"Interesting idea of time travel. I don't know I we really will be ever able to cut a sphere from our spacetime continuum, but who knows?"
We can check it experimentally. The vacuum hole is the absence of space-time, therefore vacuum holes able to cut space-time. Thus if we create vacuum holes between two objects, it must increase their isolation one from another. But we must eliminate problems with diffraction, since particles with large wavelength may overcome barrier using the wave properties. The large hole surface can shield even gravity and change the mass of objects.
"On the other hand, if the world splits, as in the Many Worlds Interpretation, disconnecting a sphere from our spacetime wouldn't allow also travel in other versions of our world too? Or in other universes, or connected components different than our own?"
The world splits at the time travel only. If an object cannot appear in our space, it appears in another universe. Hole teleportation is a sole method that allow travel to other universes. The mankind must populate all parallel universes.
"But I think that it is also very dangerous, since the time traveler may remain disconnected forever."
The teleportation in space is safe, because a traveler appear in our universe, and in the same time. But at Time travel in the past traveler may disappear forever - he remains in its new universe (if the Many Worlds Interpretation is true). Although, now I search for new solutions at the time travel in the past, the problems with travel backward in time are very complicate. I prepared this paper in a hurry because I found the FQXi page very late and I has information that essay concurs will be closed after 1 November. Therefore later I'll rewrite the paper with more detailed explanation of theory.
On the other hand, I have difficulties understanding what you mean by the change of metric during the time. Assuming a change in the metric implies another metric, a reference metric, comparing to which we measure the change.
I have proposed this hypothesis in order to solve problems with navigation in time and time scale. How we can adjust the Time Machine for a one-year jump, or one-million years jump? The main difference between past and future is the rate of cosmological expansion. But we cannot use the expansion of space for adjustment of Time Machine because the space is "nothing". Therefore I have proposed a hypothesis that the cosmological metric expansion of space could change the sizes of material objects during a large time interval that cannot be measured by a local observer because its body and measuring device changes in the same proportion. There is an (theoretical) opinion in science that nuclear forces in nuclei and chemical bonds between atoms and molecules do not permit increasing of the object sizes at cosmological expansion. But I don't know any scientific experiment able to prove or refute experimentally this idea; Nevertheless, the cosmological expansion acts even inside of nuclei and molecules. Scientists often compare the cosmological expansion of Universe with an expanding air-filled rubber balloon. If we expand a rubber balloon with the picture, it increases all the distances in the picture, including the sizes of objects. Imagine that during 1000 years all macroscopic objects on Earth increase in sizes by 1 millimeter because of cosmological expansion of space. We cannot observe this effect because it is very small and slow, and observer's body and measuring device also increase by 1 millimeter. This effect can observe the Time Traveler only, because he can compare its old meter with the new meter. In other words, Time traveler only has a reference metric, comparing to which he measures the change. If this hypothesis is true, it allows adjusting the Time Machine for jump over given time interval by contracting or expanding the objects (for example by heating or cooling). By the word, in some SF movies like "Back to the Future" the Time Machine is cooled at time travel.
There are a lot of phenomena able to change the object's sizes: Lorenz contraction, gravitational contraction, object's change at heating. The gravitational contraction is an example where space metric can change the object's sizes. Therefore cosmological metric expansion of space also may change the objects; it is a hypothesis only, common sense often deceives us. If it is a wrong idea, we must look for another way to create a Time Scale for the Time Machine. We cannot use the radiocarbon-like datting to adjust the Time Machines.
Sincerely,
Constantin Leshan