## Expanding Universe

Reference: Ian Stewart’s Cabinet of Mathematical Curiosities

The starship Indefensible starts from the center of a spherical universe of radius 1,000 light years, and travels radially at a speed of one light year per year — the speed of light. How long will it take to reach the edge of the universe? Clearly, 1000 years. Except that I forgot to tell you that this universe is expanding. Every year, the universe expands its radius instantly by precisely 1000 light years. Now, how long will it take to reach the edge? (Assume that the first such expansion happens exactly one year after the Indefensible starts the voyage, and successive expansions occur at intervals of exactly one year.)

It might seem that the Indefensible never gets to the edge, because that is receding faster than the ship can move. but at the instant that the universe expands, the ship is carried along with the space in which it sits, so its distance from the centre expands proportionately. To make these conditions clearly, let’s look at what happens for the first few years.

In the first year, the ship travels 1 light year, and there are 999 light years left to traverse. Then, the universe instantly expands to a radius of 2000 light years, and the ship moves with it. so it is then 2 light years from the center, and has 1998 left to travel.

In the next year, it travels a further light year, to a distance of 3 light years, leaving 1997. But then the universe expands to a radius of 3000 light years, multiplying its radius by 1.5 so the ship ends 4.5 light years from the center, and the remaining distance increases by 2995.5 light years.

Does the ship ever get to the edge? If so, how long does it take?

Hint:

It will be useful to know that the nth harmonic number

$H_{n}=1-1/2+1/3-1/4+ \ldots + 1/n$

is approximately equal to

$\log {n} +\gamma$

where $\gamma$ is Euler’s constant, which is roughly 0.5772156649.

More fun later,

Nalin Pithwa

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