Relativity of Simultaneity - Issue with the train car example

Question

One of the classic examples to describe that simultaneity is relative is the following:

"Imagine a freight car, traveling at a constant speed along a smooth, straight track. In the very center of the car there hangs a light bulb. When someone switches it on, the light spreads out in all directions at speed c. Because the lamp is equidistant from the two ends, and observer on the train will find that the light reaches the front end at the same instant it reaches the back end... However, to an observer on the ground these same two events are not simultaneous. For as the light travels out from the bulb( going at speed c in both directions) the train itself moves forward, so the beam going to the back has a shorter distance to travel than the one going forward. According to this observer, therefore, [the light hits the back] before [the light hits the front]." - Griffiths, Introduction to Electrodynamics

Here is a picture of the thought experiment:

My lack of understanding stems from the fact that I am thinking about how fast the distance between the wall and the light beam decreases to 0. Take for example the back wall:

In the reference frame of someone in the train, it makes sense that the distance between the back wall and the light beam decreases at the rate of c, the speed of light, because in this frame the back wall is not moving and the light is moving at the speed of light.

However in the reference of the outside observer looking into the moving train, he/she will see the back wall moving to the right (as in the picture above) and the light beam moving to the left, but the rate that the distance between the back wall and the light beam decreases can't be greater than c, because nothing can move faster than the speed of light c. (If two observers are moving towards each other at the speed of light, they don't see each other moving at twice the speed of light, rather they still observe the other moving at the speed of light). So if the light bulb and the back wall were observed to start at the same distance away in both observers' frames than they both have to see the events happening at the same time.

Where am I tripping up/ where is my logic inconsistent?

Answer 1

but the rate that the distance between the back wall and the light beam decreases can't be greater than c, because nothing can move faster than the speed of light

You are mistaken. Yes, nothing can move faster than light in a sense that physical motion is fundamentally bounded from above by $c$. But an abstract mathematical quantity, which is the distance between the light beam and the back wall of the wagon, can decrease with a slope higher than $c$. There is nothing in Special Relativity that forbids it from happening.

On the contrary, we know that the light beam is traveling to the left exactly with the speed of light $c$, because this is physical motion we are talking about. And since the wall is moving to the right, the distance decreases faster than $c$. Again, this is completely normal.

Consider another thought experiment: two particles flying in the opposite directions with almost the speed of light each. The distance between them obviously increases with speed $2c$. There is nothing wrong with this! However, in the rest frame of one of the particles, the other one travels not with $2c$, but approximately with $c$ because it is the way the relativistic velocity addition functions.

Answer 2

The two events, (light striking both the front and back of the train simultaneously) will be seen as such by any other observer in a different state of motion than the train. Any other outcome would violate causality since the back and front of the train are local to each other and to the source of light. To properly understand why, one must take into account the Lorenz contraction that the train is shortened along its direction of travel according to the view of other inertial observers.

Armand, Physics as a hobby