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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?

Jaime Caffarel
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    Gravitational force varies smoothly with distance. Gravitational potential varies smoothly with distance. Why would gravitational time dilation not do the same? – Ghoster Oct 07 '23 at 16:25
  • @Ghoster I think that gravitational time dilation varies smoothly, but not infinitely smoothly. I mean, I assume that the distance "d" from my question cannot be smaller than the Plank length, but I was thinking if it might be at least a little bit greater. Sorry if the question was not clear. – Jaime Caffarel Oct 07 '23 at 17:52
  • There is no evidence that the Planck length is the smallest length. And in General Relativity, the theory that predicts gravitational time dilation, there is no theoretical smallest length. – Ghoster Oct 07 '23 at 19:04
  • Related: Is spacetime discrete or continuous? – Ghoster Oct 07 '23 at 20:24

2 Answers2

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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)

John Doty
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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.

JEB
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