Picture a solid lead ball. Apart from the imaginary case where it's emerged in an infinite volume of water (in which case the water will not form a black hole), in which it surely will float (the gravity of the water pulls on the ball spherical symmetrically), maybe it can in another situation.
Assume the ball finds itself in a thick layer of water. The ground the water is resting on is flat and has an equal temperature everywhere, above zero degrees Celcius. The water is in thermal equilibrium with this ground, and, assuming the water surface is flat too, it's also in thermal equilibrium with the air above the water, which has a constant temperature which is lower than the bottom temperature.
It is clear that a (non-static) equilibrium develops. Heat is moving from the ground to the water surface.
But my main question is: Can a lead ball in these circumstances float in this massive and deep water or be accelerated upward at a certain depth?
Here one can find info about the compressibility of lead.
Here one can find the compressibility of water.
This is NOT a homework question.
Best Answer
There's currently an answer which observes that the density of water increases with pressure, and speculates that at some depth the density of liquid water becomes higher than the density of lead. That speculation is unwarranted. The linear approximation that liquid water's density increases by a few parts per million per atmosphere doesn't account for phase transitions. At temperatures below the liquid-vapor critical point, water becomes a solid at pressures above $10^4$ or $10^5$ atmospheres:
The "hot ice" phases, which occur at high pressure, have approximate densities of
All of these densities are much closer to the STP density of water (1 g/cm$^3$) than to the STP density of lead (11 g/cm$^3$) so it seems highly unlikely that the density of liquid water just on the low-pressure side of that phase transition is high enough to support lead. You could probably confirm that using the NIST Fluids Webbook.
So a sufficiently deep ocean on an ice-giant planet would develop a floor made of ice-VI or ice-VII, depending on the temperature there, and your lead ball would sink to this floor. If geological/hydrological/cryological dynamics made this hot-ice floor unstable, your lead ball would tend to sink into it as its surface evolved.
Also note that the highest-pressure phase, ice-XI, is less dense than ice-X. That inversion probably drives convection processes in the mantles of ice-giant planets, which would transport heavy metals like lead or iron down into the cores of those planets.