Special relativity

[zaza]

In response to the likes of Code Doc and friends, who have expressed curiosity about the nature of special relativity, I will attempt to explain it here.

How do we know if two events have happened at the same time? In principle, if a measurement is taken exactly midway between the two events, and the signal from those events arrives at the same instant, we can say that they occurred at the same time.

Consider a station with a long platform. At each end of the platform, there is a firework. Alice stands exactly halfway down the platform. At the chosen moment, the fireworks are detonated, and Alice sees the flashes from each exploding firework at precisely the same instant, confirming that the two events were simultaneous.

As the fireworks went off, a long train was rushing through the station, and the explosions left burn marks on the sides of the train. It just so happened that Bob was standing on the train exactly halfway in between the two burn marks. As he looked out of the window, however, what did he see? It takes a finite amount of time for each flash from the fireworks to reach him, and consequently during that time, he has moved position. Thus he sees the flash from the firework at the front of the train BEFORE he sees the one from the back of the train. According to Bob, the two events were not simultaneous. He knows that they were both the same distance away from him, because he can tell how far he is away from the burn marks. But he saw the flashes at different times.

Since the speed of an object is the distance it travels divided by the time it took to do so, and the separations will be measured as the same by Alice and by Bob, there are only two options.

1) The times are the same, but the speeds are different: The speed of the light coming from the firework at the front of the train is different to the speed of that coming from the back; it is more of an “approach speed” in which the velocity of the train accounts for the difference between the two. This implies that there is some “absolute frame of reference”, and in order to determine the true measurement, you need to account for the relative motion of yourself to this “absolute frame”. The question, then, is “Is Alice standing in the absolute reference frame?” Should all measurements in the Universe be taken in comparison to a girl standing at a station, possibly somewhere in Utah? Even Americans aren’t THAT self-centred. Mostly.

2) The speeds are the same, but the time intervals are different: In other words, the notion of measurement is different depending on your frame of reference.



Much of the misunderstanding around special relativity comes from the confusion of these two points. However, the notion of an approach velocity does not apply to light. It has the same velocity regardless of the frame of reference, and it is not correct to think of it in the same way that you think of moving cars, for example. I suspect that some of this confusion arises because attempts to explain it, such as mine above, necessarily place it in the context of real-world objects. Talking about trains and fireworks encourages thinking of light as “just another object”, but it can’t be treated in the same way.

The reasons for discounting explanation 1 come down to Einstein’s First Postulate of special relativity, namely “All physical laws have the same form in all inertial frames of reference”. The alternative is that physical laws can only be composed in ONE frame of reference, and corrections and modifications need to be made in order to apply them to any other.

Of primary concern is Maxwell’s wave equation, which requires that an electromagnetic wave travel at the speed of light, and which was Einstein’s original consideration. If the speed of light is NOT the same in all frames of reference, then by riding along next to a light beam you will observe a stationary sinusoidal variation in space of the electric and magnetic components of the wave. However, this is not an acceptable mathematical solution of the Wave Equation, implying that therefore this is not the correct equation to describe wave mechanics in this frame of reference. In order to preserve the laws of physics, special relativity must hold.

In a sense, this is a slightly circular argument, because the end result is that the laws of physics had to be rewritten anyway. Distance and time are no longer really distance and time, they have to be converted using what are known as Lorentz transformations. These basically scale time and space by a factor which is dependent on v/c, ie the ratio of your speed to the speed of light. In day-to-day life, v <<c, so it makes very little difference to our measurements of distance and time. However, this factor becomes more and more important as v gets larger.
The actual conversion factor is called gamma, and is as follows:

gamma = 1 / sqrt(1 – v²/c²)

When v -> c, gamma -> 1/0 = infinity.

Predictions such as length contraction and time dilation result directly from this. Relativistic predictions have been directly verified, including the well-known one in 1971 where 4 atomic clocks were flown around the Earth in both directions and compared to identical clocks kept on the ground.

In summary, special relativity accurately describes the laws of physics of the Universe in frames of reference moving at constant velocity.
To describe accelerating reference frames, Einstein needed to go further, and developed the outstanding General Theory of Relativity. But that's not for discussion here .