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A visualization of special relativity

By Kadi Runnels
This video is an entry for the Breakthrough Junior Challenge 2015 which gives a unique visualization of Special Relativity using hyperbolic geometry. This idea was inspired by the famous woodcut by M.C. Escher Circle Limit III. 

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Video transcript

The theory of Special Relativity, which is commonly associated with Albert Einstein, has proven to be one of the most successful theories in all of physics, surviving any test we can think of to try to prove it wrong. However, it has a reputation for only being possible to learn through headache inducing thought experiments where the student has to keep up with pulses of light, multiple clocks, and moving observers all while trying to adapt to a new concept of space and time. While all of this is fascinating, and I really encourage you to look into the thought-experiment development of special relativity, my goal in this video is to present you with a more direct intuition for how special relativity works without using as much math. Let’s start with explaining Galilean Relativity, where the outcome of any experiment will be the same regardless of the frame of reference—that is to say, if all relevant parts of the experiment are given some boost with the same speed and direction, the experiment will still come out the same. The concept of a boost will be important later on. For this discussion, think of a boost as a push into a new frame of reference. Imaging throwing a ball straight up in your driveway. It will fall straight down. Now imagine that you and the ball are in a car that is moving at a constant 60 mph west. From the frame of reference of the ground, if you throw the ball up, it will move upward, but also westward at 60 mph to form an arc. If I asked what the ball looks like to you, you would quickly tell me that it looks like it is moving straight up and straight down, just like in your driveway. While this seems obvious, it is only because of symmetry that we can make this assumption. The idea behind Galilean Relativity is that you can skip the step where you consider the frame of reference from the ground because of the symmetry between all frames of reference. In fact, the ground is just the Earth’s surface, which is rotating at some thousand miles per hour relative to the axis. But even the axis orbits the Sun, which in turn orbits the Galaxy. It therefore follows that all frames of reference are equally valid, or in other words, there is no way to determine if you are moving or at rest in any absolute sense, only the relative motion between different objects is physically meaningful. This is called the principle of relativity. Okay. Now we need to switch gears for a minute to talk a little bit about electromagnetism and waves. Electromagnetism is the theory describing the one unified interaction between charges, currents, magnets and light. In the nineteenth century, it was realized that the equations for electromagnetism had a class of solutions that corresponded to physical waves in the electromagnetic field. It did not take long for experiments to produce these “electromagnetic waves” in the laboratory, and confirm that light is just a small subset of these waves with a particular range of frequencies. The electromagnetic field was thought to exist in the æther, with its own state of motion, through which electromagnetic waves of light travelled at a relative speed of 300 million meters per second. This, however, poses a problem for Galilean Relativity. To see why, we are going to use an analogy—water waves. Imagine you are floating in a lake, and you drop a baseball in the water. The disturbanc e created in the height and motion of the water will propagate outward as an expanding ring. This ring will let us determine if we are rest with respect to the water. As long as you are in this lake, you can determine if you are at rest by dropping a baseball in the water and checking to see if you are remain in the center of the expanding ring-shaped wave around you. If you are moving, you will be able to see this as the center of the expanding wave moves away from you–you don’t stay in the center of the expanding ring. Giving such a test to determine if you are moving or at rest may seem to contradict the principle of relativity, but remember, you are only measuring your motion relative to the water and its associated frame of reference. So, according to the æther idea, we should be able to do the same thing with an electromagnetic or light wave as we did in the lake. Which would mean that we should be able to see our movement in the æther by measuring lightwaves. Physicists Albert Michelson and Edward Morley tried an experiment to measure this. What they found is that it does not appear that there is any motion at all through the æther, even at times of year when the earth is moving at a velocity up to 60 km/sec different. This experiment showed that the æther theory was wrong, and that the electromagnetic field did not behave like some cosmic lake with a fixed background frame. Instead, the electromagnetic field treats all frames of reference equally, which means that the speed of light is 300 million meters per second in every frame of reference. This is called the principle of constancy of the speed of light. But wait! How can this be consistent with Galilean Relativity? Galilean Relativity says that if a Galileo Galilei is moving toward you at 1 million meters per second and he shines a light at you that leaves him at 300 million meters per second, then the light will arrive to you moving at 301 million meters per second, breaking the principle of constancy of the speed of light. This inconsistency led theoretical physicists Henri Poincaré, Albert Einstein, and Hendrik Lorentz, as well as mathematician Hermann Minkowski to develop a new theory of space and time. Somehow it had to be true that there is no preferred frame of reference and the speed of light is the same in all frames of reference. Instead of following the original thinking of Einstein or Lorentz, I am going to use the fact that everyone thinks they are in the center of the universe. No, really — all observers see themselves as staying in the center of a pulse of light emitted from them, regardless of their state of motion. The resolution found by the fathers of Special Relativity required replacing Galilean Relativity. So we need symmetry that respects the principle of relativity and also the principle of constancy of the speed of light as well. Enter Lorentzian Relativity which was discovered by Lorentz and Einstein. A visualization of the principle of Relativity will require a geometry in which every point and every direction is essentially the same. There is a theorem in mathematics that there are exactly three such geometries. To begin with, let’s look at a geometry that does not satisfy this property. On the cube, not any two points are the same type. For example, while these two points are geometrically equivalent, these two are not. The geometry that does fit our criteria is the plane. Since the plane is infinite any point could call itself the center. The next one is the sphere. Since the sphere is perfectly round, any point could call itself the center or the pole. In both of these geometries each point is basically the same. The third and final geometry is one that is less familiar, since there is no perfect way to represent it in three dimensional space. It is called the hyperbolic plane, and it is what we will use to understand relativity. In our visualization, the regular plane will represent Galilean relativity and the hyperbolic plain will represent Lorentzian relativity. So here we go… Imagine a car is moving at some speed through a large parking lot. According to Newtonian physics, the car can have any speed in any direction. The speed of the car can be represented as a point on this grid of tiles. To move from one tile to an adjacent one would require a standard boost, say 10 mph. Remember that a boost is just a push into a new frame of reference. Each point on this circle represents a velocity of 10 miles per hour in a different direction, and each point on this circle represents a velocity of 20 miles per hour. This grid represents Galilean relativity. Notice that no particular tile is special: we can easily shift the grid around and put any tile in the center. Now I am going to state something that will become clear later: nothing can move faster than the speed of light. Therefore the velocities that the car can move at are represented inside of this circle. For example, this point represents half the speed of light north. If you imagine this circle is the size of a dinner plate, any speeds that would practically be obtained by a vehicle on earth stay in a tiny circle only about a thousand atoms in diameter. So, the new grid of tiles looks like this. The most important thing to understand is that the tiles become more and more compressed near the boundaries. Each tile is still the same amount of boost, but they shrink in size as you approach the edge. This explains why you can never get to the speed of light. To increase your speed, you need to undergo a certain boost, moving to the next tile. But as you get near the speed of light, each tile takes you less and less distance with the same boost. So while the speed of light is only a finite number of “meters per second” away, it is still an infinite number of tiles that you must cross to get there. So you can see when you compare our two grids that in the first one (which represents Galilean relativity), the velocities are additive — each tile adds the same amount of boost or change in velocity regardless of which frame of reference you are observing it from. In the second grid (which represents Lorentzian relativity), each tile adds the same amount of boost, but it looks completely different depending on where you are viewing it from. If you viewing a change that is far away, the boost or tile appears very small, but if you are in the center or close by, the tile appears very large. With a bit of visualization, you can imagine that the grid could be shifted so that this arc moves to this arc, with all of these tiles getting compressed over here, and these tiles expanding to fill up all of this space. So, you can see that any tile could reasonably call itself the center tile, and so it follows, each tile sees the speed of light as a circle around it. Now maybe you can see why, with Lorentzian relativity, each observer sees light emitted from them moving out in a perfect sphere regardless of their state of motion which in our visualization is represented by which tile they are on which. One last thing to note is that Galilean Relativity is an approximation to Lorentzian relativity as long as the speeds involved are small. For example, the 1000 atom diameter circe representing typical speeds of human vehicles is hardly affected by the distorting near the speed of light. The theory of special relativity found by Einstein gave rise to a whole new picture of space and time, which became incorporated into some of the most profound and successful theories in all of physics. I wish I could go on to talk about these ideas of spacetime and mass-energy, but unfortunately I am out of time. I hope I’ve given you a taste of Special Relativity, and why you can never reach the speed of light without using all of this. I want to encourage anyone interested to find material on the topic of Special Relativity and to look at the woodcut Circle Limit III by M.C. Escher which inspired this video. My next video will be on Schrödinger.