Exponential Growth in Flat Spacetime II
I've been wanting to put this up for a while, since apparently the conclusions of my last post weren't justified; I missed an important case! The bacteria *can* survive if, instead of filling space, they restrict themselves to a spherical shell!
The bacteria are able to sustain their growth while staying under the speed of light and using reasonable amounts of energy. They can even experience an unlimited amount of proper time. The downside to this is that any two bacteria in different locations will eventually lose contact with each other. This may remind you of galaxies drifting out of sight under the influence of dark energy, and that turns out to be a very similar situation.
More details below the fold. PDF forthcoming, since math in HTML is awkward.
Setup
Consider a spherical shell of bacteria with radius r, centered at the origin in an n+1-dimensional flat spacetime. Let...
- t = time for a stationary observer at the origin.
- τ = time for a bacterium in the shell.
- β = dr/dt
- γ = dt/dτ = (1 - β2)0.5
- k = some constant such that dr/dτ = kr (This is the assumption that the bacteria grow exponentially.)
Using the chain rule, we can rewrite kr:
kr = dr/dt dt/dτ = βγ = γ( 1 - 1/γ2 )0.5 = (γ2 - 1)0.5
Then γ = ( 1 + k2r2 )0.5, and from here it's easy to check that we always have β < 1, so nothing breaks yet.
Energy
You can work this out in full detail if you'd like, using the formula E = γm0c2 for total (kinetic plus rest) energy. I'll only sketch it out here.
For sufficiently large radius r, the expression above guarantees γ < 2kr. Since the surface area-and therefore the mass-of the expanding sphere of bacteria grows as rn-1, the required energy grows as rn. This is proportional to the volume enclosed by the sphere of bacteria, so with a mostly uniform density of energy throughout space and the right proportionality constant, these bacteria can sustain their growth forever.*1
Subjective Time
Do the bacteria have an infinite amount of time to do stuff? Or are they doomed to a finite existence, like a geometric sum? More precisely, this question asks whether the integral ∫ dτ/dr dr diverges for r → ∞. This comes down to an application of the chain rule:
∫ dτ/dr dr = ∫ dτ/dtdt/dr dr = ∫ 1/γβ dr = ∫ 1/kr dr
This does indeed diverge, so the bacteria on the shell will have an infinite amount of time to do what bacteria do best.
Communication I: Direct Light Signals
Suppose a bacterium at the sphere's south pole sends an e-mail to its clone at the north pole. The e-mail, encoded as a stream of photons, flies in a straight line toward the north pole. Does the e-mail ever reach its destination?
The rate at which the e-mail closes the gap between itself and the north pole is given by 1 - β. If the time integral of this diverges, then the e-mail can always catch up no matter how late it was sent. We start with a few lemmata:
- dγ/dt = kβ2
- Begin by computing dγ/dr using the formula for γ. It comes out to equal kβ. Then using the chain rule, we multiply this by dr/dt = β to obtain the desired result.
- If γ > 20.5, then (1 - 1/γ)-1 < 2(γ-1)
- Left as a not-very-interesting exercise; you can discover the steps for this one merely by working backwards.
- If γ > 1, then γ - (γ2 - 1)0.5 < γ-1
- Another exercise that can be figured out by working backwards.
We apply the lemma 1 to change the integral:
∫ 1 - β dt = ∫ (1 - β)/kβ2 dγ
With some shuffling around of symbols using β2 = 1 - 1/γ2, the integral becomes:
k-1 ∫ (1 - 1/γ)-1 (γ - (γ2 - 1)0.5) dγ
Applying lemmata 2 and 3, this integral is less than 2/k ∫ γ-2 dγ, which converges! Therefore, after enough time has passed, opposite ends of the sphere will be unable to contact each other. As the sphere continues to grow, more and more parts of it become out of the reach of even light-speed communication.
Communication II: Message Passing
If the bacteria instead pass messages along the surface of the sphere, there's a nice result limiting how far such messages can go. Start from the Minkowski metric in spherical coordinates dt2 - dr2 - r2dΩ2, where dΩ is a velocity term along the surface of the sphere. We can rewrite dt2 in terms of k, r, and dr, simplifying to get k-2r-2dr2 - r2dΩ2.
The fastest possible signals will travel along paths that make this expression zero. Setting this equal to zero and taking square roots gives us the differential equation k dΩ = r-2 dr, whose solution is k ΔΩ = rinitial-1 - rfinal-1.
From this we get the inequality k ΔΩ < rinitial-1, or equivalently, rinitial ΔΩ < k-1. The left side of this inequality is a distance along the surface of the sphere; only the bacteria that lie within this distance when a signal is sent can be reached by the signal. The right side is just the growth rate constant; the faster the growth, the smaller the accessible neighborhood.
I Might Be Making Stuff Up After This Point
Let's go back to the Minkowski metric, dt2 - dr2 - r2dΩ2. This measures the time*2 that passes for something moving with radial velocity dr/dt and tangential velocity r dΩ/dt. If dΩ is zero, this is the purely radial motion of a bacterium in the shell. Then what remains, dt2 - dr2, must measure time as experienced by the bacteria, so the metric can be written as dτ2 - r2dΩ2.
You might recognize a resemblance to the FLRW metric. In fact, you can consider the sphere of bacteria as its own little universe whose size (scale factor) r changes with time τ. To be more precise, r grows exponentially in τ, and you might recall what other kind of universe exhibits exponential growth in its size: one dominated by a cosmological constant.
Like, wow. Within a flat universe, these bacteria are simulating a universe full of dark energy.
Footnotes
- This ignores the minor detail that a sufficiently large volume of any uniform density is contained by its Schwarzschild event horizon. Wikipedia says this would manifest itself as the edge of the visible universe. I haven't tried analyzing the effects of this-presumably it requires accounting for spacetime curvature. The reader is welcome to try.
- Written this way, it's actually the square of an infinitesimal time. Normally you see metrics expressed as "dτ2 = something", but I'd already used tau.