- Our coordinate system has an $x$ space axis and a $t$ time axis.
- We have observer (1) standing still at the origin, and measures time with a variable $t$.
- We have observer (2) moving to the left with a constant velocity $v$.
- Observer (1) who is at rest sees a photon starting from the origin travelling towards the right with constant velocity $c$. The position of the photon at time $t$ is $x = c t$.
- Observer (2) also sees this photon. At time $t$, he sees the position of the photon as $x' = vt + ct$.
- From our rule of invariance, we have that $dx/dt = c = dx'/dt'$.

$\begin{aligned}
&c = \frac{dx}{dt} = \frac{dx'}{dt'} = \frac{dx'}{dt} \frac{dt}{dt'} \\
&c = \frac{d(vt + ct)}{dt}\frac{dt}{dt'} \\
&c = (v + c) \frac{dt}{dt'} \\
&\frac{c}{v+c} = \frac{dt}{dt'} \\
&dt' = (v+c)dt/c \\
&t' = (v+c)t/c = (1 + v/c) t
\end{aligned}$

So we get the relation that time elapsed for observer (2) is related
to observer (1) as $t' = (1 + v/c) t$.
- This checks out: Assume our observer is moving leftward at $v = c$. He will then see the photon move rightward at $x' = 2ct$. So if his time slows down to have $t' = 2t$, we will have that $x/t' = 2ct/2t = c$.

- However, this forumla allows us to go faster than the speed of light with no repercurssions! (It is also not the correct formula as anticipated by the usual derivation). This can be fixed.

- Now assume that Observer 2 was moving
*rightward*, not*leftward*. That is, we simply need to set $v = -v$, since this change of sign accomplishes flipping the direction of motion in 1D. This gives us the equation $t' = (1 - v/c)t$.

- According to this new equation, we are not allowed to approach the speed of light. If we attempt to do so, we will need to elapse zero time; and if we exceed the speed of light, we will need to elapse
*negative time*.

- However, these formulae lead to an absurdity. If our observer 1 and observer 2 witness two photons, one moving leftward and one moving rightward, one will need to write down the equations $t' = (1 \pm v/c)t$, which plainly leads one to contradiction.

- It is true that
*as per observer (1) standing at the origin*, the distance between observer (2) and the photon is $vt + ct$. - It is
*not*true that observer(2) sees the distance between them and the photon as $vt + ct$. - Intuitively, this equation of $x' = vt + ct$ completely ignores length contraction, and hence cannot be right.
- Alternatively, the equation of $x' = vt + ct$ imposes galilean relativity, where I am attempting to naively connect reference frames, which cannot be correct.