You can calculate that easily.
Although black holes sound very interesting and mysterious in science fiction, they are actually probably one of the least interesting celestial objects to study, simply because their actual properties are pretty minimal and their behavior at the macroscopic level is already very well characterized under General Relativity; note that as long as you are far enough away that your trajectory doesn’t pass through the ergosphere of the black hole, from a gravitational standpoint it may as well just be a really big star. Where black holes get interesting is at the interface (event horizon) where the effects of gravity and the other forces (nuclear and electromagnetic) have influence at close to the same order of magnitude, which gives rise to a number of phenomena, chiefly so-called Hawking radiation, that we cannot otherwise reproduce in any terrestrial environment. This may also give us other insight into how gravity acts at a quantum scale provided we could get close enough to observe the effects.
Unfortunately, for a stellar mass black hole that means being able to tolerate extreme gravity gradients that are unsurviveable. With a much larger supermassive black hole, like that in the center of spiral galaxies, the definition of the event horizon is much less well defined and those events only occur very infrequently compared to the area of the surface of the event horizon, hence why Chonos would select the smallest, rather than the largest black holes. Another consideration is that the radiation environment in the plane of rotation of a large black hole is very likely lethal if it has acquired enough mass to make a accretion disk; it would essentially be like being in a large plasma chamber with spontaneous generation of high energy X-rays from bremsstrahlung effects and other interactions with the local magnetic field. It would be better from an observational standpoint to be stationary above the plane of rotation of the black hole, but then you’d obviously need some way to maintain position against the acceelration of the hole’s gravity that isn’t dervied from a crappy Disney movie.
Although the energies and electrohydrodynamic conditions around a stellar mass black hole are much higher than what we can produce in a lab today, in practice it may not be nearly as useful to observe just because we would have no control over the location where specific phenomena occur. Although the energies produced by particle collisions in the Large Hadron Collider are serveral orders of magnitude below the what we see from very high energy cosmic rays in the upper atmosphere, with the LHC we can cause the collisions to occur at very specific locations where we have detectors by controlling the beam with powerful magnets. Even then, billions of collisions are required to statistically confirm the creation of the Higgs boson or pentaquark particles, which is why researchers had to wait for years of operation before announcing their discoveries. Actually what is seen is not the particles themselves, as they don’t look like anything special and last only tiny fractions of a picosecond, but the characteristic debris of their decay, which is like sifting through a landfill looking to reconstruct a package of gum wrappers. (BTW, this has resulted in computational systems and methods to search enormous datasets for very small and infrequent patterns using advance statistical procedures, which has given rise to the ‘Big Data’ revolution, massively parallel data processing, high throughput large data handling, and very sophisticated data visualization systems; basically all of the technology used in modern day Internet commerce, global climate simulation and visualization, and an entire host of other applications. Next time some ignornant fool bitches about the billions of dollars “wasted” on the LHC and the search for fundamental particles, you can point out that nearly all of the supporting technology translated directly into commercial applications that now support multi-billion dollar industries.)
Examination of effects at the ergosphere of a fast rotating black hole also may be very useful in confirming or refining some of the more esoteric predicitons arising from theoretical and computational models of Einstein gravity and general relativity theory, but since you have a teleporter that instantaneously transports you across non-local distances you’ve implicitly kind of put a kink in some of the underlying assumptions of general relativity to begin with, which is always a problem when you introduce technomagical processes into an otherwise rational universe. The intellectual gyrations you have to go through to explain why you can’t simply beam the away team back at the first sign of trouble, or better yet, just transmit disposable clones of critical crew members is just about the least of the gaping plot holes given birth by poorly thought out scriptwriting conceits.
The most intersting objects to examine from a purely scientific perspective, other than enormous large scale cosmic structures, would be neutron stars and other objects balanced on the narrow edge between gravity and particle physics as we currently understand them. While black holes are mathematically simple–almost trivial in the case of the Schwartzchild metric, and easy enough to solve in a linearized form for simple Kerr metrics–objects that are composed of very densified but discrete particles may have some phenomenally interesting properties and interactions which could allow us to better understand the early evolution of the universe.
On a side note, while having a teleporter might seem to be appealing in terms of the ability to colonize other planets, the reality is that even if terrestrial-type planets (rocky worlds in the assumed habitable zone of a F, G, or K class star) are common, the precise conditions to support human and Earth life–specifically the atmospheric composition, balance of temperature variation, and a well-governed and stable hydrologic cycle–are likely to be vanishingly rare. The notion of finding a planet that we can live on unprotected is almost wholly improbable. It makes much more sense to develop the technology and infrastructure to create artificial habitable structures in space that use materials efficiently and have a stable, self-governing thermodynamic and hydrological cycle, or alternatively, alter our own bodies to be able to tolerate a much wider variety of environmental conditions and hazards without rapidly breaking down and collapsing just because we don’t have access to piddling resources like an oxygen/nitrogen atmosphere or uncontaminated water.
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