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.
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.
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.