 Welcome to Relativity 12 – Gravitational Waves. The Laser Interferometer Gravitational Wave Observatory, or LIGO, has two stations, one in Hanford, Washington, and one in Livingston, Louisiana. On September 14, 2015, LIGO made the first reported observation of a gravitational wave. In this video, we'll see how general relativity predicts gravitational waves, what they are, and how they can be detected. First, to provide some context, we'll discuss Newtonian gravitational theory. In Newton's theory, two spherical masses, M1 and M2, separated by a distance r, feel a mutual attraction, which varies as the product of the masses and inversely as the square of the distance. This describes so-called action at a distance. The masses exert these forces without being in contact with each other, indeed without, apparently, being in contact with anything. So it's natural to ask, how is the force conveyed? What is the mechanism by which the masses pull on each other? Newton's answer was, I frame no hypothesis. It is enough that gravity does really exist and act according to the laws which we have explained. Newton is telling us to shut up and calculate. He's given us a theory that accurately predicts physical phenomena, but it doesn't give us an intuitive picture of what's going on behind the scenes, so to speak. Now, maybe it's because our bodies primarily interact with the world through mechanical means, but we tend to want a mechanistic explanation of phenomena. We want to see the ropes and pulleys responsible for making things behave the way they do. Let's look at one mechanical explanation for gravity that was put forward shortly after Newton published his theory. To start, we'll limit ourselves to a single dimension. Suppose a plate exists in space. And suppose a continuous stream of particles flows from left to right. When the particles encounter the plate, they're absorbed. The absorbed particles transfer their momentum to the plate, which produces a force pointing toward the right, analogous to wind pushing on a sail. Now, suppose particles flow in both directions. Also, assume the particles do not mutually interact, so the streams pass through each other. The plate will feel forces in both directions, which will cancel out, producing zero net force. Now, if two plates are present, the left plate will shield the right plate from the rightward stream, and the right plate will shield the left plate from the leftward stream. The result is that the left plate will feel a rightward force, and the right plate will feel a leftward force. If we are not aware of the presence of the particles, our interpretation might be that the plates are attracting each other, action at a distance. In fact, the plates interact through an intervening medium. Now consider three dimensions, with particles spread throughout space and flowing in all directions. If we focus on a particular object, the particles of interest are those which converge on the object. This view is a 2D cross-section. The converging particle circles correspond to converging spheres of particles. The object is equally pushed from all directions, so feels no net force. If a second object is present, it will absorb a portion of the converging particle stream, and the first object will feel a net force in the direction of the second. If the second object moves closer, it will absorb a larger portion of the particle stream, and the resulting net force on the first object will be greater. Working out the dependence of the force on the distance between the objects, you find precisely the inverse square relation of Newton's theory. So there is our mechanical explanation of gravity. Objects don't interact via action at a distance. They interact via a mechanical medium, and all action is local. We might call this mechanical medium the gravitational ether, except there are problems with this explanation. One was noted by Newton. The force of interaction is not a function of the body sizes, but of their masses. A small body of large density can exert a greater force than a large body of small density. Another problem is that if the gravitational particles transfer momentum, they should also transfer energy. As an object absorb particles, it would heat up. It's straightforward to show that Earth and everything on it should vaporize in a matter of seconds. We try to explain away these and other problems by assuming ever more fantastic properties for the particles. In the end, mechanical theories of gravity create more problems than they solve. Euler summed up the consensus among physicists. I shall always prefer to confess my ignorance of the cause of gravity than to have recourse to such strange hypotheses. Although no workable mechanical explanation of gravity has been found, abstract mathematical developments eventually modified the action and the distance concept. We can think of the two masses as interacting through a gravitational field. Newton's formula for the force exerted by mass 1 on mass 2 is minus g times the product of the masses over the square of the distance between them. g is the gravitational constant, and the minus sign denotes an attractive force, a force which tends to decrease the distance r. Let's write this expression as the product of two terms, g1 times m2. g1 is the gravitational field of mass m1. It has no dependence on mass m2. The distance r can refer to any point in space. Our picture is that mass m1 creates a gravitational field everywhere in space, which points toward the mass, and decreases inversely as the square of the distance. This field is not a mechanical thing we can view directly, but we can see its effects by placing a second mass anywhere in space and observing the resulting force. The physical predictions of this model are exactly the same as for action in a distance, so we might think of it as merely a mathematical tool. However, it does provide a framework in which masses do not interact directly at a distance, but rather locally through the gravitational field. We can add another level of abstraction by defining a gravitational potential. To motivate this, consider this figure where h is height or elevation above sea level and x is distance parallel to sea level, such as latitude or longitude. The gravitational field of Earth points downward with the same magnitude everywhere. Suppose we have a mass at sea level, and we move it to some other point on the ground. The theoretical work required is the mass m times the function u of x, which we call the gravitational potential. We want to know the force acting on the object parallel to the ground. If we move the object a distance delta r and record the change in potential, then the average force is minus m times the change in potential over the change in position. In the limit that the distance goes to zero and dividing by the mass, we get that the acceleration along the ground is minus the rate of change of the potential function with distance. If the ground is flat, the change in potential is zero and the force is zero. If the ground has a steep slope, the change in potential is large and the force is large. In any case, the force always points downhill. The potential concept can be applied to any gravitational field. A mass m1 creates a potential u1 throughout all space. We can abstractly visualize this as forming a type of landscape, such that other masses will accelerate in the quote downhill direction. The potential corresponding to Newton's law of gravity is minus g times mass over distance. The gravitational field is then the negative of the slope of this potential. And a mass m2 will fall, quote, downhill, which is toward mass m1. Let's visualize a star surrounded by spherical shells of constant gravitational potential, which we show here in cross-section as rings. Now let's ask, what happens to the surfaces of a constant potential if the star moves? Because potential depends only on instantaneous distance, the surfaces will move in lockstep with the star. We see that the gravitational potential concept hasn't completely replaced action at a distance. It's moved it from the interaction of two masses to the interaction of a mass with its gravitational potential. If all action is local, we might instead expect to see the moving star act on the first potential surface, followed by that surface acting on the second surface, and so on. The result would be a type of sloshing around of the surfaces of constant potential as the effects of the star's changing position propagated outward. In the 19th century, a complete theory of electromagnetism was developed. We show two views of the electric potential of a point charge as described by this theory. For a charge at rest, the electric potential falls off inversely with distance, analogous to gravity. When the charge oscillates, however, we observe the type of sloshing effect we've been talking about. But more than that, we see waves of potential peeling off and propagating away from the charge. These are electromagnetic waves.