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TheMajor comments on Utilitarianism and Relativity Realism - Less Wrong
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Comments (30)
To phrase your result in terms a physicist would use: an all-time integral of a scalar function (happiness) is not Lorentz-invariant. But rather than draw a philosophical conclusion from this I would suggest modifying the equation for total happiness. If I recall correctly the standard method is defining a happiness density (over space), so rather than state "There is X amount of happiness at this point in time" you would state "At this point in time and space there is Y amount of happiness being created/destroyed", and then define the total happiness as an all universe integral (so you integrate over spacetime).
I do hope that I'm not making some elementary mistake (imagine the embarrasment), but this thread seems to fall in the general category of 'attempting to draw philosophical conclusions from a limited understanding of modern physics'. I remember reading material here on LessWrong that warns about this, such as adding up to normality.
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.
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.
Ah. 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.