How do you get time dilation from $g_{00}$ in general and from the Schwarzchild metric in particular?

Question

1.Why is the observer at the bottom measuring the proper time rather than the coordinate time?

2.How do we go from time dilation to $g_{00}$ or from $g_{00}$ to time dilation?

I'm trying to understand the following passages.

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line.

Coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event.

-https://en.wikipedia.org/wiki/Proper_time

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes space-time in the vicinity of a non-rotating massive spherically symmetric object. The equation is

${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}=t_{f}{\sqrt {1-{\frac {r_{s}}{r}}}}=t_{f}{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}=t_{f}{\sqrt {1-\beta _{e}^{2}}}<t_{f}}$

where $t_{0}$ is the proper time between two events for an observer close to the massive sphere, i.e. deep within the gravitational field.

Are the events also close to the observer and massive object? It seems so.

$t_{f}$ is the coordinate time between the events for an observer at an arbitrarily large distance from the massive object (this assumes the far-away observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate).

-https://en.wikipedia.org/wiki/Schwarzschild_metric

Answer 1

It's most important to focus on question 2. I'll work with $c=1$.

Time dilation is computed as $\sqrt{ds^2/dt^2}$ on suitable assumptions of $dx^i/dt$. Special relativity in Minkowski space gives time dilation due to motion, say with Cartesian coordinates $x^i$ satisfying $dx^i=\beta^i dt$; using this in $ds^2=dt^2-dx^2$, $\sqrt{ds/dt}=1/\gamma$. In general relativity, time dilation can happen even to motionless bodies with $dx^i=0$, so $ds^2=g_{\mu\nu}dx^\mu dx^\nu\implies\sqrt{ds^2/dt^2}=\sqrt{g_{00}}$.