Ah, so you’re saying my initial statement would be true in Newtonian physics, but is false in quantum mechanics (for reasons which I’m struggling to grasp—my knowledge of physics is strictly that of an interested layman)?
Now that is very interesting indeed.
Does this have anything to do with holographic theory and the fact that the Bekenstein bound is defined over area rather than volume?
Would it be correct to say this is still true for an uncharged black hole, based on the existence of frames of reference in which matter evaporated just before entering the event horizon?
Ah, so you’re saying my initial statement would be true in Newtonian physics, but is false in quantum mechanics...?
No, newtonian vs QM isn’t making a difference here and wnoise’s first answer makes perfect sense in the classical setting. But I’m not sure why he didn’t say that in response to this question. Let’s transpose your example into Maxwell’s equations, for concreteness. Maxwell’s equations in vacuum are determined, but once you put in the charged body, they’re underdetermined. In potential form, they say that the wave operator applied to the four-vector-potential is equal to the four-current. In vacuum, this is four equations in four unknowns and one expects solutions to be determined by boundary values, but if you don’t constrain the four-current, it says that you can make the vector-potential anything you like (which is some constraint on E&B). You can violate causality in any direction. You can have vacuum up to time 0, and then have charge appear.
You can pin down the point particle or the field of charge by fiat, but then that’s a law of physics differing between the two versions. In real life, you need to do that by some other physical process, some other law that governs the charged particles and is coupled to Maxwell’s equations. In particular, the charged particles should get pushed around by the electromagnetic field. That’s not Maxwell’s equation! E&B and the charge tell you how much force, but you need to know the mass to determine the effect of the force. In the point particle case, you just have one mass per particle and I think that’s the end of the story. In the uniformly distributed case, you need a mass density field, plus probably other stuff to determine the motion of the charge. Plus you need other forces involved to prevent the soup from dispersing from Coulomb’s law. It’s that other force that, presumably, will be felt on the slice that is not space-like.
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?
It’s not in general true in quantum mechanics. It is true for 1-particle quantum mechanics if and only if the potential and any other interactions (e.g. form of canonical momentum for external magnetic field) are specified.
The reason is that the arena of quantum mechanics is not space with
3-dimensions, but configuration-space, with 3-n dimensions, one space
for each particle (disregarding symmetries). Having time evolution be
known lets us get rid of one spatial dimension, but we need to get rid
of one spatial dimension for each of n particles. The other thing that destroys any hope is that non-local interactions are often used to model systems.
Of course quantum mechanics is only an approximation to quantum field theory, which is nicely local in the spatial sense.
Does this have anything to do with holographic theory
You know, I asked that at a colloquium nearly a year ago, and got back the answer “no”, but without a satisfactory explanation.
Would it be correct to say this is still true for an uncharged black hole, based on the existence of frames of reference in which matter evaporated just before entering the event horizon?
I can’t quite figure out what you’re asking here, and probably couldn’t give an answer without a full theory of quantum gravity.
Ah, so you’re saying my initial statement would be true in Newtonian physics, but is false in quantum mechanics (for reasons which I’m struggling to grasp—my knowledge of physics is strictly that of an interested layman)?
Now that is very interesting indeed.
Does this have anything to do with holographic theory and the fact that the Bekenstein bound is defined over area rather than volume?
Would it be correct to say this is still true for an uncharged black hole, based on the existence of frames of reference in which matter evaporated just before entering the event horizon?
No, newtonian vs QM isn’t making a difference here and wnoise’s first answer makes perfect sense in the classical setting. But I’m not sure why he didn’t say that in response to this question. Let’s transpose your example into Maxwell’s equations, for concreteness. Maxwell’s equations in vacuum are determined, but once you put in the charged body, they’re underdetermined. In potential form, they say that the wave operator applied to the four-vector-potential is equal to the four-current. In vacuum, this is four equations in four unknowns and one expects solutions to be determined by boundary values, but if you don’t constrain the four-current, it says that you can make the vector-potential anything you like (which is some constraint on E&B). You can violate causality in any direction. You can have vacuum up to time 0, and then have charge appear.
You can pin down the point particle or the field of charge by fiat, but then that’s a law of physics differing between the two versions. In real life, you need to do that by some other physical process, some other law that governs the charged particles and is coupled to Maxwell’s equations. In particular, the charged particles should get pushed around by the electromagnetic field. That’s not Maxwell’s equation! E&B and the charge tell you how much force, but you need to know the mass to determine the effect of the force. In the point particle case, you just have one mass per particle and I think that’s the end of the story. In the uniformly distributed case, you need a mass density field, plus probably other stuff to determine the motion of the charge. Plus you need other forces involved to prevent the soup from dispersing from Coulomb’s law. It’s that other force that, presumably, will be felt on the slice that is not space-like.
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?
It’s not in general true in quantum mechanics. It is true for 1-particle quantum mechanics if and only if the potential and any other interactions (e.g. form of canonical momentum for external magnetic field) are specified.
The reason is that the arena of quantum mechanics is not space with 3-dimensions, but configuration-space, with 3-n dimensions, one space for each particle (disregarding symmetries). Having time evolution be known lets us get rid of one spatial dimension, but we need to get rid of one spatial dimension for each of n particles. The other thing that destroys any hope is that non-local interactions are often used to model systems.
Of course quantum mechanics is only an approximation to quantum field theory, which is nicely local in the spatial sense.
You know, I asked that at a colloquium nearly a year ago, and got back the answer “no”, but without a satisfactory explanation.
I can’t quite figure out what you’re asking here, and probably couldn’t give an answer without a full theory of quantum gravity.