## LESSWRONGLW

To phrase your result in terms a physicist would use: an all-time integral of a scalar function (happiness) is not Lorentz-invariant.

Yes it is, since Lorentz-transformations have determinant 1, i,e., are measure-preserving. The issue in the example is that happiness isn't a function on all of space-time, it is a function on the world lines of being capable of experiencing it.

[anonymous]6y0

You can integrate happiness over the proper time along those world lines; I suspect that's equivalent to integrating a happiness density that looks like SUM_i h_i(t) delta(x - x_i(t)) over spacetime.

0TheMajor6yAh. It's about time the assumptions were made clear. I thought that 'creation of happiness' was a function defined on spacetime, and the proposed definition defines the total happiness to be only the happiness created on the observers world line. I believe this is not Lorentz-invariant - while a scalar H(x,t) might be invariant under such a transformation we are interested in H(x,t)dt, which messes up the invariance. And I think your remark about the determinant is just a rewording of my point: the determinant of a matrix describes the change in volume of a (in this case) 4-dimensional volume, but if we integrate only in one direction our result can still change (almost) arbitrarily. And therefore introducing an all-space integral solves the problem - this quantity does deal with all four dimensions.

# -3

## Introduction

Most people on less wrong seem to be some kind of hedonic consequentialist.   They think states with less suffering and more joy are better.  Moreover, it is intuitive that if you can cause some improvement in human well-being to be achieved then (other things being equal) it is better to realize that improvement as soon as possible.  Also, most people on this site seem to be realists about special relativity.  That is they assume that any inertial reference frame is an equally valid point from which to describe reality rather than believing there is one true reference which offers a preferred description of reality.  I will point out that these beliefs (plus some innocuous assumptions) lead quickly to paradox.

## Relativity Realism

Before I continue I want to point out that empirical observations really are agnostic about the existence of a preferred reference frame.  Indeed, it's a consequence of the theory of relativity itself that it's predictions are equally well explained by postulating a single true inertial reference frame and simply using the Lorentz contraction and time dilation equations to compute behavior for all moving objects.  To see that this must be true not that if we take relativity seriously the laws of physics must work correctly in any reference frame.  In particular, if we imagine designating one reference frame to be the true reference frame then, relativity itself, tells us that applying the laws of physics in that reference frame has to give us the correct results.

In other words once we accept Einstein's equations for length contraction and time dilation with velocity we can interpret those equations as either undermining the idea of a fixed ether against which objects move (any reference frame is equally valid) or that there really is a fixed ether but objects in motion behave in such a manner that we can't empirically distinguish what is at rest.

At first blush this second result seems so jury rigged that surely the simpler assumption is that there is no preferred reference frame.  This relies on a false description of the situation.  The question isn't, "do we assign a low prior probability to the laws of physics conspiring to hide the true rest frame from us?"  Presumably we do.  The question should be, "given that the laws of physics do conspire to make a special rest frame empirically indistinguishable from any other inertial frames what probability do we assign to such a frame existing?"  After all it is a mathematical truth that the time dilation and length contraction do perfectly conspire to prevent us from measuring motion relative to some true rest frame (if it existed) so in deciding whether to believe in a preferred rest frame we aren't deciding between laws that would and wouldn't hide such a frame from us.  We are only deciding whether, given we have such laws, whether we think such an undetectable true rest frame exists.

To make it even more plausible that there is some true rest frame I will remark (but not argue) that relativity is a pretty general phenomena that can be derived from any model that conserves momentum, where the forces obey the inverse square law and all propagate at a constant speed relative to some fixed background, matter is held together in equilibrium states of these forces and time is implicitly measured via the rate it takes these forces to propagate.  In other words if you have atoms held together by EM forces and the time it takes physical processes to happen is governed by the time it takes either forces or matter to cross certain distances then relativity comes for free.  So it isn't amazing that we might have a true prefered reference frame and yet it be impossible to experimentally determine that frame.

(As an aside this interpretation of relativity, fully consistent with all observables so far, makes for much better scifi since FTL travel doesn't allow anyone to go back in time).

## A Paradox Resulting From Relativity Realism

Suppose we have two different brain implants that will be implanted in two different conscious but coma bound individuals.  After a delay of 10 minutes after implantation the first device delivers an instantaneous burst of euphoria every second.  The other delivers an instantaneous burst of discomfort every second.  I assume we would all agree that (with sufficient additional assumptions) the world is a better place if we implant just a device of the euphoria inducing kind and a worse place if we just implant a device of the second kind.  So assume the devices are appropriately calibrated so that the effect of implanting both is neutral (or very very nearly so).  So far so good.

I think we can all agree that the world would be better off if we delayed implanting the discomforting device by 10 minutes (or equivalently implanted the pleasurable device 10 minutes earlier).  If you dispute this conclusion then you get absurd results if you even admit the possibility of a universe that exists forever as in such a universe it is no better to permanently increase human welfare now than to delay that increase by 10 minutes or 10 centuries.

Now assume that the two individuals receiving the transplants are actually on spaceships moving in opposite directions at high rates of speed and the implantation is done at the instant they pass by each other.  For simplicity we assume everyone else dies at this instant (or add an irrelevance of identical outcomes assumption and note that the two ships are moving at the same velocity relative to everyone else).

From the reference frame of the individual who received the beneficial implant we can analyze the situation as follows.  Without loss of generality we can assume the ships are traveling at an appropriate speed so that for every second that pases in our reference frame only 1/2 a second passes on the other ship.  Thus in this reference frame the first experience of discomfort is delayed by 10 minutes and then only occurs every other second.  Now surely the world is no worse off because the discomfort occurs less frequently.  But ignoring the fact that the discomforting device fires less frequently this is exactly equivalent to implanting the desirable device 10 minutes before the undesirable one.  Thus, since implanting both in the same reference frame was neutral, it is actually favorable (better than not implanting them) to do so when the recipients are in fast moving reference frames moving in opposite directions.  Note the same result holds if we assume the device only creates discomfort or euphoria a single time with the minor assumption that if two worlds only differ in events before time t then what happens after time t is irrelevant to which one is preferable.

However, the same analysis done in the reference frame of the unpleasant implant gives the exact opposite conclusion.

Perhaps one might try and avoid the paradox by insisting that no experience truly occurs instantaneously.  However, this is easily seen to be futile.

Assume that each device inflicts pleasure or discomfort for duration epsilon << 1 second.  If you assume that the total badness of the uncomfortable experience is somehow mediated by changes in neurochemistry or other physical properties you are lead to the assumption that even described from the reference frame of the desirable implant the experience of 2*epsilon seconds of discomfort by the time dilated individual is really no worse than the experience of epsilon seconds of discomfort would be for someone with that implant in your reference frame.  In other words when time is dilated the experience of pain per unit time is diluted.  This leads to the exact same result as above.

On the other hand if we really do increase the weight we give to pain experienced by those undergoing time dilation an even simpler set of implants leads to paradox.  These implants start working immediately, one generating a pleasant experience for 5 minutes the other an unpleasant experience for 5 minutes again calibrated so that installing both is overall neutral.  Now by assumption from the reference frame of the beneficial implant things are overall worse (the longer duration of discomfort experienced by the other individual is overall worse than someone in the same reference frame getting the undesirable implant) and vice versa from the other reference frame.

The use of instantaneous experiences was merely a way to simplify the example but irrelevant to the underlying inequalities.  Those inequalities are a result of the implicit time discounting forced by the assumption that other things being equal it is better for improvements to occur now rather than later combined with the fact that realism about relativity renders facts about simultaneity incoherent.

Personally, I think the only decent way of avoiding this paradox is to deny realism about relativity.  Sure, it's a radical move.  However, it's also a radical move to say it's not true that it's better to cure cancer now than in 10 centuries even if the human race will continue to exist forever.  Indeed, even if you don't assume literally infinite duration of effects even an unbounded potential length of effect with probabilities that decrease sufficiently slowly is equally problematic.

## Responses

I've deliberately avoided phrasing this dilemma in terms of a formal paradox and listing the assumptions necessary to generate the paradox.  Partly this is laziness but it's also a desire to see how people are inclined to respond before I attempt to draw up formal conditions.  After all I ultimately want to capture common views in the assumptions and if I don't know what people's reactions are I can't pick the right assumptions.