Gravitational Time Dilation at Event Horizon

[mjordan2nd]

According to Wikipedia, the gravitational time dilation formula is given by

$$t_0 = t_f \sqrt{1 - \frac{2GM}{rc^2}} = t_f \sqrt{1 - \frac{r_0}{r}}$$

where

t0 is the proper time between events A and B for a slow-ticking observer within the gravitational field,

tf is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking 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),

and r is the radial coordinate of the observer.

Does this mean that if we get arbitrarily close to the event horizon we can make our time dilation factor increase arbitrarily without bound? Does this also mean that for an observer on the event horizon time never passes?

 

[Orodruin]

This is like asking if time would stand still when traveling at the speed of light. The formula you cite for gravitational time dilation assumes that the observer is stationary and there is simply no way for an observer to be stationary at the event horizon.

 

[mjordan2nd]

What about getting arbitrarily close to the event horizon? Would there be a point in space where, say, only ten years have passed since the creation of the black hole though the black hole looks billions of years old from Earth? What about on the other side of the event horizon? How does gravitational time dilation work there?

 

[Nugatory]

What about getting arbitrarily close to the event horizon? Would there be a point in space where, say, only ten years have passed since the creation of the black hole though the black hole looks billions of years old from Earth?

Yes. You cannot be stationary at the event horizon, but if you're willing to accept an arbitrarily large acceleration you can hover arbitrarily close to the event horizon and your time dilation relative to an observer at infinity will also be arbitrarily latrge.

What about on the other side of the event horizon? How does gravitational time dilation work there?

It doesn't work at all. There are no such thing as "stationary" inside the event horizon, so we have the same problem with defining time dilation as we have at the event horizon, and a further problem in that there is no possible way to compare the time on the clock inside the horizon with the time outside the horizon.