# Special Relativity

By Katie Clark, Tori Dunston, Kelly Fan, Abrianna Macklin, and McKenna Vernon

Picture a hummingbird. At any moment, it can go in any of the three dimensions it is a part of. So, it could go up and down, forwards and backwards, or left and right. But, one thing that is not taken into account is time. As it moves through space, it is also occupying time. However, we’re not used to thinking about our world in a four dimensional sense. But, as the movement of the pigeon progresses, so does time. This is known as the relationship between space and time, and it is the primary foundation that special relativity is built on. So, at any given moment, it actually can move in four dimensions at once. This can be simply modeled using a spacetime diagram. A spacetime diagram models space on the x-axis, and time on the y-axis, which is now known as the t-axis. It is often modeled in light-units, which makes the slope of something moving at light speed equivalent to one. This makes it easier to model things moving at the speed of light. Spacetime diagrams are a good way to model waves of light or sound.

The frequency of a wave describes the number of oscillations of a wave that are emitted per second. Something that emits 367 oscillations per second will have a frequency of 367 Hz. Each oscillation has its own worldline. So, an example would be imagining an ambulance going at 30 m/s, who is releasing a siren that is moving at 300 m/s and operates at a frequency of 500 Hz. If you are 300 meters away from the ambulance, you would hear the siren one second later, at the frequency of 500 Hertz. However, when another siren was released a second later, you would only be 270 meters away, because you have to factor in the speed of the ambulance coming at you. So, you would hear the siren in less than one second, but the frequency would still be 500 Hertz. That would mean that you would perceive the frequency to be about 550 Hertz, which is higher than the actual frequency. This is known as a one-way Doppler shift. When you look at the light that is emitted instead of the sound, each Doppler shift is given a name. When something is approaching the observer at a fast enough speed, it is a blue shift, because they would perceive the frequency to be higher than it actually is. When something is moving away from the observer, it would be a red shift, because the frequency would seem slower.

In this first graph, Bob is stationary and Alice is moving towards him. Bob is emitting radiation in all directions. As soon as the radiation hits Alice it bounces back at the same speed. This can be represented by f(c+v). Now alice is moving away from Bob. this can be represented by f(c-v) In the next 2 graphs Alice is emitting the radiation. In this graph Alice is moving towards Bob. The first graph can be represented by fc/c-v. The second graph shows Alice moving away from Bob. This is represented by fc/c-v.  Another cool feature of special relativity is the phenomenon known as time dilation. The speed of light is seen as one thing that is constant for all observers. Since we know the ether is not real, the speed of light should move at the same speed for everyone, because it is not going through any other medium. However, that is not the case. When someone is moving at close to light speed, they experience time faster than any observer. In the example of the twin paradox, a twin leaves the Earth going at a speed close to light speed. When they return, they will have aged less than the other twin that is remaining on the Earth. This event is explained through time dilation, which is represented by a lambda. So, one second of the twin that is in space is equivalent to one lambda for the twin on the Earth. Depending on how fast the twin is flying, the time dilation factor will be more or less.

The most famous example of time dilation is Albert Einstein’s equation E=mc squared. Everyone is familiar with the equation but most of us on Earth have no idea what that means. E=mc squared basically means “energy equals mass times the speed of light squared.” To explain this saying a bit more, the equation says that energy and mass (matter) are interchangeable; they are different forms of the same thing. Under the right conditions energy can become mass, and vice versa. One example is that a beam of light can be the same thing as a walnut. To find out if that was true, you would multiply the mass of the walnut by the speed of light to determine how much energy is bound up inside it, because whenever you convert part of a walnut or any other piece of matter to pure energy, the resulting energy is by definition moving at the speed of light. Pure energy is electromagnetic radiation whether light or x-rays or anything really and electromagnetic radiation travels at a constant speed of 300,000 km/sec. So if you may ask “Why then do you have to square the speed of light?” Well it has to do with the nature of energy. When something is moving four times as fast as something else, it doesn’t have four times the energy but rather 16 times the energy in other words, that figure is squared. So the speed of light squared is the conversion factor that decides just how much energy lies within a walnut or any other chunk of matter. And because the speed of light squared is a huge number, 90 billion meters per second, squared, the amount of energy bound up into even the smallest mass is truly mind boggling.