# Special relativity

Special relativity, a theory developed by Albert Einstein in 1905, reconciles relative motion with the unchanging speed of light. It postulates that

- all laws of physics are valid in all inertial frames of reference;
- the speed of light is constant with respect to all inertial frames of reference.

Suppose you are sitting on a chair in a room that has no windows. Which way is the earth moving? Is it even moving at all? Because of the first postulate, you *cannot tell*—all laws of physics are valid, so there is no observation you could make that would change if you were in a different inertial frame of reference. This is called *equivalence*.

Consider a train moving at half the speed of light. In *frame A*, we have an observer inside the train. In *frame B*, we have an observer outside the train, watching it zoom by. Observer A switches on a flashlight:

According to equivalence, observer A neither knows nor cares whether the train is moving. He switches on the flashlight and observes a ray of light travelling $\displaystyle 2 \Delta{} d_{\text{A}}$ in some amount of time that we’ll call $\displaystyle t_{0}$. Since the speed of light is constant in all inertial frames, we are confident that

$\displaystyle v = \frac{\Delta{} d}{\Delta{} t} \qquad \implies \qquad c = \frac{2 \Delta{} d_{\text{A}}}{t_{0}}$.

So far so good. But what about observer B? She notices that the ray of light is actually moving horizontally as well as vertically. She records the distance $\displaystyle 2 \Delta{} d_{\text{B}}$ and she uses a stopwatch to measure the time it takes for the ray to travel that distance, which we’ll call $\displaystyle t '$. Now she tries calculating the speed of light using the same method as observer A:

$\displaystyle c = \frac{2 \Delta{} d_{\text{B}}}{t '}$.

Wait a minute. It’s obvious that
$\displaystyle t_{0} = t '$,
so observer B must get a different answer than observer A! That’s impossible, because the second postulate of special relativity tells us that the speed of light is constant with respect to *all* frames of reference. If we were talking about a ball, this would all be fine, but we aren’t—this is light, so the observed speed must be *c* for both of them. If *c* remains constant, and
$\displaystyle 2 \Delta{} d_{\text{A}} \ne 2 \Delta{} d_{\text{B}}$,
then how can those equations give the same answer?

*Time*. It’s counterintuitive, but
$\displaystyle t_{0} \ne t '$.
The ray of light takes *different amounts of time* to make its trip depending on the frame of reference. Who’s right, observer A or observer B? Both are—for their respective frames. Their is no universal frame of reference, and there is no absolute universal time either. Time is dependent on the frame of reference and spatial position. It therefore does not make sense to say that two events happen at the same time in different parts of space. The simultaneity of two events depends on the observer’s frame of reference.

One consequence of special relativity is *time dilation*. A clock running on a spaceship moving near the speed of light will run slower than an identical clock on Earth. A person in that spaceship will age slower than an identical twin on Earth. For the observer in the spaceship, everything is completely normal—it is the people on Earth that have *fast* clocks.

If an observer B travels at a speed *v* for the proper time
$\displaystyle t_{0}$ and
returns to observer A, who has been at rest all along, then observer A will have experienced a longer amount of time, the relativistic time
$\displaystyle t '$, given by

$\displaystyle t ' = \frac{t_{0}}{\sqrt{1 - \left ( \frac{v}{c} \right )^{2}}}$.

Notice that the denominator rounds to 1.0 unless the speed is very large. This is why we don’t notice special relativity in our everyday lives.

Another consequence of special relativity is *length contraction*. As an object approaches the speed of light, it will seem to get shorter. We can get the relativistic length
$\displaystyle L '$ from the proper length
$\displaystyle L_{0}$
with

$\displaystyle L ' = L_{0} \sqrt{1 - \left ( \frac{v}{c} \right )^{2}}$.

Yet another bizarre consequence of special relativity: faster objects act as if they have more mass. The space shuttle might have a *rest mass* of two million kilograms, but its actual mast approaches infinity as the space shuttle approaches the speed of light. We can calculate this mass with

$\displaystyle m ' = \frac{m_{0}}{\sqrt{1 - \left ( \frac{v}{c} \right )^{2}}}$.

Einstein also applied his theory of relativity to *matter*. In chemical reactions, mass is conserved because the atoms are just being shuffled around. In nuclear reactions, mass changes because elements change into different elements. This is accounted for by a change in energy according to the mass–energy equivalence equation,

$\displaystyle E = m c^{2}$.