On this post,
Time according to the gravity of Sagittarius A?
They found the factor of time on the varying distances from the SMBH, I am curious on how to turn those factors into relatable terms such as: At 1 km from the event horizon ____ time passes on earth in an hour.
I'm just not sure how to figure this out.
I am curious on how to turn those factors into relatable terms such as: At 1 km from the event horizon ____ time passes on earth in an hour.
The key here is to understand what the formula relates to. The (all important) observer is in this case a notional observer so distant that curvature (and hence gravitational effects) are negligible.
That observer is the "base clock" we reference. Everything is dilated relative to that clock.
That means that the dilation on Earth (not strictly speaking an ideal distant observer) has to be taken into account to get an answer in the form you mean. So starting with the formula you had :
$$t = t_\infty \sqrt{1 - \frac{r_s}{r}}$$
We want this relative to Earth so we need :
$$t_e = t_\infty \sqrt{1 - \frac{r_s}{r_e}}$$
And combining youy get :
$$t = t_e \sqrt{ \frac {1 - \frac {r_s} r} {1 - \frac {r_s} {r_e}} } = t_e \sqrt { \frac { r_e(r-r_s) } { r(r_e-r_s) } }$$
Now in the specific case of Earth and Sagitarius A*, $r_e$ is huge so the dilation factor for Earth is as close to one as makes no difference in practice, so you don't really need to go to all that trouble and you can use $t_\infty \approx t_e$.
