I was contemplating computronium the other day and it occurred to me that there might be a point at which the computronium becomes so dense that gravitational time dilation effects would cause it to lose "efficiency" as expressed in calculations per second measured by an outside observer.

I think I've found the (approximate) density value where the maximum cps is and it's in the ballpark of $1.07 \times 10^{26}$ kg/m${}^3$. This is a MUCH higher density than either a neutron star (between $8 \times 10^{16}$ kg/m${}^3$ and $2 \times 10^{18}$ kg/m${}^3$ according to Wolfram Alpha) or a proton (which I think is around $3.14 \times 10^{18}$ kg/m${}^3$, right?) so this is obviously just a thought experiment.

Anyway, is there anything fundamental about this theoretical "computational limit"? I think it's been pretty well established that information and computation are fundamental and can be converted to and from energy (Szilard's Engine, Jarzynski Equality, Landauer's Principle, etc.), so it's definitely fundamental in that sense, but it seems like Bremermann's limit and/or the Bekenstein bound might conflict with my calculated value...

Am I making any sense, or am I just spouting gibberish?

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    $\begingroup$ Could you show the calculation leading to the density estimate? In particular, are you assuming all of the computronium has the same potential? $\endgroup$ – Anders Sandberg May 23 '18 at 5:46
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    $\begingroup$ Time dilation is not just a function of density, it depends also on geometry of your computronium (size, volume, form etc.), Moreover, if we add rotation, calculations would be completely different. So, which model did you use for your estimates? $\endgroup$ – A.V.S. May 23 '18 at 6:19
  • $\begingroup$ I was assuming a spherical, non-rotating mass of computronium with uniform potential, a radius of 1 meter and a total mass of 4.5×10^26. The time dilation factor I used was for a point on the surface of the aforementioned sphere, so computronium further inside the sphere would have been experiencing more time dilation. I did not take that into account and I'm not sure how it would influence the final result. $\endgroup$ – Thor May 23 '18 at 13:32

Let us assume a ball of computronium with constant density $\rho$ and radius $R$, where the total amount of computations per second if we ignore gravity is proportional to the total mass, $C\propto M$.

The simple way of calculating an estimate is to assume everything is at the same gravitational potential. This is wrong since the inner parts of a heavy sphere will experience more time dilation; this is why the core of the Earth is a few years younger than the surface. Still, as a first approximation we can use the gravitational time dilation formula $$\omega(r)=\omega_0 \sqrt{1-\frac{2GM}{c^2r}}$$ where $\omega_0$ is the rate of time at infinity and $r$ the location of the clock. If we just use the surface rate the total number of computations per tick as seen by infinity will be $\propto \omega(R)M = \omega_0 \sqrt{M^2-\frac{2GM^3}{c^2R}}$ and if we plug in $M=4\pi\rho R^3/3$ (this is again slightly off, since the volume at radius $R$ in a curved spacetime is not the same as in flat space) we get $$ \omega_0 \sqrt{(4\pi/3)^2 R^6\rho^2-\left(\frac{16G \pi^3}{27c^2}\right ) R^8\rho^3}$$ This expression has a maximum for a given $R$ or $\rho$. In particular, for constant size this is $\rho=2c^2/ G\pi R^2$, or for $R=1$ m $\approx 8.5729\times 10^{26}$ kg/m$^3$ (if I have done the algebra right). Not too far away from the ballpark estimate in the original question. One could also arrive at this through dimensional analysis. Note however that this value depends on $R$: there is nothing fundamental about it, since if you want a bigger or smaller value you can just select a different $R$.

Now, we can do the same in more detail by actually carefully integrate the computational contributions of different layers by (1) using the time dilation formula and volume integrals that take the actual volumes into account, or (2) do the hardcore integration for an interior solution with constant density. That might be a good evening project, but it is pretty qualitatively clear that there will be a maximal density or radius of computronium computational output. At first it must increase as $\propto R^3$ but eventually start levelling off and decreasing to zero as we approach the black hole or Buchdahl limit.

I do not think this is quite the same kind of fundamental limit as the other listed, but it does correspond to an optimum curve in $(R,\rho)$ space that presumably meets the other limits in interesting points.


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