Socio-economic effects of a black hole on a civilization [closed]

Question

Suppose a "solar system", or "black-holar system"?, in which the star is replaced by a black hole. There is no source of energy, I know, but somehow an advanced civilization managed to live in this place. This civilization is spread among all the planets in this system. Due to the strong differences in the curvature of space-time, each planet has very different conditions when we look at the system as a whole yet they are small enough such that locally things are not very different between one another. One of the effects is that the time runs much slower in the innermost planets than in the outermost planets. Yet this civilization inhabits both planets and they interact and they trade, etc.

So imagine for a while that such a thing is possible (maybe there is also a star orbiting, I don't know), I want to discuss about the socio-economic effects rather than the physiological aspect, which has been discussed in other questions.

E.g. time running at very different rates implies that perhaps time itself is a valuable good to trade. From the point of view of someone who lives in the innermost planet, people in the outermost planets can do things in a very short time, so probably manufacturing industries would be in the outermost planets. From the point of view of someone who is in the outermost planet, things run very slow in the innermost planet so he can send things to store there with much less aging than in his own planet. So people in the innermost planet could sell this no-aging-storing service.

As another example: Tourism will probably be highly affected. People in the innermost planets can vacation in their own planet as we do on Earth. But someone in the outermost planets cannot go to the innermost planet for vacation as his/her/its boss will get very angry if he/she/it takes a lot of vacations. On the other way around, people in the innermost planets would be happy to vacation in the outermost planets as they can make, e.g., a single day in their jobs equivalent to one or two weeks in the outermost planets. If they take two weeks in their jobs, they can spend two or three months in the outermost planets in vacation.

What other effects would this have in economy and society?

Answer 1

TL;DR: for a planet in a stable orbit around a supermassive black hole, time dilation won't get any more dramatic than about 20% slower than a distant observer. Larger time dilations are only associated with unstable orbits, which end in death by spaghettification or being flung out into space. It is impossible to get high time dilations on something the size of a planet around a stellar mass black hole.

Ultimately, this is good for your question, because questions of the form "given weird circumstance X, please list every possible socioeconomic consequence" are far too broad for this site and will be closed!

Welp, there it goes.

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Here's the problem. You can work out gravitational time dilation in the simple case using this formula: $$d_f = \sqrt{1-\frac{2GM}{rc^2}}$$

where $G$ is the gravitational constant, $M$ is the mass of the body of interest and $r$ is the distance from the barycenter.$d_f$ is always less than or equal to 1, and greater than zero. We experience gravitational time dilation from the sun of the order of 1 part in 100 million (compared to flat space), for example... $d_f$ would be 0.99999999.

If you already know your desired dilation factor $d_f$, you can rearrange this formula to get the orbital radius at which you'd experience that time dilation: $$r = \frac{2GM}{c^2(1 - d_f^2)}$$

(note that as $d_f$ tends to zero, time dilation becomes infinite, and $r$ becomes equal to the Schwarzschild radius, which means you've hit the event horizon and all bets are off)

If the black hole had the mass of our Sun, and your desired time dilation factor was 1/14 (to get your one day on the inner world equals 14 days on the outer world), you end up with an orbital radius of 2.96km, which is obviously a little problematic for something that's the size of a planet.

If the black hole were more like Saggitarius A*, which is millions of times heavier, you get an orbital radius of more like 11.87 million kilometres. This is a little larger than the Schwarzschild radius which would be more like 11.81 million kilometres.

Unfortunately, this is very much inside the limit of the Innermost Stable Circular Orbit, which is no smaller than three times the Schwarzschild radius. This means either your planet is either going to spiral in to the event horizon and inevitable doom, or will be flung out into space. In no case can you have a planet which anyone could visit with a 14x time dilation.

Now, for all black holes which have a Schwarzschild metric, the time dilation at a given radius $r$ where $r$ is expressed in units of Schwarzschild radii is the same: $d_f = \sqrt{1-\frac{1}{r}}$. The 3-Schwarzschild radius limit would therefore have a time dilation factor of ~0.817, ie. time will appear to run about 20% more slowly than for a distant observer. This would be the maximum gravitational time dilation in a stable orbit for a non-rotating Schwarzschild black hole of any mass.

Answer 2

For the gravitational time dilation to have any visible effect, the black hole would have to be very massive. Since the star has been replaced with the black hole, its mass must remain about the same (otherwise the orbits of the planets would be destabilized). Consider that the Schwarzschild radius of one solar mass is about 3km; stable orbits are possible beyond 9km radius and at the Mercury's distance, the time dilation will be on the order of $10^{-7}$, i.e. about a second per (terrestrial) year - quite invisible in common life, though we already need better timekeeping than that - expect a lot of corrections factors in low level networking protocols, to make intra-system communication and navigation work.