Conditions at black hole's event horizon

This question had, at least partially, been discussed here before, but I feel that the record has not been set straight. There seem to be lack of agreement regarding conditions (like gravitational acceleration and time dilation) at the event horizon.

There seem to be one school of thinking stating that gravity and time dilation at the event horizon is infinite, and also other spacetime metrics can stop making any common sense. My feeling is that this is incorrect, because it points to physical singularity at the event horizon. While this singularity is predicted by the Schwartzshild metric, later works have demonstrated that it was an artifact of Schwartzshild's coordinate system and does not really exist.

Second school of thought is leaning towards the idea that gravity (and time dilation etc.) are becoming infinite only in central singularity, while anywhere else, including the event horizon, these parameters are actually finite. However, I could not find any numbers or formulas that can help a layperson like me understand the physical conditions at (or slightly above) the event horizon.

• What gravity looks like at the apparent event horizon is observer dependent. That's no different from what gravity looks like at Earth's surface: a free falling observer wouldn't detect any gravity (to first order), at all. They have no way of telling whether they are falling towards Earth or Jupiter or a black hole or if they aren't falling, at all. To second order there are substantial tidal forces. At the Schwarzschild radius of a stellar black hole these would be very "uncomfortable", but for a galactic black hole that would not be a problem for a human observer. – CuriousOne Jan 29 '16 at 22:06
• The event horizon is a coordinate singulaity which appears to be a singularity in a stationary refernce frame with respect to the black hole, away from the black hole. It ceases to be a coordinate singularity in the reference frame of a body falling into a black hole. The central singularity in a black hole is a physical singularity. – Peter R Jan 30 '16 at 0:22

Both schools, as described by you, are wrong.

There seem to be lack of agreement regarding conditions (like gravitational acceleration and time dilation) at the event horizon.

Gravitational acceleration isn't a thing. The metric evolves according to the Einstein equation, test particles move along geodesics determined by their tangents, and the stress-energy tensor evolves according to it's own evolution rules.

And time dilation is always between two things. In general, the time between two events is the proper time along a curve between two events so will depend on the shape of the curve as well as the regions the curve passes through.

There seem to be one school of thinking stating that gravity and time dilation at the event horizon is infinite,

At an event horizon, the curvature could be slight (it is smaller for larger event horizons), but the time dilation between the event horizon and spatial infinity (two regions) is actually straight up infinite. So the curvature could be small if you are far from the singularity, but the time dilation could be larger.

Think of time dilation as a function of how deep in a gravitational well you are at and think of curvature as a function of how curved things are. When the event horizon is bigger then you are farther from the singularity and have less curvature but the overall well is deeper and you are deeper in it (there was lots more curvature outside all spread out and you went through a lot of curvature already when getting this close).

When there was a big black hole, and you dealt with a lot of spacetime that was curved to get close enough to get to the event horizon.

any numbers or formulas that can help a layperson like me understand the physical conditions at (or slightly above) the event horizon

Outside a nonrotating uncharged spherically symmetric black hole with no gravitational radiation and parameter $M,$ the time dilation between a shell of surface area $4\pi r^2$ and spatial infinity is $\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}}.$ And you can see it approaches infinity as $r\rightarrow 2GM/c^2$ which is the event horizon.

But this time dilation is between these two regions, locally you don't notice it as locally everything is experiencing the same dilation. It's like you are moving slow, but so are your clocks and every atom around you. All you would notice is that things farther away seem blue and fast and the things closer in seem red and slow.

• Thank you for your detailed answer. Looks like the situation is more complicated than I originally described, but it seems like "first school" description is more correct - there are singularity-like conditions at the event horizon. Thinking of a "gravity well" - it seems like there are two wells - one (dot-shaped) at the center of black hole, and one, bubble-shaped, at the event horizon. And if Peter R is correct, bubble-shaped singularity does not behave exactly like a true singularity. – Alexander Feb 2 '16 at 0:57