[brazzy]

As long as the measure of productivity corresponds to something physical (which even bits on a harddisk are), then an exponential increase in productivity (as we have seen in the past) would eventually require humanity and its products to form a sphere whose radius grows faster than the speed of light.

Do you agree that that is a pretty hard limit?

[TheSOB88]

I think you may have skipped some steps... Could you fill in the middle a bit? I'm confused.

[Retric]

Let's suppose the amount of physical goods produced increases 1% each year. Let's add that as physical objects they take up some space minimum amount of time. Now let's add that humanity is spreading out in all directions at the speed of light. If humans cover 1 light year of space they can expand faster than the new amount of stuff. However Volume = 4/3 pi * radius ^ 3 so what happens when humanity covers 1,000 light years. Well, next year they need to put 1% more stuff into 1002^3 / 1000^3 = 0.6% more space which means they need to increase the density which at some point means you have a black hole.

PS: 1% growth per year seems slow and sustainable but in 20,000 years your talking about 2.7 with 86 zeros after it. Start with 1 atom for one second and compare it with every atom in the earth for a year and your only at 4.2 with 57 zeros. So if the total economic output of humanity was 1 atom we could not sustain 1% economic growth per year for 30,000 years.

Edit: This assumes that the economy is based on physical goods, if we assume intangibles can take up 99.99...9% of the economy then 1% growth is possible for long periods of time even if it's somewhat meaningless.

[brazzy]

In out three-dimensional space, volume grows with radius at r^3. If something physical grows exponentially (2^t), then the radius of its volume needs to increase over time at an accelerating rate, which means the growth of that radius will eventually surpass c.