# Is it possible to extract the relation between the flow of time and the density of space from GTR? [closed]

Is it even possible to formulate such relation?

• What you mean with "density of space"?
– JanG
Commented Dec 31, 2021 at 15:52
• There is gravitational time dilation. In analogy, I think of gravitational length contraction. Commented Dec 31, 2021 at 15:57
• To understand you properly, can define the meaning "flow of time"?
– JanG
Commented Dec 31, 2021 at 17:09
• I mean the pace of flow of time, that is slower on the surface of the planet in comparison to the intergalactic space. Commented Dec 31, 2021 at 18:19
• 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!
– JanG
Commented Dec 31, 2021 at 18:45

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. $$$$d\tau=\sqrt{g_{00}}~dtdr~.\tag{1}$$$$ In particular, for Schwarzschild vacuum metric, it yields $$$$d\tau=\sqrt{1-r_{S}/r}\cdot dtdr,\tag{1}$$$$ where $$\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 .