• What is Special Relativity?
  • The Fundamental Postulates
  • Notation
    • Standard Configuration
    • Four vectors
    • Lorentz Transformation
  • Time Contraction
  • Relativistic Mechanics
    • Example: Compton Spread

When we read What is LHC:ATLAS?, I mentioned that since the speed of the particles approaches the speed of light, we have to account for the relativistic effects.

"However, we do have Galilean relativity which has worked perfectly fine for me" you might inject, and you'd be right. Galilean relativity serves us just well in most of our lives. In fact, for speeds far below the speed of light Einstein relativity and Galilean relativity are nearly indistinguishable from each other. ^[1]

However, in certain circumstances such as when you're using a particle accelerator, or even something as mundane as using GPS, Galilean relativity is not enough, and you have to use Einstein's relativity.

"Okay, so I need this Einstein's relativity. What is it?" you ask begrudgingly.

Einstein's special relativity (refered to just as special relativity from here)^[2] can be derived entirely from two fundamental postulates.

The Fundamental Postulates

1. All intertial reference frames are equivalent.

This version of the postulate, however, follows a somewhat circular argument as an object which does not accelerate does not have a net force, however, the only way we know that it doesn't have a net force is by checking if the object accelerates.

This inconsistency lead Einstein to develop his general theory of relativity which solves the problem by working for any reference system, and not just inertial systems. However, if you ignore this detail, you'll be fine.

2. The speed of light in vacuum, $c$, is a universal constant in all reference frames.

While today, we know this to be true as we have measured it empirically,^[3][4] Einstein couldn't possibly have known this at his time, you rebut. And you'd be right. In fact, it's possible to derive that given that light is a electromagnetic wave that light must travel at $c$ from Maxwell's equations. However, Einstein describes in his notes from 1949 himself finding it as a solution to an apparent paradox he stumbled upon when he was 16. This approach leads to the stronger claim that $c$ is constant in all reference frames.

Notation

Standard Configuration

The standard configuration is a configuration of coordinate systems often used in special relativity to ease, and somewhat normalize the notation.

The standard configuration consists of an rest system $S$ with coordinates $(x,y)$^[5], and another system $S'$ with coordinates $(x',y')$ moving with $v$ along the $x$-axis with respect to $S$.^[6] Furthermore, at $t=0$, we set $x=x'=0$.

This setup is illustrated below.

Four vectors

Since handling the individual dimensions separately can quickly become a mess, it's an often used convention to combine the $3+1$ dimensions into a four vector with the unit vectors:

$\textbf{e}_t,\textbf{e}_x,\textbf{e}_y,\textbf{e}_z$

The index of the time dimension can wary between authors. One convention is to use the zeroth dimension as the time dimension, so the unit vectors become

$\textbf{e}_0,\textbf{e}_1,\textbf{e}_2,\textbf{e}_3$

which gives us the four vector

$\textbf{a}=\sum_{a=0}^3 a^\alpha \textbf{e}_\alpha.$

By using Einstein notation, we get

$\textbf{a}=a^\alpha \textbf{e}_\alpha$

where it's assumed we sum over repeated greek letters.

We have adopted the convention where we denote four vectors with bold face such as, $\textbf{a}$, while we continue to denote three vectors with an arrow such as, $\vec{a}$.

For some more syntatic sugar, let

$\eta_{\alpha \beta} = \textbf{e}_\alpha \cdot \textbf{e}_\beta = \begin{bmatrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}.$

In which case

$\textbf{a}\cdot \textbf{b}=\eta_{\alpha \beta} a^\alpha b^\beta.$

The reason we have a $-1$, and not $1$ in the time dimension is to avoid messing up causality which the philosophers tell me is a good thing.

One neat thing about using four vectors is that the dot product between any two four vectors (or a chain of them) is invariant. So

$\textbf{a} \cdot \textbf{b} = \textbf{a}' \cdot \textbf{b}' .$

Lorentz Transformation

In the standard configuration, the componentwise Lorentz transformation is given by

$\begin{aligned} &a'^{0} = \gamma(a^{0} - \beta a^1 ), \\ &a'^{1} = \gamma(a^{1} - \beta a^0 ), \\ &a'^{2} = a^2, \\ &a'^{3} = a^3. \\ \end{aligned}$

Time Contraction

While deriving every formula of special relativity is beyond the scope of this project, deriving time contraction seems worthwhile due to it's broad applicability.

Let's consider a light-clock consisting of two perfect mirrors put parallel with a distance $L$ to each other, and let a foton travelling at speed $c$ bounce between the mirrors.

Obviously, the time it takes for the foton to bounce from the bottom mirror to the top mirror and back again is going to be

$t = \frac{2L}{c}$

because time equals distance over speed.

This can be rewritten as

$L = \frac{ct}{2}$

Now, let's consider the same system, but this time the mirrors move along the $x$-axis with velocity $v$ while the foton continous to bounce between the mirrors.

Let the time it takes for the foton to complete one full bounce be $t'$, and let the distance the mirrors have moved before the foton comes back be $2s$. Then we know that

$2s = v\cdot t'.$

So we can write $s=\frac{v\cdot t'}{2}$.

We know that the foton travels along $h$ twice for a total distance of $2h$, so we find that

$2h=c\cdot t.$

We now have all the sides in a right-angled triangled, and per pythagoras, we know that

$\big(\frac{vt'}{2}\big)^2 + \big(\frac{ct}{2}\big)^2 = \big(\frac{ct'}{2}\big)^2.$

Which gives us

$c^2t^2 = c^2 t'^2 - v^2t'^2.$

By dividing by $c^2$, we find

$t^2 = t'^2 - \frac{v^2t'^2}{c^2}=t'^2 \big( 1 - \frac{v^2}{c^2} \big).$

Isolating $t'$, we find

$t' = t \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$

Or when $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$,

$t' = t \gamma.$

Relativistic Mechanics

Just like classcal mechanics, we still have conservation of momemtum and total energy in relativistic collisions.

Example: Compton Spread

We consider a system consisting of a photon and an electron. In the initial state, the photon has a four momentum vector $\textbf{p}_\gamma$, and the electron is at rest, $\textbf{p}_e = m_e c(1,\vec{0})$.

In the final state, the photon has the four momentum vector $\bar{\textbf{p}}_\gamma$, and the electron has the four momentum vector $\bar{\textbf{p}}_e$.

The setup is illustrated below.

This is an example of Compton spread. The angle of spreading $\theta$ depends on the wavelength of the incoming photon with the relation

$\bar{\lambda}=\lambda+\lambda_c(1-\cos(\theta))$

where $\lambda_c = \frac{h}{m_e c}$.

This chapter has leaned on the content from "Speciel Relativitetsteori" by Ulrik Uggerhøj whenever a source has not been cited.