How do physicists think about Zeno's arrow?

[Roboramma]

I think the question is something like this:

If time is incremental and step-like, rather than continuous, such that it can be divided into moments, what is the difference at one moment between a moving arrow and a stationary arrow such that one arrow "knows" to move in the next instant of time whereas the other arrow "knows" to stay put?

The question has pretty much the same meaning if it is continuously divisible, but is perhaps a little harder to think about.

*"knows" here can simply mean "what property causes it to..."

 

[marplots]

I think the question is something like this:

If time is incremental and step-like, rather than continuous, such that it can be divided into moments, what is the difference at one moment between a moving arrow and a stationary arrow such that one arrow "knows" to move in the next instant of time whereas the other arrow "knows" to stay put?

The question has pretty much the same meaning if it is continuously divisible, but is perhaps a little harder to think about.

*"knows" here can simply mean "what property causes it to..."

That is a very nice way of putting it. Thank you.

 

[Roboramma]

Physics really only tells us "if a system is in a particular state at a particular point in time, what will happen next?". In that sense, I think TB answered the question well in the his first post: the difference between the two arrows is that their momentum is different.

What is momentum? Like every other property of anything, it's just a variable that can help us answer that question at the beginning of my post.

But everything is like this: what is mass? What is energy? What is time? etc.
These are all things that we can measure, in as much as a property is fundamental it can't be described as an emergent property of something else, and we end up with just a number. Yes, we can model it's behaviour with the laws of physics, but I get the feeling that you are looking for a more fundamental answer, and I don't really think there is one.

What I think may help, though, is to look at the question experimentally. TB did a very good job of that when he said:

you could arrange it so that the result could be read without needing the arrow to move. E.g. the kinetic energy means the arrow exerts additional graviational force, if you had a dynamometer above each arrow as you froze them, they would give (incredibly slightly) different readings.

 

[marplots]

I don't mind if they think about it as "having" momentum. I just wanted to know how they captured the situation in their own heads.

 

[DevilsAdvocate]

Assuming an ideal arrow, so there are no effects on its shape or dimensions due to the firing or air resistance, and assuming we can measure length exactly to any precision in this frozen state, then relativistic length contraction will tell us the exact speed and direction of motion, but not the sign of the direction.

That is what I was thinking. Wouldn’t the arrow in flight be shorter?

To get the arrow moving, it has a force applied to the back of the arrow, presumably from a bow string. That force pushes on the atoms at the back of the arrow. Those atoms, in turn, push the atoms in front of them, and so on. The arrow gets compressed because the atoms at the back move forward before the atoms at the front of the arrow. So the arrow is shorter, and more compact, than it was before it was shot.

Because we know that the arrows are identical, by measuring the precise length we can conclude that the shorter arrow is in flight. Although I’m not sure we could tell whether the arrow is flying forwards or backwards.

Does that make sense?

 

[DevilsAdvocate]

Physics really only tells us "if a system is in a particular state at a particular point in time, what will happen next?". In that sense, I think TB answered the question well in the his first post: the difference between the two arrows is that their momentum is different.

What is momentum? Like every other property of anything, it's just a variable that can help us answer that question at the beginning of my post.

That seems to be just begging the question. Saying that the difference is that one has momentum and the other does not is no different from saying that one arrow is in flight and the other is not.

The question is whether there is, at that instant, any physical difference in the material properties of the arrows. If there is not, then the property of “momentum” of “being in motion” or “being in flight” is some non-material property. If the arrows are the same, the only thing that causes one to move in the next instant and the other to not move Is some mystical, unmeasurable property of “momentum”.

With a large object like an arrow, it doesn’t seem like that would be the case. Wouldn’t the arrow in flight have some atoms squished uncomfortably together that in the next instant are going to repel and cause motion?

With quantum particles, doesn’t this then become a problem because then “momentum” truly is undetectable and undeterminable? In that case, the property of “momentum” gets explained (however unsatisfactorily) by probability fields, where the probability is a function of time.

 

[Salerio]

I want to ask this question too: are there an infinite amount of steps between 1 and 2
(f.e.:1.999999999999999, 1.9999999999999991 etc to infinity). Or are there a finite amount of steps between 1 and 2 in mathematics?

That's been proven to be non-provable. Cantor thought there was an infinite number of reals between any two given numbers, Godel proved this couldn't be proven.

 

[Aepervius]

Zeno's paradoxes seem to revolve around trying to divide zero by zero.

You missed a nice pun you should have said :
trying to divide zero by zeno

 

[TubbaBlubba]

That's been proven to be non-provable. Cantor thought there was an infinite number of reals between any two given numbers, Godel proved this couldn't be proven.

Huh? The proof that there are an uncountable number of irrationals in any interval is fairly straightforward...

 

[ehcks]

Huh? The proof that there are an uncountable number of irrationals in any interval is fairly straightforward...

Yeah, I thought you just took the average of the two points and you always come up with a new number. You could then take the average of that new number and either of the previous points forever.

 

[Pulvinar]

Here's a version that captures my query:
If you had an arrow sitting in front of you, motionless, and someone shot a second arrow past you, you could arrange it so that at some instant, the first and second arrows were in alignment and at that instant perfectly indistinguishable by observation. Yet the second arrow happily continues on toward it's target when time resumes.

So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?

That property is relative velocity. It's as much a real, first-order property of any physical object as its relative position is. Our problem here is in imagining the arrow frozen in time and yet still being able to observe it. In reality, if we don't observe its instantaneous velocity, we're simply not fully observing it.

 

[TubbaBlubba]

Yeah, I thought you just took the average of the two points and you always come up with a new number. You could then take the average of that new number and either of the previous points forever.

That, roughly speaking, works for a countably infinite number of rationals (due to the Archimedean property). And you can then show that any real interval containing a countably infinite number of rationals also contains an uncountably infinite number of irrationals, completing the proof.

 

[Roboramma]

With a large object like an arrow, it doesn’t seem like that would be the case. Wouldn’t the arrow in flight have some atoms squished uncomfortably together that in the next instant are going to repel and cause motion?

An arrow undergoing acceleration by, for instance, a bowstring, would. An arrow with some constant velocity wouldn't, nor would one in a gravitational field (excluding tidal forces).

Think of it this way: the "stationary arrow" is moving relative to the "moving arrow". Which one do you expect to have atoms squished uncomfortably together, or whatever particular property you're looking for to explain the motion of the arrow?

(Of course, there are some properties that will distinguish them, but how you measure them will depend on which reference frame your measuring equipment is in)

 

[Darwin123]

That property is relative velocity. It's as much a real, first-order property of any physical object as its relative position is. Our problem here is in imagining the arrow frozen in time and yet still being able to observe it. In reality, if we don't observe its instantaneous velocity, we're simply not fully observing it.

You and the OP may be starting from different definitions of 'material property'. The OP may be hypothesizing that a 'material' property must be objective. You may be hypothesizing that a 'material' property must be conserved.

Part of the OP's problem may be that the word 'material' is ambiguous. I think that the responders are divided between those who think 'material' means 'subjective' and those who think 'material' means 'conserved'.

Subjective and conserved are not equivalent. The horizontal momentum of the arrow may be conserved aside from a perturbation caused by air resistance. However, the horizontal momentum is not subjective. A turtle will see a zero momentum for the stationary arrow and nonzero momentum for the moving arrow. However, suppose Achilles is running to keep up with the arrow. Achilles will observer a nonzero momentum for the stationary arrow and a zero momentum for the moving arrow.

Sometimes, the a measurable property can be both. The number of atoms in an arrow can be conserved. Further, the number of atoms in the arrow are objective. All observers would count the same number of atoms in both arrows.
The turtle will count a specific number of atoms in the arrow. Achilles will count the same number of atoms in the arrow.

The same goes for momentum and kinetic energy. I would call them 'material properties' since they are measurable. To me, any measurable quantity that is conserved is a 'material quantity.'


The measurement of a subjective quantity will vary with the observer by definition. Hence, they are subjective. The kinetic energy, the relative velocity, the linear momentum, and the angular momentum are are also subjective. The reference has to be determined from the context of the physical situation.

The language of Zeno is spoken from the viewpoint of the surface of the earth. Zeno did not tell us whether the archer was on the ground or riding in a fast moving chariot. Therefore, all the other Greeks took the default reference, which is the archer is standing on the ground. If Zeno had said that the archer was standing in a moving chariot, then your answer would be definitely correct. However, specifying an archer on the ground would have confused the issue even more.

The difference between a moving arrow and a stationary arrow has to be subjective. The number of atoms does not determine whether the arrow is moving. The linear momentum relative to the earths surface does determine whether the arrow is moving.

Note that it doesn't matter whether Newton or Einstein are addressing the question. Newton and Einstein would agree that the physical properties that distinguish moving from not moving are subjective. The physics of Newton implies that the forces of the universe are invariant to a Galilean transformation. The physics of Einstein imply that the forces of the universe are Lorentz invariant. As long as the forces of the universe are invariant to any transformation, the movement of the arrow is 'subjective'.

The reason that the Jesuits banned the concept of infinitesimals is because infinitesimals are subjective. An infinitesimal is invariant to scale transformations. An infinitesimal looks the same through a microscope as it does through a telescope. So infinitesimals are intrinsically subjective. In the Jesuit mind, subjective meant immaterial.

Therefore, stop discussing the Zeno problem lest you all be DAMNED FOREVER and AYE!!!

 

[marplots]

Very nice. Nommed.

 

[Modified]

Think of it this way: the "stationary arrow" is moving relative to the "moving arrow". Which one do you expect to have atoms squished uncomfortably together, or whatever particular property you're looking for to explain the motion of the arrow?

I would expect measurements to be done in the frame of the stationary arrow. That is what we mean by "stationary" and "moving".

Freezing time is also frame-relative. The reference frames of the stationary and moving arrows don't agree on the timing or order of all events. The arrows and their surroundings will be in different states depending on which frame is selected for the freezing.

 

[Roboramma]

I would expect measurements to be done in the frame of the stationary arrow. That is what we mean by "stationary" and "moving".

Freezing time is also frame-relative. The reference frames of the stationary and moving arrows don't agree on the timing or order of all events. The arrows and their surroundings will be in different states depending on which frame is selected for the freezing.

Right. But he wasn't suggesting looking for some property that is dependent upon reference frame. I'm pretty sure he wasn't talking about length contraction, considering that he said:

Wouldn’t the arrow in flight have some atoms squished uncomfortably together that in the next instant are going to repel and cause motion?

My point being that if that were the case it would also be the case with the stationary arrow when viewed from a different frame. But I think he will intuitively see that that isn't the case.

 

[TheAdversary]

To recap the original question : If one arrow is standing still, and the other is moving, what property distinguishes it from the still arrow
when we freeze time?

The problem is that you say 'freeze time', but the concept of motion doesn't make any sense without time.
If we define velocity as the change in position divided by the change in time, and the change in time is zero, then you're dividing by zero.
Bang, you're dead. Paradox galore.
This is why we use differential equations since it neatly bypasses this problem.
Ultimately all of those paradoxes manifest themselves because the notion of continuity must first be defined in a way that avoids paradox.

 

[schrodingasdawg]

So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?

Since there are many well-trained (even if not degreed) physics afficionados on this forum, I'd like to know, not necessarily what the resolution to the original "paradox" is, but how someone with a better understanding of physics (and more schooling) than me views the situation - how do you think about it?

I have two ways of looking at the arrow.

The first is in line with Aristotle's original refutation of the paradox, which is that it's meaningless to talk about the arrow being frozen or at rest in an instant of time. Velocity is only meaningfully spoken of when you look at a finite duration of time, and you examine the motion that the arrow takes in that finite duration. Quite some time after Aristotle, some mathematicians figured out how to define an "instantaneous velocity" which could be ascribed to an object during an instant of time, but this definition still relies on Aristotle's observation. The instantaneous velocity is a limit: you consider the average velocity of the arrow over some interval of time [t,t+h], and find the limit of this average velocity as you make h arbitrarily small. In other words, the definition of instantaneous velocity requires one to consider the motion of the arrow over a finite duration of time, as Aristotle pointed out.

The second perspective involves a reformulation of classical physics due to Hamilton. You can talk about a physical object in terms of its position x(t) and its instantaneous velocity v(t), but you can also make a Legendre transform and change the description you use to one of position x(t) and momentum p(t). The momentum in the Hamiltonian formalism is no longer conceived of as the instantaneous motion of the arrow, but is instead another coordinate in a larger space called phase space. (If the arrow moves through three-dimensional space, phase space is six-dimensional; there is a momentum coordinate for each position coordinate.) Since an arrow in motion and an arrow at rest are situated at different places in phase space, you can distinguish them within the formalism even if you insist on considering the arrow as frozen in time.

Another way of looking at the second perspective is that the moving arrow carries some quantity of momentum, and rather than the momentum being an empirical description of the fact that the arrow has so much mass and is moving so fast, the fact that it possesses momentum is what causes it to move. (Hamilton's equations seem to insist on this, as I see them.)

 

[GlennB]

Ultimately all of those paradoxes manifest themselves because the notion of continuity must first be defined in a way that avoids paradox.

Yes. If you think you've found a paradox then you just haven't thought it through, or need to know more.