What's the resolution of gravitational time dilation?

Question

Time dilation near a gravitating mass depends on the distance to that mass. So I assume that time at 1 meter from a very massive object goes slightly slower than at 1.01 meters, and also slower than at 1.000001 meters. Is there a (theoretical) minimum distance limit at which time dilation remains the same?

I mean, is there any distance d, at any point in space p (near a very massive object) at which anything situated at +-d from the point would experience the same time dilation experienced at p?

Answer 1

Experimentally, there's no evidence of a minimum distance limit. Last year, JILA researchers published the results of a observations of gravitational time dilation from a 1mm altitude difference in Earth gravity.

Bothwell et al. Nature volume 602, pages 420–424 (2022)

Answer 2

For special relativity:

$$ \gamma = \frac 1 {\sqrt{1-v^2/c^2}} = \frac 1 {\sqrt{1 - 2E/mc^2}}$$

where $E = \frac 1 2 mv^2$ is the newtonian kinetic energy term.

Gravitational time dilation is similar:

$$ t_f/t_0 = \frac 1 {\sqrt{1-2GM/rc^2}} = \frac 1 {\sqrt{1-2U/mc^2}}$$

where $U = GMm/r$ is the Newtonian potential energy term.

So now you can wing it: A neutron star has around $a=10^{12}g$ gravity, so with

$$ v = \sqrt{2ah} $$

you can go from there.