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I've recently become interested in the equation for gravitational time dilation, which is as follows:

$$ t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}=t_{f}{\sqrt {1-{\frac {r_{s}}{r}}}} $$

I understand the concepts of proper and coordinate time, along with the Schwarzchild radius. However, Schwarzchild's other contribution is griefing me, which is the $r$, or "the radial coordinate of the observer" based on the Schwarzchild metric.

I've tried researching this r, but everything I find makes it seem more confusing. If it is really a "sphere centered around the body", then it is not necessarily a physical entity, as I understand it. If that is so, then why does its circumference / radius have any bearing on time dilation?

Could someone direct me to the answer of what it is, and what it's purpose is?

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  • $\begingroup$ @CountTo10: It is ??? Do you have a reference for that? I thought it was the radius at which the circumference was $2 \pi r$. Space is warped around a massive object so measuring the distance to the center doesn't actually give you $r$. (Of course, for objects like the earth, which aren't neutron stars or black holes, $r$ is indeed approximately the difference to the center of the object.) $\endgroup$ – Peter Shor Nov 29 '16 at 3:16
  • $\begingroup$ Any coordinate in a metric is just that: a coordinate, which may or may not have a direct physical interpretation. Conceptually $r$ is the radius from the center. Quantitatively, it's the coordinate radius from the center, and behaves quantitatively very similarly to a normal radius, and identically at $r \gg r_s$ $\endgroup$ – DilithiumMatrix Nov 29 '16 at 3:27
  • $\begingroup$ Related: physics.stackexchange.com/questions/243103/… as @PeterShor points out in his comment above, thinking of r in terrestrial distance terms is incorrect in the region around a black hole. $\endgroup$ – user108787 Nov 29 '16 at 3:38
  • $\begingroup$ @PeterShor Yeah, the full quote was "the radial coordinate, measured as the circumference, divided by 2π, of a sphere centered around the massive body". But it was that last part that through me for a loop. But what exactly do you mean by "measuring to the center doesn't actually give you r"? $\endgroup$ – A. Forty Nov 29 '16 at 3:46
  • $\begingroup$ In Euclidean space, the circumference of a circle is $2 \pi r$. In curved space, it isn't necessarily. In Schwarzchild coordinates, the coordinate $r$ increases when you move farther from the center of mass, but the coordinate is defined by the circumference at that distance from the center, and not by the distance to the center. $\endgroup$ – Peter Shor Nov 29 '16 at 3:57