 At the end of the second segment on general relativity, I pointed out that Einstein's prediction that accelerating masses can create gravitational waves had yet to be discovered. As soon as I released the book, they were discovered. So I have added this chapter to the video book to go into gravitational waves. What they are. What is a ripple in space-time? What creates them? What can create a gravitational wave large enough for us to detect? And how did we actually detect it? Now we used interferometers, a Michelson interferometer, to actually detect the first gravitational wave. I'll end by discussing the impact on cosmology that the discovery of gravitational waves means. But first, what is a ripple in space-time? Here, on Earth, far from an event that could create a gravitational wave, we have a relatively flat space with a Euclidean metric, G, that isn't changing with time. A ripple represents small deviations from this flat space-time metric. We use H to represent these deviations. Transactions to Einstein's equations show that a gravitational wave's metric oscillates sinusoidally, just like light. And it travels at the same speed as light. As the wave moves down the z-axis, planes, at different times, experience different values for the metric used to measure distance on the plane. This makes the wave a transverse wave, just like light. We see two possible polarizations for a gravitational wave. We call one H plus for the action along the x and y-axis. We call the other H cross for action along the diagonal. To see what an oscillating H plus metric does, we'll measure the changes in the distance between two points on the plane when a gravitational wave passes. Here we have an x-y plane with the wave passing into the page. We mark two points on the x-axis, one meter apart, in Euclidean flat space, where H is zero. When H is greater than zero, the distance between the two points on the x-axis become longer than one meter, by an amount equal to H times the original distance. At the same time, a one meter distance on the y-axis will shrink to less than one meter by the same amount. When H returns to zero, the distance between these points returns to one meter. When H is less than zero, the distance between the two points on the x-axis will become shorter than one meter, and the distance between the two points on the y-axis will become longer than one meter. Here's an exaggerated look at what an oscillating H plus polarized gravitational wave does to a square plate it passes through. Again the wave is passing into the page. For an H cross polarized wave, the effect would be similar, but shifted 45 degrees. In describing a gravitational wave, we can now be more precise than it's a ripple in space time. A gravitational wave is an oscillating polarized metric that operates in the plane perpendicular to the direction of the wave as it moves through space at the speed of light. And we have seen what this means for the objects that encounter such a wave. They are stretched and squeezed in various directions. We'll now turn our attention to the kinds of massive accelerating objects that can create such a wave. In order to generate a gravitational wave, you need a non-spherically symmetric rotating system. For example, here's a binary star system with two masses revolving in a circular orbit around a common center of gravity. The star's acceleration creates gravitational radiation that travels out from the system in all directions just like the light they are generating. The gravitational wave solutions show that the frequency of the created gravitational wave is twice the rotation rate of the binary system. We also see that the polarization and maximum gravitational wave amplitude depend on the masses of the two objects. The distance between them, their rotational velocity, the viewing angle, and how far away the system is from the observer. There is one more key factor to consider when it comes to binary systems, namely that the gravitational waves carry energy and momentum away from the system. We call this gravitational luminosity. Newton and Kepler provided the mechanics for understanding what happens to the orbit when gravitational energy is lost. Because binding energy is negative, a loss of energy will make it a larger negative. This has the effect of reducing the distance between the two objects. This in turn increases their velocity. A shorter circumference and faster velocity reduces the time it takes for a full orbit and therefore increases the frequency of rotation and therefore the frequency of the gravitational wave. And the wave equations show that the amplitude of the gravitational wave will increase with the frequency. The rate that the frequency is changing is called the chirp. It gives us the ability to express the amplitude of the gravitational wave in terms of the frequency and the rate the frequency is changing. Whether the masses and the distance between the masses. This is crucial because for most cases we will have no way of knowing directly what the masses are or how far apart they are. But measuring the frequencies might be possible. If we can also measure the amplitude, we can even calculate the distance to the binary system. Because this distance is based on gravitational wave luminosity, it is called the luminosity distance. For most all gravitational wave sources, this will be the only way to figure out how far away they are. With a decaying orbit, the objects will eventually collide and coalesce. The resulting wave form, called a coalescing wave form, serves as a signature for this kind of gravitational wave source. It has three phases, the in spiral, the merger and the ring down to an object that is no longer asymmetric and therefore no longer radiating gravitational waves. To get an idea on the expected amplitudes and frequencies for gravitational waves created by a system like this one, let's put in some numbers. Suppose this system is a hundred light years away and each star is the mass and size of our sun. At the point where they are about to touch, we would see the maximum amplitude. In this example, we get 10 to the minus 21. This is a very small number. It is approximately the ratio of the width of a human hair to the distance to Alpha Centauri four light years away. Here is where this data point fits on a graph with wavelength decreasing along the x-axis and amplitude increasing along the y-axis. Binary systems like this one are plentiful and all around us. There are literally billions of them sending gravitational waves our way from every direction. But the gravitational waves they create are weak and totally indistinguishable from one another. They just wind up contributing to a background noise level. In our sensitivity graph, we see that in order to detect a gravitational wave, a binary system will have to create waves with greater amplitudes and higher frequencies to generate smaller wavelengths than the noise level marked in green. To stand out, a binary system is needed that can achieve much higher velocities. And as we have seen from our example, the large diameters of stars prevents them from ever getting close enough to reach the needed velocities. But rotating neutron stars might be small enough to achieve the needed speed. Here's a system with two equal mass neutron stars that have reached the point where they are whirling around each other 10,000 times a second. The stars merge in a few milliseconds, sending out a burst of gravitational waves and a brief intense gamma ray burst. You can see the three phases, the in spiral, the coalescence or merger, and the ring down to an object most likely a curr black hole that is no longer asymmetric and therefore no longer radiating gravitational waves. If we fed the waveform into an audio generator, it would sound like this. We call it the chirp. The mass of a typical neutron star is 1.5 times the mass of the sun, with a radius of only 10 kilometers. If the system is 33,000 light years away, an average distance for a Milky Way object, it would give us a theoretically detectable wavelength and amplitude. But coalescing neutron stars are not common events. Scientists estimate that there might be one of these neutron star mergers every 50 years inside the Milky Way. To get a higher rate, we have to move outside the galaxy into the Virgo supercluster, our local supercluster that we covered and how far away is it video book. Within a 50 million light-year radius, we expect to have as many as 10 or more neutron star mergers per year, because we're including thousands of galaxies. Unfortunately, at this extended distance, the amplitude drops to the 10 to the minus 21 range. Stellar mass black holes can't get as close as neutron stars because their Schwarzschild radius is larger than the radius of a neutron star, but their mass alone can create larger gravitational wave amplitudes. Here we see a black hole merger simulation. If each black hole has a mass of 7 suns, the Schwarzschild radius is 20 kilometers, twice the radius of the neutron stars. As the orbital radius shrinks to twice the Schwarzschild radius and the black holes are approaching each other's photon sphere, their velocities approach 70% of the speed of light. This produces a shorter gravitational wavelength and a larger amplitude, putting this kind of event well into the theoretically detectable area. A number of other major cosmological events can also create gravitational waves. Here is a chart of some of the events and their expected wavelengths and amplitudes. Supernova, binary mergers like the ones we've been analyzing. Black hole mergers. And remnants from the Big Bang. In 1974, 58 years after Einstein predicted the existence of gravitational waves, two radio astronomers, Joe Taylor and Russell Holtz, were looking for new pulsars using the 305 meter Ericebo radio telescope in Puerto Rico. They found one. It's named PSR B1913 plus 16. And it led to the first indirect verification of Einstein's prediction. You'll recall from the globular clusters in supernova chapter, in the How Far Away is it video book, that a pulsar is a rapidly rotating neutron star with a powerful magnetic field. The result is a sort of magnetic lighthouse, which, if aligned correctly, flashes in our direction twice each cycle. These signals are highly regular, in fact, pulsars are some of the best clocks in nature. And this allows extremely precise measurements of their motion. This one was pulsing every 59 milliseconds, indicating that the pulsar rotates 17 times per second. But Holtz and Taylor noticed that the pulsars varied regularly every 7 and 3 quarters hours, with pulses arriving three seconds earlier at some times relative to others. This meant that the pulsar was in an elliptical orbit with another neutron star. This was the first binary neutron star ever discovered. In the orbital motion, they calculated the star's masses, their closest approach, called a peristron, and their furthest distance apart, called the apistron, as well as the system's inclination. With this information and the gravitational wave equations, they were able to calculate the amount of gravitational radiation, the expected decay of the orbit due to the lost gravitational energy, and the corresponding reduction in the time it takes per orbit. This graph maps the accumulated reduction in orbital periods against time, assuming that Einstein's equations are correct. Holtz, Taylor, and others have studied this binary system for 40 years now. This graph records their measurements. We see that the measurements fit the theory perfectly. This gave scientists confidence that Einstein's gravitational waves do indeed exist. But direct detection remained tricky for two main reasons. One is that the amplitude of the waves are so small, and the other is that the measuring sticks you might use to measure a change in length are changed themselves. In other words, the changed length will still read out as one meter. The stretching and squeezing does put a strain on the plate, and that can be measured with an attached wire that acts as a resistor. It's called a strain gauge. If we attach wires along the plate instead of a meter stick, we can measure changes in the resistance of the wire as it is stretched and squeezed. A longer, thinner wire will provide more resistance to an electric current, and a shorter, fatter wire will provide less resistance to an electric current, thus giving us a measure of the strain. Unfortunately, this technique is literally millions of times too insensitive to measure the tiny gravitational wave amplitudes, H. But this technique is why we call H a measure of strain. Michelson interferometers look like the best chance to detect these waves. You recall that we covered the interferometers in the first chapter of this video book. The arms on that one were 11 meters long, and its sensitivity was nowhere near what is needed for gravitational waves. Today we have LIGO, the Laser Interferometer Gravitational Wave Observatory that has built two identical interferometers, 3,000 kilometers apart, with one near Hartford, Washington, and the other near Livingston, Louisiana. Here are the L-shaped LIGO instrument components. It has a powerful near-infrared laser with an output after amplification that reaches 200 watts of 1064 nanometer light. The beam splitter and mirrors that act as test masses are 40 kilogram objects suspended via fused silica glass fibers to minimize noise due to vibrations. Additional internal and external active vibration minimization technologies eliminate the effects of everything from nearby traffic to lunar tidal forces. The four kilometer arms are 10,000 cubic centimeters of ultra-high vacuum, equal to one trillionth of an atmosphere. In addition, each arm contains reflective mirrors that route the light back and forth inside the arms 280 times before it hits the exits for recombination. The photodetector is a state-of-the-art indium gallium arsenide photodiode array with a high quantum efficiency designed to detect extremely small amounts of light at a wavelength of 1064 nanometers. The laser light is split and sent to the two mirrors. On return they are recombined and sent to the photodetector. The beams returning from the two arms are kept out of phase so that when the arms are both in sync, as when there is no gravitational wave passing through, their light waves subtract and no light arrives at the photodetector. When a gravitational wave passes through the interferometer, the distance along the arms of the interferometer are shortened and lengthened, causing the beams to become slightly out of sync, hence some light arrives at the photodetector indicating a signal. When LIGO's extra 280 passes through the tube, a gravitational wave strain amplitude of 10 to the minus 21 would displace the mirrors by 10 to the minus 18th meters. That's one thousandth the diameter of a proton. On our sensitivity graph we see where LIGO's characteristics fit. This is a range where powerful binary system mergers within the Virgo supercluster should be detectable. At 9, 50, and 45 seconds, coordinated universal time, on the 14th of September 2015, a signal was detected by the LIGO detector in Livingston and 6.9 milliseconds later in Hanford. It was a chirp signal that lasted just over two-tenths of a second. When we route the wave into a sound generator, here's what it sounds like. This plot combines the data from both sites. The waveform is consistent with coalescing masses, with a 10-cycle 200 millisecond in spiral that gives us the frequency, the rate of change of the frequency, and a peak wave amplitude. A merger that takes around two milliseconds, and a ring down as the coalesced objects cease to radiate gravitational energy. Detector noise introduces errors into all the calculations based on these figures. That's why we'll provide a range for each item. The amplitude and frequency data points give us the luminosity distance. It is important to note that gravitational waves experience red shifting as they travel across the cosmos, just like light does. Having traveled around a billion light-years, this wave would have experienced a red shift near 0.1. So the frequency we see here is a bit smaller than the frequency at the start of the wave's journey here. The frequency data also gives us the chirp mass. Taking the red shift information gleamed from the merger and ring down portions of the waveform, we get the binary system masses. These masses are too large for neutron stars that are only a few times the mass of the sun. So we must be witnessing the merger of two large stellar black holes. During the last 200 milliseconds of their in spiral, the orbiting velocity of the black holes increased from 30% the speed of light to 60% of the speed of light. Over the same period, the distance between the two black holes went from around a thousand kilometers to just under 200 kilometers when their event horizons made contact. Modeling the final ring down shows that the mass of the resulting curve black hole is around 62 solar masses. That's three solar masses less than the sum of the masses of the two inspiring black holes. This mass was converted to the radiated gravitational energy. In other words, during the final 20 milliseconds of the merger, the power of the radiated gravitational waves peaked at about 3.6 times 10 to the 49th watts. Let's take a second to get a feel for how large this number is. In our How Far Away Is It segment on nearby stars, we found that the sun converts 4.26 metric tons of matter into energy every second. The resulting power output is equal to 4 billion hydrogen bombs exploding every second. The sun is an average star, so we can use this as an average stellar power output. From our segment on local superclusters, we saw that there are 250,000 trillion stars within 1 billion light years. This represented around 7% of the total number of stars in the universe. We get the total power emitted by all the stars in the visible universe by multiplying the average watts per star times the number of stars. The power generated by this merger of the two stellar mass black holes is 26,000 times greater than the combined power of all the light radiated by all the stars in the universe. That's the signal we saw in September 2015, a billion years after it happened. The wave information does not tell us in which direction it came from because each interferometer is a whole sky monitor with very little directional information. But having two detectors does give us some directional data. For example, if the wave came in parallel to the line between the two sites, the signals would have registered at the exact same time. If the wave was perpendicular to the line, we would have seen a time delay of 10 milliseconds because the wave travels 3,002 kilometers through the Earth at the speed of light. What we detected was a wave that came in at an angle that caused a delay of 6.9 milliseconds. The dotted line represents the distance the wave had to travel for a piece of it to reach the Hanford interferometer. A little trigonometry gives us the angle. Of course this angle gives us a circle of possible directions. Interferometers are most sensitive to waves that come in perpendicular to their two arms. Sensitivity drops off as the incidence direction departs from the perpendicular. The curvature of the Earth gives the two LIGO interferometers an angle difference of around 27 degrees. This creates slight amplitude and phase inconsistencies across the two detectors that enable a narrowing of the probabilities to a smaller portion of the sky. Here are the most probable directions as seen from Earth. The best way to increase the accuracy of our directional findings is to use a third gravitational wave detector to triangulate the source. Several are under construction or being upgraded to do just that. Here is a map of current and future gravitational wave observatories on the Earth's surface. One of the greatest opportunities we have now that we can detect gravitational waves will be the ability to observe events that happened before light was traveling across the cosmos. The first 380,000 years after the Big Bang are known as the Dark Ages. As you can imagine, there is a lot of guesswork that goes into figuring out what happened during that period in the universe's history. Gravitational waves created within the Dark Ages may help us untangle that mystery. More time will tell.