If a particle was at position X for zero time, was it there?

[ddjj77]

A moving particle has been at position X for zero time. Was it ever at position X?
Can zero time be considered as never, as in "I was in Rome for zero amount of time."?
It seems like it would have been at position X if time passed in pieces the size of Plank time.

 

[anorlunda]

I think you should read this article https://en.wikipedia.org/wiki/Zeno's_paradoxes

 

[sophiecentaur]

You can either go very deep into Maths Analysis and get familiar with the 'real answer' to your question or you can think in terms of measuring the progress of the particle in tiny steps. (Like with crude measuring equipment). You can then allow the particle to be somewhere between X and X+0.00001 for 0.0035 seconds, between X+0001 andX+0.0002 in the next 0.035 seconds etc, etc. and then work towards smaller and smaller steps. That will always yield a value for a velocity, however short the steps are. This is the basis of (the immensely useful) Differential Calculus.
On a philosophical level, unless the particle actually disappears at point X, then it has to have been at X.
Do a search on Zeno's Paradox if you want to add interest to the question.

 

[Dale]

A moving particle has been at position X for zero time. Was it ever at position X?

Of course yes. You stipulated that it was at position X so by stipulation it was at X. The fact that it was only there momentarily doesn’t change your stipulation nor is it logically self contradictory.

 

[ddjj77]

"was only there momentarily"

So momentarily and zero time are equivalent?

 

[ddjj77]

I need help with understanding this: "The current standard treatment or so-called "Standard Solution" implies Zeno was correct to conclude that a runner's path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts." (From Internet Encyclopedia of Philosophy)
If infinity is too many, should the path contain a finite number of parts? As many Planck distances as needed?

 

[jbriggs444]

I need help with understanding this: "The current standard treatment or so-called "Standard Solution" implies Zeno was correct to conclude that a runner's path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts." (From Internet Encyclopedia of Philosophy)
If infinity is too many, should the path contain a finite number of parts? As many Planck distances as needed?

The standard resolution from the mathematics perspective is that actual infinity is not a problem.

From the physics perspective, another resolution is to notice that the notions of an infinitely precise "position" and a continuous "path" are not physically realized in the first place.

The Plank distance is not the granularity of the universe. It is simply a distance unit.

 

[Dale]

So momentarily and zero time are equivalent?

Yes. Although it should be zero duration.

If infinity is too many,

Read it more carefully. Infinity is not too many.

 

[A.T.]

A moving particle has been at position X for zero time. Was it ever at position X?

It was there, but didn't loiter.

 

[ddjj77]

Yes. Although it should be zero duration.

Read it more carefully. Infinity is not too many.

Oops, thanks for noticing that.

 

[ddjj77]

"From the physics perspective, another resolution is to notice that the notions of an infinitely precise "position" and a continuous "path" are not physically realized in the first place."

I assume that speed would also not be physically realized? But momentum would be?

 

[Herman Trivilino]

A moving particle has been at position X for zero time. Was it ever at position X?

Yes. If it spent a nonzero amount of time at position X it would have had to stop there, but since it's a moving particle it doesn't stop.

Note that the same behavior is exhibited by clocks. Where I am now it will soon be 11:00 pm. But for how long will it be exactly 11:00 pm? The answer is zero. If the clock spent any nonzero amount of time at any clock-reading, then the clock would have to stop. Accurate clocks don't stop.

Can zero time be considered as never, as in "I was in Rome for zero amount of time."?

You are not a particle because you have a nonzero size. And Rome is not a position because it has a nonzero size. Imagine walking into and through Rome. Choose one edge of your body, say the front edge, and one edge of Rome, say the south edge. The leading edge of your body spends no time on the southern edge of Rome as you walk through it. If it spent a nonzero amount of time there you would have to stop there for that amount of time. But you don't stop, you walk.

This issue was not resolved until the calculus was invented in the 17th century.

An object can occupy a position for zero time. For example, when a ball is tossed upward and reaches its highest point, how much time does it spend there? It has a zero velocity but a nonzero acceleration at that time.

It seems like it would have been at position X if time passed in pieces the size of Plank time.

Yeah, but there is no evidence to suggest that time behaves that way. As far as we know it's continuous, not discrete.

 

[bahamagreen]

An object can occupy a position for zero time. For example, when a ball is tossed upward and reaches its highest point, how much time does it spend there? It has a zero velocity but a nonzero acceleration at that time.

Don't both the velocity and acceleration go to zero at the highest point?
If the ball's acceleration is positive and decreasing while approaching the highest point, and negative and increasing while subsequently retreating from the highest point, doesn't the highest point coincide with the point in time when the acceleration is zero as it changes from positive to negative?

That nonzero acceleration is not with respect to the gravitation of the Earth, is it? Just simple coordinate acceleration with respect to the Earth's surface...?

 

[Herman Trivilino]

Don't both the velocity and acceleration go to zero at the highest point?

When the acceleration is zero it means the velocity isn't changing. Therefore the velocity would continue to be zero, meaning the ball would remain at its highest point. But it doesn't. As soon as it reaches its highest point it starts to fall.

If the ball's acceleration is positive and decreasing while approaching the highest point,

The ball's acceleration is constant if the effects of friction are negligible.

and negative and increasing while subsequently retreating from the highest point,

The ball's acceleration would continue to be positive and constant as it falls.

doesn't the highest point coincide with the point in time when the acceleration is zero as it changes from positive to negative?

No. There is no change in the acceleration. If it's +9.8 m/s² on the way up it's +9.8 m/s² on the way down.

Suppose the ball's initial velocity is -19.6 m/s. One second later it's -9.8 m/s. Another second later it's zero. Another second later it's +9.8 m/s and another second after that it's +19.6 m/s. Note that the velocity increased by 9.8 m/s every second. In other words, a constant acceleration of +9.8 m/s².

 

[Herman Trivilino]

Note that if you were closed in a box so that you couldn't see outside it, and the box did just what the ball did as described in the previous post, you would feel nothing different when you were at the highest point.

 

[A.T.]

Don't both the velocity and acceleration go to zero at the highest point?
If the ball's acceleration is positive and decreasing while approaching the highest point, and negative and increasing while subsequently retreating from the highest point, doesn't the highest point coincide with the point in time when the acceleration is zero as it changes from positive to negative?

You are confusing acceleration with velocity.

 

[bahamagreen]

Sorry, yes; the acceleration is constant (gravity) and acts throughout the entire passage of the ball, including the high point.

Is there a reason why you chose to assign a negative value to the ball's initial velocity (and so a positive value to the acceleration)? It does not matter to the math if consistent, but the convention for that example is the opposite, isn't it?

 

[jbriggs444]

I assume that speed would also not be physically realized? But momentum would be?

An infinitely precise momentum is also not physical.

 

[Herman Trivilino]

Is there a reason why you chose to assign a negative value to the ball's initial velocity (and so a positive value to the acceleration)?

The OP chose the acceleration to be positive, and therefore the velocity would have to be negative when the ball is rising.

It does not matter to the math if consistent, but the convention for that example is the opposite, isn't it?

I wouldn't call it a convention. It's probably more popular to refer to the upward direction as positive, but in a situation where something is always moving downward it might be more popular to choose the downward direction as positive because then both the acceleration and the velocity are positive.

 

[Jon Richfield]

This paradox only arises if one takes the idealistically mathematical view that gave rise to Zeno's paradoxes in the first place. In this view there are such concepts as infinity, as a point, as a line, a locus, and many others. Mathematically all such concepts are valid in suitable formal axiomatic structures, and such structures are convenient as bases for applied or empirical descriptions of aspects of reality.

There are however two (at least) aspects of empirical "reality" that conflict with idealised formal descriptions. In those terms the formal mathematical view is simplistic. (I suspect that to describe reality more realistically, we need a fuzzy-mathematical description or something related, but I am not equipped to deal with the associated concepts. Good luck anyone who is ahead of me )

One such aspect is that of our quantum view of nature, in which there is a limit to how precisely you can specify either the particle's location (it is not a "mathematical", a "Euclidean" point) or the time in which the particle might in principle be observed in that location with zero probability. Those coordinates in space/time/probability are non-zero in magnitude or measure, or whatever concept is relevant, and accordingly do not match Zeno's conceptions. The particle remains there until it is space-like and time-like separate from that region: 1e-40 sec? 1e-100 sec, who knows, but not zero!

And that was the good (bad?) news. Even if we weren't in a quantum-based universe, we still are in an information-based, information-limited universe. Information is a physical reality, small for sure (like log relationships) compared to energy but still non-zero. Now, the smaller the interval of time or space one wishes to determine, the greater the information one requires, and to specify even a single true physical point, would require infinite information, for which our physical universe has not enough scope. In other words, the Euclidean axiom that one can select any particular point on a line, is only an idealisation, not a physical fact. One cannot even pick one point anywhere, let alone find it again thereafter. Accordingly, even without QM, we can at best pick a fuzziness, not a zero-measure point of location or time.

And very, very important, this has nothing to do with you or me getting to work with a ruler or stopwatch; it applies equally strongly to the interactions between entities such as particles when a proton falls in a forest when no one is watching. In other words what we might call "reality".

Without the slightest disparagement of Zeno in his day, we have to (are in a position to) think more analytically in terms of recent changes of point of view of physical realities.

 

[Chris Miller]

You can either go very deep into Maths Analysis and get familiar with the 'real answer' to your question or you can think in terms of measuring the progress of the particle in tiny steps. (Like with crude measuring equipment). You can then allow the particle to be somewhere between X and X+0.00001 for 0.0035 seconds, between X+0001 andX+0.0002 in the next 0.035 seconds etc, etc. and then work towards smaller and smaller steps. That will always yield a value for a velocity, however short the steps are. This is the basis of (the immensely useful) Differential Calculus.
On a philosophical level, unless the particle actually disappears at point X, then it has to have been at X.
Do a search on Zeno's Paradox if you want to add interest to the question.

Never cared for Zeno's paradox. A point has zero dimension, and so physically does not exist (only conceptually). Also, the question of How long does it take Mary to get from point A to point B at velocity V is not one for differential calculus.

 

[Michaela SJ]

From a non-scientist's view: if the particle in question exists then it exists in time and zero time is included in time.

 

[Trust10]

Let us take this as an experiment to measure the particle at position X at zero time. If the time T of the particle is zero, then the velocity of the particle should be at infinity, which in real sense is impossible. Plugging this information into the uncertainty principle, the more accurate we measure the position X, the less accurate we get the velocity. Since from the question, the position X was already certain at zero time, the velocity should be uncertain at that time.

Was it ever at position X? . To answer to this is NO because If you try to check if the particle was ever at X position at zero time, you will find out the condition required for this to be possible is if the particles velocity remains uncertain even at infinity, which in real sense is impossible unless you know how to measure infinity.

 

[jbriggs444]

Was it ever at position X?

See response #4 above. This is a question about language and maybe mathematics, not experimental physics.

 

[votingmachine]

A moving particle has been at position X for zero time. Was it ever at position X?
Can zero time be considered as never, as in "I was in Rome for zero amount of time."?
It seems like it would have been at position X if time passed in pieces the size of Plank time.

If you start driving at 60 mph from mile sign 0, you pass mile sign #1 at 1 minute and mile sign #2 at 2 minutes.. You are between the two for 1 minute (speed = 60 mph = delta-X/delta-T). You might notice that you were between mile #1 and the sign at mile #1.5 for 30 seconds. And you were between #1 and #1.25 for 15 seconds.

The problem you are having is when you drop the distance interval to 0, (delta-X is then zero). The time interval is also zero. You are at any position for zero time, although you go thru all the infinite points at 60 mph.

You might also consider that if one is never at any position, then a wall in the road would not matter. After all, the moment you hit is of time interval zero. And by your reasoning, that means you are never there.

 

[Daniel Bolden]

take the particle sitting or moving through x, now get out your stopwatch. and as the particle gets
to, through...or stays at x, measure zero time elapsed. yes, the particle was at x.
however, the smallest measurement of time that has any meaning, and is equal to 10^-43 seconds.
No smaller division of time has any meaning. according to wiki, thanks wiki.

for me, i'd say that on some infinitely small interval, too small to be an official particle,
that it at least exists as some potentiality of information, some quasi energy in the process of
being expressed in the field.

a particle, even mass-less particles disturbs the field, so it's creating a 'history' the history
isn't wiped out between intervals.

 

[jbriggs444]

take the particle sitting or moving through x, now get out your stopwatch. and as the particle gets
to, through...or stays at x, measure zero time elapsed. yes, the particle was at x.
however, the smallest measurement of time that has any meaning, and is equal to 10^-43 seconds.
No smaller division of time has any meaning. according to wiki, thanks wiki.

If you are in the realm where the Planck time, approximately 5 x 10^-44 seconds, is relevant then you are well beyond the realm where particles have a precisely defined position at all, much less a well defined position as a continuous real-valued function of continuous real-valued time.

 

[Daniel Bolden]

If you are in the realm where the Planck time, approximately 5 x 10^-44 seconds, is relevant then you are well beyond the realm where particles have a precisely defined position at all, much less a well defined position as a continuous real-valued function of continuous real-valued time.

that's the reason i was saying, since histories aren't deleted, then something existed, though we humans may have no clear definition of what that something is.

 

[jbriggs444]

that's the reason i was saying, since histories aren't deleted, then something existed, though we humans may have no clear definition of what that something is.

And I am saying that the notion that something existed at a precise position at a precise time is not physical. We have (idealized) models that may assume such, but those models are wrong at that level of detail.

 

[Daniel Bolden]

And I am saying that the notion that something existed at a precise position at a precise time is not physical. We have (idealized) models that may assume such, but those models are wrong at that level of detail.

ok i got you're meaning. well said.
i thought of another answer more on the math side.
take matrix A, then add a zero matrix. the resulting matrix = matrix A.

 

[sophiecentaur]

IMO, until one has a pretty thorough familiarity of the formalities of Calculus, one can’t start introducing Planck Time as an answer to questions like this one. There’s no proof that Planck units represent actual discrete steps so they just do not help.

 

[Dale]

however, the smallest measurement of time that has any meaning, and is equal to 10-43 seconds.

There is no indication yet that this is correct. It may eventually turn out to be correct, but we are quite far away from being able to justify this claim.

 

[Daniel Bolden]

There is no indication yet that this is correct. It may eventually turn out to be correct, but we are quite far away from being able to justify this claim.

i was just quoting wiki. i was implying that i believe, that the particle would at least exists in some form. though it may not be observable ,
or measurable, nor its state understood. but a zero interval = no change.
matrix A + zero matrix = matrix A. but there are a lot of great responses on this thread that are more sophisticated. i think it was a fun question.

 

[Dale]

i was just quoting wiki.

And I was just pointing out that the wiki comment is not based on any evidence. Wiki can be a good reference, but it also can get things wrong.

 

[Daniel Bolden]

if the particle was nudging its way up to x...almost there

A moving particle has been at position X for zero time. Was it ever at position X?
Can zero time be considered as never, as in "I was in Rome for zero amount of time."?
It seems like it would have been at position X if time passed in pieces the size of Plank time.

I've re-read the question. It seems to be saying the particle was moving in field Y for some time then passed onto field X for zero seconds. It never arrived into field X or position x. same as saying I drank zero cups of coffee this morning.

 

[Daniel Bolden]

And I was just pointing out that the wiki comment is not based on any evidence. Wiki can be a good reference, but it also can get things wrong.

my apologies. that was a bit of reaching. you caught that fast.

 

[ddjj77]

if the particle was nudging its way up to x...almost thereI've re-read the question. It seems to be saying the particle was moving in field Y for some time then passed onto field X for zero seconds. It never arrived into field X or position x. same as saying I drank zero cups of coffee this morning.

Why fields and not positions? Fields are infinite.
A moving particle has varying positions and has been at infinite positions for zero time at each position.

 

[Daniel Bolden]

Why fields and not positions? Fields are infinite.
A moving particle has varying positions and has been at infinite positions for zero time at each position.

you have been at zero positions if you have spent zero time anywhere. if a particle has passed through a gas cylinder for 2 thousandths of a second,
then it has spent 2 thousandths of a second passing through positions.

 

[ddjj77]

you have been at zero positions if you have spent zero time anywhere. if a particle has passed through a gas cylinder for 2 thousandths of a second,
then it has spent 2 thousandths of a second passing through positions.

That's my position too, but I want to find out if that's true. Has the particle been at any position?
Maybe the answer is in Heisenberg's uncertainty principle. If the particle is moving, then we can know its momentum, but not its position.

 

[Dale]

you have been at zero positions if you have spent zero time anywhere

That's my position too,

Your position is not self consistent then. In the opening post you specified that it had been at a position for zero time, which is inconsistent with it not being at that position.

 

[A.T.]

Maybe the answer is in Heisenberg's uncertainty principle...

Yes, but it's not really needed here. All you need is to realize that particles are not points in the mathematical sense.

 

[ddjj77]

Your position is not self consistent then. In the opening post you specified that it had been at a position for zero time, which is inconsistent with it not being at that position.

My opening was the question if being at a position for zero time amounts to having been at that position at all.

 

[Dale]

My opening was the question if being at a position for zero time amounts to having been at that position at all.

And clearly and obviously if it is at the location for 0 duration then it was at the location.

 

[ddjj77]

Yes, but it's not really needed here. All you need is to realize that particles are not points in the mathematical sense.

So, if a particle has a finite size (Planck diameter?), it would take a finite time for it to enter into and exit from position X, therefore it has been at position X?

 

[ddjj77]

And clearly and obviously if it is at the location for 0 duration then it was at the location.

Doesn't zero time mean never?

 

[Dale]

Doesn't zero time mean never?

No. Why would it?

 

[ddjj77]

No. Why would it?

Like zero speed means no motion.
Maybe there's no such thing as zero time. Maybe the shortest time is Planck time?
I'll accept your view for now.

 

[olgerm]

When the acceleration is zero it means the velocity isn't changing. Therefore the velocity would continue to be zero, meaning the ball would remain at its highest point. But it doesn't. As soon as it reaches its highest point it starts to fall.².

Consider particle with trajectory $x(t)=t^3$ at time t=0 it's acceleration and speed are 0, but it does not remain in point x=0.

 

[olgerm]

It is wrong to claim, that particle was not in some location $\vec{X}$ if it was there duration 0, because you can choose frame of reference where it is always moving(is never in rest). But answer to the question whether it was in that location must be same in all frames of reference.

 

[sysprog]

A moving particle has been at position X for zero time. Was it ever at position X?

A consistent description would depend on the size of the particle, on the size of position X, and on the speed of the particle.

If you imagine a photon that traverses a path from A to B, somewhere along which path is position X, then the answer to the question "was it ever at position X", is "yes", and the answer to the question "for how long a time", is "for however long it takes a photon to get from one side of position X to the other".

If position X is a mathematical point, with zero size, no particle of non-zero size could ever be entirely contained within its scope, so in that sense, that of the ordinary locational meaning of "at", the particle would not ever be "at" position X; however, the particle could be said to occupy position X for as long as any part of it were traversing position X.

If a photon, be it wave, field, or particle, has a positive finite size, and travels at speed of light, that size and speed will determine the size of a positive non-zero time interval over which the occupancy occurs.

It seems like it would have been at position X if time passed in pieces the size of Plank time.

The Plank time seems like a reasonable candidate for a minimum non-zero temporal duration.

If time is not continuous, but occurs in discrete instants, or chronons (Cornell U pdf article), then there is a minimum (non-zero) size of a (non-empty) time interval, and a particle that could reasonably be said to have ever arrived at a location would have to have arrived there at some, (at minimum single), chronon, but could presumably be posited to have remained at that location for zero chronons prior to the onset of its departure therefrom.

I think something like that is about as close as one can get to a consistent notion of a minimum size of a time interval over which, or during which, a moving particle could acquire a history of ever having been at position X.

Can zero time be considered as never, as in "I was in Rome for zero amount of time."?

In plain English it can.