The short answer is, yes, you do meaningfully experience the interior of a black hole (and then inevitably meet the singularity at the center). It is also true that one of the dimensions of space and the dimension of time switch roles inside, but there's some confusion as to what that means, which I'll describe in more detail.
What happens is that inside the horizon the inward direction starts acting like the future direction in regular flat space-time. What this means for someone going through the horizon is that they are "forced" to move further inward, in much the same way as you right now are being "forced" to move toward tomorrow. In other words, your future lies at the singularity -- once inside the horizon you cannot avoid it.
To give some explanation for why this is the case, let's start with what we call the "space-time interval". The space-time interval is a way to define distances in space-time, just like how the familiar Pythagorean theorem defines distances in 2D Euclidean space. In 2D space we may write the squared distance asNow, how to do apply this in space-time? Isn't time totally different from space, even with the units? The trick is to convert the time into a distance, and we can do that by multiplying by the speed of light. (speed)x(time) = (distance). Since the speed of light is the same in all reference frames, this conversion factor of c works the same for everybody. Then, with time and space now having the same units, we define the distance in space-time asIt's basically the same thing as the Pythagorean theorem, except for the notable difference that there is a minus sign instead of a plus sign. And it doesn't really matter if you say it is x^2 - (ct)^2, or (ct)^2 - x^2. All that matters is that you choose one of terms to have the opposite sign as the other, and then stay consistent with that choice. Then what relativity says is that although different observers will disagree on what they measure for distances x and times t between events, everyone agrees on the space-time interval, s^2. And this little equation basically contains everything about special relativity, including the effects of time dilation and length contraction.
Going back to that minus sign, this is what makes one of the directions of space-time act differently than the others. Whichever direction has the opposite sign of the others is the "time-like" direction, while the other dimensions are "space-like".
What a gravitational field does is distort the shape of space-time, and inside a black hole the signs end up switching places. Let's write the space-time interval for a black hole to see why. Yes, this looks scary, but stay with me and don't panic.
The first term has t and represents the time coordinate. The second term has r and represents the radial coordinate (toward or away from the singularity), and rs
is the specific radius of the event horizon. Third is θ which is like latitude. Finally, Φ is like longitude.
Outside the black hole, r is greater than rs
, so the first time has 1 minus a fraction less than 1, which is positive, and then a minus sign in front which makes the entire first term for the time coordinate negative. The second term for the radial coordinate will be positive, and so will the third and fourth coordinate directions. So the time coordinate is timelike and the three spatial coordinates are spacelike, just like in regular space-time. But once we go within the horizon, r is smaller than rs
, and so the first term will become positive while the second term becomes negative. That's why the behavior of these two directions switches, and the radial direction acts like time. It doesn't mean we are free to move in time, but rather that we are forced to move inward. Moving outward is as impossible as moving backward in time.
One last principle of relativity we can call on here is that physics in all inertial frames of reference works the same. You cannot tell the difference between floating in a spaceship which is far from any gravitational field, or floating in a spaceship which is falling near the Earth. And so too, with freefall into a black hole. If you fell into a very large (supermassive) black hole inside a small box, then for the entire trip to and even inside the event horizon you would notice nothing
unusual. Just weightlessness the whole time.
(There is a caveat in that close to the singularity, tidal forces become important, and eventually you will feel and be destroyed by them. But for a very large black hole these forces are negligible at the horizon.)
So in a real sense, the horizon of a black hole, and even most of its interior, is not a very strange place as far as physics and space-time are concerned, provided that your view is limited to only a small region which is falling into it along with you -- like the inside of the small box. If you extend your view very far then the effects of the tidal forces start becoming noticible, just as you would notice things falling in different directions if you compare very distant locations on the Earth. General relativity teaches that these tidal forces are
the direct manifestation of curved space-time, and thus are the natural way to measure a gravitational field.