Special Relativity

When Maxwell proved that light was actually electromagnetic waves, everyone assumed that they were waves that were traveling through some sort of medium. It was assumed that light was undulations in a medium like waves on the ocean. They called this medium "ether". In Ancient Greece, there was a theory that everything on Earth was composed of the four elements, but that celestial bodies were made of a fifth element called ether. Galileo was the first person to try to measure the speed of light. When Victorian scientists tried to measure the speed of light, they found to their astonishment that it was the same whether you were measuring in the same direction as the Earth's rotation or the opposite direction. Some people tried to explain this by saying that the Earth was enveloped in a shell of ether that was motionless with respect to the Earth. However, that complex theory contradicted how you observed astronomical phenomena. The great question plaguing everyone was, "Why is it that the speed of light appears to be the same for everyone?" Everyone wracked their brains trying to think of the answer to that question. Albert Einstein did not actually answer that question, but instead solved the puzzle in an ingenious way by simply saying, "Why not just say that the speed of light really is the same for everyone?"

In 1905, the 26 year old Albert Einstein put forward his special theory of relativity. At that time, Einstein was a Technical Expert (Third Class) in the Swiss Patent Office, working on physics in his spare time and in what has been termed "splendid isolation" from physicists in the academic community. Relativity has a reputation among those who have not studied it, as a difficult subject. It is not mathematical complexity that stands in the way of understanding. If you can solve a quadratic equation, you are overqualified. The difficulty lies entirely with the fact that relativity forces us to reexamine critically our ideas of space and time. Our life experiences are restricted in that we have no direct experiences with tangible objects moving faster than a tiny fraction of the speed of light. It is no wonder that our ideas of space and time, molded by this restricted experience, are also restricted.

Special Relativity rests on two basic postulates. The first postulate is that the laws of physics are the same for all observers in all inertial reference frames. No frame is singled out as preferred. The second postulate states that the speed of light in free space has the same value c in all directions, and in all reference frames. Galileo assumed that the laws of mechanics were the same in all inertial reference frames. Einstein extended this idea to include all the laws of physics, including electromagnetism and optics. This is not to say that the measured values of quantities are the same. The speed of light is the ultimate speed. The speed of light is the speed of a massless particle as measured from the reference frame of a particle with mass. The more energy you impart to a particle with mass, the closer it will go the speed of light, although it will never go that speed. The closer a particle is to massless, the closer its speed will be to the speed of light with a given amount of force imparted to it, although it will never go that speed. No particle with mass can go the speed of light. No massless particle can go any speed other than the speed of light. The speed of light is 2.99792458 x 108 m/s.

Some people seem to be under the impression that speeds faster than the speed of light somehow exist but are somehow forbidden to us. They fail to understand relativity. There is no such thing as speed faster than light, even one that's somehow forbidden to us. If you're capable of imagining such a thing as a speed faster than light, then you're just showing that you don't understand relativity. Unfortunately, many people seem capable of imagining this since the idea is rampant through science-fiction and UFO belief. The phrase "faster than light" is as meaningless as the phrase "south of Antarctica". To say that if you were sufficiently technologically advanced, you could go faster than light, is like saying if you were sufficiently technologically advanced, you could go south of Antarctica.

An event is something for which you can assign three space coordinates, x, y, and z, and one time coordinate t. Let’s imagine the following thought experiment as to how someone might measure these coordinates for an event. Let’s say the observer's coordinate system is fitted with a close packed three-dimensional array of measuring rods, one set of rods parallel to each of the three coordinate axes. Thus you need only read the three space coordinates at the location of the event. For the time coordinate, we imagine that every point of interaction of the array of measuring rods has a tiny clock. The observer reads the clock at the time and location of the event. The array of clocks must be synchronized properly. Here's one way you can do that. The observer has a helper at each clock. The observer stands at the origin, and sends out a light signal when the origin clock reads t = 0. When the light pulse reaches each helper, that helper sets his clock to read r/c, where r is the distance to the origin. All these procedures refer to a single observer in a single reference frame. All other observers would have to do the same thing.

Let's say one observer, Sam, notes that two independent events, say flashes of red and blue light, called event Red and event Blue, occur at the same time. Let's say there is another observer, Sally, who is moving at a constant velocity v with respect to Sam, also records these two events. Sally would record the two events to have not occurred at the same time. If two observers are in relative motion, they will not, in general, agree as to whether two events are simultaneous. If one observer finds them to be simultaneous, the other will not, and conversely. Simultaneity is not an absolute concept but a relative one, depending on the state of motion of the observer. Of course, if the relative speed of the observers is very much less than the speed of light, the measured departures from simultaneity become so small as to not be noticeable.

There is a famous thought experiment illustrating this using trains. Let's say Sam and Sally are each on a different train on parallel tracks. Sally's is moving forward while Sam's is standing still. Then a red light flashes in front of them, and a blue light flashes behind them. The light travels away from each light bulb in all directions at the speed of light. The light from both bulbs reaches Sam at the same time. The light from the red bulb reaches Sally before the light from the blue bulb reaches her. Sam would say that the two events were simultaneous while Sally would say they were not.

The relativity of simultaneity is closely related to the relativity of time. If two observers measure the time interval between a given pair of events, they will not in general agree as to how long that interval is. Let’s say Sally is on a moving train, she has a light bulb, a clock, and mirror attached to the ceiling. Sam is standing outside the train with his own clocks. Sally flashes the light bulb, and then uses the clock to measure the time it takes for the light to bounce off the mirror, and return to her. Event 1 is the turning on of the light bulb. Event 2 is the arrival of the light back at its source after reflection from the mirror on the ceiling. For the time interval between the two events, Sally finds that Δt = 2D/c, where D is the distance between the source and the mirror. For Sally, the two events occur at the same place, and she can measure the events with a single clock. A time interval measured by a single resting clock is called the proper time interval, Δt0. Let's say Sam is watching this, and measures the time interval himself. The speed of the light pulse is the same for Sam as it is for Sally, but it travels a larger distance for Sam. Imagine that for Sally the path of the light is a line segment, and for Sam, it's the sides of a triangle, as he watching the train pass by. For Sam, it travels a distance of 2L. The time interval measured by Sam between these two events is Δt = 2L/c where L = squareroot of((v/2)Δt)2 + D2 so therefore L = squareroot of((v/2)Δt)2 + ((c/2)Δt0)2). If we take these last two equations, eliminate L, and solve for Δt, we find that Δt = Δt0/squareroot of (1-(v/c)2). For Sam, the two events occurred at different places, so he had to use two synchronized clocks.

Δt = Δt0/squareroot of(1 - β2)

where β = v/c is the speed parameter. The time measured by two clocks at different locations is always greater than that measured by one clock at one location. This is called the time dilation effect. You can also write this in the form

Δt = γΔt

where γ is the Lorentz factor.

γ = 1/squareroot of (1-β2)

If you want to measure the length of a rod that is at rest with respect to you, you can, at your leisure, note the position of its end points on a long stationary scale, and subtract the two readings. If the rod is moving, however, you must note the positions of the end points simultaneously, in your reference frame, or your measurement isn't the length. Because simultaneity is relative, and it enters into length measurements, length is a relative quantity also. If the length of an object at rest in your reference frame is L0, called the proper length, the length you would measure if the rod is moving past you, parallel to itself, at speed v = βc is given by

L = L0 squareroot of (1-β2) = L0

This is called the length contraction effect.

Before relativity, the Galilean transformation equations converted coordinates from one reference frame to another. They are close to be accurate at low speeds.

x' = x – vt

t' = t

In relativity, the Lorentz transformation equations convert coordinates from one reference frame to another.

x' = γ(x - vt)

t' = γ(t - vx/(c2))

Dutch physicist H. A. Lorentz actually thought up these equations before Einstein. It's similar to how Wallace actually thought up evolution completely independently of Darwin, who is now viewed as the greatest biologist. Why was Lorentz's name not used in Broadway musicals? ("...Circe and her swine, your brain will never deflate the great Einstein...") When Lorentz came up with this, he did not fully believe that this was an actual description of the actual Universe.

You can use the Lorentz equations to compare the velocities that two observers in different inertial reference frames S and S' would measure for the same moving particle.

v = (v' + u)/(1 + uv'/c2)

where u is the velocity of S' with respect to S.

The relativistic definition of momentum is p = γmv. The relativistic definition of the total energy of a particle is E2 = (pc)2 + (mc2)2.

One of the most famous things in special relativity is the supposed twin paradox. Let's say you have two identical twins, and one of them moves away from the other at high speed. Each will appear younger to the other. If the traveling twin returns home, then how can they each appear younger to the other? The solution is this. If they don't reunite, but could see each other through a telescope, each would appear younger to the other. That might seem odd, but there's no paradox there. Up until this point, their experiences are the same. However, if the traveling twin were to turn around, and start coming back, their experiences would stop being the same. In order to turn around, the traveling twin would feel an acceleration that the other does not. Their situations would stop being symmetrical. From the point of view of the stay at home twin, the other would age much slower, and appear younger, and be that way when they return. From the point of view of the twin in the rocket, the other would appear to age slower on the two legs of the journey, leaving and returning, but while turning around, would age rapidly, which would more than counteract the two legs of the trip. Thus when they meet up again, the two twins will both agree that the twin that was on the rocket would appear younger. There are many different ways of explaining the so-called paradox. In all these versions, the acceleration appears to be key. However, some have suggested that that is not the crucial point. For instance, you could imagine the traveling twin traveling through a cylindrical universe, and thus returning without undergoing acceleration.