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).

Friday, May 20, 2011


Half of college grads regret their choice of major. If you're not going on to grad school, then here is relationship between major and employment.

Thursday, May 12, 2011

A deceptive average

I continue to learn from WeatherSpark. Based on the average, you might think Laredo is hot and humid. Actually, Laredo is usually hot and dry in the day, and as the day goes on, it gets hotter and drier until around supper time; it gets humid and cooler through the night. This I much prefer over hot and humid daytime climates.

In other news, thunderclouds are making it dark at lunchtime. The campus sidewalk lamps were just turned on, and months of waiting for rain are over. My office window is very distracting right now.

Tuesday, May 10, 2011

Making money at zero cost

Last fall, I became convinced we need to get US nominal GDP back up to its trend level. Nothing has changed since then, except that my impatience with the Fed has overcome my blogging energy barrier.

Let's do a helicopter-drop style thought experiment. Congress raises the debt limit. The US Treasury sells the Federal Reserve Bank $3.1T in special, 999-year bonds. To "pay" for the bonds, the Federal Reserve Bank simply types "new account... balance: 3,100,000,000,000.00" into a computer. As always, the Fed remits all its profits to the Treasury, so the Treasury gets $3.1T in exchange for approximately nothing.

Each US citizen then receives a check for $10K. (More realistically, the payments are staggered: perhaps people get ten monthly payments of $1K, or some people get their $10K sooner than others, etc.) The US Mint will need to ramp up production in anticipation of temporary demand to carry more cash around (especially from the unbanked). Also, it'll cost a small amount to print and mail all those checks. However, the primary initial result of sending out all these checks is to flip bits on computers: a person deposits his check at his bank, and some number stored on a computer increments by 10,000.00.

Many people put their $10K into savings and keep it there for years. Most people spend at least some of it soon. Many spend all of it soon. For some things, the price responds to new demand only slowly. For other things, especially globally traded commodities, nominal prices in dollars quickly increase to compensate for the sudden jump in demand by Americans to trade dollars for things. (Increased American demand would also increase the real prices of some goods. This effect is harder to estimate, and could vary a lot by good.)

Nominal wages change very slowly (e.g., I only get a cost-of-living-adjustment once a year). Therefore, in the short term, American real wages decline. Aggregated over all US producers, labor is the biggest cost, so we expect short-term declining real costs for US producers overall. Pair declining domestic producer costs with a sudden increase in domestic demand, and US producers of domestic goods and services will have a big new incentive to expand production.

(There will be international effects too. US exporters have a new cost-side incentive to expand (but no new demand-side incentive). Foreign producers of US imports have a new demand-side incentive to expand (but no new cost-side incentive).)

This expanding production will be self-limiting (in the short term) because it will increase the demand for the inputs to production. This increase will drive up the real prices for many non-labor inputs until it is not profitable to expand any further. [Added: It's not just a supply-side limiting effect; even after a $3.1T shift income, demand still slopes down, so after an initial demand spike increases a price of a good/service, the price would still go down as producers expanded (until further expansion was unprofitable), even if producers did not experience increasing input prices.] However, with 9% unemployment, most US industries will not need to raise real wages to attract new hires. (McDonald's hired 62,000 people last month, out of 1,000,000 applicants!) Also because of high unemployment, for many pre-existing jobs, a real wage increase will not be necessary to retain the employee.

Bottom line: Printing more money right now would be a free lunch for the US economy. The caveat is that this is a statement about an aggregate net gain, not necessarily a net gain for everyone. Fixed-rate mortgage lenders are obvious potential losers, as are bond holders in general. Also, if have you job security, you might be more worried about my talk of declining short-term real wages (long-term real wages are much harder to predict) than about the plight of the unemployed.

It is certainly possible to print too much money. The slack in the labor market is finite; past some point printing more money will just increase all prices, including wages, without increasing production. Moreover, the Fed must credibly commit to keep nominal GDP near its trend level in the long term, not to push it higher. Otherwise, we risk high and unpredictable inflation that will impose real costs by making it hard to plan for the future and diverting business people's time toward more frequent price and wage adjustments. However, we're nowhere close to having too much money. Inflation is below 2%.

Monday, May 09, 2011

In which I am less cryptic

...compared to my last post. US Monetary policy is too tight. QE2 worked. More is needed.

V? No. U? No, L.

L as in "down, then sideways." When will I stop seeing charts like this? We need QE3.

Wednesday, May 04, 2011



How can China consume half the world's iron ore production? Tim Worstall:
"The highest percentage of any metal resource that China consumes is iron ore, at a barely comprehensible 47% of world consumption."

...Note that the second largest US steel company on our list, Nucor... only make[s] new steel from old. Further, everyone does this to some extent, but in the western/industrialised world we’ve got to the point that no one, ever again, is going to build another blast furnace (yes, this is the generally accepted industry view).

...I’m pointing just at that 47% of world consumption number: this is an artefact of the way that the western (OK, industrialised) world is recycling steel rather than making new.

Tuesday, May 03, 2011

FICA as flat tax

Karl Smith has a clever idea here (and more here): transition (more) toward a flat payroll tax by (1) uncapping the SS tax, (2) raising SS tax rates, and (3) cutting income tax rate. (The Medicare tax has already been uncapped.)

Now begin my criticisms. Smith argues that it's politically expedient to wait for SS taxes to produce a budget surplus before cutting income taxes, but that could result in an indefinitely long interim period of significantly top higher marginal rates. Would this be worth it in the long run? Even if the income tax really did wither away, what's to stop SS rates from being progressivized? For Medicare tax rates, this has has already begun (read the last two sentences of footnote c).

In the near-term, it actually looks more politically feasible to do (1), (2), and (3) simultaneously. For Obama to maintain at least the appearance of no tax increases for those making under 250K, he cannot allow (1) or (2) to happen without compensating income tax reductions (at least for incomes under 250K). The problem with this scheme is that these income tax reductions might get phased out above 250K, resulting in high marginal rates.

This is all rather pie-in-the-sky, but there is an important underlying fact that motivates people to come up with these schemes: the median voter doesn't know calculus. In particular, (s)he doesn't know what d(tax)/d(income) is, nor how d(tax)/d(income) differs from tax/income.