/: Absolute Zero
Don't think too hard on this one. I can't come up with an elegant way to ask this.
Objects with mass have potential energy due to gravity fields they are exposed to. This potential energy is usually calculated as relative to some zero, which could potentially be infinitely far away from all other gravitational fields, establishing a sort of absolute potential energy, which ironically will always be negative. Another way to think of it is how energy in debt the object is compared to what it needs to achieve escape velocity.
Now, if this object is very near the event horizon of a black hole (not that I know of any other kind), it's absolute potential energy will be enormous. Will it be comparable to the energy the rest mass contains? If so, then from this perspective, the object must contain no energy and thus ceases to exist if it crosses the event horizon. Unfortunately, I don't recall the equations well enough to work this out, so I'm being lazy and seeing if the LJ community can solve it for me.