A friend pointed me at a very old paper on Paul Krugman's blog in which the then assistant professor discussed the economics of interstellar trade. The economics is straightforward if unexciting and the writing entertaining, but there is one puzzle. Early in the paper Krugman writes:
“The remainder of this paper is, will be, or has been, depending on the reader’s inertial frame, divided into three sections.”
This implies that the order in time of the sections–whether the order in which they were written or are being read is not clear–depends on the reference frame of the observer. That would be true if the events in question were on a spacelike trajectory, if their separation in distance was greater than the speed of light times their separation in time. It is not true, however, if they are on a timelike trajectory, since a shift from one inertial frame to another does not change the ordering of events that are timelike with respect to each other.It follows that either the article was written while the author was moving faster than the speed of light or else he expects it to be read by a reader moving faster than the speed of light. This appears inconsistent with the statement a little further down in the article that “travel must occur at less than light speed.”
Because a spacelike four-vector reverses its direction in time with a suitable change in the observer's reference frame, a procedure that sends a message faster than light is, in some reference frame, sending it backwards in time. It follows that the technology used to move the author (or reader) along a spacelike trajectory could also be used to send message from future to past, raising a variety of familiar problems with causality. This may explain why Professor Krugman, in his model, assumed that the planetary futures markets were perfectly informed.
10 comments:
My God, that's the funniest thing I've read in several weeks. Brilliant! Now I just want to see Ken MacLeod use it in a novel about the futures markets of interstellar empires. . . .
You're just like all the other famous economists with PhD's in physics. Big showoffs.
Nice observation! I'm embarrassed I missed that since I'm studying relativity atm.
Although what you've pointed out is amusing, I'm afraid I disagree with your reading of the quote. It's not about the order in which the sections were written but about when the writing (and thus the division) took place. Interpreted this way, it's simply pointing out the fact that, even as one reads the paper, there may exist other (potential future) readers in whose inertial frame the paper has not yet been written. Indeed, at a point in spacetime far spacelike separated from ourselves, there might be three future readers, in whose respective frames the paper has already been written, is being written, and has yet to be written.
Faster than light communication does not demonstrate any violation in causality or constitute time travel.
You may know of distant things sooner, but that isn't time travel.
Did you fail to notice the the statement at the start of the paper that said this research was funded by the committee to re-elect senator William Proxmire?
Proxmire being the senator who thought all space travel was boondoggle and waste of taxpayer money (but liked cheese subsidies being from Wisconsin?)
I agree with anonymous. I don't think the quote referred to the ordering of the chapters but to their existence, in which case it is accurate.
Another comment he makes is that general relativity is required to handle planets that aren't in the same inertial frame.
Special relativity can handle it too and based on his theorem that time dialation doesn't matter in practice, Newton's laws could be used to handle it as well.
One thing he didn't consider was the possible effects of time dialation on both interest rates. In effect, cheap interstellar travels provides a method for people to slow down aging.
If ships are cheap to construct, more and more people would put their money in a bank and go for a 1-2 century trip. The net effect would be to reduce the interest rate. No matter what the interest rate, traveling for 1000 years is better than staying on a planet.
The overall effect would be to shift the balance towards labour and away from capital.
A person who is willing to work/manage a company would end up being very well paid. Only a small fraction of a company's costs would be to pay for capital.
Ofc, 1000 years is along time. The longer you wait, the greater the risk of returning after the bank went bankrupt (or more dangerously, the society decided that all 'travelers' were parasites and canceled all their assets, eliminating the benefits of spreading your investment between multiple banks.)t
"Interpreted this way, it's simply pointing out the fact that, even as one reads the paper, there may exist other (potential future) readers in whose inertial frame the paper has not yet been written."
I don't know what "even as one reads the paper" is supposed to mean--simultaneity isn't well defined unless you specify the reference frame.
It is, however, well defined for two events in the same place. So the obvious meaning is:
"The event of writing this sentence is before the event of writing the later passages (or perhaps of reading them) in some reference frames, after it in others."
But that requires the two events to be spacelike wrt each other, which requires either that the author was moving faster than the speed of light when writing it or that the text was transmitted to the reader faster than the speed of light.
David, by "even as one reads the paper..." I meant: when I read the paper there exists at the same time (measured in my inertial frame) another reader. In his inertial frame, the book has not yet been written. I admit I am being loose with the term "reader," meaning by it anyone who has or will ever read the paper -- but I argue such looseness is important. If "reader" means "someone who is reading the paper *now*" then distant observers may disagree over whether or not someone is a reader. Naturally--as you correctly point out--the event of writing is necessarily timelike separated from any act of reading, and in that order. So the other reader (as he exists at this time in my frame) has not read it yet.
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