Albert Einstein revolutionized the world of physics when he discovered the special theory of relativity. It is only possible due to his contributions that valuable technologies like GPS function. But what is it exactly?

Dilation and Contraction

The theory behind special relativity is two-fold:

1. All inertial frames of reference of equivalent.

2. Observers must measure the same value for the speed of light in a vacuum.

This means that if even an observer has a huge difference in relative velocity, the speed of light is the same. This causes knock-on effects on the experience of time, and time dilation:

$$ Δt = \frac{Δt_0}{\sqrt{1-\frac{v^2}{c^2}}} $$

The proper time \(Δt_0\) is the time interval with respect to an observer at rest. The dilated time is the time when the observer is at velocity \(v\).

In the same way that time can be dilated, length can also be contracted:

$$ L = L_0\sqrt{1-\frac{v^2}{c^2}} $$

The relationship here is inverse! Time can only be dilated, and length can only be contracted. If the velocity was the speed of light (impossible) then the time interval would be infinite, and the length of an object would be 0.

Relativistic Addition

Two objects traveling at different relativistic velocities further complicate these calculations. If we want to calculate the proper velocity \(u\) of an object from its relativistic velocity \(u'\) relative to an object at proper velocity \(v\) or vice versa, then we can use these equations:

$$ u = \frac{u' + v}{1 + \frac{vu'}{c^2}} $$

$$ u' = \frac{u - v}{1 - \frac{vu}{c^2}} $$

As you can see, these are very similar equations, but their differences are important.

Mass and Energy

It's the moment you've been waiting for, the famous equation!

$$ E = \frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}} $$

Oh, it looks a little bit different. That's because just like length and time are relativistic, mass is as well. In order to calculate total energy from rest mass, we must apply the same expression to it. There's another way to calculate \(E\):

$$ E = KE + E_0 $$

where \(E_0\) comes from the classic \(E = mc^2\) equation. We can also find the kinetic energy by rearranging these equations as

$$ KE = (m - m_0)c^2 $$

from finding the difference between relativistic mass and proper mass.

One more important attribute affected by relativity is momentum. As it depends on mass, it looks similar to the equation for \(E\):

$$ p = \frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}} $$