 Since electromagnetic theory predicts vibrations in the field of electromagnetic potential, and, classically, gravitational interactions can be described by a gravitational potential, it's natural to ask if gravitational waves might be predicted by general relativity. And if so, what do they represent? And, how might they be observed? A challenge to be aware of from the outset is that gravity is very weak. Consider two electrons separated by some distance. Since they have mass, they exert a gravitational force on each other. The magnitude of that force is the gravitational constant G times the square of electron mass divided by the square of the distance. The electron mass is very small, as is the gravitational constant, so this force is small. How small? Electrons have electric charge, so they repel each other with an electrical force. That force is the charge squared divided by 4 pi times a small constant, epsilon 0, times the square of the distance. The electron charge is numerically much larger in magnitude than the electron mass, since both force expressions vary as the inverse square of distance. Their ratio is a dimensionless constant, which works out to 2.4 times 10 to the minus 43. So, the gravitational force is about 43 orders of magnitude smaller than the electrical force. The only reason we are aware of the gravitational force is that most matter is electrically neutral, with equal amounts of positive and negative charge. Repulsive and attractive electrical forces tend to cancel out, leaving zero net electrical force. Gravity, on the other hand, is always attractive and not canceled by a repulsive form of gravity. A classic demonstration of the weakness of gravity is shown here. You can use a comb to push around a piece of paper, but you can't use the gravitational attraction between the two to lift it from the table. Running the comb through your hair leaves it with a small static electric charge. When you bring it near to, but not touching, the paper, it picks the paper up. The piece of paper feels more force from a bit of static charge on the comb than it does from the gravitational attraction of the entire Earth. Electromagnetic radiation is produced by accelerating charges. The force on an electron of charged Q in an electric field E is Q times E. By Newton's second law, this equals the electron's mass times its acceleration. Solving for acceleration, we get A equals Q over M times the electric field. The ratio Q over M for an electron is numerically very large. So it's easy to accelerate an electron with a modest electric field. With little power, a radio transmitter can jiggle electrons around billions of times per second, producing a significant electric field. The force on a particle of mass M in a gravitational field G is M times G. Setting this equal to M times A and solving for A, we get simply A equals G. Instead of acceleration depending on a large charge to mass ratio, it depends on a mass to mass ratio, which is just one. It takes a lot of mass to generate a significant gravitational field, and it takes a very strong gravitational field to accelerate masses significantly. The prospect for detecting gravitational waves seems pretty dim, and we definitely aren't going to be building gravitational wave radios anytime soon. In spite of these sobering considerations, let's go ahead and see if general relativity allows for at least the theoretical possibility of gravitational waves. Let's recall the basic recipe of general relativity. We specify three coordinates of space. For our present purposes, we take these to be standard rectangular coordinates. And we have one coordinate of time. At every point of spacetime, there exists a field, which is the metric tensor. The variation of this field through spacetime determines the equations of motion. In that sense, the metric tensor is roughly analogous to the gravitational potential, although it has 16 components due to symmetry, only 10 are unique. If we have two spacetime events separated by infinitesimal displacements, dx1, dx2, dx3, and dx4, we can calculate ds squared equals the sum over i and j of gij times dxi times dxj. If this number is negative, then we define the negative of that, which will be positive, to be d sigma squared. If the first expression is positive, our interval is time-like, and ds is the proper time between the events. If the second expression is positive, our interval is space-like, and d sigma is the proper distance between the events. So if gravitational waves exist, they might manifest as fluctuations in the measured distance between points in space. The equations of general relativity are quite complicated, and it is difficult to use them directly to search for wave solutions. Shortly after publishing the general theory, Einstein presented a simplified approximate form of his equations, we'll call the linearized theory. This is the form he used to predict gravitational waves. We'll use Greek letter eta to denote the flat spacetime metric of special relativity. Calling our coordinates x, y, z, and t, the proper time and distance formulas are as shown here. The key assumption of the linearized theory is that the metric deviates only slightly from the flat spacetime form. We write Gij equals eta ij plus h ij, where the h components are very small. A product of two or more of the h components is negligible, and we drop these from our calculations. We'll now look for solutions where the h components are a small constant a times the sine of k times t minus z, all times an array of constant numbers, Bij. These constants determine which coordinates are affected by the wave. The sine factor corresponds to a wave traveling at the speed of light along the z-axis with a wavelength determined by the constant k. Let's represent the sine wave amplitude by the vertical position of a ball. You might imagine the ball floating on a body of water. The z-axis points from left to right. As time goes on, the ball, at z equals zero, oscillates. Placing balls at different z values, we see that they all oscillate in the same manner, but with varying time offsets. The result is a wave that travels left to right in the z direction. Now we modify our previous general relativity maxima code to test this form of solution. In the first line, we enter the flat spacetime metric. In the second line, we add a sine wave times an array of constants. These constants specify which components of the metric tensor will oscillate. Here we guess that only the first diagonal component does, and we get the full metric tensor shown in the third line. Einstein's equation of gravity in empty space is that the Ricci tensor vanishes. We developed the Ricci tensor in video seven. We impose a small deviation assumption by telling the computer to expand the Ricci tensor in a so-called Taylor series in the parameter a. The result is not vanish, so this is not a solution. Continuing to guess and test, we find that a one for the first and a minus one for the second diagonal component satisfies the equation and so is a solution. We also find that ones for the first two off diagonals gives a solution. Here's our first solution. The first array represents the flat spacetime background. The second array, with its sine wave amplitude, represents the gravitational wave. We want to find out what observable effects this wave might have. Here's our second solution. As we'll see later, this is identical to the first solution with a 45-degree rotation. We call these the two possible gravitational wave polarizations. One more point. The waveform does not have to be sinusoidal. With a bit more work, you can show that any wave of the form f of t minus z is a solution, or f is an arbitrary function. Let's work out the physical significance of our first solution using the proper distance formula and limiting consideration to the plane z equals zero. Let's consider points on the curve x squared plus y squared equals one. In flat spacetime, this is a circle of radius one, but in the presence of a gravitational wave, we can't assume the x and y coordinates have to quote Einstein an immediate metrical meaning. Let's label the proper distance from the origin to the point x equals one, y equals zero, as sigma x. And the proper distance from the origin to the point x equals zero, y equals one, as sigma y. For z and t constant, the proper distance formula becomes d sigma squared equals one minus a sine kt dx squared plus one plus a sine kt dy squared. The coefficients don't explicitly depend on x and y, so we don't have to limit ourselves to differential displacements, and we can drop the d's. Plugging in x equals one, y equals zero, we get that sigma x equals square root one minus a sine kt. Since a is small, an excellent approximation to the square root is one minus one half a sine kt. For x equals zero, y equals one, we get sigma y equals one plus one half a sine kt. In flat spacetime, the distance from the origin to x equals one, y equals zero, would be a constant value sigma x equals one. But these formulas tell us that in the presence of a gravitational wave, the distance would oscillate in time, likewise for sigma y. When sigma x decreases, sigma y increases and vice versa. Let's plot the curve x squared plus y squared equals one in terms of sigma x and sigma y. At t equals zero, sine kt equals zero, and both distances are one. The curve x squared plus y squared equals one is then a physical circle. As t increases, sine kt increases to one. At this point, sigma x equals one minus one half a, and sigma y equals one plus one half a. The curve x squared plus y squared equals one is now a physical ellipse elongated vertically. As t increases further, sine kt decreases back to zero. The sine continues decreasing until it reaches its minimum value of minus one. The curve x squared plus y squared equals one is now a physical ellipse elongated horizontally. At a later time, we again have sine kt equals zero, and we're back to a circle. The process repeats as long as the gravitational wave is present. What's strange is that for all times, the coordinate description of these changing shapes remains x squared plus y squared equals one. If the coordinate description of the curve doesn't change, but the shape of the curve oscillates, then the coordinate system itself must, in some sense, be oscillating. Indeed, the coordinates do not have an immediate metrical meaning. They simply provide an abstract address for points in spacetime. Could it be that this gravitational wave is just a mathematical artifact without physical significance? Maybe if we put particles at rest on the x squared plus y squared equals one curve, we'd find that as time goes by, they move off it in such a way that there's no oscillation of any physical structure. To answer this, we look at the equations of motion. Here, u one through four are coordinate velocities, and the four equations give the coordinate accelerations. If the first two velocities are zero, these equations tell us that none of the velocities ever change. So particles can indeed remain on the x squared plus y squared equals one curve indefinitely, even in the presence of a gravitational wave. Therefore, they must physically oscillate. Let's visualize this by imagining a circular arrangement of spherical dust clouds in the z equals zero plane. Position on the screen does not correspond to coordinate values, but to proper distance values. The spheres are not solid. They can deform arbitrarily. In the presence of a gravitational wave, the structure is squeezed and pulled alternately in the horizontal and vertical directions. To visualize the third spatial dimension, imagine we place copies of this ring structure at different values of the z coordinate, and we view the entire structure from on the z axis. We can now see the physical effects of a gravitational wave pulse propagating toward us. Here's the same scenario viewed from the side. The z coordinate increases from left to right. We've been visualizing our first solution. The second solution is identical, but rotated 45 degrees about the z axis. We can form superpositions of these two fundamental polarizations to get other solutions. If we change one of the signs to a cosine and add the two, we get a circularly polarized wave, which has the shape of a rotating ellipse. Viewed in three dimensions, the wave spirals through space. We can imagine how such a wave could arise from the orbit of two large masses about their common center of mass. Additionally, we see that the wave completes a full period of its oscillation when the objects go through only half of their orbit. The period of the gravitational radiation will therefore be half the orbital period. Equivalently, the gravitational radiation frequency will be twice the orbital frequency.