Help with relativity simultaneity paradox

Question

To begin I will explain briefly a classic thought experiment used to demonstrate simultaneity;

There is a 100m long train rushing towards a 100m tunnel at a fraction of the speed of light, from an observers perspective the train is shorter than the tunnel and so the observer can safely trigger guillotines at either end of the tunnel to briefly fire and not hit the train when the train is inside, however from the trains perspective the guillotine in front fires first and the one behind afterwards as to not hit the train.

However I have thought of a slight modification to the experiment that appears to break relativity;

Imagine a sensor on the front of the train which detects when a guillotine passes infront of it, this then sends an electrical signal down the train to a light at the back of the train which, I think, would turn on at the same time as the back guillotine drops, from the trains perspective this is fine however from the observers perspective information has travelled instantly, the firing of the first guillotine has turned on the light with no delay. What is the resolution to this? One possible resolution to this that I have thought of is the the simultaneous events of the light and the back guillotine in the trains perspective is not simultaneous in the observers perspective, however this doesn't seem to make sense as its a simultaneity paradox within a simultaneity paradox and I can't see any reason for the two events to be non simultaneous (they're in the same place moving at the same speed)

Answer 1

Imagine a sensor on the front of the train which detects when a guillotine passes infront of it, this then sends an electrical signal down the train to a light at the back of the train which, I think, would turn on at the same time as the back guillotine drops, from the trains perspective

That's not possible.

Instead of a detector on the front of the train, imagine that a photon is emitted at the instant the front gate closes. Since the gate and photon emitter are co-located, all observers agree that the photon is emitted at the instant the front gate is closed.

The photon propagates at $c$ to the location of the back gate and is detected there. Clearly, the photon is detected there after the front gate closes.

Since, in the tunnel's frame of reference, the two gates closed simultaneously, the back gate has already closed when the photon is detected.

And, since the back gate and photon detector are co-located, all observers agree that the back gate is already closed when the photon is detected.

Thus, your reasoning (which you did not provide) with which you arrived at your conclusion is flawed.

As always, try drawing a spacetime diagram and, I think, the flaw will become clear.

Answer 2

from the observers perspective information has travelled instantly, the firing of the first guillotine has turned on the light with no delay.

Herein lies the flaw in your reasoning. The electrical signal doesn't instantly propagate, it propagates at the speed of light. From the perspective of every observer, the electrical signal from the sensor reaches the back of the train at the same time as the interruption of light from the dropping guillotine, so no new information is available from the addition of the sensor and light bulb.

Answer 3

The only way the guillotine drop and the light turning on are simultaneous from an observer's perspective is if the situation is designed such that at the time the electrical signal reaches the light, the light traveling to the observer from the guillotine (when it started to drop) and the light being emitted from the train are equidistant to the observer, so that the light from each reaches the observer at the same time. Given that, the only way that light reaches an observer at the same time is if the train were moving at the speed of light, which violates relativity.