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What is the most time dilation possible outside of the event horizon of a black hole with the reference to us?

I know that time dilation is stronger if you are near an extreme amout of mass. It is also important how close you are to this mass.

So my first thought was: more massive black hole = more time dilation. But if a black hole gets more massive, the event horizon will also get bigger so you would also have to move away from the singularity. So more mass = more distance to the mass. Less mass = less distance to the mass.

Also: I don't need to survive this or having a stable orbit. Let's say I can just hover at a fixed distance from the event horizon with a timer equiped.

So my question again: What would be the greatest time dilation possible outside of the event horizon? If 1 minute has passed on earth, how many secound would have passed on a watch on the position of greatest time dilation if I look through an telescope with infinit resolution? (Also to counter the effect of lightspeed limitation, I start the timer on earth when I see that the timer on this position is startet)

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Time dilation with respect to a far away observer tends to infinity as you get close to the event horizon.

$$dt = {1 \over \sqrt{1- {r_s \over r}} }d\tau$$

Here $t$ is the time of a far away observer, $\tau$ is the time of an observer that hovers above the black hole at distance $r$, while $r_s$ is the Schwarzshild radius.

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  • $\begingroup$ It might be worth pointing out that this makes time dilation approach infinity as $r=r_s$ is approached. $\endgroup$ Sep 24 at 9:05
  • $\begingroup$ @AndersSandberg I though I had said this in my first sentence. Is it not clear? $\endgroup$
    – Prallax
    Sep 24 at 10:35
  • $\begingroup$ Just for me but isn't the gravitation force stronger inside the event horizon if you get closer to the singularity? shouldn't this mean that time dilation would also be stronger? But higher than infinity isn't possible. Does this make sense? $\endgroup$ Sep 24 at 11:28
  • $\begingroup$ @somedude324334 gravitational force is not a well defined concept in GR, so I cannot say whether it is stronger inside. Anyway, no observer can "hover" inside the event horizon (everything must fall towards the singularity), so it makes no sense to talk about the time dilation for a hovering observer inside the event horizon $\endgroup$
    – Prallax
    Sep 24 at 14:37
  • $\begingroup$ Keep in mind that the formula I gave in the answer only works for hovering observers. If someone is orbiting the black hole, or falling towards it, the time dilation will be different, depending on the actual trajectory. If you like, I can also cover these two cases in the answer $\endgroup$
    – Prallax
    Sep 24 at 14:39
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Time dilation is dependent on Two ways

  • SPEED
  • Gravity And is variable...

In regards to speed, the faster acceleration determines perceptable time movement.

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