Special Relativity

Previously, we briefly discussed General Relativity and how it expands Newtonian Mechanics to include the interaction between mass and space(time). Today, we are going to talk a little bit about Special Relativity in that what happens to physics when we approach velocities near the speed of light.

There are two postulates that Albert Einstein proposed make up Special Relativity.

1.      All Laws of Physics are invariant in all inertial systems - basically, the laws of physics must remain the same for all reference frames that are not accelerating.

2.      The speed of light in a vacuum is the same for all observers, regardless of the motion of the observers. In other words, the speed of light, c, is the same for a stationary person and a person moving at any speed, up to the speed of light (which is impossible).

One of the amazing things about Special Relativity is the idea of time dilation. It means that for any body travelling at any speed, time is not constant. To a stationary observer, it would appear that the clock for the moving object slows down, but for the observer in motion, the clock of the stationary person speeds up. In both cases, each observer sees the clock in his or her reference frame as moving normally. How does this work?

We have to use math to show this. There is an equation that describes how time is relative using the Lorentz transformation:

Where  is the time for the moving observer,  is the time for the stationary observer, and  is the Lorentz factor given by , where v is speed of the moving observer and c is the speed of light (3x10⁸ m/s). Since v is always less than the speed of light, is always greater than one, therefore Δt’ > Δt. For the stationary observer, the clock of the moving person is slow, while for the moving observer, the stationary person has a fast clock. For velocities much smaller than the speed of light, γ is virtually 1 and the times are equivalent. However, we have seen for objects moving near the speed of light (particles with little mass), this actually holds true. An example is a muon. A stationary muon would decay in only 2.2 μsec; however, when a muon is travelling near the speed of light will last much longer than 2.2 μsec (to a stationary observer).

Another strange feature of special relativity is the idea of length contraction. In other words, a moving object will appear shorter than it would be if it were stationary.

Where Δx’ is the length of the moving object and Δx is the length of the stationary object. This again leads to another strange phenomenon: relativity of simultaneity.

Relativity of simultaneity means that something that happens simultaneously in one inertial frame is not necessarily simultaneous in another. The best example given is called the ladder paradox.

Imagine a ladder and a barn. The ladder is just a little bit longer than the length of the barn. Now imagine that the ladder is moving at a relativistic speed. Someone in the barn reference frame will see the ladder shorter than its stationary length, and can close doors at both ends with ladder inside simultaneously. However, to keep the ladder from crashing through the end door, the doors must open up again. For the ladder, however, it sees the end door close first, then the front door close secondly, with the end door opening up before the ladder reaches it. The ladder does not see both doors closed at the same time.

A third phenomenon of special relativity is the idea of infinite mass. For a moving mass, Δm’=γΔm. As v approaches c, you can see that γ nears infinity (v/c approaches 1 and the denominator in the Lorentz factor approaches 0). This is why nothing can reach the speed of light as its mass will become infinite, which leads to another famous equation: E=mc², the energy-mass equivalency. This means in order to accelerate a mass to the speed of light, the amount of energy required goes to infinity.