Why does the "gravitational potential" - something defined by convention - rule time dilation and length contraction in GR?

Question

Assume that we have an exactly uniform gravitational field like that occur for an infinitely large plate, yet with a finite mass. As we know, two similar clocks located in a specific alignment in the field with different distances away from the plate, and at rest WRT the plate, undergo similar gravity, and thus the clocks are expected to run at the same rate. Even all the experiments performed in the compartments within which the clocks are located have the same outcomes.

However, according to GR, the clock which is nearer to the plate runs slower as viewed by the other clock located farther from the plate just because the nearer clock is in a lower (more negative) gravitational potential regardless of the strength of the gravitational field. Why this the case?

If the gravitational potential is something determined by convention, why and how it has become so important, rather than acceleration with real physical impacts, in affecting some real phenomena such as time dilation and length contraction? As far as I think, those physical qualities determined by convention are somehow apparent. Therefore, I think it is as if we claim that because the apparent size of the farther clock is smaller, thus this apparent phenomenon affects time rates or the length measurements!

Where is the problem?

Added according to @JohnRennie 's answer:

The gravitational potential is not a fundamental property. The fundamental property is the geometry, and given some choice of a coordinate system the gravitational potential emerges from the geodesic motion.

Likewise the time dilation is a consequence of the geometry. So both the potential and the time dilation are a result of the geometry. It is not the case that the potential causes the time dilation.

I think these statements are somehow self-referential. We can also claim that it is the G-potential that determines the geometry around a G-mass as well as vice versa, which especially can be justified considering the fact that the coefficients in the Schwarzschild metric can easily be reformulated as functions of the G-potential.

Note that what matters here is the difference in the gravitational potential energy $\Delta\phi$ i.e. we set $\phi=0$ at the origin of our coordinate system then take the difference in the GPE relative to this point. The absolute value of the potential is not a physical observable.

You are right. However, the mentioned difference can be interpreted as the work done on a unit mass (the clock) to move it from infinity to the surface of the planet. I just cannot perceive how this work plays a decisive role in the clock rate.

For better understanding my issue, assume that we have a massive spherical shell. The G-acceleration is zero inside the shell as well as in infinity. The Schwarzschild observer located at infinity measures the rate of the clock located on the surface of the shell smaller than the same clock in his own hand. However, the observer at the center of the shell with similar feelings (zero G-field) to those experienced by the Schwarzschild observer, detects no change in the clock rate located on the shell compared to his because the potential difference is zero. This is slightly strange to me!

On the other hand, if there is an authenticity with the work done on the clock in GR, why general relativity predicts no change for the clock rates in E-fields (E-potentials) for charged clocks? That is, if we consider a massless shell though highly electrically charged and if use a charged clock, we may have to do the same work as we did on the chargeless clock in the previous example. However, this work can not affect the time rates for the clock located on the charged shell from the viewpoint of the Schwarzschild observer. Why this is the case?

Answer 1

The gravitational potential is not a fundamental property. The fundamental property is the geometry, and given some choice of a coordinate system the gravitational potential emerges from the geodesic motion. That is, for any particular choice of coordinates the geodesic equation gives a coordinate acceleration that can be integrated to give a gravitational potential. Note that different choices for the coordinate system will give different gravitational potentials.

Likewise the time dilation is a consequence of the geometry. So both the potential and the time dilation are a result of the geometry. It is not the case that the potential causes the time dilation.

But as you say the time dilation is certainly correlated with the difference in gravitational potential energy. This happens because in the weak field limit the metric becomes:

$$ \mathrm ds^2 \approx -\left( 1 + \frac{2\Delta\phi}{c^2}\right) c^2~\mathrm dt^2 + \frac{1}{1 + 2\Delta\phi/c^2}\left(\mathrm dx^2 +\mathrm dy^2 + \mathrm dz^2\right) \tag{1} $$

You'll find derivations of this on the Internet, or there is a nice derivation for the specific case of the Schwarzschild metric in Derivation for the weak field Newtonian metric around Earth.

We get the time dilation by considering a stationary observer, i.e. $dx = dy = dz = 0$ in which case equation (1) gives relationship between the proper time for the observer and our coordinate time:

$$ \mathrm d\tau^2 = \left( 1 + \frac{2\Delta\phi}{c^2}\right)dt^2 \tag{2} $$

giving the time dilation:

$$ \frac{d\tau}{dt} = \sqrt{1 + \frac{2\Delta\phi}{c^2}} \tag{3} $$

Note that what matters here is the difference in the gravitational potential energy $\Delta\phi$ i.e. we set $\phi=0$ at the origin of our coordinate system then take the difference in the GPE relative to this point. The absolute value of the potential is not a physical observable.

Answer 2

The question is tricky – which doesn't mean uninteresting – because it contains some assumptions that are actually impossible in general relativity, or only approximate in specific situations; and because it also uses notions that are not generally defined in general relativity, but can only appear as approximations in specific situations. So one would need to reformulate the question by first defining the situation and approximation more precisely, and checking if they are actually possible.

This is not meant to be an answer – sorry, feel free to downvote – but to point out some crucial issues in the original versions of the question.

• Uniform, indefinitely extended gravitational fields are not possible in general relativity. See e.g. § 7.6 (among others) in Crowell's book. But maybe the question may be reformulated without this assumption.

• Ideal clocks are usually massless objects, so the "work" done in moving them is zero. Considering a massive clock brings nonlinearity and non-stationarity, and the notion of a "gravitational potential" breaks down. Maybe it's possible to salvage something by taking approximations and limits as the clock's mass goes to zero – one has to check whether that's possible.

• The notions of gravitational work and energy quickly become iffy in general relativity. Baez gives a very nice overview with references. From one point of view, a clock naturally falls from infinity onto a spherical shell: there's no work involved, its motion is a geodesic.

• The discrepancy in ticking rate between a standard clock's ticks, and the ticks received by that clock from another, distant standard clock depends on how the worldlines of the tick-signals "diverge", which in turn depends on the curvature of the region where the worldlines lie. So it's natural that the clock at infinity in the Scharzschild metric measures a discrepancy with the ticks received from a clock on a spherical shell: the tick-signals have worldlines in a curved geometry. It's also natural that the clock at the centre of the shell doesn't measure any discrepancy, because the tick-signals have worldlines in a flat space.

The crucial point is that there are no "potentials" in general relativity, which would allow you to compare things at different places without worrying about what's in between. In general relativity this kind of phenomena is path-dependent.

In fact, a standard clock could even receive ticks from another standard clocks twice and at two different rates, owing to gravitational lensing.