In that generality, this is false. Not all differential equations are causal in all directions.
I don’t doubt that pathological examples exist. I don’t suppose you
have any handy? I really would be interested. I do doubt that physical
examples happen (except perhaps along null vectors).
The prototypical 4-d wave equation is f_tt—f_xx—f_yy—f_zz = 0. I don’t see how rearranging that to
f_tt = f_xx + f_yy + f_zz provides any more predictive power in the t direction than
f_xx = f_tt—f_yy—f_zz provides in the x direction. (There are numerical stability issues, it’s true.)
In particular, everyone I’ve ever heard talk about reconstruction in GR mentioned space-like hypersurfaces.
Well, that’s partially an artifact of that being the sort of question
we tend to be interested in: given this starting condition (e.g. two
orbiting black holes), what happens? But this is only a partial answer.
In GR we can only extend in space so long as we know the mass densities
at those locations as well. Extending QFT solutions should do that.
The problem is that we don’t know how to combine QFT and GR, so we use
classical mechanics, which is indeed only causal in the time direction.
But for source-free (no mass density) solutions to GR, we really can
extend a 2+1 dimensional slice in the remaining spatial direction.
No, I think you’re right that directions in which solutions to differential equations are determined by their boundary values are generic.
But I think it is reasonable (and maybe common) to identify causality with well-posedness. So I think that rwallace’s definition is salvageable.
I’m confused about GR, though, since I remember one guy who put in caveats to make it clear he was talking about uniqueness, not constructive extension, yet still insisted on spacelike boundary.
I don’t doubt that pathological examples exist. I don’t suppose you have any handy? I really would be interested. I do doubt that physical examples happen (except perhaps along null vectors).
The prototypical 4-d wave equation is f_tt—f_xx—f_yy—f_zz = 0. I don’t see how rearranging that to f_tt = f_xx + f_yy + f_zz provides any more predictive power in the t direction than f_xx = f_tt—f_yy—f_zz provides in the x direction. (There are numerical stability issues, it’s true.)
Well, that’s partially an artifact of that being the sort of question we tend to be interested in: given this starting condition (e.g. two orbiting black holes), what happens? But this is only a partial answer. In GR we can only extend in space so long as we know the mass densities at those locations as well. Extending QFT solutions should do that. The problem is that we don’t know how to combine QFT and GR, so we use classical mechanics, which is indeed only causal in the time direction. But for source-free (no mass density) solutions to GR, we really can extend a 2+1 dimensional slice in the remaining spatial direction.
No, I think you’re right that directions in which solutions to differential equations are determined by their boundary values are generic.
But I think it is reasonable (and maybe common) to identify causality with well-posedness. So I think that rwallace’s definition is salvageable.
I’m confused about GR, though, since I remember one guy who put in caveats to make it clear he was talking about uniqueness, not constructive extension, yet still insisted on spacelike boundary.