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I'll start by admiting that I'm a cockroach compared to Einstein or Hawking, and I think that due to Hawking's great mind, maybe he wrote the book in a way he found conspicuous. But I don't have Hawking's IQ, and for me is definitely harder to understand the concept of "Time Dilation" that Hawking was trying to explain with his thought experiment. Could you please help me folks?

Here is my doubt fully explained:

In his book, A Briefer History of Time - Chapter 6 (Curved Space), Hawking explains how gravity affects time through a thought experiment. It starts this way:

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Imagine a rocket ship out in space. For convenience, imagine that the rocket ship is so long that light takes one second to traverse it from top to bottom. Finally, suppose there is an observer at the ceiling of the rocket ship and another at the floor, each with identical clocks that tick once each second.

Suppose the ceiling observer w aits for the clock to tick, and then immediately sends a light signal down to the floor observer. The ceiling observer does this once more the next time the clock ticks. According to this setup, each signal travels for one second and then is received by the floor observer. So just as the ceiling observer sends two light signals a second apart, the floor observer receives two, one second apart.

How would this situation differ if the rocket ship were resting on earth, under the influence of gravity, instead of floating freely out in space? According to Newton’s theory, gravity has no effect on this situation. If the observer on the ceiling sends signals one second apart, the observer will receive them one second apart. But the principle of equivalence does not make the same prediction. We can see what happens, that principle tells us, by considering the effect of uniform acceleration instead of the effect of gravity. This is an example of the way Einstein used the principle of equivalence to create his new theory of gravity.

So let’s now suppose the rocket ship is accelerating. (We will imagine that it is accelerating slowly, so we don’t approach the speed of light!) Since the rocket ship is moving upward, the first signal will have less distance to travel than before and so will arrive sooner than one second later. If the rocket ship were moving at a constant speed, the second signal would arrive exactly the same amount of time sooner, so the time between the two signals would remain one second. But due to the acceleration, the rocket ship will be moving even faster when the second signal is sent than it was when the first signal was sent, so the second signal will have even less distance to traverse than the first and will arrive in even less time. The observer on the floor will therefore measure less than one second between the signals, disagreeing with the ceiling observer, who claims to have sent them exactly one second apart.

This is probably not startling in the case of the accelerating rocket ship—after all, we just explained it! But remember, the principle of equivalence says that it also applies to a rocket ship at rest in a gravitational field. That means that even if the rocket ship is not accelerating but, say, is sitting on a launching pad on the earth’s surface, if the ceiling observer sends signals toward the floor at intervals of one each second (according to his clock), the floor observer will receive the signals at shorter intervals (according to his clock). That is startling!

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To put it simply, please recall that in the accelerating Rocket case, a "Doppler-Effect-like" phenomenon was active: the accelerating Rocket was approaching the light receiver (floor observer) towards the light emitter (ceiling observer).

Thanks to this "Doppler-Effect-like" phenomenon, each time the Rocket accelerated, it shortened the path needed for light to traverse from ceiling to floor, so each subsequent signal would arrive in less time (faster), and give the floor observer "the impression" that the light signal had been emitted "earlier in time".

Since the floor observer would be receiving light signals in a time period shorter than what he would have expected (1 second), he would then have been left no chance but to think that "events were occurring faster than his clock's ticks", and hence experience a "Time Dilation" effect.

But in order to experience a "Time Dilation" effect, the Rocket at rest would need a "Doppler-Effect-like" phenomenon, that is, the light emitter (ceiling observer) would need to be approaching the light receiver (floor observer) each passing second, and since there is not any acceleration to shorten the light-traverse-path, I can't understand how can Time dilate on Earth.

Is it just me, or did someone else extract the same understanding from the experiment's speech?

What part am I missing that could make me understand this experiment?

Thanks in advance!

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