Some moving considerations
In Tim Maudlin's class we've been going through theories of space-time. I had a worry last week that he talked me out of, but now I'm back to thinking there's something worth saying. First some background.
First player: "Newtonian space-time". Space is made out of lots of points that persist through time. So I can pick out a point at 1:00, call it A, and then at 2:00 I can meaningfully ask of any point in space whether it's the same point as A (and in general, how far it is from A). We can either think of the points as enduring through time (the way Newton did), so a point B being the same point as A means that A=B—it's numerically the same object. Or we can think of the points as being points in space-time, and there's some special "sameness" relation that holds between space-time point A (which only exists at 1:00) and space-time point B (which only exists at 2:00).
Second player: "Neo-Newtonian space-time". Space-time is made out of lots of points. The points don't persist through time, so in general it isn't meaningful to ask of a point at 1:00 whether it's the same as a point at 2:00. (Instead, there's a weaker kind of structure, called "affine structure". If I have a point A at 1:00, a point B at 2:00, and a point C at 3:00, it makes sense to ask whether A, B, and C are "in a straight line", which physically means that an object could travel through A, B, and C without accelerating at all.)
Nobody wants to believe in Newtonian space-time: it posits more structure than Newtonian physics needs. For instance, in Newtonian space-time we can ask whether the universe is drifting at some uniform rate, but this kind of motion wouldn't have any physical consequences. Nobody believes in Neo-Newtonian space-time nowadays either, but it's the kind of space-time we would want to believe in if we still believed in Newtonian physics. And I think what I'm going to say applies to the sort of space-time people do generally believe in, too.
Third player: the "at-at" theory of motion. This is a theory of what it is for an object to move. And the simple theory goes like this: an object moves if (and only if) it is at different places at different times. It's a nice theory.
Finally, here's my worry: it looks to me like the at-at theory requires full Newtonian space-time. The theory doesn't make sense if it isn't possible for an object to be at the same place at different times. And this kind of structure isn't around in the Neo-Newtonian universe. So in NN-world the at-at theory makes motion meaningless.
First response: that's right! It doesn't make sense to ask, in absolute terms, whether an object is at different places at different times, and so there isn't any absolute motion. Instead, there's only relative motion. We can ask about point A at 1:00 and point B at 2:00 whether they're the same point relative to a frame of reference. If I pick New Brunswick as a frame of reference, then there is a definite fact of the matter whether a car at the corner of Suydam and Nichol at 1:00 is at the (relatively) same point at 2:00. And so there is a definite fact of the matter whether the car moved relative to this frame of reference. And (the response goes) those are the only kind of definite motion facts.
The problem for this response is that there is absolute motion in Newtonian physics. This is a fact that perplexed everybody for centuries. Imagine two balls, attached by a chain, alone in the universe. Are they spinning or not? There's a way to find out the answer: measure the tension in the chain. If they aren't spinning, there's no tension. If they are spinning, there is some. This is a frame-independent, absolute fact. And if the ball-chain assembly is rotating (absolutely), then the balls must be moving (absolutely). In NN-world the at-at theory can't account for this.
Second response (this is my gloss on what Tim said in class): ok, simple at-at was wrong. But we don't need to abandon the heart and soul of the at-at theory. And the heart and soul is this: there is nothing to moving above and beyond positions and times, no primitive motion properties. In short: motion supervenes on the space-time trajectory. That is to say, if you know all of the space-time points an object occupies through its lifetime, you know everything there is to know about the object's motion.
The revised at-at theory seems fine. I have but three remarks.
First remark. On the simple at-at theory, position was primary, velocity secondary (it reduces to change in position), and acceleration tertiary (it reduces to change in velocity). On the revised at-at theory, this isn't true. Acceleration isn't primitive, but neither does it reduce to velocity-facts or position-facts. It reduces to space-time trajectory facts and affine structure facts, those alone and those directly.
Second remark. In the olden days, the at-at theory offered an account of motion that was independent of the structure of space-time. You can do this if you have genuinely enduring places, points in space that are numerically the same from moment to moment. Then you can tell if an object is moving by looking purely that the numerical identity or distinctness of the points it occupies over time. When we switched to the four-dimensional picture, though, we had to appeal to space-time structures to make sense of motion. In Newtonian space, the structure exactly mirrors the endurance of points. In Neo-Newtonian space it's weaker than that. In either case, the mere occupancy relations doesn't tell you whether or not an object moves: it's the occupancy relations plus the cross-time spatial structure, the "links" between 1:00 points and 2:00 points.
Third remark. Some people took the simple at-at theory to be an account of the meaning of "move". Revised at-at, though, looks nothing like an account of the meaning of "move". And I can't see any plausible meaning account in its vicinity.
(One attempt: "to move is to have a non-inertial space-time trajectory." But that doesn't capture merely relative motions. So should we say "move" is ambiguous between relative and absolute motion, with totally different definitions for each? That doesn't sound like plausible semantics to me.)
First player: "Newtonian space-time". Space is made out of lots of points that persist through time. So I can pick out a point at 1:00, call it A, and then at 2:00 I can meaningfully ask of any point in space whether it's the same point as A (and in general, how far it is from A). We can either think of the points as enduring through time (the way Newton did), so a point B being the same point as A means that A=B—it's numerically the same object. Or we can think of the points as being points in space-time, and there's some special "sameness" relation that holds between space-time point A (which only exists at 1:00) and space-time point B (which only exists at 2:00).
Second player: "Neo-Newtonian space-time". Space-time is made out of lots of points. The points don't persist through time, so in general it isn't meaningful to ask of a point at 1:00 whether it's the same as a point at 2:00. (Instead, there's a weaker kind of structure, called "affine structure". If I have a point A at 1:00, a point B at 2:00, and a point C at 3:00, it makes sense to ask whether A, B, and C are "in a straight line", which physically means that an object could travel through A, B, and C without accelerating at all.)
Nobody wants to believe in Newtonian space-time: it posits more structure than Newtonian physics needs. For instance, in Newtonian space-time we can ask whether the universe is drifting at some uniform rate, but this kind of motion wouldn't have any physical consequences. Nobody believes in Neo-Newtonian space-time nowadays either, but it's the kind of space-time we would want to believe in if we still believed in Newtonian physics. And I think what I'm going to say applies to the sort of space-time people do generally believe in, too.
Third player: the "at-at" theory of motion. This is a theory of what it is for an object to move. And the simple theory goes like this: an object moves if (and only if) it is at different places at different times. It's a nice theory.
Finally, here's my worry: it looks to me like the at-at theory requires full Newtonian space-time. The theory doesn't make sense if it isn't possible for an object to be at the same place at different times. And this kind of structure isn't around in the Neo-Newtonian universe. So in NN-world the at-at theory makes motion meaningless.
First response: that's right! It doesn't make sense to ask, in absolute terms, whether an object is at different places at different times, and so there isn't any absolute motion. Instead, there's only relative motion. We can ask about point A at 1:00 and point B at 2:00 whether they're the same point relative to a frame of reference. If I pick New Brunswick as a frame of reference, then there is a definite fact of the matter whether a car at the corner of Suydam and Nichol at 1:00 is at the (relatively) same point at 2:00. And so there is a definite fact of the matter whether the car moved relative to this frame of reference. And (the response goes) those are the only kind of definite motion facts.
The problem for this response is that there is absolute motion in Newtonian physics. This is a fact that perplexed everybody for centuries. Imagine two balls, attached by a chain, alone in the universe. Are they spinning or not? There's a way to find out the answer: measure the tension in the chain. If they aren't spinning, there's no tension. If they are spinning, there is some. This is a frame-independent, absolute fact. And if the ball-chain assembly is rotating (absolutely), then the balls must be moving (absolutely). In NN-world the at-at theory can't account for this.
Second response (this is my gloss on what Tim said in class): ok, simple at-at was wrong. But we don't need to abandon the heart and soul of the at-at theory. And the heart and soul is this: there is nothing to moving above and beyond positions and times, no primitive motion properties. In short: motion supervenes on the space-time trajectory. That is to say, if you know all of the space-time points an object occupies through its lifetime, you know everything there is to know about the object's motion.
The revised at-at theory seems fine. I have but three remarks.
First remark. On the simple at-at theory, position was primary, velocity secondary (it reduces to change in position), and acceleration tertiary (it reduces to change in velocity). On the revised at-at theory, this isn't true. Acceleration isn't primitive, but neither does it reduce to velocity-facts or position-facts. It reduces to space-time trajectory facts and affine structure facts, those alone and those directly.
Second remark. In the olden days, the at-at theory offered an account of motion that was independent of the structure of space-time. You can do this if you have genuinely enduring places, points in space that are numerically the same from moment to moment. Then you can tell if an object is moving by looking purely that the numerical identity or distinctness of the points it occupies over time. When we switched to the four-dimensional picture, though, we had to appeal to space-time structures to make sense of motion. In Newtonian space, the structure exactly mirrors the endurance of points. In Neo-Newtonian space it's weaker than that. In either case, the mere occupancy relations doesn't tell you whether or not an object moves: it's the occupancy relations plus the cross-time spatial structure, the "links" between 1:00 points and 2:00 points.
Third remark. Some people took the simple at-at theory to be an account of the meaning of "move". Revised at-at, though, looks nothing like an account of the meaning of "move". And I can't see any plausible meaning account in its vicinity.
(One attempt: "to move is to have a non-inertial space-time trajectory." But that doesn't capture merely relative motions. So should we say "move" is ambiguous between relative and absolute motion, with totally different definitions for each? That doesn't sound like plausible semantics to me.)
Posted by Jeff on Friday, March 02, 2007
2 comments
