Why do atomic clocks measure different elapsed times at different gravitational field strengths if their internal rate is stable quantum transitions

Question

Atomic clocks rely on quantum transitions governed by fundamental constants which do not change with (are independent of) gravity. Does this suggest that their internal ticking rate should remain stable regardless of location despite what we observe in the experiments that show that clocks in stronger gravitational fields tick more slowly than those in weaker fields?

Answer 1

In modern physics there are two distinct concepts of time: proper time and coordinate time.

Proper time is the time that is physically measured by a clock. It is only defined at the point where the clock is at any given moment. (More technically it is defined only on the worldline of the clock).

Coordinate time is the mathematical time that is a coordinate in certain common coordinate systems. It is defined over an entire region of spacetime. (More technically on an open subset of the spacetime manifold).

Any good and functioning clock will measure (to within experimental precision) the same rate of proper time as any other. However, since each clock can only measure proper time where it is located, the proper time of two distant clocks cannot be compared without adopting some simultaneity convention. That is, without using coordinate time.

So time dilation is not the comparison of two proper times on two clocks. It is the comparison of one clock’s physical proper time to some specified mathematical coordinate time.

Atomic clocks are not particularly special in this regard. They are just used because their experimental precision is so good. But any clock’s proper time measures the passage of physical time at the clock’s location.

When you say that a clock has slowed down, that means that your mathematical coordinate time is marking more passage of time than physical clocks at that location.

Answer 2

Does this suggests that their internal ticking rate should remain stable regardless of location despite what we observe in the experiments that show that clocks in stronger gravitational fields tick more slowly than those in weaker fields?

No. I observe those quantum transitions to take place at a different rate, most importantly while suspended either next to me ($\gamma_\text{clock}/\gamma_\text{me}=1$), near the Sun where gravity is stronger ($\gamma_\text{clock}/\gamma_\text{me}>1$), and in interstellar space where gravity is weaker ($\gamma_\text{clock}/\gamma_\text{me}<1$). Further more complex changes are observed if the clocks are moving as well.

One of the only reference-frame-invariant quantities in the Universe is the speed of light $c$ (along with the traces of a few tensors that are physically-important). Two observers at any distance from a gravity well will always agree that a photon moves at $c$ exactly (times a factor—the key thing is that everyone agrees on it). Similarly they will also always agree on the total spacetime-length $\text ds^2=g_{\mu\nu}\text dX^\mu\text dX^\nu$ the light travelled along a geodesic. There is no such invariance for the rate of quantum transitions, i.e. the tick rate of atomic clocks.

The constants governing those transitions might be invariant, but the energy scales at which the transitions occur (and even things like the mass of transitioning matter within the clock) are not invariant under coordinate transformation or movement around a gravity well. Those constants are not the only determining factor for the tick rate.

Answer 3

It is a common misconception that relativistic effects cause clocks to run more slowly. Good clocks will tick-off a second every second regardless of whether they are time dilated in some way. The real cause of the observed effects is the geometry of spacetime, which causes the time between two events to be dependent on the path followed between them.

To illustrate the point, imagine that I drive directly from A to B while you take some meandering route. If we compare odometer readings, mine will have clocked-up fewer miles than yours- that is not because my odometer is counting up the miles at a slower rate to yours, but because the actual distance I have travelled is less. The same applies to clocks. If a time dilated clock records a shorter time it is not because it is running slow but because it has actually travelled a shorter distance through time.

Answer 4

When a clock or any object is moved into lower gravity, or moved at speed, its time itself changes. This has nothing at all to do with how the clock works, and it is entirely unavoidable.

Your clock display changes by one second every second. But that is the clocks second according to its speed and its gravity. It is not my second.

(If you had an ancient pendulum clock that would most likely run slower in lower gravity; I can’t see it being affected by speed on a spaceship except for centrifugal power. And the effect would be more than relativity).