 With physical arguments convincing most physicists that gravitational waves must exist and carry energy, our attention turns to the question, can gravitational waves be observed? We consider a likely source of gravitational waves, two massive objects following circular orbits about their common center of mass. Einstein presented results for such a system in 1918. Assume the spherical objects have masses M1 and M2, and separation R. We'll work in standard units. If the orbits are well described by Newtonian mechanics, the orbital period is 2 pi times the square root of the distance cubed over the gravitational constant G times the sum of the masses. As we've discussed previously, the gravitational radiation period will be half the orbital period. The power carried away by gravitational radiation is a constant times the product of the masses squared times the sum of masses over the fifth power of the separation. The predicted gravitational wave amplitude is a constant times the product of the masses over the separation times the distance between the masses and the observer. Plugging in the mass of the Sun, the mass of the Earth, and the separation between them, we find 190 watts of gravitational radiation about the power exerted by a bicyclist. Absolutely negligible. For Jupiter, we find about 5 kilowatts, still negligible. Gravitational waves play no significant role in our solar system. From the power formula, we see that to get substantial gravitational waves, we need two large masses with a small separation, and hence a small orbital period. There's reason to be optimistic about finding such a system. The orbital energy is minus G times the product of masses over 2 times their separation. For this energy to decrease, the separation must decrease. As energy is radiated away in gravitational waves, orbital energy decreases, separation decreases, orbital period decreases, and radiated power increases. This causes an even greater decrease in orbital energy and so on. This should eventually lead to a runaway process with ever stronger and more rapidly oscillating gravitational waves being emitted until the two masses ultimately collide. If enough binary star systems exist, then at any given time there might be a reasonable chance of finding one in the final stages of this process. In 1974, Hulse and Taylor observed the first compelling evidence for gravitational waves. They discovered a new pulsar, a rapidly rotating neutron star, whose large magnetic field focuses a beam of electromagnetic radiation. The periodic sweeping of that beam past Earth generates a sequence of radio pulses. They noticed something unusual about this pulse sequence. The pulses occur, on average, about 17 times a second, but for a few hours they move slightly closer together, and then for a few hours they move slightly farther apart. Clearly the pulsar is periodically moving toward and away from Earth, indicating that it is orbiting as part of a binary system. Detailed observations and modeling have revealed that the orbital period is 7.75 hours, that the mean separation is about 2 billion meters, only about 3 times the radius of our Sun, and that both objects are neutron stars, with masses about 1.4 times the Sun's mass. Plugging these parameters into our formula for radiated gravitational wave power, we find a value of 6 times 10 to the 23 watts. Actually, the objects are in highly elliptical and not circular orbits. Correcting for this gives an estimated power about 10 times larger. This is on the order of 2% of the electromagnetic power radiated by our Sun, a significant amount that should have an observable effect on the orbital period. In this figure, the blue curve shows the cumulative shift in the orbital period due to gravitational wave radiation, as predicted by general relativity. The red dots are observed values. The agreement between theory and observation is excellent. Although not a direct observation of gravitational waves, this is very strong circumstantial evidence for their existence. For their work, Holson-Taylor were awarded the 1993 Nobel Prize in Physics. The Holst-Taylor binary is about 21,000 light-years away. The amplitude of its gravitational waves at Earth are predicted to be around 10 to the minus 22 or so. Given this strong circumstantial evidence, we now consider the possibility of observing gravitational waves directly, which leads to the questions, what exactly would we observe, and how would we make the observation? We've been visualizing a gravitational wave as causing a ring of freely falling dust clouds to periodically compress and stretch along horizontal and vertical axes. Let's focus on the horizontal axis and the left and right most dust clouds. If we're floating in space and observe two objects accelerating in opposite directions, we infer that they are subject to oppositely directed forces. Because the accelerations are independent of mass, we conclude that these are gravitational forces. In this scenario, a gravitational wave is simply a gravitational force that varies through space and oscillates in time. The response of a physical system to a gravitational wave can be calculated by simply adding this force to the other forces acting on the system. Let's draw an analogy by noting that objects on Earth are, of course, acted on by Earth's gravitational field, but they are also acted on by gravitational forces from the Moon, as well as from the Sun. The gravitational pull of the Moon is directed toward the Moon's center and is stronger at points on Earth that are closer to the Moon. Earth as a whole falls toward the Moon with an acceleration equal to the Moon's gravitational field at Earth's center. The difference of this force at a point on Earth's surface and center is an extra-gravitational force that objects at that point will feel. To the extent that Earth's crust is rigid, it will experience this force as a stress but will not deform. At the same time, the non-rigid ocean will deform and in doing so will move relative to the crust. Ocean tides are the result. Appropriately, we call these tidal forces. More generally, the term can be applied to the spatially varying component of any gravitational force. Therefore, one way to think of gravitational waves is as tidal forces propagating through space. Objects in freefall will accelerate in response to these forces. While solid objects will experience stresses, but to the extent they are rigid, will not deform. So, in principle, we could observe gravitational wave tidal forces by their contribution to Earth's tides. To get a sense for how much they might contribute, consider that Earth is about 13,000 kilometers across and imagine our dust cloud array is this size. A gravitational wave with amplitude 10 to the minus 21 would cause a deformation of about 1.3 times 10 to the minus 14 meters. This is a tiny fraction of the size of an atom. About the width of seven protons laid side to side. Completely negligible. And yet, gravitational waves have been observed by LIGO, the Laser Interferometer Gravitational Wave Observatory. Using instruments that are only four kilometers in length. To do this, required the construction of the most sensitive measurement instrument ever built. Let's examine just how sensitive LIGO is by considering two freely falling test masses, four kilometers apart. In the presence of a gravitational wave, the distance between the masses oscillates with a period TGW, the period of the gravitational wave, and with an amplitude equal to the amplitude of the gravitational wave, H0, times the unperturbed distance L. The rate of change of distance is relative velocity and the rate of change of that is relative acceleration. We interpret this as a difference of gravitational field between the masses. We find the amplitude of this difference is H0L, times the square of 2 pi over TGW. Let's consider a gravitational wave with an amplitude of 10 to the minus 21, the amplitude of the first gravitational wave detected by LIGO, and a period of four hours, roughly that of the waves emitted by the Holst-Taylor binary. Plugging in those values, we find a maximum difference in gravitational field for the two masses of 7.62 times 10 to the minus 25 meters per second squared. Earth's surface gravity, 1G, is 9.8 meters per second squared. So the effect of the Holst-Taylor binary gravitational wave is less than a tenth of a trillionth of a trillionth of a G. To appreciate how small this is, assume we place a mass M four kilometers from one of the test masses and eight kilometers from the other. We can use Newton's law of gravity to calculate the difference of this mass's gravitational field at the two test masses. Equating this to the value above, we find a mass of 2.44 times 10 to the minus 7 kilograms, less than a milligram, about one-tenth the mass of a mosquito. In other words, if it flew in a circle around LIGO every eight hours, the effects of a mosquito's gravitational field on the LIGO test masses would overwhelm the effects of the Holst-Taylor binary's gravitational wave. A key to LIGO's success is that it looks for gravitational waves with a much shorter period on the order of one-hundredth of a second. A gravitational wave with the same amplitude but this much shorter period produces a much larger effect, although still less than a trillionth of a G. The equivalent gravitational field corresponds to a mass of about five-hundred tons. That's about the mass of a fully loaded Airbus A380 jumbo jet. The gravitational waves detected by LIGO produce roughly the same effect as the gravitational field of an A380 flying around LIGO in a six kilometer circle. So, how do we know what LIGO saw wasn't just a five-hundred ton terrestrial mass moving around a circle? Because for the period to be one-hundredth of a second, the mass would have to be going around the circle about fifty times per second. The required velocity is thousands of times the speed of sound, almost one percent the speed of light. Five-hundred ton terrestrial objects don't do that. Here's a figure showing the noise level in the LIGO detector, the vertical axis, as a function of gravitational wave frequency, the horizontal axis. The black curve is total noise, while various colored curves show the contributions from different phenomena. The solid green curve, labeled gravity gradients, is the effect of gravitational variations due to terrestrial objects. As the frequency decreases, this rises steeply. On Earth, it gets increasingly difficult to distinguish longer period gravitational waves from gravitational fluctuations due to Earth-based objects. What is sometimes referred to as Newtonian noise. But for gravitational waves of short enough period, LIGO's signal is clear and unambiguous. And, of course, the fact that the same signal is detected by two instruments separated by three thousand kilometers is further evident of its extra terrestrial origin.