# Spaceship orbiting a black hole under the event horizon - a prison where no one escapes

The relationship between the distance, $d$, of the event horizon from a black hole's center and the tidal acceleration, $a$, experienced by an object near a black hole allows for scenarios in which a human could - assuming no other forces than gravity - cross the event horizon un-spaghettified.

In fact, the tidal acceleration experience by a 2 m tall person near (100 km from) the event horizon of a black hole of 100 million solar masses is only:

$$a =2\cdot(6.67*10^{-8})\cdot(1.9*10^{-41})\cdot{200}/{(2.95*10^{13})^{3}} = 0.00020 \text{ cm/s}^2$$

(The event horizon has a radius of $295$ million km)

My question is the following:

Given the black hole at the center of S5 0014+81, with an estimated mass of $40$ billion solar masses, would it be feasible to have a spaceship survive, for a time, in a decaying orbit below the event horizon?

If so, what would it be like to be inside that spaceship, and how long would that orbit last? (in their FOR)

I want this spaceship to be an ULTRA high security prison where there is no hope of escape due to the laws of the universe.

Fun fact,

This black hole is estimated to live to be somewhere around $1.342*10^{99}$ years old before it dissipates by the Hawking Radiation. (wow!)

• If you want a prison that is absolutely, positively, impossible to escape from based on the laws of physics, why bother with a prison at all? – a CVn Jan 16 '18 at 23:03
• Once they are beyond the event horizon, they have been imprisoned beyond all possibility of escape according to our understanding of the laws of physics. I'm not even sure it is possible to have a discussion as to what it would be like on a spaceship "below" the event horizon. – Thucydides Jan 16 '18 at 23:05
• You're describing a death sentence. If you want to send people to die in a black hole, just toss them in. Why would you waste the money on a prison station? – kingledion Jan 17 '18 at 0:11
• There is no such thing as an orbit "under" the event horizon. The event horizon is the boundary where the direction towards the black hole becomes timelike; once the event horizon is crossed, the direction towards the black hole becomes "after" the event horizon. This is why the event horizon cannot be crossed back: it lies in the past. – AlexP Jan 17 '18 at 1:12
• I'm not usually one to worry about the premise of a question... but your society either (a) has one whomping religious conviction along the lines of "thou shalt not kill" or (b) they're a society of psychopaths who want people to suffer as long as astrologically possible. – JBH Jan 17 '18 at 2:49

# Calculations

In this question, I solved for the stresses on an spacecraft passing close to a black hole's event horizon. There isn't some magical barrier that you go through around an event horizon, you just can't get out; you probably wouldn't even notice. I can use the same process to calculate the stresses that a prison slightly inside the event horizon would face.

### Event horizon of a black hole

The event horizon is the distance from a black hole where the escape velocity is equal to $c$. Escape velocity is

$$v_e = \sqrt{\frac{2GM}{r}}.$$

A black hole with 40 billion solar masses will have mass $8\times10^{40}$ kg. Solving for r when $v_e=c$ gives $$r = \frac{2GM}{c^2} = \frac{2\cdot6.7\times10^{-11}\cdot2\times10^{38}}{\left(3\times10^{8}\right)^2} = 1\times10^{14} \text{ meters}.$$

### Gravity as a function of distance from the black hole

A person is 2 m tall, and 'orbiting' just inside the event horizon at $1\times10^{14}$ m from a black hole of mass $8\times10^{40}$ kg.

The tidal acceleration between the head and feet of a 2 meter tall person due to the gravity of the black hole is \begin{align}a &= \frac{m_{hole}G}{(r+2)^2}-\frac{m_{hole}G}{r^2} \\ &= 6.7\times10^{-11}\cdot8\times10^{40}\frac{1}{\left(100000000000002\right)^2}-\frac{1}{\left(1\times10^{14}\right)^2}\\ &=-2\times10^{-11} \frac{\text{m}}{\text{s}^2} \end{align}

### What would that do to a 1 km long cylinder?

Conveniently, in my other question, I calculated the tension on a 1km long cylindrical object. Conveniently, this could be a pretty reasonable prison space station. Near the ergosphere of a black hole, the tension forces would destroy any known object, but what about at the even horizon?

Following the same math in the other question, and with the same structural assumptions, I get the differential stress on any slice of the prison/station:

$$\frac{dF_{slice}}{dl} = \frac{2\times10^{35}}{(1\times10^{14}+l)^2}.$$

Total net force on the rod is $2\times10^{10}$ N, and maximum stress of about $1\times10^{12}$ N. Working backwards using the equation for gravity, we see that the assumption is that the station has a mass of 40000 tons. Depending on the cross-sectional area of the load bearing parts of your station, we can calculate the stresses. If your station is a cylinder 100m in radius, and 1/10 of the available area is taken up by load bearing structures, then the maximum stress on the 3000 m$^2$ of load bearing structure is about 6 MPa. A common structural steel has a yield strength in tension of about 250 MPA, so this isn't too much. If you have the technology to build space stations inside a black hole, then it is reasonable that you could construct it out of materials that won't fall apart.

The second question is how long you can maintain your orbit. Using simple Newtonian mechanics (Warning! Not valid near a singularity!) the gravitational pull of $2\times10^{10}$ N will have to be counter-acted by thrust. Now, that is a lot of thrust, about three orders of magnitude greater than a Saturn V. I suppose it really depends what sort of propulsion system you have. Thrust as a function of mass flow rate is given as $T = v\frac{dm}{dt}$. Assuming exhaust at the speed of light from some magical propulsion system, you still need 70 kg tons of propellant passed every second to keep from falling into the black hole.

# Conclusions

Given the small tidal acceleration, even a large object (1 km long, 25000 tons) could reasonably be kept together with known materials at the event horizon of such a large black hole.

As for keeping such an object in orbit, for any propulsion system with reaction mass, the propellant usage would be very large (about 250 tons per hour, as calculated above). Given that the mass of the whole station is 40,000 tons, you would burn through the entire station's mass in a week. Just like the tyranny of the rocket equations, the tyranny of a black hole's gravity is oppressive: the more propellant you keep on board, the harder you are pulled in and the more propellant you need. I suppose you could be refueled with propellant, but that is a lot of money to be literally throwing into a black hole.

For some sort of reaction-less system, well, I don't know how to measure that. You can't gain momentum out of black hole, so I don't know how you could calculate the thrust given off by a photonic engine. In any case, the 'not falling into the hole' part seems to be the catch, with any propellant based system. Doesn't seem very reasonable, given relatively hard science constraints.

• You can actually orbit the singularity of a rotating black hole without it decaying, or so I hear. – PyRulez Jan 17 '18 at 17:26
• The innermost stable orbit for a black hole is the photon sphere. Any orbit below that is unstable for non-rotating black holes. For rotating black holes, the lowest stable orbit approaches the event horizon (seeing that the object is travelling in the same direction as the rotation) However, no orbit of a singularity in any conditions (as far as i'm aware) can be stable below the event horizon. – Austin A Jan 17 '18 at 17:32
• "There isn't some magical barrier that you go through around an event horizon" — that is, unless the firewall hypothesis is true. – celtschk Jan 19 '18 at 16:52
• All future directions lead to the singularity. There is no amount of thrust that will keep you from falling in in a finite amount of time. – PyRulez Feb 12 '18 at 4:19

In order to maintain an orbit you need to be going fast enough to not plummet towards the singularity. Orbiting after all is basically just falling so fast you miss the planet(or given celestial body).

Now in order to be below the event horizon and not be sucked into the singularity you'd have to be orbiting extraordinarily fast. You'd probably have to go faster than light in order to maintain such an orbit considering not even light has the capability of escaping the event horizon. And light doesn't even have to with the mass that you're having to drag up! I'm sure you're aware that objects that have mass cannot achieve the speed of light. E=mc^2 and whathaveyou.

So in short, no, it's not really feasible to have even a decaying orbit below the event horizon.

But let's say for a second you've invented faster than light travel and are somehow able to not immediately fall towards the center and also everyone on the ship isn't dying for some reason.

What the crew would experience is something called Gravitational Time Dilation. What it says is that the amount of time that has elapsed differs between two observers depending on how far each of them are from a gravity well. The closer one is, the slower time passes. Now this doesn't mean the crew and prisoners would feel like they're going in slow motion, it means that if they could see outside somehow, time would appear to be passing faster. This would only be exacerbated by the fact that they are going faster than light which would itself cause its own form of time dilation. The end result is that what could only be an hour to the people in the prison could be eons to the people outside.

As for the laws of the universe, well, if the only way of having even a decaying orbit is go faster than light then hypothetically the prisoners could riot, hijack the prison ship, and high tail it out of there.

My final notes would be that however long the orbit lasts depends on how wide the orbit is and how much fuel you have to maintain it. Although since a singularity takes up no space at some point the ship will just appear to be spinning around the center for an eternity before it hits the center.

https://en.wikipedia.org/wiki/Gravitational_time_dilation

https://en.wikipedia.org/wiki/Time_dilation

It is not terribly plausible, but it is mathematically possible for a ship to survive indefinitely in the proper sort of black hole.

If we're talking about a basic Schwarzschild black hole--uncharged and non-rotating--then no orbits are possible. Once you cross the event horizon, you will hit the singularity in finite time. Now, that in and of itself is not necessarily a problem--after all, if the fall time were longer than the inmates' expected lifespans, such that they'd all die of natural causes before hitting the singularity anyway, that sounds like a perfectly good deal. And it turns out that, the bigger the black hole, the longer you have to live, and furthermore that there are things you can do with rockets to extend your subject time.

Unfortunately, the maximum lifetime you get even with supermassive galactic core black holes is on the order of hours, not decades.

So, we need a different kind of black hole. Black holes that are charged, rotating, or both contain a second inner horizon, where the radial coordinate switches from timelike back to spacelike again. As a result, while you will inevitably cross from the outer horizon to the inner horizon in a finite time, once you have crossed the inner horizon you can in principle avoid the central singularity. It is well established that photons can have stable (spirally, non-circular) orbits in this region, and work by Russian physicist Vyacheslav Dokuchaev indicates that there is no fundamental reason why massive particles--including spaceships or even whole planets--could not also maintain stable (non-equatorial, non-elliptical) orbits around the singularity, below the inner horizon.

Of course, those predictions are based on idealized cases; actually adding extra massive particles to the universe besides the black hole singularity itself complicates the metric, and the inner horizon is not particularly stable. As such, not everyone agrees with Dr. Dokuchaev that an inner habitable region would actually exist within any real black hole. But, it's good enough for sci-fi!

So, your prison ship's orbit can theoretically last indefinitely. What would it be like? Well, if you have windows, the exterior view would be trippy. Space is pretty warped in there, to the point that you don't just get "normal" gravitational lensing--photons follow weird spiralling paths, so where you see other objects through the window bears very little relation at all to where anything actually is outside. On human scales, inside the ship, however, space would still be geometrically flat enough not to cause any real problems. The inmates would experience tidal gravity that would pull them towards the ends of the ship pointing towards and away from the singularity, the precise strength and direction of which would change over the course of the non-planar orbit; but, those forces would be weaker near the center of gravity of the ship than towards its extreme ends. If the ship rotates to provide spin gravity, and controls it orientation so that it's rotation axis is always aligned with the line towards the singularity, those effects could be almost entirely eliminated. And, you can make them as small as you want anyway by just picking a suitably heavy black hole, rotating suitably quickly, so that the inner horizon is at a great enough distance from the singularity to allow for orbits at a great enough distance to make the tides negligible on the scale of the ship.

It is generally agreed that crossing event horizon by itself should be harmless to the traveler. However, conditions inside the event horizon are very much up to the debate. Physical equations point to some very weird results, like space inside the event horizon starts to behave like time, and time starts to behave like space.

So, the spaceship can possibly cross into event horizon without immediate destruction. But it is not possible to tell what would happen to it inside, and how soon something can happen to it, either from outside observer's or traveler's point of view.

Our real knowledge of the conditions inside a black hole is so limited, that the depiction of them in the movie "Interstellar" can very much be true. "Why not"?

The answer is your own hands. If you can calculate the tidal forces experienced over two metres at a distance of one hundred kilometres above the event horizon. Surely you do a similar calculation for the tidal forces over a distance of two metres, say, one hundred kilometres below the event horizon.

In which, you would know the situation for persons on spaceship orbiting the centre of mass of the supermassive black hole. Remember spaghettification only occurs when the tidal forces are capable of rendering people and things asunder over short distances.

One thing about supermassive black holes often overlooked is that a spaceship can pass through the event horizon without realizing it had done so. The event horizon isn't a physical barrier, it is a surface of gravitational potential where escape velocity is equal to the speed of light. Spaceships should be able to orbit just inside the event horizon as if it was a normal orbital vehicle. It's just that it can never escape from the interior of the black hole.

One caveat: This would be an excessively expensive method imprisoning anyone. It would be a major expenditure for a galactic economy. It would have to be more of a grand symbolic gesture than an effective prison system.

I remember sth that time and space are so dilated the traveller would never get very far in the black hole. Not sure if that was a scientific opinion.

Anyway most black holes are small, so the gradient in gravitation between someone's head and feet could be large enough to rip him into pieces long before he reaches the event horizon from the outside. Inside, I guess this problem will grow worse. Although that could take a long time if what I say above is correct. Cruel and unusual? :-/

• Not so. Supermassive black holes (SMBH) have diameters of light hours. Yes imprisoning someone inside even a SMBH does constitute cruel and unusual punishment. – a4android Jan 17 '18 at 1:42
• @a4android OK, I rather thought about your neighbouring stellar black hole, which is anyway more accessible. – Karl Jan 17 '18 at 7:25
• I know what you mean. The question isn't concerned with accessibility. The supermassive black hole S5 0014+81 is 3.7 gigaparsecs distant. Getting there would be a major undertaking in its own right. There must be easier ways of imprisoning someone or something. – a4android Jan 17 '18 at 11:38
• I get from the other answers that I'm basically right, but quantiatively off by a few orders of magnitude. Still, for a novel ... ;-) – Karl Jan 18 '18 at 10:47