 Welcome to video 9 of this series on the theory of relativity. Here we're going to examine a brief 1905 paper by Einstein having a title in the form of a question. Does the inertia of a body depend upon its energy content? Einstein presented a thought experiment that argued the answer is yes. Mass is a measure of the energy content of a body, expressed in arguably the most famous equation of physics, E equals MC squared. Here are the key points of his argument. One, energy is conserved. Two, a moving body has kinetic energy K that depends on its mass M and velocity V. From there follows three. If kinetic energy changes and velocity does not change, then mass must change. Four, the energy of a light pulse, a photon, is proportional to its frequency. And five, a light pulse has different frequencies, hence different energies for different moving observers, the so-called Doppler effect. The idea of kinetic energy is illustrated by our experience of lifting an object above the ground and letting it fall. An object on the ground has weight due to its mass M and Earth's gravity. To lift at a height H, we need to do work, that is, exert some energy W, which is the object's weight, MG, times the height H. Letting it drop, we find that when it returns to the ground, it has a speed U, such that the kinetic energy of one half M times U squared is equal to the energy we exerted to lift the object. And in general, kinetic energy should depend only on the mass and velocity and should be proportional to mass. Because, for example, lifting two identical objects requires twice the work and will produce, hence, twice the kinetic energy. A pulse of light clearly carries energy as evidenced by solar cells. In its 1905 special relativity paper, Einstein showed that the energy of a light pulse depends on its frequency. Indeed, this is a foundational principle of quantum mechanics where we say that a photon carries energy E, which is a constant H, times its frequency F. Now we consider the Doppler shift phenomenon. We start with the classical version. A particle at rest in a particular frame emits light waves of a given frequency. Two observers are on either side of the particle in a frame that moves to the right with velocity V. In a given time interval, the approaching observer passes through more oscillations of the wave than the receding observer. More oscillations per time means a higher frequency. More specifically, the particle has mass M and emits two photons of frequency F0, one to the left and one to the right. A frame that moves to the right with velocity V carries two observers. As throughout this series, we use units in which the speed of light is one light second per second, so V represents the velocity as a fraction of the speed of light. The observers are positioned so that one approaches the source while the other recedes. The classical Doppler effect predicts that the approaching observer will see the frequency increased by a factor of 1 plus V, while the receding observer will see it decreased by a factor of 1 minus V. Accordingly, the observers will see the photon energies increased or decreased by the same factors. Now let E represent the energy of the particle in the rest frame and H the energy of the particle in the moving frame. The subscript 0 will denote energies before the photons are emitted. The difference of the moving and rest frame energies is just the kinetic energy of the particle due to its velocity V relative to the observers. The subscript 1 will denote energies after the photons are emitted. Again, the difference is the kinetic energy of the particle. Now, the difference of the before and after energies in the rest frame is simply the energy carried away by the two photons, 2HF0. Likewise, in the moving frame, there we have HF0 times 1 minus V plus 1 plus V. The plus and minus V's cancel and the resulting total photon energy is the same as in the rest frame, 2HF0. The difference in the before and after kinetic energies is the difference in photon energies in the two frames, which is 0. So there's no change in kinetic energy. Since there's no change in velocity, there is therefore no change in mass. That's the classical result. Now we'll look at the prediction of relativity. As in videos 1 and 2, we'll use Latin letters for the rest frame and Greek letters for the moving frame. The particle is at x equals 0 and the observers are at xi equals plus and minus 1. The picture relativity paints in the rest frame looks identical to the classical case. The subtle difference is that as we've seen, the observers will have a different perception of time. Following the graphical approach we developed in use in videos 2 and 5, we draw a spacetime diagram. The particle is at x equals 0, the thick horizontal blue line. The receding observer is at xi equals 1, the top thick red line. And the approaching observer is at xi equals minus 1, the bottom thick red line. The tilt of the red lines is determined by the velocity v. In both systems, light moves with speed 1. Light emitted at t equals 0 travels towards the receding observer along the green line x equals t and towards the approaching observer along the green line x equals minus t. Light emitted one second later travels along x equals plus or minus t minus 1. Using these facts and the special relativity relations between the Latin and Greek coordinates, we can solve for the time interval between light signals perceived by the receding observer, call this delta tau 1, and likewise for the approaching observer, delta tau 2. We find that the frequencies, which are 1 over these time intervals, differ from the classical predictions. For the receding observer, 1 minus v is replaced by the square root of 1 minus v over 1 plus v. For the approaching observer, 1 plus v is replaced by the square root of 1 plus v over 1 minus v. Here we see the Doppler shifts predicted by classical theory and relativity. They agree well for low velocities, but at high velocities, relativity predicts more frequency increase for the approaching observer, the top curves, and less frequency decrease for the receding observer, the bottom curves. The change in light energy is due to the sum of the two Doppler shifts. In the classical theory, the sum is zero, but relativity predicts a positive sum that increases with velocity. That is, relativity predicts the moving observers will measure more emitted energy. More emitted energy leaves less energy behind in the form of kinetic energy. Since the velocity v doesn't change, that must mean the mass of the particle decreases. Let's see the details. Here are the Doppler shifts, the relativity values on the left and the classical values on the right. The squiggly lines indicate that they agree for small velocities. If we again look at the energy differences, the before and after energy difference in the moving frame is now hf0 times the sum of the square root factors. The Doppler shifts no longer cancel out, leaving a factor of 2. Instead, we find a factor of 2 divided by the square root of 1 minus v squared. That is, 2 times the dilation factor beta, the same factor responsible for the twin paradox. Relativity predicts the moving frame. We'll see the emitted light energy as a factor of beta larger than the rest frame season. The difference in the before and after kinetic energies will no longer be zero, but instead the expression shown here. Again, a change in kinetic energy with no change in velocity requires a change in mass. Now, notice the 2hf0 factor. That's simply the energy lost by the mass in its rest frame. We'll call that delta e. The decrease in kinetic energy is therefore delta e times the expression in parentheses. Now, Einstein points out that the expression reduces to 1 half v squared for small velocities, where we know classical physics works well. Here's a graph that shows the excellent agreement for small velocities. To make the connection with classical kinetic energy, we write the velocity v is the ratio of the velocity in meters per second. u to the speed of light in meters per second c. A classical kinetic energy is 1 half mass in kilograms times the velocity in meters per second squared. As we've argued, the change in kinetic energy must be due to a change in mass. So we equate 1 half delta e u over c squared to 1 half delta mass m times u squared. Canceling common factors and rearranging. We arrive at change in energy equals change in mass times the speed of light squared. Einstein's conclusion is the mass of a body is a measure of its energy content. If the energy changes by delta e, the mass changes in the same sense by delta e over the speed of light squared. Now, I think it's really important at this point to step back and remember what it is that makes a good scientific theory. Here's consummate physicist Richard Feynman describing the scientific method. In general, we look for new law by the following process. First, we guess it. Then we comp- Well, don't laugh. That's the really true. Then we compute the consequences of the guess to see what if this is right, if this law that we guessed is right, we see what it would imply. And then we compare those computation results to nature. Or we say compare to experiment or experience. Compare it directly with observations to see if it if it works. If it disagrees with experiment, it's wrong. In that simple statement is the key to science. It doesn't make a difference how beautiful your guess is. It doesn't make a difference how smart you are who made the guess or what his name is. If it disagrees with experiment, it's wrong. That's all there is to it. Yep. Everything we've described so far is really just a guess. An educated guess, but a guess, nonetheless. It's not a valid theory just because some guy named Einstein proposed it. I suppose that's why he titled the paper with a question. As he himself concludes, it is not impossible that with bodies whose energy content is variable to a high degree, the theory may be successfully put to the test. If the theory corresponds to the facts, radiation conveys a inertia between the emitting and absorbing bodies. Of course, one of the reasons Einstein is so famous is that he had a knack for making good guesses about radical ideas. Sure enough, 40 years later, delta E equals delta MC squared was demonstrated quite dramatically. A somewhat less intense test presented on the Centennial of Einstein's paper verified his prediction to a high level of precision. Now, how far can you take the change of mass into energy? Can you change all the mass into energy? Well, we now know the answer is yes, you can, as in the case of matter, antimatter, annihilation. So we are justified in simply writing E equals MC squared. Energy and mass are equivalent. The speed of light squared is huge, so we find that one kilogram of mass is equivalent to a huge amount of energy, about two and a half billion dollars worth if you could convert it to electrical power. If you use units in which the speed of light is one, as we generally do in this series, you get the very elegant result that energy simply equals mass. Back to our expression for the change in kinetic energy. This expression strongly suggests that the relativity formula for kinetic energy should be MC squared, the intrinsic energy of a mass m, times one over the square root of one minus V squared minus one, and that's just MC squared times the dilation factor beta minus one. And notice if V equals one, that is the speed of light, the square root is zero and one over zero is infinity. If you plot the relation between kinetic energy and velocity, you get the blue curve shown here. Newtonian theory predicts the red line. The two theories agree for small velocities, but Einstein predicts that as the kinetic energy of an object increases towards infinity, the speed approaches but never reaches the speed of light. No matter how much energy you exert trying, you will never get an object to reach the speed of light, let alone exceed it. The speed of light is an absolute speed limit. This is another radical prediction of relativity. Is it true? Well, a classic test was performed by Bertosi in 1964. He fired electrons at an aluminum plate. Measuring the time they took to travel a given distance, he determined the velocity, and measuring the temperature increase of the plate, he determined their kinetic energy. When he plotted his results, they agreed quite well with Einstein's predictions to within his experimental uncertainty. Indeed, every particle accelerator that has ever operated has provided overwhelming confirmation of Einstein's incredible predictions regarding the equivalence of mass and energy.