No. The state of affairs along the slice of space passing through Earth’s equator, for example, does not uniquely determine the state of affairs at 1° north latitude. But the state of affairs now, does determine the state of affairs one second in the future. (Relativistic motion can tilt the axes somewhat, but not enough to interchange space and time.)
All our physical models are described by local partial differential equations. Given the data on an (n-1) dimensional slice (including derivatives, of course), we can propagate that to cover the whole space. (there are complications once GR is in the picture making the notion of global slices questionable, but the same result holds “locally”.)
If the data at the slice doesn’t include derivatives, you can’t propagate in time either.
All our physical models are described by local partial differential equations. Given the data on an (n-1) dimensional slice (including derivatives, of course), we can propagate that to cover the whole space.
In that generality, this is false. Not all differential equations are causal in all directions. I doubt that it’s true of most physical examples. In particular, everyone I’ve ever heard talk about reconstruction in GR mentioned space-like hypersurfaces.
UPDATE: Actually, it’s true. Until I redefine causal.
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.
How does that square with e.g. the fact that the gravity of a spherically symmetric object from the outside is the same as that of the same mass compressed into a point at the same center of gravity?
The short answer is that there’s more there besides the gravitational field[1] (in the approximation that we can think of it as that). There are the various elementary particle fields. These will have their own values[2] and derivatives, which are part of a giant system
of PDEs intertwining them. Two different spherically symmetric objects with the same gravitational field will have different particle fields.
If there were only gravity, we would have something like the Schwarzschild solution—which is uniquely determined by mass.
These are not number-valued fields, but operator valued, and usually spinor-operator and vector-operator valued at that.
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.
All our physical models are described by local partial differential equations. Given the data on an (n-1) dimensional slice (including derivatives, of course), we can propagate that to cover the whole space.
In that generality, this is false. Not all differential equations are causal in all directions. I doubt that it’s true of most physical examples. In particular, everyone I’ve ever heard talk about reconstruction in GR mentioned space-like hypersurfaces.
No. The state of affairs along the slice of space passing through Earth’s equator, for example, does not uniquely determine the state of affairs at 1° north latitude. But the state of affairs now, does determine the state of affairs one second in the future. (Relativistic motion can tilt the axes somewhat, but not enough to interchange space and time.)
All our physical models are described by local partial differential equations. Given the data on an (n-1) dimensional slice (including derivatives, of course), we can propagate that to cover the whole space. (there are complications once GR is in the picture making the notion of global slices questionable, but the same result holds “locally”.)
If the data at the slice doesn’t include derivatives, you can’t propagate in time either.
In that generality, this is false. Not all differential equations are causal in all directions. I doubt that it’s true of most physical examples. In particular, everyone I’ve ever heard talk about reconstruction in GR mentioned space-like hypersurfaces.
UPDATE: Actually, it’s true. Until I redefine causal.
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.
How does that square with e.g. the fact that the gravity of a spherically symmetric object from the outside is the same as that of the same mass compressed into a point at the same center of gravity?
The short answer is that there’s more there besides the gravitational field[1] (in the approximation that we can think of it as that). There are the various elementary particle fields. These will have their own values[2] and derivatives, which are part of a giant system of PDEs intertwining them. Two different spherically symmetric objects with the same gravitational field will have different particle fields.
If there were only gravity, we would have something like the Schwarzschild solution—which is uniquely determined by mass.
These are not number-valued fields, but operator valued, and usually spinor-operator and vector-operator valued at that.
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.
Can I have your input at the “marketplace of insights and issues”? Please? On any of the two questions I posed there for physicists.
In that generality, this is false. Not all differential equations are causal in all directions. I doubt that it’s true of most physical examples. In particular, everyone I’ve ever heard talk about reconstruction in GR mentioned space-like hypersurfaces.