With no immortality, population growth is exponential. With everyone immortal, population growth is a slightly faster exponential. It’s a Problem that has to be solved no matter what, in either case, or else things get Bad. In both cases, the minimum sufficient measure is artificially capping reproduction. With immortality, the cap (as measured in children per parent) is somewhat lower.
Exponential growth is okay for awhile, but there will always come a point when it’s not. There are hard limits imposed on our expansion by the speed of light, and they are only cubic, not exponential.
here are hard limits imposed on our expansion by the speed of light, and they are only cubic, not exponential.
Not if space is hyperbolic. The volume of a hyperbolic sphere is exponential in its radius.
Actually, even if space is Euclidean, the effects of special relativity still mean that we can expand exponentially (for as long as we’re able to maintain some constant amount of acceleration in any direction we choose, that is). In Minkowski spacetime, the subset of the future light cone consisting of points where a fixed amount of proper time has elapsed is in fact a hyperbolic 3-space.
(But for all practical purposes, you’re right about the unsustainability of exponential growth.)
Actually, even if space is Euclidean, the effects of special relativity still mean that we can expand exponentially (for as long as we’re able to maintain some constant amount of acceleration in any direction we choose, that is).
I’m curious what would be the requirements in terms of energy and mass to maintain this motion under special relativity. In Newtonian physics, constant acceleration of a constant mass requires power increasing linearly with time—are things substantially different under SR? Also, even if free energy is somehow given, is it possible to maintain constant acceleration without constantly losing mass in the other direction due to momentum conservation (assuming you’re not moving along infinite-mass rails of some sort)?
Well, the amount of energy needed seems to be linear in the Lorentz factor, which is rate of change of co-ordinate time with respect to proper time. But the latter is increasing exponentially, so I think you need power increasing exponentially with time to maintain constant acceleration.
Also, even if free energy is somehow given, is it possible to maintain constant acceleration without constantly losing mass in the other direction due to momentum conservation (assuming you’re not moving along infinite-mass rails of some sort)?
No—you’d need to somehow keep on propelling yourself. I suppose in theory you could harvest forward momentum from any stars and galaxies you meet, no matter how rapidly you’re whizzing past them. That’s what’s going to need exponentially increasing power.
A lot of this depends on the subtleties of the laws of physics. I think the best solution is to end death now; if the math doesn’t work out in our favour, we can always just slow down subjective time and build our own exponentials.
A lot of this depends on the subtleties of the laws of physics.
Oh sure. In any case, the “exponential growth in a Euclidean space” scenario wouldn’t work because we wouldn’t be able to maintain constant forward acceleration indefinitely, if everything around us was moving backwards at close to light speed.
I think the best solution is to end death now; if the math doesn’t work out in our favour, we can always just slow down subjective time and build our own exponentials.
The last bit may not be possible. There’s a finite limit to the amount of information one can store in a given region of space (or else the amount of mass-energy needed to represent it would collapse into a black hole). So to have exponentially many people, you’ll always need an exponential amount of space, whatever you do to subjective time.
If we slow down subjective time at an always increasing rate, we could ensure that the amount of people that we can support increases at the same rate as the amount of people we need to support.
I see. That does the keep space requirements down, but subjectively we’d be moving along this graphical timeline at constant speed. I don’t how much computation is possible in the ‘Dark Era’. (Perhaps only a finite amount?)
I don’t know if that’s more or less of a problem than death or limiting births. That is a question for a FAI; while we may be able to solve it, only an AI would need to. I doubt that the concept of personal identity will even survive the singularity, so this could easily end up not mattering.
Oh sure. I regard the above purely as a mathematical curiosity.
It doesn’t provide exponential growth for very long, because the people at the frontier of this expanding hyperbolic sphere would reach the heat death of the universe alarmingly soon. At least, I think that’s what would happen, assuming the universe will end in a heat death (as opposed to a Big Rip or Big Crunch).
ETA: Actually, that only defeats the ‘exponential growth in a Euclidean space’ scenario. But still, I agree that we can’t blithely assume current physics.
With no immortality, population growth is exponential. With everyone immortal, population growth is a slightly faster exponential. It’s a Problem that has to be solved no matter what, in either case, or else things get Bad. In both cases, the minimum sufficient measure is artificially capping reproduction. With immortality, the cap (as measured in children per parent) is somewhat lower.
Exponential growth is okay for awhile, but there will always come a point when it’s not. There are hard limits imposed on our expansion by the speed of light, and they are only cubic, not exponential.
Not if space is hyperbolic. The volume of a hyperbolic sphere is exponential in its radius.
Actually, even if space is Euclidean, the effects of special relativity still mean that we can expand exponentially (for as long as we’re able to maintain some constant amount of acceleration in any direction we choose, that is). In Minkowski spacetime, the subset of the future light cone consisting of points where a fixed amount of proper time has elapsed is in fact a hyperbolic 3-space.
(But for all practical purposes, you’re right about the unsustainability of exponential growth.)
I’m curious what would be the requirements in terms of energy and mass to maintain this motion under special relativity. In Newtonian physics, constant acceleration of a constant mass requires power increasing linearly with time—are things substantially different under SR? Also, even if free energy is somehow given, is it possible to maintain constant acceleration without constantly losing mass in the other direction due to momentum conservation (assuming you’re not moving along infinite-mass rails of some sort)?
Well, the amount of energy needed seems to be linear in the Lorentz factor, which is rate of change of co-ordinate time with respect to proper time. But the latter is increasing exponentially, so I think you need power increasing exponentially with time to maintain constant acceleration.
No—you’d need to somehow keep on propelling yourself. I suppose in theory you could harvest forward momentum from any stars and galaxies you meet, no matter how rapidly you’re whizzing past them. That’s what’s going to need exponentially increasing power.
(Somehow I don’t think this is going to work!)
A lot of this depends on the subtleties of the laws of physics. I think the best solution is to end death now; if the math doesn’t work out in our favour, we can always just slow down subjective time and build our own exponentials.
Oh sure. In any case, the “exponential growth in a Euclidean space” scenario wouldn’t work because we wouldn’t be able to maintain constant forward acceleration indefinitely, if everything around us was moving backwards at close to light speed.
The last bit may not be possible. There’s a finite limit to the amount of information one can store in a given region of space (or else the amount of mass-energy needed to represent it would collapse into a black hole). So to have exponentially many people, you’ll always need an exponential amount of space, whatever you do to subjective time.
If we slow down subjective time at an always increasing rate, we could ensure that the amount of people that we can support increases at the same rate as the amount of people we need to support.
I see. That does the keep space requirements down, but subjectively we’d be moving along this graphical timeline at constant speed. I don’t how much computation is possible in the ‘Dark Era’. (Perhaps only a finite amount?)
I don’t know if that’s more or less of a problem than death or limiting births. That is a question for a FAI; while we may be able to solve it, only an AI would need to. I doubt that the concept of personal identity will even survive the singularity, so this could easily end up not mattering.
Oh sure. I regard the above purely as a mathematical curiosity.
It doesn’t provide exponential growth for very long, because the people at the frontier of this expanding hyperbolic sphere would reach the heat death of the universe alarmingly soon. At least, I think that’s what would happen, assuming the universe will end in a heat death (as opposed to a Big Rip or Big Crunch).
ETA: Actually, that only defeats the ‘exponential growth in a Euclidean space’ scenario. But still, I agree that we can’t blithely assume current physics.