The economics of relativity

An excerpt from a work in progress (by me) on the appropriate choice of discount rate:

Looking to physics, Einstein’s theory of relativity suggests there is no fact of the matter as to what time it is. Any measurement of time (when is “now”?) is relative to the perspective of an observer, and to the velocity of that observer, relative to the speed of light. In other words, if you are traveling very fast, you are moving into the future at an especially rapid rate. Yet it seems odd, to say the least, to discount the well-being of people as their velocity increases. Should we pay less attention to the safety of our spacecraft, and thus the welfare of our astronauts, the faster those vehicles go? If for instance we sent off a spacecraft at near the velocity of light, the astronauts would return to earth, hardly aged, many millions of years hence. Should we – because of positive discounting — not give them enough fuel to make a safe landing? And if you decline to condemn them to death, how are they different from other “residents” in the distant future?

I am told that L. Ron Hubbard considered the related question of what a currency should be worth in a very fast spaceship.  What about the term structure of interest rates?  Uncovered interest parity?  To be fully general, must we add a "speed" term in addition to the usual "risk premium"?  Comments are open.

Comments

Here's a postulate worth considering, which strikes me as being intuitively obvious:

A utility function should transform as a scalar under the Lorentz transformation.

I don't have time to check the algebra rigourously, but I think this should imply that:

(Observed discount rate) * (Observed elapsed time) = (Proper discount rate) * (Proper elapsed time)

[Or r' t' = r t, for short]

Using the standard result from the Lorentz transformation that:

t' = t/sqrt(1-v^2/c^2)

we then get

r' = r * sqrt(1 - v^2/c^2)

[v and c, being the speed of the traveller and the speed of light, respectively]

So, in other words, as the astronaut's speed approaches the speed of light, his discount rate, as observed from earth, will approach zero.

Your comments are not quite correct. There is a fact of the matter as to what time is. It is another dimension. There is no fact of the matter as to when 'now' is. There is no fact of the matter as to whether events with a certain relationship happened before one another -- but there are events where it is true that no matter your speed you will agree that one happened before the other. (Two events which are timelike separate -- that is, when a line drawn between the two on a graph of spacetime could be the worldline of a particle -- are always seen as happening in the same order by any observers. Two events which are spacelike separated -- that is, when a line connecting the two points on a spacetime graph could not be the worldline of a particle or a photon -- are such that people can disagree on their temporal order based upon their frame of reference.) And the comment about "moving into the future faster" seems . . . very odd. The actual affect of inertial motion is that other clocks slow down, not speed up. The "fly away really fast and come back and everybody is really old" affect only happens because you have to accelerate really rapidly. If we lived in the Pac-Man universe (ie- some kind of 4-D donut such that you could go around the universe without accelerating), I'm not sure you'd have the affect then.
For a particularly good discussion of the topic, you should look at "General Relativity from A to B" by Geroch. (http://www.amazon.com/gp/product/0226288641/sr=1-3/qid=1138370133/ref=sr_1_3/103-6951743-2955027?%5Fencoding=UTF8)

When we are deciding what near-light-speed missions to fund, we're going to make decisions using our earthbound frame utility function. Interstellar trade is not an attractive prospect for a quarterly earnings-driven company, whatever unfathomable magnitude the centuries-later payoff might have. If a spaceship company did its business on up-front payments from passengers, one would indeed not expect them to worry particularly about safety at the end of the journey. Talk about agency costs!

The people going on the mission will discount the payoff from the journey differently from the spaceship company, of course, but they surely can't realistically predict the payout beforehand--for example, any destination would be perceived at a lag of many light years when they're making the decision whether to embark. Maybe aliens just blew up the planet.

I expect if the travelers plan to stay on uninhabited Planet X, they can use a discount rate equal to their current discount rate, from the time expected to spend in their reference frame (presumably not much time at all) and their expectation of the awesomeness of Planet X.

But if they're planning on coming back to Earth, many centuries later, they'll need to consider what their journey will net them on Earth--and it's hard to guess what goods Earth will pay for a thousand years from now. Then there's the hard-to-guess probability that Earth gets blown up before they get back.

Hey, maybe in the future humanity will have gone socialist and the returning travelers will be able to share in the plenty for no charge! More plausibly, they can work part-time in a museum and earn a decent living--assuming returning near-light-speed travelers are rare at any point in far-future time. But the market would be responsive to the demands of exotic 80th century consumers, not 22nd century relics, so I wouldn't bet on outrageously increasing expected utility by time-shifting to the future. (Even a mere 2.5 centuries is no sure thing--would Mozart enjoy being zapped to the present? Well, possibly. He could surely become a top hip-hop producer; I suspect he might find the lifestyle to his taste.)

Basically, I am not bullish on interstellar trade.

Only the least risk averse among us would trust the stability of everything from the individuals to a system of governance while millions of miles away for thousands of years.

Those that effectively travel to the future have little to give to that future, especially considering the trends of technological progress.

Such a mission is selfish (without the pejorative).

In Orson Scott Card's Ender novels, one of the main characters grows very rich and lives for thousands of years by spending most of his time in interstellar travel at relativistic speeds. It seemed to me like it was so easy to earn fantastic sums through compound interest that more people would adopt that tactic and interest rates would plummet so that it no longer worked. The cost of course is that of travelling into the future and thus being left behind by social change, having your friends and family age and die, etc. An interesting item to ponder.

Re: Doug

Well gold would likely be a real loser, there's lots of the stuff floating around in asteroids and such. Uranium would probably be a better bet, but it seems to me that more than likely most of what a space-based economy would "need" is energy, and since there's no real way to carry more energy into the future in any reliable sense, that seems to be a real loser of a proposition. Especially if said economy harvests lots of power from interstellar bodies like our sun.

It is odd that the comments today about relativistic astronauts are so negative compared to the posts on cryonauts the other day. Is it because we think the cryonauts will wake in 100 years and the relativistic astronauts will return in 10,000? is that rational? it is just as easy, indeed, easier for the astronauts to return in 100 years than in 10,000.

I'd rather fly to the future. The view of the universe will be better than the view inside your freezer.

I don't quite understand your comment: "In other words, if you are traveling very fast, you are moving into the future at an especially rapid rate. Yet it seems odd, to say the least, to discount the well-being of people as their velocity increases." No one else here seems to have suggested a similar idea.

The twin paradox says that someone in a spaceship which returns to earth will be younger than his twin who stayed behind. I don't see how you would get from this that we would discount the well-being of the traveller.

Well, here is a sense in which it might be true. Suppose we could give one of them comfortable conditions and require the other to endure discomfort. Then it would make more sense to have the traveller be the one who is uncomfortable, since he will have to endure it for a shorter subjective period. In that sense we might be said to discount his well-being. Is that what you had in mind?

In other comments here, some have suggested that astronauts could be thought of as having low (temporal) discount rates, in that they care about the future much more than we stay-behinds do. This could be seen as an advantage to them - they could invest a few dollars and come back from their trip to find it worth millions. In this way, having a low discount rate is seen as advantageous.

Indeed, some philosophers argue that we should not discount the future at all, on rational and even moral grounds. You can make a pretty good case that your total life happiness will be optimal if you have a very low discount rate.

If you could get a brain implant or take a drug to lower your discount rate, would you do so? Change yourself so that future happiness was as valuable as happiness today? Saving money for the future would be easy, or putting in time to improve your skills for future benefit. It would arguably make your expected total lifetime happiness higher, for the typical person.

I think you should have a physicist check up on your paper, it is easy to go wrong when discussing relativity. Anyway, here are the comments I have as an enginering student who has read a little but not much special relativity.

Your statement that you travel towards the future with an increased speed when you go fast is wrong. If I see you pass by with great speed I will observe that time is going slower for you. In the same way, you will observe that time is going slower for me. (It is important not to confuse the effects of acceleration with the effects of speed. You and I both agree whom is accelerating of us but not who is moving at what speed.)

Also I am sceptical of the first comment. In particular what is meant by the observed discount rate should be specified.

I am afraid you are not taking into account the relatavistic changes in the marginal propensity to consume (If the number of Angels dancing is (Ax) and the circumference of the pinhead is (Hubbardx)then the equation is (Ax)=(MHubbardx)squared.

To Andy Wood: Yes that makes things clear. Thank you for the explanation.

The reason you treat the astronaut differently from your distant descendants is that you are directly responsible for the astronaut's situation. He can't get more fuel, or a different fuel, and he can't reduce his fuel use (without going *splat* at the end). We are expecting our great grandchildren to have time to implement one of those solutions.

why travel speed of light when you could travel a certain time of light and possibly age faster?

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