How is strong time dilation consistent with weak tidal forces? Nolan's latest film, Interstellar, takes pains to explain to lay audience members that the passage of time slows in the presence of strong gravitational fields (as per Einstein's theory of General Relativity). While this certainly makes for an innovative plot device, the way this conceit is portrayed in the film seems inconsistent. Specifically, how can the gravity near Gargantua be so enormous as to produce order-of-magnitude differences in the passage of time, yet at once be so insignificant as to effect the astronauts in no other way whatsoever? At the very least, shouldn't the astronauts be crushed and/or ripped to shreds by tidal forces? How is it that they can saunter around Miller so easily?
 A: The simplest reason for this is the fact that gravitational time dilation is governed, to leading order (in the zero-spin case for simplicity), by the factor $\sqrt{1 -\frac{2GM}{c^{2}r}}$.  Now, just to make our measurements easier, let's rewrite the mass in terms of the radius of the event horizon:
$$r_{0} = \frac{2GM}{c^{2}}$$
Now, our time dilation factor is: $\sqrt{1 -\frac{r_{0}}{r}}$
Meanwhile, for a head-on course with the black hole, and to leading order, we get that the radial acceleration and the tidal forces are proportional to $\frac{GM}{r^{2}} = \frac{r_{0}c^{2}}{2r^{2}}$.  Note that when $r \rightarrow \infty$, you have the time dilation factor tending toward 1 and the tidal forces tending to zero, as you'd expect.  
But, here's the thing.  If I double my distance from the black hole, so $r \rightarrow 2r$, then the force will be reduced by a factor of 4, while the time dilation will be reduced much more slowly ${}^{1}$.  So, this means that it is perfectly possible for time dilation to be a strong effect while tidal forces are still weak, just because the time dilation effect falls off more slowly.
${}^{1}$let's say I go from $2_{r_{0}}$ to $4 r_{0}$, then the time dilation factor will change from $\sqrt{\frac{1}{2}}$ to $\sqrt{\frac{3}{4}}$, for a change factor of $\frac{1}{\sqrt{3}}$, which is certainly a much less aggresive factor than dividing by 4.
A: time slows in the presence of strong gravitational fields
It's not the gravitational field that determines time dilation, it's the gravitational potential. The Newtonian approximation really isn't correct here, but let's use it anyway for insight:


*

*The potential falls off like $1/r$ with distance $r$.

*The field falls off like $1/r^2$.

*Tidal effects go like $1/r^3$.
To get a large potential and small tidal effects, you clearly want a big $r$. That's what you have with a supermassive black hole, since its event horizon is big.
