# Interdependence of time, the speed of light and distance in General Relativity

In the relation

$$\text{time}\cdot\text{speed of light} = \text{distance}\,,$$

by definition, the speed of light is constant. When bringing a clock, like in a GPS satellite, up into orbit, i.e. into a higher gravitational potential, we say

• time (the clock) runs faster
• the dimensions of the satellite have not changed.

If, instead, we had brought a light clock into orbit, we would still say

• time runs faster
• the length of the light clock has not changed.

Yet with the light clock, we might as well declare

• time runs at the same pace, yet the clock seems to run faster because
• the clock's length has shrunk.

Obviously, different explanations for the two different clocks would be nonsense. But if the GPS's clock does, intricately, depend on the dimensions of the clock itself, the second explanation would hold for it too: the satellite could have shrunk, or rather, we could interpret it that way.

Question: Is there anything in the theory of GR that forces us to prefer

• c constant, distance constant => time changes

over

• c constant, (flow of) time constant => length changes

with gravitational potential? It seems as if we cannot really independently measure time and distance, they are always connected by the speed of light.

Except if, for example, we can measure time in a fundamentally different way from a light clock in that it does not depend on the dimension of the apparatus down to the details of atomic scales, that would be an argument. Can we? Is there another argument that lets us prefer the "time changes" over the "length changes" explanation --- other than maybe mathematical convenience?

• Would you think the shrinking would happen independently of orientation with respect of the gravitational field? If not you could easily discern between the two declarations. – Alchimista Dec 30 '18 at 13:44
• There is nothing known of GPS clocks running differently depending on the satellite orientation. But this does mean nothing for the question of whether they are fundamentally different form a light clock, i.e. independent of any predefined length. – Harald Dec 30 '18 at 14:53
• Shouldn't be a light clock being a trajectory of light? I am saying change the orientation of your light beam/clock. Would you expect something independently of orientation? In other words your doubt is about measuring in toto. Not about GR. – Alchimista Dec 30 '18 at 15:18