Wednesday, June 23, 2010

Kip Thorne and the Twin Paradox

Here is a beautiful little calculation by Kip Thorne that I just can't resist.

First of all, let me thank Thorne for his massive tomb, Gravitation, which all physicists know as "Misner, Thorne, and Wheeler." Is it the big black book that all physicists tote around for the rest of their lives -- you can often see it behind them when they do on-camera interviews. It's heavy enough to nearly form a black hole of its own.

Gravitation (Physics Series)If  you're a graduate student physics, you have to buy it. It now costs $123, at least on Amazon. I think I got mine for $40, and I still read it a few times a year. Someday I hope to even understand it.

In any case, you all know about the twin paradox: if your twin were to fly away from Earth, while you remained here, he/she would experience time differently than you -- in particular, they would record the length of their trip as much less than yours.

How much less?

Good question. And a complicated one. In particular, your twin would not leave Earth at a constant velocity, but would accelerate (and then decelerate on the turnaround) and so calculations based on special relativity (with that ubiquitous factor of gamma) do not apply.

But here Thorne gives the results of a nice little calculation that is probably on the qualifying exams of his graduate students:

\[ T_ M = (2 c/g) [exp(gT_ F/4c)- exp(-gT_ F/4c)]. \]

This assumes your twin accelerates away from Earth at g, then (at the midpoint of her journey) switches to a deceleration of -g. Then they turn around and do the same on the way home.

Here, the twin's age is T_M. Your age is T_F.

Thorne gives a nice plot of this calculation here. Here's the bottom line, where "Florence" is your twin and "Methuselah" is you, remaining on Earth:

If Florence's clocks and aging report a round trip time of 10 years, Methuselah will have aged by 25 years. If Florence aged 30 years, Methuselah will have aged 4,500 years. If Florence aged 88 years, Methuselah will have aged 14 billion years, which is the current age of our Universe! Unfortunately, no known rocket fuel, not even thermonuclear fusion, is capable of producing the sustained multi-year-long acceleration required for such a trip.

Isn't physics beautiful?

1 comment:

E R E said...

kind of unrelated, but this seems like grist for your mill

(irritating, innit? and from fellow oregonians, no less!)