How do physicists think about Zeno's arrow?

[TubbaBlubba]

Not sure about that, but some of them rely on the assumption that an infinite series of finite numbers cannot add up to a finite number. That is simply false, but 2500 years ago people did not know that.

Actually Archimedes did and used precisely this to counter the paradoxes.

 

[GlennB]

Do you mean there would always be some internal tensions, some material difference captured by the molecules of the moving arrow? It seems to me to just shift the problem's scale, but not attack the root idea.

The root idea is: "So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?"

I'm suggesting a moving arrow will show signs of movement that a static arrow won't. High-speed photos of the very same arrow at rest and in flight will show differences. Flexing too - watch a javelin in flight at an athletics meeting and that 'rigid' javelin will be visibly flexing.

 

[TubbaBlubba]

You could think of this as an ordinary differential equation. We know the arrow had some initial velocity, and we know its position at some later point in time. With Newton's laws we can thereby compute how it will behave.

In a sense that's really all there is to it. At any instant it has a position and velocity, and we can compute where it will be at some other instant.

 

[marplots]

The root idea is: "So what property does the moving arrow have that the static arrow does not, and especially in that instant of time?"

I'm suggesting a moving arrow will show signs of movement that a static arrow won't. High-speed photos of the very same arrow at rest and in flight will show differences. Flexing too - watch a javelin in flight at an athletics meeting and that 'rigid' javelin will be visibly flexing.

It seems to me we could overcome this in two ways. The first is to design the arrow in flight so that it would deform (while in flight) in a manner to match the stationary arrow. The second is to look at some particular atom in the moving and stationary arrows as a proxy for the same argument.

 

[marplots]

You could think of this as an ordinary differential equation. We know the arrow had some initial velocity, and we know its position at some later point in time. With Newton's laws we can thereby compute how it will behave.

In a sense that's really all there is to it. At any instant it has a position and velocity, and we can compute where it will be at some other instant.

It sounds like you are invoking a descriptive element instead of assigning particular properties to each arrow. This is OK, but leaves me dissatisfied.

Do you take the position that physics is no more than a good description of how things act, instead of capturing some underlying essential bit about how things actually are?

I do get that you have answered the question, I'm just trying to pin down the type of answer which satisfies you. I am more interested in how the situation is perceived by someone who is skilled at this stuff than I am in getting the "right" answer.

 

[TubbaBlubba]

It sounds like you are invoking a descriptive element instead of assigning particular properties to each arrow. This is OK, but leaves me dissatisfied.

Do you take the position that physics is no more than a good description of how things act, instead of capturing some underlying essential bit about how things actually are?

I do get that you have answered the question, I'm just trying to pin down the type of answer which satisfies you. I am more interested in how the situation is perceived by someone who is skilled at this stuff than I am in getting the "right" answer.

The problem as I see it is that your "freezing of time" introduces a distinctly non-physical aspect in the problem (especially apparent if we consider relativistic implications). "Frozen time" only really shows up in contexts such as solving differential equations.

 

[marplots]

The problem as I see it is that your "freezing of time" introduces a distinctly non-physical aspect in the problem (especially apparent if we consider relativistic implications). "Frozen time" only really shows up in contexts such as solving differential equations.

Does it matter if, instead of freezing, we make the time interval extremely short? Is there, under current theory, some "smallest time" possible?

Does anything change if we make the moving arrow move extremely slowly?

It seems to me we could generate the same situation by making both arrows move, with some "nearly nothing" difference between their rates of motion. Then, just pick the slower one as your reference frame to generate the same situation.

 

[TubbaBlubba]

Does it matter if, instead of freezing, we make the time interval extremely short? Is there, under current theory, some "smallest time" possible?

Does anything change if we make the moving arrow move extremely slowly?

It seems to me we could generate the same situation by making both arrows move, with some "nearly nothing" difference between their rates of motion. Then, just pick the slower one as your reference frame to generate the same situation.

Then what changes is that over that time frame, the arrow's position, relative to some frame of reference is some function of time - that is to say, it has momentum. Momentum, of course, depends on your frame of reference. What matters, in order to model the system, is how momentum is transferred between parts of the system over time. (One can model a system similarly using energy).

If you want to get into the nitty-gritty details of all the consequences of this, general relativity is your best bet.

Stuff moves. It's fundamental, simple, and has a lot of non-simple consequences.

 

[ehcks]

Does it matter if, instead of freezing, we make the time interval extremely short? Is there, under current theory, some "smallest time" possible?

Does anything change if we make the moving arrow move extremely slowly?

It seems to me we could generate the same situation by making both arrows move, with some "nearly nothing" difference between their rates of motion. Then, just pick the slower one as your reference frame to generate the same situation.

You can't physically stop time. You can mathematically stop time, but even then, at the instant you're measuring from, one arrow has velocity and the other doesn't.

It's like most other paradoxes. It's only a problem when you think about it too hard. Everything works just fine otherwise.

 

[marplots]

You can't physically stop time. You can mathematically stop time, but even then, at the instant you're measuring from, one arrow has velocity and the other doesn't.

It's like most other paradoxes. It's only a problem when you think about it too hard. Everything works just fine otherwise.

As I said in the OP, my interest was reengaged from the pre-Socratics. I'm happy that ideas they thought about still have the ability to inspire questions in me. This is not universally true, but some of it has stood the test of time well.

 

[lomiller]

I really see no need to go beyond classical mechanics in this case.

There are explanations in classical physics and even ones that predate classical physics but the way I read the OP I think the answer he's really looking for is the quantum mechanics one.

Does it matter if, instead of freezing, we make the time interval extremely short? Is there, under current theory, some "smallest time" possible?

Sort of. The Uncertainty Principle specifies a minimum uncertainty in the product of location and momentum. What this means is that if you define the arrows velocity, which you do by saying they are traveling at different velocities, this forces a fuzziness in it's location that never goes away no matter how small an instant in time you try to look at.

Since the locations are "fuzzy" you can't actually identify an instant or very small period of time where they are lines up exactly, it's actually physically impossible to do so. Similarly if you look at a moving object over to short a period of time the fuzziness of it's location prevent you from seeing it move, instead what you have is an increased probability that it's slightly farther along.

 

[marplots]

Sort of. The Uncertainty Principle specifies a minimum uncertainty in the product of location and momentum. What this means is that if you define the arrows velocity, which you do by saying they are traveling at different velocities, this forces a fuzziness in it's location that never goes away no matter how small an instant in time you try to look at.

Since the locations are "fuzzy" you can't actually identify an instant or very small period of time where they are lines up exactly, it's actually physically impossible to do so. Similarly if you look at a moving object over to short a period of time the fuzziness of it's location prevent you from seeing it move, instead what you have is an increased probability that it's slightly farther along.

It sounds like both arrows would suffer from the same defect, so that trying the experiment, at some level of fidelity, you wouldn't be able to detect a difference between the two. Or, is that wrong, and they wouldn't be equally "fuzzy?"

In other words, the fact that they can't be measured well seems to imply the differences between the two fade away - which, while it might help ruin the question, makes it even more mysterious that one still retains the "moving" identity and the other does not.

I'm pretty sure I've made an error in those two paragraphs, but I'm not sure what the error actually is.

 

[Darwin123]

Does it matter if, instead of freezing, we make the time interval extremely short? Is there, under current theory, some "smallest time" possible?

Your question is a mortal sin! Seek not an answer unless you want to jeapardize your soul!

The Jesuits banned this quest almost five hundred years ago! The Roman Catholic Church has never recalled this taboo! They have even apologized for condemning Galileo's theory of heliocentricity. However, they NEVER apologized to Galileo or anyone else for banning the concept of infinitesimals!

'Infinitesimals: How a Dangerous Mathematical Theory Shaped the Modern World', by Amir Alexander (Scientific American Publications, 2014)
"On August 10, 1632, Five Men in Flowing Black Robes convened in a somber Roman palazzo to pass Judgement a simple preposition: that a continuous line is composed of distinct and infinitely tiny parts. The doctrine would be come the foundation of calculus, but on that fateful day the Jesuit fathers ruled that it was FORBIDDEN. With a stroke of the pen they launched a war for the SOUL of the modern world."

The judgement was never rescinded, hence it is still active. Discussion of infinitesimals is still heresy and a MORTAL SIN on the level of -birth control!!

This ABOMINABLE HERESY was started by Zeno the Eleatic circa 500 BC described on pages 9, 10, 12, 142, 146, 259 and 303 of the book that I cited above. Religious leaders have fought the Idols of the Infinitesimal with righteous energy! However, Infinitesimals have continued to corrupt the Materialist Scientists of this Sinful World!

So don't read the book above! Don't buy it from Amazon for a mere $16.00 USA or $18.50 Canadian! If you dare read this profane history of the infinitesimal concept, YOU WILL BE DAMNED!

 

[TubbaBlubba]

There are explanations in classical physics and even ones that predate classical physics but the way I read the OP I think the answer he's really looking for is the quantum mechanics one.

How do you figure? Honestly, quantum mechanics, in principle, only really absolutely have to come into play once you start asking really weird questions about potential energy, electrodynamics, and the like.

 

[acementhead]

"Is there, under current theory, some "smallest time" possible? "

Yes of course, 10^-43 seconds. Some people call it Plank time.

Zeno's paraox is a misnomer; there is no paradox. It is simply and obviously wrong.



Edited "teime" to read the intended "time"

 

[MaartenVergu]

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?

 

[acementhead]

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?

Both. Whichever of the two you prefer for your intended use. Duh.

 

[alexi_drago]

What's frozen? Are the arrows frozen but light still travels or is light frozen too?

Say an observer is viewing and the arrows are 0.5m long and 1m away from the observer and perpendicular to the line of sight and in the centre of vision.
If there are lines marked right on the centre of the length of the arrows that's where it would appear on the stationary arrow but if the moving arrow is traveling at 100m/s then the centre line would appear to be about 1/100mm towards the front tip of the arrow, if my maths is right, which im not totally sure about.

 

[marplots]

"Is there, under current theory, some "smallest time" possible? "

Yes of course, 10^-43 seconds. Some people call it Plank time.

Zeno's paraox is a misnomer; there is no paradox. It is simply and obviously wrong.

Yes, it is wrong that, "hence, motion is impossible" - but Zeno knew that too. He was pointing out there is some problem with the way the philosophers of his time thought about such things. Which is why I wondered how modern physicists thought about the same things.

The question isn't about it being wrong, but the "why it's wrong." Maybe there is no good reason why, or maybe some of the reasons already given are satisfactory.

 

[Modified]

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.

Red/blue shift could tell us whole story, but that feels sort of like cheating as it's a property of radiation the arrow has emitted rather than of the arrow itself.