Is it even possible to formulate such relation?
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1$\begingroup$ What you mean with "density of space"? $\endgroup$– JanGCommented Dec 31, 2021 at 15:52
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$\begingroup$ There is gravitational time dilation. In analogy, I think of gravitational length contraction. $\endgroup$– MarcinCommented Dec 31, 2021 at 15:57
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2$\begingroup$ To understand you properly, can define the meaning "flow of time"? $\endgroup$– JanGCommented Dec 31, 2021 at 17:09
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$\begingroup$ I mean the pace of flow of time, that is slower on the surface of the planet in comparison to the intergalactic space. $\endgroup$– MarcinCommented Dec 31, 2021 at 18:19
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$\begingroup$ The answer to your question is yes. If you don't mind I will try to explain it clearly tomorrow. In the meantime I wish you a Happy New Year! $\endgroup$– JanGCommented Dec 31, 2021 at 18:45
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The gravitational time dilation and length contraction are analogues to that in special relativity theory (SR). For example, in static spherically spacetime the first one is related to square root of metric component $g_{00}$ and the second, to square root of metric component $g_{rr}$, i.e. \begin{equation} d\tau=\sqrt{g_{00}}~dt<dt,~~~~~dl=\sqrt{g_{rr}}~dr>dr~.\tag{1} \end{equation} In particular, for Schwarzschild vacuum metric, it yields \begin{equation} d\tau=\sqrt{1-r_{S}/r}\cdot dt<dt,~~~~~dl=1/\sqrt{1-r_{S}/r}\cdot dr>dr,\tag{1} \end{equation} where $\tau$ is called proper time, and $l$ proper distance or length. A very lucid explanation of the connection between $\gamma$ factor in SR and metric factors in GR one can find on page 4 in S. M. Blinder's paper "Centennial of General Relativity (1915-2015); The Schwarzschild Solution and Black Holes", https://arxiv.org/abs/1512.02061 .
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$\begingroup$ Thank you so much for this explanation and the Blinder's paper. Is it a heresy to multiply the last two equations by sides to shorten the gamma factor and interpret the result? If not, how would you interpret it? $\endgroup$– MarcinCommented Jan 1, 2022 at 14:14
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$\begingroup$ @Marcin, no it is not a heresy. In SR its meaning is just that length contraction can be derived from time dilation (see en.wikipedia.org/wiki/Length_contraction, section Using time dilation). Generally, in SR and GR time dilation means always length contraction. $\endgroup$– JanGCommented Jan 1, 2022 at 15:39
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$\begingroup$ I get it and I agree, so I'm asking directly - is this correct? physics.stackexchange.com/questions/685237/… $\endgroup$– MarcinCommented Jan 1, 2022 at 15:46
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$\begingroup$ @safesphere Thank you, very helpful. Maybe you know if there is a justified postulate of the elementary and quantized spacetime volume? $\endgroup$– MarcinCommented Jan 1, 2022 at 16:27
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1$\begingroup$ @safesphere, thanks a lot for your corrections! $\endgroup$– JanGCommented Jan 1, 2022 at 16:55