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Suppose there are two identical drones in a weak and uniform gravitational field that are initially at the same place and with synchronized clocks.

Assume drone A first rises vertically for 10 seconds of proper time and waits for 100 seconds. Drone B first waits 100 seconds and then rises for 10 seconds of proper time in exactly the same way as drone A did. Do the two drones end up at the same space-time location?

It is known that the rate with which a clock ticks increases with the altitude in such a field. As the drone rises its clock will tick faster then.

So intuitively, I would say that the two drones would not end up at the same space-time location. However, I'm not quite sure I understood mathematically how to solve this problem. Also, would they end up in the same space-time location if a drone would first wait for 10 seconds and then rise for 10 seconds (and inversely for the other drone)? What would change?

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If drone A rises higher where it is less affected by the gravitational field (weighs less), then its clock will run faster than the lower drone. If the other drone then later rises, it's clock will differ. the longer they spend at different heights in the gravitational field, the greater the difference will be. However in a weak gravitational field the difference will be extremely small.

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You are asking about gravitational time dilation.

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The higher the gravitational potential (the farther the clock is from the source of gravitation), the faster time passes.

https://en.wikipedia.org/wiki/Gravitational_time_dilation

It is a misconception that mass creates gravitational time dilation, in reality it is stress-energy, that is the difference in stress-energy between two points in space. Now in your case, the two drones are at the same distance from the center of gravity, so their clocks (let's put a clock on them) can be synchronized. For the intents of purposes, let's synchronize them.

Now in your case, drone A rises for 10 secs and then stays there for 100 secs (where its clock ticks faster). Meanwhile, drone B is waiting for 100 secs at a lower altitude (where its clock ticks slower), and then rises.

Now it is that time, while drone A is already higher (away from the center of gravity), but drone B is still lower (closer to the center of gravity), while drone A's clock is ticking slower then drone B's clock.

Let's disregard the time while they are rising.

So for at least 90 seconds, drone A's clock will tick slower.

This will mean, that when the two drones meet again, that is they are at the same altitude again, their clocks will start running at the same rate again. But there is already a difference between the two clocks.

Drone A's clock ticked less.

You are talking about being at the same space-time location.

I assume you mean that they are at the same (close) space location (meet again at the same altitude). And that they are there at the same time.

That is true, they will meet again. But their clocks show different times.

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