And again, I am surprised, and perhaps missing something.
In classical mechanics with Maxwell’s equations, suppose you have a non-spacelike slice, and 1 meter to the north there is a spherical charged object with a given charge and mass. Is it not the case that the electromagnetic and gravitational effects of this object are just the same as they would be if the same charge and mass were in the form of a point particle? How could you tell the difference, even in principle, just by looking at what is happening at the slice?
Another question: if a 3d slice is enough to determine what happens in a 4d volume of space-time, even when the slice is not spacelike, does relative size matter? Suppose the slice is a spherical shell surrounding a small volume… say for example, a piece of fairy cake… presumably this is enough to determine what happens inside the volume. Is it also enough to determine what happens in the rest of the universe?
In no direction is a 3d slice enough to determine what happens with, say, Maxwell’s equations. This is true in a space-like direction, as in your example, and it is true in a time-like direction, as in the case of an electron and a positron appearing out of the vacuum. Throwing in Newton’s laws isn’t enough to change this. You need to know what governs the creation of particles or what holds together the uniform sphere of charge.
Certainly you need to know those things, but I’m not clear on how that relates to the 3dness of the slice; suppose you add rules like particles can neither be created nor destroyed and spherical charged particles hold together by fiat, doesn’t that solve that problem?
And again, I am surprised, and perhaps missing something.
In classical mechanics with Maxwell’s equations, suppose you have a non-spacelike slice, and 1 meter to the north there is a spherical charged object with a given charge and mass. Is it not the case that the electromagnetic and gravitational effects of this object are just the same as they would be if the same charge and mass were in the form of a point particle? How could you tell the difference, even in principle, just by looking at what is happening at the slice?
Another question: if a 3d slice is enough to determine what happens in a 4d volume of space-time, even when the slice is not spacelike, does relative size matter? Suppose the slice is a spherical shell surrounding a small volume… say for example, a piece of fairy cake… presumably this is enough to determine what happens inside the volume. Is it also enough to determine what happens in the rest of the universe?
In no direction is a 3d slice enough to determine what happens with, say, Maxwell’s equations. This is true in a space-like direction, as in your example, and it is true in a time-like direction, as in the case of an electron and a positron appearing out of the vacuum. Throwing in Newton’s laws isn’t enough to change this. You need to know what governs the creation of particles or what holds together the uniform sphere of charge.
Certainly you need to know those things, but I’m not clear on how that relates to the 3dness of the slice; suppose you add rules like particles can neither be created nor destroyed and spherical charged particles hold together by fiat, doesn’t that solve that problem?