 From what we've done so far, it seems like there should be no doubt that gravitational waves exist, at least in principle. And of course, by now, they've actually been observed. But this question, do gravitational waves exist, was vigorously debated for decades. Let's just consider Einstein's opinion on the matter. At one point in 1916, his answer was no. In a letter to Carl Schwarzschild, he wrote, Later in 1916, his answer was yes. He'd found wave solutions for his linear theory, and then developed a formula for their energy. However, after publication, an error was found in his derivation. In 1918, he published a paper correcting that error. He established that three types of gravitational waves exist, one being the type we've solved for previously. But in 1922, Arthur Eddington showed that the other two types are non-physical. They represent an abstract oscillation of a coordinate system, and they don't correspond to anything physically observable. By 1936, Einstein had changed his mind again. In a letter to Max Born, he wrote, Einstein and Rosen submitted their findings to the journal Physical Review. It was returned with a request to address reviewer comments. Einstein, who wasn't happy about having his results questioned, withdrew the paper and submitted it to another journal. Later in 1936, Einstein changed his answer to maybe. According to Leopold Infeld, Einstein told him, I found a mistake in my paper, and he told the lecture audience, if you ask me whether there are gravitational waves or not, I must answer, I do not know. Still later in 1936, he changed his answer back to, yes. He corrected the mistake he had found, and revised his and Rosen's paper to reflect this new conclusion. Rosen, who was out of the country at the time, did not agree with Einstein's revision, and as late as the 1970s, continued to argue that gravitational waves do not physically exist. Now, no one doubted that there were wave-like mathematical solutions of the equations of general relativity. The controversy was whether or not these describe physically observable phenomena. A bit of the flavor of this controversy can be gleaned from Eddington's 1922 paper. He wrote, I think that Einstein really left the question of the speed of propagation rather indefinite. The propagation of the absolute physical condition, the altered curvature of spacetime, has not hitherto been discussed. After referencing the three types of waves Einstein had found, he continued, waves of the first and second types have no fixed velocity. A result which rouses suspicion as to their objective existence. And he noted that Einstein had also become suspicious of these waves for another reason, because he found that they convey no energy. So, there are two problems in trying to answer the gravitational wave question. First, in general relativity, coordinate systems don't have immediate metrical meaning. A wave oscillation might represent an actual physical phenomenon, or it might be a mathematical artifact due to the coordinate system. It can be difficult to distinguish between these cases, which was one point of Eddington's paper. Second, general relativity provides no unambiguous way to define the energy density of a gravitational field. If a wave carries no energy, it can have no observable effects, and it does not exist in any physical sense. Here's one simple physical argument against the existence of gravitational waves. Suppose a building on Earth has two bowling balls resting on its roof. The bowling balls feel their own weight due to Earth's gravitational field. Do the bowling balls emit gravitational waves? Probably not. By analogy with electromagnetic, we might expect that only accelerating masses would emit waves. On the other hand, according to relativity, a gravitational field is equivalent to an accelerating reference frame. But it seems ridiculous to assume that objects at rest on Earth's surface would radiate gravitational waves. Now suppose one of the balls falls off the roof. It will accelerate toward the ground. Are gravitational waves produced in this case? There is an accelerating mass in this scenario, so we might be tempted to answer yes, the falling ball radiates gravitational waves. But consider a reference frame that falls with the ball. In that frame, the ball is not accelerating. It's floating weightlessly. In fact, the ball in the roof is accelerating upward. So, a freefall observer might conclude that the falling ball emits no gravitational waves, while the ball in the roof does. These inconsistent conclusions seem to cast doubt on the possibility of gravitational wave radiation. In general relativity, the question, what and where is gravitational energy, is rather messy. Let's first consider the question, what and where is mass energy? Suppose we have a particle, which is at rest in a green-eyed observer's reference frame. She can enclose it within an imaginary box of dimensions delta x, delta y, and delta z. She can calculate the volume of the box and the energy of the mass. We'll use standard units, so E equals mc squared, where m is the rest mass of the particle. Now she can calculate the energy density inside the box, as the energy divided by the volume. Suppose a brown-eyed observer also views the particle. And in the green coordinates, he's traveling to the left with velocity v. So, in his coordinates, the particle is moving to the right with velocity v. We'll put a bar over values measured in the brown coordinates. According to relativity, he will see the x dimension of the box contracted by a factor of square root 1 minus v over c squared. See video 5b in this series for a discussion of length contraction. Therefore, the brown-eyed observer will measure a smaller volume. He will also measure a larger energy, because in his reference frame, the particle also has kinetic energy. Now, if he calculates the energy density inside the box, he will get a larger value than the green-eyed observer. We've gotten used to relativity predicting different measurements of time, length, and energy for different observers. It's no problem. The relationships between the measurements are well defined, and any observer can apply a coordinate transformation to predict another observer's results. However, if the energy is zero in the green frame, it has to be zero in the brown frame, and conversely. In other words, if there is no particle in the box in one frame, there cannot be a particle in the box in another frame. Simply changing coordinates can't cause a physical object to pop in or out of existence. We might say that matter energy can be localized. In general relativity, there's a similar result for the energy of electromagnetic fields. Electromagnetic energy can be localized. If one observer measures non-zero electromagnetic energy at some point of spacetime, then other observers will also. If one observer measures zero, then all other observers must also. However, this is not true for the energy of the gravitational field. To see why, imagine that in our reference frame, there is a particle initially at rest with mass M1 and charge Q1. We measure its acceleration, A1. And we write Newton's law of motion for the particle. Force equals mass times acceleration. There are two possible sources of force. Gravitational force, mass M1 times the gravitational field G, and electric force, charge Q1 times the electric field E. Dividing through by M1, we find that the particle's acceleration is the gravitational field plus the particle's charge-to-mass ratio times the electric field. Based on this alone, we cannot separate out the gravity and electric field contributions. Suppose a second particle initially at rest at essentially the same place has mass M2 and zero charge. And we measure its acceleration, A2. Since it has no charge, the force on it is due to gravity alone. So, its acceleration is simply the gravitational field G. Taking the difference A1 minus A2, the common G terms cancel, leaving only the electric force on particle one. This gives us an unambiguous procedure for determining the electric field. There is no analogous procedure for the gravitational field, however. Now it's true that the acceleration of particle two equals the gravitational field. But imagine a second reference frame initially at rest with respect to the first frame, that falls freely in the gravitational field. The measured particle accelerations in the second frame will be those measured in the first frame, lest the second frame's own acceleration G. The result is that the particle accelerations in the freefall frame have no gravitational component. In a freefall frame, there is no gravity. However, there is still an electric field. At a single point in space, gravity cannot produce relative acceleration between two particles, an electric field can. The result is that if I claim that at a certain point in space, at a certain time, the gravitational field has a certain value, and therefore there is a certain density of gravitational energy according to some formula I've developed, you can always dispute this by referring to the freefall frame in which there is no gravity. Any definition of gravitational energy is necessarily dependent on the choice of coordinates. Here's another way to look at the problem. Suppose I'm inside a small box floating in deep space. I claim that an electromagnetic wave passes through my box. Let's assume this wave consists of two pulses. During the first pulse, the electric field points upward, and during the second it points downward. I can demonstrate the existence of this wave by extracting energy from it. I place two equal masses on either end of a spring. The masses have opposite charges, and the positive charge is on top. Before the wave arrives, the spring is relaxed. When the upward pointing electric field is present, the positive charge is pulled upward, and the negative charge is pulled downward. The spring will stretch until these electrical forces are offset. Energy has been extracted from the electric field and stored in the compressed spring. If I try to do the same thing with a gravitational wave, I'm going to use the same energy as I do with the propeller. The positive charge is pushed upward, and the negative charge is pushed upward. The spring will compress until these electrical forces are offset. Energy has been extracted from the electric field and stored in the compressed spring. form. If I try to do the same thing with a gravitational wave, I am bound to fail. With masses on the spring ends, the upward-pointing gravitational field will cause the entire mass spring system to fall upward as a whole. The spring will be neither stretched nor compressed. Likewise, the downward-pointing gravitational field will cause the entire system to fall downward as a whole, with no change in the spring length. At no point has energy been transferred to the spring. Without being able to extract energy from the gravitational field, in what sense am I justified in claiming a gravitational wave exists? Now, if we observe masses at separate points in space, in a non-uniform gravitational field, it is possible to see relative acceleration. If these objects were connected by a spring, that spring would be stretched. Energy would be transferred from the gravitational field to the spring. And, this relative acceleration cannot be transformed away by a change of coordinates. As we discussed in video 4, this relative acceleration is what leads us to say that space-time is curved by massive objects. This curvature is, in Eddington's words, an absolute physical condition. It can be observed and quantified by any observer in any coordinate system. Indeed, this is what is described by the Riemann tensor, which we introduced in video 7. The challenge of quantifying local gravitational energy is summed up in the classic text Gravitation by Misner Thorn and Wheeler as follows. Regarding local electromagnetic energy, and the same can be said of local mass energy, there is one and only one formula for this quantity. It has weight. It curves space. It is observable. Regarding local gravitational energy, there is no unique formula for it. It has no weight. It does not curve space. It is not observable. At issue is not the existence of gravitational energy, but the localizability of gravitational energy. It is not localizable. Enormous time and effort were devoted in the past to trying to answer this question, how to describe gravitational energy density, before investigators realized the futility of the enterprise. And so, due to such mathematical and conceptual subtleties, the existence of gravitational waves remained controversial for decades. In the 1950s, some physicists began to consider the problem from a less abstract and more physical point of view. A 1956 paper by Pirani, titled on the physical significance of the Riemann tensor, was an important step in this direction. He wrote, A difficulty in general relativity theory is the lack of what might be called the theory of measurement. One does not learn systematically how to choose the appropriate coordinate system in which to calculate this or that quantity to be compared with observation. The result is fruitless controversy. In this vein, at a 1957 conference on gravitation, Richard Feynman presented his so-called sticky beat argument. Consider a gravitational wave of the type we've been visualizing throughout this video. Each of the dust clouds is freely falling in the gravitational field. Consider the left and right most dust balls and assume they are instead solid spherical beads. The beads will not deform significantly, but will still experience relative acceleration. Assume holes are drilled in the beads and they slide along a rod with some stickiness or friction. An object moving against friction generates heat, which is a form of energy. The rod plus bead system is therefore extracting energy from the gravitational wave. Therefore, the gravitational wave must carry energy. This simple thought experiment leapfrogged over abstract mathematical arguments and eventually played an important role in convincing most physicists that gravitational waves are real physical phenomena. Feynman later wrote, I was surprised to find that a whole day of the conference was spent on this issue and that experts were confused. That's what happens when one is considering energy conservation tensors, etc. Instead of questioning, can waves do work?