# 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

## 1 Answer

In general relativity a macroscopic object is exposed to the tidal force due to the curvature of the spacetime geometry. However as long as the structure of the clock, either mechanic or electronic or atomic or light or whatever, is not compromised by the consequent vertical stretching/horizontal compression, the clock measures reliably the proper time in the rest frame of the object. A ruler in the rest frame, unless the tidal force is extreme, measures reliably the proper distance. Hence, time and distance can be measured independently.

The gravitational time dilation is observed in weak gravitational fields where the tidal force is negligible.

As per above, your second option "length change" has no fundament.

• How exactly do you define the ruler? As far as I can tell (no kidding), walking into the wood, grabbing a stick and defining it to be the unit length called 1 qrix would do. The you travel near and far into more or less deep gravitational wells and always call the stick length 1 qrix. And you're happy that light traveling the length of 1 qrix always takes the same time as measured by, aaahoooo, well, the clock made by defining that one clock tick is what a light beam takes to travel along 1 qrix. I don't understand of how to get out of this circular reasoning. – Harald Dec 30 '18 at 11:37
• What you measure in that way is the proper time in an arbitrary reference frame, however here we are comparing proper times in different frames. To observe the gravitational time dilation you measure the gravitational redshift of a radiation emitted closer to a gravitational mass and received farther. The frequency is inversely proportional to the period, so you measure time. – Michele Grosso Dec 31 '18 at 17:36