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## Main Question or Discussion Point

I'm reading in a textbook (Gravity by J. Hartle) that gravitational time dilation is implied by the equivalence principle. The following thought experiment is described. A vertical rocket at rest in a uniform gravitational field (no tidal effects) is compared to a rocket constantly accelerating (at non-relativistic velocities) far from any source of gravitation. Alice in the rocket's nose emits light pulses at equal intervals on a clock at her height. Bob in the tail measures the time interval between receipt of the signals on an identical clock at his location. Bob receives light pulses at shorter intervals than they are emitted because the accelerating tail is always catching up with the signals.

In this thought experiment, it seems clear that Bob's clock does not run at a slower rate than Alice's clock. After all, the differences in velocity and acceleration as between the two clocks are both zero.

Intuitively, Bob can never receive more total pulses than Alice emits, no matter how much (non-relativistically) the rocket's velocity increases with the passage of time and no matter how long the acceleration continues. The only way Bob can experience shorter reception intervals than the emission intervals is by progressively shortening the proper flight path length between Alice and Bob at a given time. For example, let's say that before the rocket starts accelerating, 100 pulses at a time are "in-flight" along the total path between Alice and Bob. Then, when the rocket begins accelerating, no matter over how long a (finite) time period the acceleration continues, Bob can never receive more than 100 pulses in excess of the absolute number emitted by Alice during that time period. If Alice emits 100 billion pulses over the duration of the acceleration, Bob cannot receive more than 100 billion + 100 pulses regardless of the maximum acceleration rate. It seems to me that at first the reception intervals will get shorter with time, but eventually, by the time Bob has received almost all 100 of the extra original "in-flight" pulses, the reception interval must start increasing, asymptotically reverting to the original emission interval.

Moreover, when the period of acceleration ends, Bob and Alice can compare their clocks and see that exactly the same cumulative amount of time has elapsed on both clocks. No cumulative time dilation has occurred.

The same will be true even if the rate of acceleration increases or decreases over time.

This thought experiment does not illustrate actual time dilation. It illustrates only an

Is this just a case of a weak thought experiment? Is there a more convincing way to use the equivalence principle to demonstrate true gravitational time dilation?

This thought experiment suggests that gravity creates its own

In the case of an emitter and observer in concentric orbits close to the event horizon of a black hole, this causation limit presumably could be reached fairly quickly.

In this thought experiment, it seems clear that Bob's clock does not run at a slower rate than Alice's clock. After all, the differences in velocity and acceleration as between the two clocks are both zero.

Intuitively, Bob can never receive more total pulses than Alice emits, no matter how much (non-relativistically) the rocket's velocity increases with the passage of time and no matter how long the acceleration continues. The only way Bob can experience shorter reception intervals than the emission intervals is by progressively shortening the proper flight path length between Alice and Bob at a given time. For example, let's say that before the rocket starts accelerating, 100 pulses at a time are "in-flight" along the total path between Alice and Bob. Then, when the rocket begins accelerating, no matter over how long a (finite) time period the acceleration continues, Bob can never receive more than 100 pulses in excess of the absolute number emitted by Alice during that time period. If Alice emits 100 billion pulses over the duration of the acceleration, Bob cannot receive more than 100 billion + 100 pulses regardless of the maximum acceleration rate. It seems to me that at first the reception intervals will get shorter with time, but eventually, by the time Bob has received almost all 100 of the extra original "in-flight" pulses, the reception interval must start increasing, asymptotically reverting to the original emission interval.

Moreover, when the period of acceleration ends, Bob and Alice can compare their clocks and see that exactly the same cumulative amount of time has elapsed on both clocks. No cumulative time dilation has occurred.

The same will be true even if the rate of acceleration increases or decreases over time.

This thought experiment does not illustrate actual time dilation. It illustrates only an

*illusion*of time dilation which is caused by the observer receiving subsequent light pulses after progressively shorter and shorter proper flight lengths.Is this just a case of a weak thought experiment? Is there a more convincing way to use the equivalence principle to demonstrate true gravitational time dilation?

This thought experiment suggests that gravity creates its own

*illusion*of time dilation (perhaps by shortening the flight time, i.e. by progressively increasing the speed of light relative to the emitter). It seems that the same limitation should apply: That the total number of "extra" light pulses received at the gravitating body in a finite period of time due to shortening the flight time can never exceed the absolute number of pulses that are "in-flight" at any instant in time. If the light emissions continue for a duration approaching infinity, eventually the interval between pulses measured by the observer must start increasing again. Causation would be violated if the gravitating body receives more total pulses than the emitter emits during ANY finite period (measured by the emitter's clock) + the number of pulses that were "in-flight" before the counting period started.In the case of an emitter and observer in concentric orbits close to the event horizon of a black hole, this causation limit presumably could be reached fairly quickly.

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