back
2 comments
Could we realistically get so accurate that we can measure time changes due to (human scale) mass movements near a clock?
My calculations says that moving 1cm up or down earths gravity well (at the surface) changes the acceleration of gravity about 5x more than the acceleration you'd feel from a 100kg mass 1m away.

Assuming my math is correct, it's already affected by nearby human scale masses, for certain values of "near".

I believe the time dilation is caused by differences in gravitational potential, not gravitational acceleration, so it would be even worse than that.
Huh, I was thinking that it's accelerated gravitational frames that cause the dilation, and I've encountered a lot of statements that argue the same. This is from wikipedia: "This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects."

However, according to that logic, an object located in a cavity in the center of earth should experience no more dilation than an object outside the earth's potential well, because the gravitational forces / curvature gradient cancels out, and should be zero. But that isn't the case according to the same sources, for example, Wikipedia says' "Relative to Earth's age in billions of years, Earth's core is in effect 2.5 years younger than its surface."

Something's not right about how we verbalize this story about gravity

It's a very subtle point! The trick here is that you need to be careful when talking about the reference frames.

To an observer at the infinity, a clock at the core of the Earth will tick slower than a clock on the surface of the Earth because the "core clock" is sitting in a more curved space, and that's it.

The difference between the clock on the surface of the Earth and the clock at the core is that the surface clock can't follow the "straight lines" (geodesics) in that curved space. So it experiences acceleration due to the force of inertia. And the thing preventing that movement is the repulsive force between atoms that make up the bulk of the Earth.

If this repulsive force magically disappears, then the Earth's atoms will immediately start moving at the straight lines, in trajectories that will lead them all into a point at the center of the Earth.

To add: the force of inertia due to moving in curved lines instead of geodesics depends on the "steepness" of the curved space. Which decreases as you reach the center of the Earth. So you get essentially the same result as with the classic Newtonian gravity, but through an entirely different path.

A clock at the center of the planet should experience no net force by the mass of the planet.
that's the argument, yes

no net force, but net potential energy - thus gravitational dilation

Think about gravitational redshift. This is a direct sign that time is running more slowly for the emitter that is at a deeper gravitational potential.
That would be an amazing proximity sensor. "Looks like time slowed down again, there must be someone close by."
Submarine detection would be interesting with a few such precise clocks.
Given that the mass of a submarine is ~the same as the water it displaces, this wouldn't actually work.
> [a] submerged neutrally buoyant submarine produces no first order gravitational anomaly; however, because the distribution of mass throughout the submarine is not uniform, second and higher order effects can be produced. [...] For a submarine to be in stable equilibrium while submerged it is necessary that its center of mass be below its center of buoyancy (in submarine-fixed coordinates). That is, the mass of the lower half of a neutrally buoyant submarine's volume (including ballast) must be greater than the mass of the water it displaces whereas the mass of the upper half must be smaller than the mass of water it displaces. Therefore, a submerged submarine may be considered to possess a net vertical gravitational dipole moment having "negative mass" above and "positive mass" below.

https://apps.dtic.mil/sti/pdfs/AD1012150.pdf

Though, that 1989 paper concludes that because gravimeters would need a sensitivity of at least one part in 10^13 for practical usage, far beyond what was capable at the time, "[t]he concept of detecting submarines by means of detecting gravitational anomalies they produce, should be abandoned."

Yes and no. A moving submarine has a bow wave, a combination of compressed water and an actual wave manifest at the surface above. So there is a transiting bunch of extra mass to detect, at least so long as the sub is moving.
I like the TLDR at the end of the paper, but could be abbreviated further by just writing:

No.

I'm curious if any sci-fi authors were knowledgable and prescient enough to write this into their world building?

If not, it'd make for a pretty cool plot device if done well.

There's an article, I think on wired.com, years ago about exactly this. It talked about using a vast array of accurate clocks as a kind of radar. Seems plausible only with a few more orders of magnitude accuracy and miniaturization.
From what I understand, the relativistic effects are a lot less sensitive to nearby mass than the acceleration due to gravity (basically a 1/r relationship instead of 1/r^2). So while a sensitive enough gravimeter can pick up a nearby fairly heavy mass moving and such an elevation change, an atomic clock is going to be much more sensitive to elevation changes than nearby changes in density.
Asked this and related questions to o3. I do not vouch for the answers at all but you may find it interesting. https://chatgpt.com/share/6876cdd1-dfbc-8011-a55f-6915a90275...
I'd love to hear what the kids remember about this trip. It's been awhile!