In the most widely known black hole theory, what you are supposed to get at the centre is a singularity.  If you can describe the physical significance and empirical experience of such a singularity, please go ahead; a lot of us will be listening respectfully.

But think of a mass in free fall, roughly a radially symmetrical sphere, perhaps our planet. At its surface the gravitational field attraction is the mass of the planet times your mass, divided by the square of the radius. Dig down some distance, and the new gravitational field attraction is the mass of the sphere below your feet times your mass, divided by the square of the radius of the sphere below your feet. 

Simple, right?

And depending on the successive densities of the layers, your weight will variously rise or fall, but eventually it must fall, because at the centre the attraction of the ball beneath your feet is zero, and that will not generally happen all at once.

And definitely, at no point do we find that the attraction suddenly goes negative. At no point does something gravitational suddenly start pushing us upwards as a south magnetic pole would push away another south pole, right? 

So at no point is there a spike to your gravitational well, as you have described. 

In other words, when you reach the centre and find yourself in effectively zero gravity, you are not on a spike, but on a level floor of zero area, or more reasonably, at the low point of a curve that I guess would approximate a paraboloid if you were inside a symmetrical sphere of constant density. Anyone who would like to work out and explain the maths is welcome!  :-) 

Now, that black hole singularity seem to me unlikely to change anything unless we assume that the density of the inside reaches infinity. But why should it?

You tell me!

Jon

My interpretation of these gravity maps is that the depth represents the total energy required to climb out of the well at any point (search for Escape Velocity in TheLastWord for a related topic). They do not represent the gravitational field at any point, but rather the integration of that field during approach to the object down the well.

In that case, reaching the object and getting zero gravity gradient just adds zeros to the value at that point - it does not reverse the summated energy.

I think the confusion may lie in mistakenly thinking of the depth of the well as representing gravitational field. It's the gradient of the well which causes the marble to accelerate and so would represent gravity. The shape of the rubber sheet would be a cubic rotation above the surface of the body and a paraboloid below the surface. The gradient would be zero at the centre so there is no need for a spike.

Within a black hole there would be no paraboloid and the cubic rotation would go right to the central singularity where the gradient would be infinite representing infinite gravity and spacetime curvature. What happens within a Planck length of the singularity is anybody's guess!