Tuesday, May 24, 2011

Reflections on Relativity

Kevin Brown has written a wonderful book (and made it freely available on his website).

Since I already know a lot about the book's subject, I got the most out of the last chapter, where the author's mathematical and metamathematical depth are brought to bear on foundational physics questions. For example, given three spacetime events A, B, and C, it can be the case that A is close to B, B is close to C, yet A is not close to C. The most famous example of this is the twin paradox.

  • A: One twin leaves for a very high-speed trip to a distant star.
  • B: This twin reaches the opposite end of the galaxy after traveling at constant acceleration 9.8 m/s^2 (thereby simulating gravity) for 25 years. Also, the galaxy appears to have shrunk because he has reached the opposite end of it traveling at less than the speed of light in 25 years. So, his (subjective) measurement of distance traveled is less than 25 light-years. The twin's spaceship now changes the direction of its acceleration in order to head back home.
  • C: This twin completes his 25-year return trip to Earth. Total trip time: 50 years. Total trip distance: almost 50 light-years.

Upon return, the space traveler finds that his earthbound twin is long dead; about 200,000 years have passed on Earth! Moreover, from Earth's point of view, he traveled almost 200,000 light-years in distance, not almost 50 light-years. More to the point, from everyone's point of view, the Minkowski spacetime pseudo-distance sqrt((c*dt)^2-dx^2) from A to C is about 200,000 light-years, while the spacetime distances from A to B and from B to C are each 25 light-years.

In a more extreme version of the experiment, a photon leaves Earth, bounces off a planet in the Andromeda galaxy, and returns to Earth. From the galactic center's point of view, the round-trip elapsed time is 5 million years, and the net distance traveled is 2000 light-years (due to the sun's rotation around the galactic center). From the photon's "point of view," the elapsed time and distance are both exactly 0 for each leg of the trip, plus some very small amount of time and distance between the absorption and re-emission events on the distant planet. In Minkowski pseudometric, which all observers agree on, the spacetime distance from A to B and from B to C are both zero, while from A to C is almost 5 million light-years. Thus, even infinite closeness is not transitive in spacetime. (Technically, I shouldn't even talk about "the photon." There's no meaningful way to talk about whether the returning photon is "the same photon" as the one emitted.)

In fancier terminology, if we take the Minkowski pseudometric seriously, then shouldn't we model the universe with a non-transitive topology? This is discussed in mathematical detail in section 9.1.; here's a shorter version of the idea from section 9.9:

Indeed, even before the advent of quantum mechanics and the tests of Bell's inequality, we should have learned from special relativity that locality is not transitive, and this should have led us to expect non-Euclidean connections and correlations between events, not just metrically, but topologically as well. From this point of view, many of the seeming paradoxes associated with quantum mechanics and locality are really just manifestations of the non-intuitive fact that the manifold we inhabit does not obey the triangle inequality (which is one of our most basic spatio-intuitions), and that elementary processes are temporally reversible.
See also section 9.7, where Chaitin's theorem is applied to questions about determinism and predictability:
On this basis it might seem that we could eventually assert with certainty that the universe is inherently unpredictable (on some level of experience), i.e., that the length of the shortest Turing machine required to duplicate the results grows in proportion with the number of observations. In a sense, this is what the "no hidden variables" theorems try to do.

However, we can never reach such a conclusion, as shown by Chaitin's proof that there exists an integer k such that it's impossible to prove that the complexity of any specific string of binary bits exceeds k (where "complexity" is defined as the length of the smallest Turing program that generates the string).


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