Both Bobs see the lightning at the same time, and experience a light bubble radiating out at the speed of light.

But, counter to our intuition, even though lightning actually struck the platform, Train-Bob's light bubble stays centered a few feet in front of him. In contrast, the dust cloud, behaving as expected, remains centered on (and moves away with) Platform-Bob.

So one event, one dust cloud, but two light bubbles. How come?

1. As early as 1676, based on celestial observations, good Ole Roemer figured out that the speed of light was finite. He provided an estimate that was accurate in the order of magnitude. Over the centuries, his initial estimate was refined to its presently accepted value of $3 \times 10^8 m/s$.
2. Around 1865, Maxwell laid down the equations of electromagnetism, one of which is:
   $$ \nabla \times B = \mu_{0}{J} + \mu_{0}\epsilon_{0}\frac{\partial{E}}{\partial{t}}\\ $$ where,
       $ \nabla \times B $ is the 'curl' of a magnetic field,
       $ J $ is the density of a current in the neighborhood,
       $ {\partial{E}}/{\partial{t}} $ is the rate of change of an electric field in the same hood,
       $ \mu_{0} $ is the magnetic permeability of vacuum, a constant and
       $ \epsilon_{0} $ is the electrical permittivity of vacuum, also a constant.
   
   
   In English, this states that a magnetic field can be produced by an electric current OR by motion of a conductor through an electric field OR both. Of specific note is the product of the constants $\mu_{0}\text{ and }\epsilon_{0}$. By Maxwell's choice, ${1}/\sqrt{\mu_{0}\epsilon_{0}}$ is equal to the speed of light in vacuum. As ${1}/\sqrt{\mu_{0}\epsilon_{0}}$ was quite a mouthful, physicists started referring to this as $c$.
3. Popular readings on relativity quote the Michelson Morley experiment (MMX) as the basis for constant $c$. However (to me) that only confirms part of the story i.e. $c$ is constant in all directions for one observer with their own light source.
4. Einstein himself refers to William De Sitter's observations on light from Double Stars as evidence that $c$ does not depend on the relative motion of light source or observer.

The point of this long walk is to establish that $c$ is constant in the laws of physics, based on practical observations as well as theory. According to the classical principle of relativity, the laws of physics apply equally for all observers in constant motion relative to each other. Therefore, $c$ must be constant for all such observers.

Let us apply this knowledge to make sense of our gedanken. First the 4Ws:

• What happened? Lightning struck the platform. A cloud of dust was kicked up.
• Who was watching? Train-Bob and Platform-Bob.
• Where did it happen? At the origin ($x=0$) of Platform-Bob's reference frame and (close to) the origin ($x'=0$) of Train-Bob's reference frame.
• When did it happen? At the exact moment of crossing ($t=0$ on the platform). If time were universal (as it was thought to be, before Einstein), then time on the train would also be $t$.

Once emitted, light from the lightning strike, must travel at the same speed to all points to the left and right of each Bob in their own frame of reference. So each Bob must get their own light bubble, that follows a set of equations dictated by the principle of relativity.

• On the platform, the dust cloud remains at $x=0$, the train is moving away with $x = -vt$, and the light bubble is radiating out with $x={\pm}ct$.
• On the train, the dust cloud and platform should move away at $x'=vt$ and $x'=vt$. Light should radiate out at $x'=vt{\pm}ct$ according to the classical principle of relativity, but per the requirement of constant $c$, it should radiate at $x'={\pm}ct$.
• As $ct$ cannot be equal to $vt{\pm}ct$, except at the origin $t=0$, Einstein proposed that time progresses differently on the train as compared to the platform i.e. at $t'{\neq}t$.
• This allows us to restate the train frame equations of motion as, $x' = vt'$ for dust cloud and platform, and $x' = {\pm}ct'$ for light bubble, allowing $ct'{\neq}vt{\pm}ct$.
• We explore this notion of time being different for observers and their reference frames in the next section.