Distances

John Baez

January 24, 2012

You can learn a surprisingly large amount of physics just thinking about how big various things are. So, here is a tour of distance scales, from the very smallest known to the very largest. For a better explanation of the Planck length, classical electron radius, Compton wavelength of the electron and Bohr radius of the hydrogen atom, try my related webpage on length scales in physics.

Most (though not all) of the numbers here are approximate. This is especially true of the very large distances.

Here we go:

Warning: Nobody knows what the shortest possible distance is, if there is one. It could be much bigger than the Planck length, or much smaller — or there could be no shortest distance. All we know is that if it exists, it must be smaller than about 10-18 meters.

Warning: Nobody knows if the whole universe is finite or infinite in size. The further we look, the older stuff we see, so the radius of the "observable universe" is limited by the age of the universe (about 13.7 billion years), and we'll never see much farther — except by waiting.

Warning: The above figures for distances don't take the expansion of the universe into account. This only matters much for the really big distances. So, when I say the farthest observable galaxies are 5 billion light years away, I really mean that the light we see from them took 5 billion years to get here. They're farther away now!

Similarly, when I say the radius of the observable universe is about 14 billion light years, I really just mean that the universe is about 14 billion years old. If we look at something that old and ask how far it would be now, we'd get a figure of about 46 billion light years, thanks to the expansion of the universe. If you find this confusing, you're not alone. The ultimate cure is to learn more physics.

Warning: The diameter is twice the radius. So, if the observable universe has now expanded to a radius of 46 billion light years, its diameter is about 92 billion light years.

Since the observable universe has expanded to a diameter of 92 billion light years, or about 8.7 × 1026 meters, while the Planck length is a measly 1.6 × 10-35 meters, we can say that the current radius of the observable universe is roughly 5.4 × 1061 Planck lengths. That's

54000000000000000000000000000000000000000000000000000000000000

Planck lengths! A large but finite number.

Of course, we haven't really explored distances down the Planck length; there might not be anything that small. A more conservative unit might be the proton diameter, about 10-15 meters. So, if you lined up protons across the current diameter of the observed universe, you could line up about 8.7 × 1041 of them. Try imagining a line of

870000000000000000000000000000000000000000

protons! The universe is a big place.


© 2012 John Baez
baez@math.removethis.ucr.andthis.edu

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