Faster than the speed of light is relatively fast, part deux

(Turns out I had used that title on a previous blog post here, but I want to use it again, and what are you going to do about it?)

I've been pondering lately about cosmology, which I know just barely enough about to confuse myself, and by a nifty manifestation of what a friend of mine calls the "Tab Hunter Effect," I just saw a video from the fine folks at Veritasium that talks about the very thing I was pondering:

And just last night I ran across a scientific paper on the very same topic. The details are over my head, but at least the conclusions are in reasonably plain English.

Now here's something that's simple enough I worked it out by myself:

Suppose you're in a car driving 60 miles per hour and you're now 60 miles from where you started. How long have you been driving? Well, duh, 60 miles at 60 miles an hour; you've been driving an hour, right? And if you're 300 miles from where you started and you've been driving 60 miles an hour, you've been driving five hours. Or if you've gone 300 miles at 30 miles an hour, you've been driving 10 hours. And so on. In general, elapsed time equals distance divided by speed. Again, duh.

Now imagine there's an explosion in space sending fragments zipping outward in all directions and speeds, and with nothing to slow them down, each bit keeps zipping along at the same speed it had to start with.

The same relationship between time, distance, and speed holds true for those flying particles as for a car driving on a highway. If a particular piece of shrapnel has traveled 100 miles at 50 miles an hour, for example, the explosion must have happened two hours ago.

This is too easy. How do we make it sound more impressive? Well, let's measure speed in kilometers per second and distance in megaparsecs (an astronomical unit equal to a little under 3.3 million lightyears). And let's talk about speed divided by distance rather than distance divided by speed, so the result isn't the elapsed time but 1 divided by the elapsed time. And finally, despite the fact that the value pretty obviously changes with time, let's call it a "constant" just to mess with people.

Now, back in the 1920s an astronomer named Edwin Hubble -- the guy the Hubble Space Telescope is named after -- observed that galaxies (previously thought to be clouds of gas in our own galaxy) appeared to be moving away from us, and the more distant a galaxy was, the faster it was moving, and the ratio of speed to distance seemed to be pretty consistent, though Hubble's initial estimates have since been adjusted to a much smaller number.

Today it's estimated that a galaxy a million parsecs away is receding from us at about 70 kilometers per second. (That is, the "Hubble constant" is currently estimated as about 70 km/s/Mpc.) A galaxy that's two megaparsecs away is receding at 140 km/s, and so on. Since the universe is expanding everywhere, we can take any point (such as this blog) to be the center of the expansion.

If the Big Bang explosion was as simple as the simple-minded explosion of my example, then the Hubble constant should work out to be 1 divided by the age of the universe. So let's work it out and see what we get.

Type the words 1 megaparsec in kilometers into Google and it will tell you that a megaparsec is 3.08567758 × 1019 kilometers. Dividing that by 70 should give you the age of the universe in seconds. You can work that out in Google as well: Just type 3.09 * 10^19 / 70 into a Google search and you'll get 4.4142857e+17, which happens to mean 4.4142857 × 1017), where 1017 means 10 followed by 17 zeroes. Now type 4.4142857e17 seconds in years, and you'll get 1.39883e10 years, or about 14 billion years, which happens to be very close to the estimated age of the universe (roughly 13.8 billions years). Close enough for Star Fleet work anyway.

In fact, it seems too good to be true. The matter in the universe is attracting the other matter in the universe, which should be slowing the expansion down, and it also appears that for the last however many billion years, dark energy (which is a mystery) has been speeding the expansion up. Maybe they just cancel each other out enough for 1 over the Hubble constant to give us the age of the universe despite the naive simplicity of the calculation.

(Or maybe it has to do with this: If you compute the circumference of the cosmic microwave background based on our estimates for the expected size of its slight irregularities and divide that by our best estimate of the radius, you get a quotient of 2 pi. That may not seems so surprising, since it's exactly what you'd expect in Euclidean space. But one of the few things I think I know about general relativity is that gravity curves space into something not Euclidean (most noticeably close to a black hole). If space is overall Euclidean, that might imply that there's not much net gravitational influence on a cosmic scale. Perhaps helps explain why the calculation works. I don't know.)

Well, enough of that. I'm not sure whether anybody else finds my amateur stabs at amateur cosmology interesting, but hey, it's my blog.

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