift. Einstein developed the concept for this using the elevator thought experiment. Consider the elevator at rest with a light emitter fixed to the floor and a receiver fixed directly above it on the ceiling at a known distance. The emitter sends photons with a controlled wavelength to the receiver, where the arriving wavelength is measured. Here the measured wavelength of the light will be the same as the wavelength of the light emitted. Now put the elevator into a constant acceleration. Note that the receiver, at the time the light is observed, is further away from the point where the light was transmitted than it was in the static case. In other words, the receiver has acquired a velocity with respect to the light. And like the train whistle moving away, its wavelength is increased, shifted to the red. By the equivalence principle, the same result must hold in a gravitational field. But to calculate the effect as light moves away from a massive object, we need to take into account that the acceleration due to gravity is not constant. It decreases with distance as the light travels through the curved space around the object. The Schwarzschild metric that we used in the first two tests on Mercury's orbit and light bending around the Sun gives us the equation. We see that the amount of gravitational redshift for light from the surface of a massive object reaching a distant observer is proportional to the object's mass divided by its radius. Here's the gravitational redshift for the Earth and the Sun with triple the Earth's mass to radius ratio. These are very small, hard-to-measure shifts on the order of a tenth of a nanometer. Churning matter on the Sun's surface can have up to a thousand times the radial velocity equivalent to this redshift, making it impossible to measure gravitational redshift. Astronomers concluded that in order to measure this effect, they need a star with a calmer surface and larger mass to radius ratio. That would be a white dwarf. For that reason, they focused on the nearby Sirius binary star system with its giant star Sirius A and its orbiting white dwarf star Sirius B to test Einstein's theory. This binary system's orbital period is 50 years. In the 1920s, when the first measurements of Sirius B's gravitational redshift were made, the two stars were close together on the sky and the results were said to be contaminated by light from Sirius A. It wasn't until the 1960s that they were far enough apart to significantly reduce this contamination. At that time, astronomer Jesse Greenstein, working out of the Mount Wilson Observatory, measured the gravitational redshift effect to be 81 kilometers per second. Not far from the theoretical 81.3 kilometers per second, but the number of variables remained too large and difficulties separating out shift due to actual receding velocity made the results less than conclusive for testing Einstein's theory. But two physicists in a lab did prove Einstein correct. We'll cover their experiment in the next segment. 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 re-established. 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 redshift 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 redshift 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 redshift, 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. One of the most dramatic consequences of General Relativity is how gravitational redshift leads directly to the conclusion that a gravitational field slows time. We'll use the elevator thought experiment to illustrate how clocks closer to the source of gravity run slower than those further away. Picture a wave sent from the bottom to the top. Let the leading edge of the wave mark the start of a time interval. And let the trailing edge of the wave mark the end of the time interval. At the receiving end, the viewer sees that because the length of the wave has been stretched due to gravitational redshift, the length of time observed is slower than the viewer's clock. The lower clock's time is dilated. 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 the 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 whirl 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. In 1971, Joseph Halfill, a physicist, and Richard Keating, an astronomer, took four cesium atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another and their differences were consistent with gravitational time dilation. Today, we see this with our GPS 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 satellites' 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. You may recall from our segment on special relativity that the time dilation, due to velocity, creates a paradox. It goes like this. Suppose two 20-year-old twins start out together on the Earth. One of them gets into a spaceship or a trip to Vega, traveling at 99% of the speed of light. The person on the Earth sees the trip taking just over 25 years and the trip back taking the same amount of time. She is over 70 years old when a ship carrying her twin sister arrives back on Earth. But she also observes that her twin's clock ran a good deal slower than hers during the trip. Her twin is aging more slowly than she is. At 99% of the speed of light, time dilation would have the twin at just over 27 years old on her return. But from the point of view of the twin on the spaceship, she is motionless in her own reference frame. And the twin on the Earth is moving away and back. In addition, she sees the distance to Vega at only 3.5 light years due to space contraction. She also sees the twin on the ground aging slower than her over the seven-year journey. By her observations, her sister will be only one year older on her return due to time dilation. That's six years younger than she is, not 27 years older. How can it be that they are both older than the other? This is the paradox. But there is at least one point where the twin in the rocket is not in an inertial reference frame. As the spaceship approaches Vega, it decelerates to a stop and then reaccelerates back to Earth. The traveling twin finds that she is in a gravitational field and gravitational time dilation needs to be taken into account. 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 reaccelerate back to 99% of the speed of light. Ravitational 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. The general theory of relativity is now 100 years old. All the basic tests have shown it to be an accurate description of nature as we find it. The implications for astronomy have been enormous. In the next chapter, we'll cover gravitational lensing and how it enables us to see deeper into space than anyone ever thought possible.