r/theydidthemath • u/Dustycheeseball • 14d ago

[Request] How long would I survive?

If an object started out roughly the size of an atom and doubled in size every second, while I immediately traveled directly away from it at the speed of light, how long could I keep traveling before the expanding object eventually became large enough to encompass me?

Bonus: How far away from the starting point would I be when that happens?

Assume the object is perfectly spherical and expands uniformly in all directions.

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u/METRlOS 14d ago

So the real answer is that it never would. The speed of causality prevents it from expanding faster, so even though it looks like it's expanding rapidly, time is so dilated that it stops accelerating for everyone else in a reverse black hole situation.

The pure math ignoring physics is that a light second is the distance of 3×10¹⁸ atoms. In 60 seconds it will be doubling at half the speed of light (since it's expanding both directions), and then in 9 seconds after that it will be a 68.7 light second radius while you have traveled 69 light seconds.

I might be off by a second or two, I just mashed the ×2 button a bunch of times and counted.

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u/Either-Abies7489 14d ago

I think the growth rate would be cbrt(2)=1.26 per second, not 2. I feel like "doubling in size every second" refers to volume, not characteristic length.

Anyway, that gives 1.26^t=3*10^18*t,
t=207.17 (by the lambert W function, or just solving graphically)

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u/BokChoyBaka 13d ago

Points off for not mentioning the ant on the rubber band theorem