Physics with Calculus/Modern/Special Relativity

Special Relativity
One of Einstein's most famous theories was the theory of special relativity. Relativity explains the dynamics of systems in motion relative to one other.

One popular example is that of a person on a train watching a stationary person on the ground. When you view the problem in one light, you may think that the person on the train is moving (this is from the stationary persons vantage point). Now, think of the problem from the viewpoint of the person on the moving train--he thinks that the person on the ground is moving. Ultimately, you may choose to view the problem in either way--there is no right or wrong answer. Everything just depends on your viewpoint. All of special relativity can be derived from two principles : 1) Physics is the same in any inertial reference frame. If you're riding in your car on the highway (with the windows up) and you drop a ball it falls in the same way as if you were stopped at a stop sign and dropped the same ball.  2) Nothing can travel faster than the speed of light (which should really be called the speed of a massless particle). This speed is approximately $$3x10^8 m/s$$

Strictly speaking, information cannot travel faster than the speed of light. It can be shown that if a signal travels faster than light, in some reference frame, the information is received before it is sent, which is nonsense. However, there is no restriction on other things. For example, if I had a projector very far away from a screen, and I wave my hand in front of it very quickly, the shadow will travel faster than light. Also, the phase velocity of light waves in some circumstances will be greater than the speed of light. However, it is impossible to transmit any signal faster than light with these objects.

We could derive all of relativity from the two postulates, but that would not be very enlightening, because it is a rather technical proof. Instead, consider that Alice is in a frame moving at velocity v with respect to Bob. Let's call Alice's frame A', and Bob's frame A. It follows that

$$x' = \frac{x - vt}{\sqrt{1 - v^2/c^2}}$$ $$y' = y$$ $$z' = z$$ $$t' = \frac{t - vx/c^2}{\sqrt{1 - v^2/c^2}}$$.

Compare this to the Galilean transform,

$$x' = x - vt$$ $$y' = y$$ $$z' = z$$ $$t' = t$$.

The startling thing about the first set of equations, called the Lorentz transform, is that if substituted into Maxwell's equations, they leave them unchanged! If you try to substitute the second set of equations in, you get different equations. Anywhere there is electricity and magnetism, the Galilean transform must be wrong.

It is possible to derive all the results in relativity from the Lorentz transforms, much like all of non-relativistic mechanics comes from the Gallilean transform and Newton's laws.

Let's look at some properties of the Lorentz transform. First, the inverse of the transform (solving for the unprimed components in terms of the primed ones) gives exactly the same thing except with v replaced with -v, as we would expect.

If there is a burst of light, defined by $$ x = c t$$ it remains unchanged.

Four Vectors
Note that

$$x' = \frac{x - (v/c) ct}{\sqrt{1 - v^2/c^2}}$$ $$y' = y$$ $$z' = z$$ $$ct' = \frac{ct - (v/c)x}{\sqrt{1 - v^2/c^2}}$$.

With new notation,

$$x' = \gamma (x - \beta ct)$$ $$y' = y$$ $$z' = z$$ $$ct' = \gamma (ct - \beta x)$$.

Define a four-vector to be a 4-tuple with components such that when transformed to a new frame with characteristic $$\beta$$, then the components transform as above. For example, (x,y,z,ct) is a four vector. If $$ x = (a,b,c,d)$$ is a four vector, then when changing frames (when $$\beta$$ is in the x direction), we have $$ x' = ( \gamma (a - \beta d), b, c, \gamma(d - \beta a))$$.