# Discounting in a relativistic universe

I’ve heard that when starting a new sketchbook, you should begin by drawing some silly doodles on the first page to break the paralysis that a fresh book full of beautiful blank pages can induce. So for my first blog post I chose the silliest topic that came to mind, namely the intersection of ethics, economics and special relativity.

Discounting is the idea that obtaining value $$V$$ some time $$\Delta t$$ into the future is worth only $$f(\Delta t)V$$ now where $$f(\Delta t) < 1$$ is the discount factor. What exactly “value” means depends on the context. For now, we will talk about money as an example, but we will get back to this point later.

This definition of discounting raises an obvious problem: the distance in time to future events is not invariant under Lorentz boosts, so by discounting like this, your value assignments become dependent on your frame of reference. Now, as long as you never accelerate, your frame of reference will stay the same and this isn’t a practical problem (though you may still have objections on aesthetic grounds). But as soon as you change your state of motion, you’ll run into problems.

Imagine that you’re about to leave for your vacation in the Alpha Centauri system, taking the new Starline 90C moving at 90% the speed of light. Suddenly, Omega comes along and offers you a deal: it will pay you $\$90$right now but in return you will have to pay$\$100$ once you arrive at Alpha Centauri in $4.34 / 0.9 = 4.82$ years (as seen from your current frame of reference on earth). This sounds like a great deal to you: you discount at 3% per year, so the $\$100$you’ll have to pay are only worth$\$100 \cdot 0.97^{4.34} = \$86.35$to you now. So you accept the deal, board your spaceship and begin accelerating towards Alpha Centauri. But as you do, you feel your value assignments shifting – or rather you realize that you will be on Alpha Centauri in only$4.82 \cdot \sqrt{1 - (0.9)^2} = 2.10$years in your new reference frame because of time dilation. This means that you suddenly value the$\$100$ you will have to pay on arrival at $\$100 \cdot 0.97^{2.1} = \$93.80$, just because you stepped into a spaceship and took off.

So clearly, improper discounting is an important financial hazard for space tourists. But what should you do instead, if you want to keep your normal discounting procedures while on earth?

Now we need to get back to what we meant by “value”. If value refers to money, then discounting is closely related to the fact that you can invest money you already have now, so getting money at a later point in time is less valuable. The amount of time used for discounting calculations should then be the proper time of the money, so it depends on whether you were going to leave most of your money invested on earth (in which case you should discount with the 4.82 years in earth’s frame of reference) or whether you were going to invest it aboard the Starline 90C (in which case you should discount with the travel time of 2.10 years).

But what if you want to discount pure utilities? In that case the question is no longer one of economics but of ethics. We are looking for a discount function $$f$$ that satisfies the following criteria:

1. $$f$$ assigns some discount factor to each point in your future light cone – all of those are events that you might be able to influence and therefore need to take into account for utility calculations.
2. On the world line of your current frame of reference, these factors coincide with the old discounting factor $$f(t)$$ – “It all adds up to normality”.
3. $$f$$ is invariant under Lorentz boosts in the sense that if your velocity suddenly changes, and you recalculate all discount factors, they would remain the same. Essentially, your ethical judgements don’t change just because you take a flight to Alpha Centauri at relativistic speeds.

I think there is only one way of discounting that satisfies all of these desiderata1: use the spacetime interval instead of the time as measured in your current reference frame.

The spacetime interval between two points is

\begin{equation} \Delta s = \sqrt{\left(c\Delta t\right)^2 - \left(\Delta x\right)^2 - \left(\Delta y\right)^2 - \left(\Delta z\right)^2}, \end{equation}

where $$\Delta t$$ is their time difference (what we used for discounting before) and $$\Delta x, \Delta y, \Delta z$$ are the spatial distances. The nice thing about $$\Delta s$$ is that it is invariant under Lorentz transformations, so if instead of discounting with $$f(\Delta t)$$, you discount with $$f(\Delta s)$$, then your value assignments won’t change when you change frames of reference.

What consequences does this have? For small spatial distances, not much changes. The $$c$$ in the equation above means that as long as you could reach an event while travelling much slower than the speed of light, $$\Delta s \approx \Delta t$$. On the other hand, events that are close to the edges of your future light cone have $$\Delta s$$ close to 0, meaning they are discounted only very weakly. So you’d care about things that happen in 4.3 years (earth frame) on Alpha Centauri almost as much as about what happens on earth right now – and much more than about things that happen on earth 4.3 years into the future.

If you think this is absurd, I completely agree. One way to get around this is to give up desideratum 3 above. Maybe if your velocity changes very suddenly, it’s justified for your value judgements to also change very suddenly? But there is of course another way to avoid these consequences: if $$f(\Delta s)$$ is independent of $$\Delta s$$, then there is no weird bias towards things that happen close to the edge of your light cone either. In other words: don’t discount pure utilities at all.

For me, the second option is much more appealing. There are already good reasons against discounting utilities, this simply adds yet another one. It’s by no means an argument showing that you can’t consistently discount events in the far future in a relativistic universe. But if you do, there will be some consequences that I find rather unintuitive. Either you care a lot about what happens on faraway star systems in the far future, or how much you value different things changes whenever you change your velocity.

1. because every point in your future light cone lies on the world line of some reference frame that you can reach by a Lorentz boost ↩︎