Quantum particles

[Upisoft]

Δt cannot be arbitrarily set to 0. Δt represents the instantaneous change in time.

And you were arguing against it. If it is instantaneous change in time, then it is quant of time.

This is the whole function of taking the limit. Remember, the limit equation is the whole "infinitely small slice" idea.

Here your teacher wouldn't be proud of you. There is no such thing as infinetely small. If Δt is small positive value, no matter how small it is, I always can find SMALLER value. For example, Δt/2, but it can be any number between zero and Δt.

It does not require two moments to have a value, it's whole point is to NOT require two moments for calculation.

Of course. Instead you must know the behaviour of the function in time for arbitrary small, but not zero length, part of the function containing that point. Again, that assumes some knowlege in time.

In fact, separating an arrow from the flow of time would likely only possible if the arrow was traveling at the speed of light.

Why?

Moving at the speed of light would, of course, mean that it had quite a bit of momentum and would also require the more detailed momentum calculation.

No calculations are required. The momentum would be infinite, but you can't reach the speed of light to achieve that.

If we are dipping into this level of physics, we will have to also deal with the uncertainty principle. now you're fighting all of calculus!

No, I don't fight all the calculus. It's quite good tool. What I'm arguing is that you can use it on a snapshot of the arrow in some moment. What you can measure on that snapshot is position and time. There is no speed nor momentum. You can tell the difference by measuring arrow's lenght, but that, of course, means you already know what is the length of the arrow when it was in rest. And that is my point. You cannot observe speed or momentum with one measurement. It is differential value, so you have to have extra measurements.

any equation can be brought to 0/0 if dividing by 0 is allowed. That's why dividing by 0 isn't allowed.

Let me give an example:
A=X set some number X equal to A
A+X=2X add x to both sides
X-A=2(X-A) subtract 2A from both sides
1=2 divide both sides by (X-A)

I don't argue about that.

 

[Upisoft]

Upisoft, you're totally missing the point. You attempted to argue that an object does not have a property called velocity at one instant in time. This is false. It is false because there are effects of velocity (more specifically momentum) that can be observed at single instants in time.

You failed to provide valid measurement technique. All of provided techniques required extra knowledge that had to be obtained by different measurement(s).

It doesn't matter that one has to know other things about the object at different times to make sense of the measurement of the instantaneous velocity of the object in question.

Yes, it matters. Because you can't apply your techniques on something new. Let's say that real UFO appears above USA. The scientists want to measure its momentum in particular moment of time, but unfortunately they don't have any data about the UFO. No mass, no dimensions, no original spectrum(to use redshift/blueshift). You argue that they will be able to obtain knowledge about the momentum in a single measurement. How?

Your argument was that one cannot measure the position more than once at the same point in time and infer any information about the velocity. There are other ways to measure the momentum/velocity that are completely independent of position. Even if these methods require more than one measurement of the object at different times, they aren't comparing positions at different times, and thus are genuinely observing instantaneous velocity.

You didn't understand what I'm arguing about. I'm arguing that you need time.

 

[Chalnoth]

You failed to provide valid measurement technique. All of provided techniques required extra knowledge that had to be obtained by different measurement(s).

So what? How is that invalid?

Yes, it matters. Because you can't apply your techniques on something new. Let's say that real UFO appears above USA. The scientists want to measure its momentum in particular moment of time, but unfortunately they don't have any data about the UFO. No mass, no dimensions, no original spectrum(to use redshift/blueshift). You argue that they will be able to obtain knowledge about the momentum in a single measurement. How?

Redshifts typically work by molecular/atomic spectra, by the way, so as long as the UFO had some significant spectral lines, yes, we could obtain knowledge about the momentum in the direction towards/away from us that way.

But again, so what? Even if we couldn't find evidence as to the rest frame spectrum, your objections just don't make any sense whatsoever. It seems to me you just don't want to be wrong.

You didn't understand what I'm arguing about. I'm arguing that you need time.

Well, we have time. So why are you even trying to argue that point? How does it even apply to discretized space-time?

 

[Upisoft]

So what? How is that invalid?

It is not invalid. It is inapplicable. I'm arguing that there is no motion in a single moment and you try to prove the opposite using information from different moment.

Redshifts typically work by molecular/atomic spectra, by the way, so as long as the UFO had some significant spectral lines, yes, we could obtain knowledge about the momentum in the direction towards/away from us that way.

That is right, if the UFO spectrum behaves as you want it to behave.

But again, so what? Even if we couldn't find evidence as to the rest frame spectrum, your objections just don't make any sense whatsoever. It seems to me you just don't want to be wrong.

Neither you want to be wrong. That is quite human.

Well, we have time. So why are you even trying to argue that point? How does it even apply to discretized space-time?

Yes, we have time. But there is no time in a single moment. My point is that such thing as "zero length moment" does not exist. If the time is continuous, there is zero length moment. If the time is quantized there is a minimum length of time, a quant of time.

 

[[serious]]

And you were arguing against it. If it is instantaneous change in time, then it is quant of time.

rate=/=quatity. Δt is a rate, as such, the amount of time elapsed is irrelevant.

Here your teacher wouldn't be proud of you. There is no such thing as infinetely small. If Δt is small positive value, no matter how small it is, I always can find SMALLER value. For example, Δt/2, but it can be any number between zero and Δt.

That's why you are taking a limit. It allows the calculation of that infinitely small slice. This is the whole point of calculus

Of course. Instead you must know the behaviour of the function in time for arbitrary small, but not zero length, part of the function containing that point. Again, that assumes some knowlege in time.

We know an arrows behavior in time. The question is whether it has the property of momentum at any given time.

Why?

it is likely the only way of getting the rate of change of t to equal 0

No calculations are required. The momentum would be infinite, but you can't reach the speed of light to achieve that.

but if you reached the speedc of light, time would stop for the arrow. If time stopping for the arrow can still allow it to have momentum, the arrow's momentum is not removed by time stopping.

No, I don't fight all the calculus. It's quite good tool. What I'm arguing is that you can use it on a snapshot of the arrow in some moment. What you can measure on that snapshot is position and time. There is no speed nor momentum. You can tell the difference by measuring arrow's lenght, but that, of course, means you already know what is the length of the arrow when it was in rest. And that is my point. You cannot observe speed or momentum with one measurement. It is differential value, so you have to have extra measurements.

The argument is not whether you can tell if an arrow is moving at an instant in time (that is well known and would not be a paradox of any sort) it's whether the arrow is actually moving. Further more, since the original "paradox" stated that the arrow was in flight, thus, we know it is in motion. This is something we would not be able to know if we were actually being asked to ignore all information outside of that moment.

Another thing is that Zeno's arrow paradox rests on the assumption of a finite set of times during which the arrow is motionless. Since quantum time is what you are proposing, this would only be a paradox for you.

Another way of looking at it. If you define motion as being at different points at two moments in time, lack of motion must be being in the same point at two moments in time. Thus, you rational that the arrow is stationary becomes prone to the same "paradox" that you use to challenge the arrow's motion.

 

[Chalnoth]

Yes, we have time. But there is no time in a single moment. My point is that such thing as "zero length moment" does not exist. If the time is continuous, there is zero length moment. If the time is quantized there is a minimum length of time, a quant of time.

Yes. And using any of the methods mentioned previously, we could measure the momentum or velocity of a particle at such an instant of time.

 

[Upisoft]

rate=/=quatity. Δt is a rate, as such, the amount of time elapsed is irrelevant.

No. It is change in time, not rate. And it is elapsed time.

That's why you are taking a limit. It allows the calculation of that infinitely small slice.

And when you do it, Δt never reaches zero. Limits are approached but never reached. For example, what is the limit of 1/|x| when x->0?

This is the whole point of calculus We know an arrows behavior in time.

That's what I say. We know it in time. If there is no time, as in a single moment, we don't know anything about its "behaviour".

if you reached the speedc of light, time would stop for the arrow. If time stopping for the arrow can still allow it to have momentum, the arrow's momentum is not removed by time stopping.

Arrows can't reach the speed of light according current beliefs in science. So, analyzing this situation is pointless.

The argument is not whether you can tell if an arrow is moving at an instant in time (that is well known and would not be a paradox of any sort) it's whether the arrow is actually moving. Further more, since the original "paradox" stated that the arrow was in flight, thus, we know it is in motion. This is something we would not be able to know if we were actually being asked to ignore all information outside of that moment.

Or if we don't have any information regarding the arrow outside of the moment.

Another way of looking at it. If you define motion as being at different points at two moments in time, lack of motion must be being in the same point at two moments in time. Thus, you rational that the arrow is stationary becomes prone to the same "paradox" that you use to challenge the arrow's motion.

I will agree here. Zeno created his paradox by trying to exclude the time and then show that there is no motion without time. However there are more important questions that follow. Does such thing like "moment" exist? Any slice of time will have infinite number of moments then. If a slice of 1 sec. has infinite number of moments and slice of 5 sec. also has infinite number of moments, then what makes them different?

 

[Upisoft]

Yes. And using any of the methods mentioned previously, we could measure the momentum or velocity of a particle at such an instant of time.

No, you can't. They are differential values and need two measurements. You need the length, the spectrum or whatever you use when the object is in rest. That is separate measurement and it doesn't matter when you made it, of course if dad isn't right.

 

[Chalnoth]

No, you can't. They are differential values and need two measurements. You need the length, the spectrum or whatever you use when the object is in rest. That is separate measurement and it doesn't matter when you made it, of course if dad isn't right.

I don't understand why you are continuing with this point. It's just completely irrelevant. Once again, we were talking about discretized space-time, and whether or not an object has a property called velocity or momentum at a single instant in time. It does, so you're wrong. Why do you continue to argue this irrelevant point that one needs to compare measurements of objects at rest to observe velocity?

 

[Upisoft]

I don't understand why you are continuing with this point. It's just completely irrelevant. Once again, we were talking about discretized space-time, and whether or not an object has a property called velocity or momentum at a single instant in time. It does, so you're wrong. Why do you continue to argue this irrelevant point that one needs to compare measurements of objects at rest to observe velocity?

Because it's not irrelevant. The whole point is that you can't show the existance of velocity and momentum using information obtained only from that single moment. You need more informeatin from another moment. Therefore, those properties are not related to the single moment only.

 

[[serious]]

And when you do it, Δt never reaches zero. Limits are approached but never reached. For example, what is the limit of 1/|x| when x->0?

Then why did you use the limit equation if you are saying that you can't use the limit equation?

That's what I say. We know it in time. If there is no time, as in a single moment, we don't know anything about its "behaviour".Arrows can't reach the speed of light according current beliefs in science.

thus they cannot be removed from time, that's sort of my whole point. We can look at them in a given instant, but at that instant, they are still in time.

So, analyzing this situation is pointless.Or if we don't have any information regarding the arrow outside of the moment.

The debate is not what we can determine, but what is. You are stating that you don't think that the instantanious momentum of an arrow can be known, I'm stating that simple physics can calculate momentum for any given moment. Saying that something cannot be know, and is therefore known to be exactly zero is a logical fallacy. Yes, it is possible to not have enough information to calculate the momentum of an arrow. It is not possible to then conclude that the arrow must be at rest.

I will agree here. Zeno created his paradox by trying to exclude the time and then show that there is no motion without time. However there are more important questions that follow. Does such thing like "moment" exist? Any slice of time will have infinite number of moments then. If a slice of 1 sec. has infinite number of moments and slice of 5 sec. also has infinite number of moments, then what makes them different?

Quantity makes them different. Our number system is also infinitely divisible. Between 0 and 1 are an infinite number of numbers. Between 0 and 5 there are an infinite number of numbers. What's the difference between 1 and 5? 4 What's the difference between 1 second and 5 seconds? 4 seconds. No quantum number system and no quantum time are required.

 

[Upisoft]

Then why did you use the limit equation if you are saying that you can't use the limit equation?

Because velocity is defined this way.

thus they cannot be removed from time, that's sort of my whole point.

I don't know what is possible to done with time. Science did some things considered impossible in the past. Maybe we will be able to stop the time in the future, who knows?

We can look at them in a given instant, but at that instant, they are still in time. The debate is not what we can determine, but what is. You are stating that you don't think that the instantanious momentum of an arrow can be known, ...

No. I said it is not defined for the moment. I tried to explain it several times, but it looks like you can't get it. You can know it, but it is no property of the moment, because you always need extra information outside the moment.

I'm stating that simple physics can calculate momentum for any given moment.

And I don't disagree with that. What I've already said several times is that you need information that does not belong to the moment to make the calculation. And that means this property (momentum) is not property of the moment in time. And I am right, because it is not even property of the object, because two observers, one traveling with the same speed as the object and another traveling with 9/10c relative to the first observer will measure different values.

Saying that something cannot be know, and is therefore known to be exactly zero is a logical fallacy.

No, not zero. It is not defined, unexistent.

Yes, it is possible to not have enough information to calculate the momentum of an arrow. It is not possible to then conclude that the arrow must be at rest.

Do it then. Show how it can be done.

Quantity makes them different. Our number system is also infinitely divisible. Between 0 and 1 are an infinite number of numbers. Between 0 and 5 there are an infinite number of numbers. What's the difference between 1 and 5? 4 What's the difference between 1 second and 5 seconds? 4 seconds. No quantum number system and no quantum time are required.

The same time interval could be 1 second for one observer and 5 seconds for another. Where did the 4 seconds, you say are the difference, come from?

 

[[serious]]

Because velocity is defined this way.

Can the equation for calculating the instantaneous velocity for any given instant calculate the instantaneous velocity for any given instant?

I don't know what is possible to done with time. Science did some things considered impossible in the past. Maybe we will be able to stop the time in the future, who knows?

will stopping time cause them to instantly lose all momentum? Time is stopped for photons yet they still move so I would assume no. As such, stopping time for an arrow, even if it were possible, would not remove the property of momentum from them. Also, I very much doubt Zeno was commenting on theories that had not even been invented/discovered.

No. I said it is not defined for the moment. I tried to explain it several times, but it looks like you can't get it. You can know it, but it is no property of the moment, because you always need extra information outside the moment.

So what does this have to do with zeno's arrow paradox? His argument wasn't that you couldn't tell it was that it wasn't moving.

And I don't disagree with that. What I've already said several times is that you need information that does not belong to the moment to make the calculation. And that means this property (momentum) is not property of the moment in time. And I am right, because it is not even property of the object, because two observers, one traveling with the same speed as the object and another traveling with 9/10c relative to the first observer will measure different values.

The properties of a moment are not limited to just those we can observe in an isolated moment. When you get a christmas present, you are unable to tell that there is a striped shirt in box X at moment Y, yet it can still a property of Box X to contain a striped shirt at moment Y

No, not zero. It is not defined, unexistent.

So if arrows have no momentum in any given moment, they must either:
A. Have no momentum at all (if they exist in a series of moments either discrete or continuous)
B. Have a state of being that has momentum without being in the direct flow of time (having momentum only when they are not in a moment, requires discrete moments with "gaps")

Since we know that arrows have momentum, we can rule out A. I know of no theory that would allow an arrow to exist outside of a definable moment, so I would reject B until a theory is proposed that would allow it.

Do it then. Show how it can be done.

How we can not have enough information to know it's momentum at a given time? An arrow is launched at 2:34 PM in standard gravity with an angle of 15 degrees from the horizontal. What is it's momentum 15 seconds later. We have everything we need to know except the velocity at launch (or any other time since we could calculate from there).

If you are asking for how we could have enough data to calculate it without another reference moment, we could look at length measurement from the inertial reference frame of the arrow and an independent observer. From this we could calculate velocity and that with mass would give us momentum. none of the givens have a time component whatsoever.

The same time interval could be 1 second for one observer and 5 seconds for another. Where did the 4 seconds, you say are the difference, come from?

In that case, the difference would have to be velocity at the time of measurement (to place the two observers in differently moving time streams) or a period of acceleration for the observer of the 1 second. It is possible an intense gravity well could be the difference as well. At any rate, the reason for the difference in 1 second from 5 seconds is irrelevant to the presence of a difference between 1 second and 5 seconds.

 

[Upisoft]

Can the equation for calculating the instantaneous velocity for any given instant calculate the instantaneous velocity for any given instant?

You try it. You have following information about the instant as function s(t): s(0) = 7m. Now, find v for t=0 sec. Oh, and the object has mass m=8 kg. You can now find the momentum, if you can find the velocity.

will stopping time cause them to instantly lose all momentum? Time is stopped for photons yet they still move so I would assume no.

Is time stopped for them? So all infinite number of instants that a photon traveling from Sun to Earth, about 8 minutes, manage somehow to be one single instant for the photons. How?

As such, stopping time for an arrow, even if it were possible, would not remove the property of momentum from them.

Maybe not. But it sill wouldn't be a property of that instant.

Also, I very much doubt Zeno was commenting on theories that had not even been invented/discovered.

Don't doubt. He wasn't commenting. But I am.

So what does this have to do with zeno's arrow paradox?

I tried to show it, but you can't see it. Maybe it's my fault that I can't express myself better.

His argument wasn't that you couldn't tell it was that it wasn't moving.

I know. His argument was that there is no such thing as movement in an instant.

The properties of a moment are not limited to just those we can observe in an isolated moment.

And then how an isolated moment can exist, if it does not have all its properties? Can you give an example of anything else that exist, but it is not a carrier of its own properties?

When you get a christmas present, you are unable to tell that there is a striped shirt in box X at moment Y, yet it can still a property of Box X to contain a striped shirt at moment YSo if arrows have no momentum in any given moment, they must either:
A. Have no momentum at all (if they exist in a series of moments either discrete or continuous)
B. Have a state of being that has momentum without being in the direct flow of time (having momentum only when they are not in a moment, requires discrete moments with "gaps")

Or:
C. There is no such thing as "moment" or "instant", having no length. The time is quantized, Δt is never zero and momenum can be defined in the quant of time.

If you are asking for how we could have enough data to calculate it without another reference moment, we could look at length measurement from the inertial reference frame of the arrow and an independent observer.

In other words you already have to know the arrow's speed just to make sure that your second observer is in rest with the arrow.

At any rate, the reason for the difference in 1 second from 5 seconds is irrelevant to the presence of a difference between 1 second and 5 seconds.

Nice.

 

[[serious]]

Is time stopped for them? So all infinite number of instants that a photon traveling from Sun to Earth, about 8 minutes, manage somehow to be one single instant for the photons. How?

Relativity. Due to the contraction of space/time at high velocities, a photon would see every point along it's path as a single point and every moment along it's path as a single moment. Basically, as things go faster, time and space get shorter. The speed of light is where time and space = 0 length. For more detailed equations, look up the lorentz transformations.

Maybe not. But it sill wouldn't be a property of that instant.

It is a property of the arrow at any given instant.

And then how an isolated moment can exist, if it does not have all its properties? Can you give an example of anything else that exist, but it is not a carrier of its own properties?

huh? I'm saying that it has it's properties regardless of whether or not we have the requisite tools to observe them.

Or:
C. There is no such thing as "moment" or "instant", having no length. The time is quantized, Δt is never zero and momenum can be defined in the quant of time.

quantized time is only required if one assumes quantized space. A specific location in time would have a specific location in space. There is motion if that location in time has a unique position in space from proximal locations. No movement occurs within a given location but the property of motion, in the form of velocity, is still contained by the object at the given location in time. Likewise, the property of motion through time is contained in the object for all objects traveling at less than the speed of light.

Try thinnking of it this way: Say you have a stream of water coming from a faucet. Now, at a certain distance from the faucet, you imagine a 2 dimensional plane. The quantity of water in that plane is zero because the plane has no volume. However, there is water at that plane and it can be measured in terms of area and flow rate.

Likewise, an arrow in any given moment of time has the properties of velocity and momentum despite no change in location occurring "in" that moment.

In other words you already have to know the arrow's speed just to make sure that your second observer is in rest with the arrow.

Not at all. One need only have an observer in the same inertial reference frame as the arrow, or one could treat the arrow itself as an observer. You could even have two observers in different inertial reference frames from the arrow and calculate it out from the difference in perceptions of the two observers that way. The math gets a little bit harder, but you can still do it. No velocity need be given to calculate it out.

Nice.

Why thank you

 

[Upisoft]

Relativity. Due to the contraction of space/time at high velocities, a photon would see every point along it's path as a single point and every moment along it's path as a single moment. Basically, as things go faster, time and space get shorter. The speed of light is where time and space = 0 length. For more detailed equations, look up the lorentz transformations.

Bingo! Now be consistent and tell me how a photon would see its own momentum and speed.

It is a property of the arrow at any given instant.

You say that, because you define it this way or because there is reason to be this way?

huh? I'm saying that it has it's properties regardless of whether or not we have the requisite tools to observe them.

Ahh, yes, of course. We also don't have the tools to observe the pink unicorns. I wonder why you don't want to accept them so easily either.

Try thinnking of it this way: Say you have a stream of water coming from a faucet. Now, at a certain distance from the faucet, you imagine a 2 dimensional plane. The quantity of water in that plane is zero because the plane has no volume. However, there is water at that plane and it can be measured in terms of area and flow rate.

I don't argue about that. You can measure it.But you can't do it within specific instant of time. Again, external information is required. The flow rate is not property of the instant.

Not at all. One need only have an observer in the same inertial reference frame as the arrow, or one could treat the arrow itself as an observer. You could even have two observers in different inertial reference frames from the arrow and calculate it out from the difference in perceptions of the two observers that way.

Of course, you can do that. But now you have your initial problem doubled. To calculate the velocity of the arrow, now you have to observe the velocity of the observers or you can't do anything without knowing their reference frames.

The math gets a little bit harder, but you can still do it. No velocity need be given to calculate it out.

You need to know their speeds. They do not depend on your speed. They will measure the same values regardless if you're traveling with 10 m/s or near the speed of light. So, basically, you can't do anything about it. You need their velocities relative to you.

 

[[serious]]

Bingo! Now be consistent and tell me how a photon would see its own momentum and speed.

Assuming what inertial reference frame?The photon would see everything around it moving at the speed of light and would could thus assume it was moving at the speed of light in respect to any given point.

You say that, because you define it this way or because there is reason to be this way?

I say that because momentum is conserved. Without a force acting on the arrow to remove it's momentum, it will maintain the same momentum throughout the flight. Thus, in any given moment, it will have momentum.

Ahh, yes, of course. We also don't have the tools to observe the pink unicorns. I wonder why you don't want to accept them so easily either.

Again, show me a formula of scientific theory (read, makes testable predictions) that would indicate the presence of such.

I don't argue about that. You can measure it.But you can't do it within specific instant of time. Again, external information is required. The flow rate is not property of the instant.

Time, being just another dimension, would bear no special status that would make such observation any different than measuring it at another singular point along, say , the Z axis. As such, measuring it as I've specified is not different than measuring it at an instant of time apart from our own biases in how experience time.

Of course, you can do that. But now you have your initial problem doubled. To calculate the velocity of the arrow, now you have to observe the velocity of the observers or you can't do anything without knowing their reference frames.

Their reference frames are what they are. All reference frames can be calculated in respect to each other.

You need to know their speeds. They do not depend on your speed. They will measure the same values regardless if you're traveling with 10 m/s or near the speed of light. So, basically, you can't do anything about it. You need their velocities relative to you.

Why would you need their velocities in respect to me? That would just simplify matters as I could act as an observer as well. More data points and we can choose the easiest observer's observations to calculate from. If we wanted the arrow's velocity in respect to me, it would make sense to include me as one of the observers as it would zero one of the observers out in respect to the chosen inertial reference frame. That just makes the math easier again.

 

[Upisoft]

Relativity. Due to the contraction of space/time at high velocities, a photon would see every point along it's path as a single point and every moment along it's path as a single moment. Basically, as things go faster, time and space get shorter. The speed of light is where time and space = 0 length. For more detailed equations, look up the lorentz transformations.

Bingo! Now be consistent and tell me how a photon would see its own momentum and speed.

Assuming what inertial reference frame?The photon would see everything around it moving at the speed of light and would could thus assume it was moving at the speed of light in respect to any given point.

So, how this is consistent? You said that the photon will travel zero space for zero time. It could not possibly see any movement, because everything, including all the universe has length 0 in the direction it travels. Also the photon has zero time to make that observation. Therefore, everything along its path is in one point. Including its startring and its ending point.

I say that because momentum is conserved.

The moment is conserved in an inertial reference frame.

Without a force acting on the arrow to remove it's momentum, it will maintain the same momentum throughout the flight.

That's called inertia and it's property of an inertial reference frame.

Thus, in any given moment, it will have momentum.

No. Because two observers in different inertial reference frames would measure different values.

Again, show me a formula of scientific theory (read, makes testable predictions) that would indicate the presence of such.

There is no need of formula. Virtual particles exist according QM, which is testable theory, isn't it? If they exists, then it is possible to exist groups of virtual particles. The groups of the virtual particles have very low probability to form an invisible pink unicorn, but it's not zero. Each moment an "attempt" to create it is being made. But there are infinite number of moments, according your theory of time continuum. Therefore in any time slice the probability of existence of an invisible pink unicorn is 1, because you have finite probability and infinite number of attempts. Besides, that is valid for any point of space, therefore infinite number of such unicorns exist in any instant. Of course, such structure is too complex, so it would exist for extremely short time.

Time, being just another dimension, would bear no special status that would make such observation any different than measuring it at another singular point along, say , the Z axis. As such, measuring it as I've specified is not different than measuring it at an instant of time apart from our own biases in how experience time.

Time is not "just" another dimension. In any other dimension I can turn around and start walking in the opposite direction. I can't do that with time.

Their reference frames are what they are. All reference frames can be calculated in respect to each other.
Why would you need their velocities in respect to me?

You have an arrow moving in the space (no graviy wells around for simplicity). There are two observers. You don't know anything about their reference frames, except they are inertial reference frames. They measure something about the arrow that can be measured and send the information to you. Show me how you can calculate the momentum of the arrow from the data observed.

 

[[serious]]

So, how this is consistent? You said that the photon will travel zero space for zero time. It could not possibly see any movement, because everything, including all the universe has length 0 in the direction it travels. Also the photon has zero time to make that observation. Therefore, everything along its path is in one point. Including its startring and its ending point.

yup, but you are the one saying that disallows measurement of velocity, not me. The photon would see all matter with a velocity of c and all matter with a complete space/time contraction in the direction of travel.

The moment is conserved in an inertial reference frame.

I'm not sure what you are claiming here. You seem to be saying that the entirety of an inertial reference frame must occur in a single moment. This makes no sense though so I assume you are trying to say something else. If I am traveling in a straight line at a constant velocity, I remain in the same inertial reference frame for that length of time. That length of time contains multiple moments regardless of continuous or discrete time.

That's called inertia and it's property of an inertial reference frame.

yup

No. Because two observers in different inertial reference frames would measure different values.

and both measurements will be accurate. That does not change the inertia the arrow has, it just leaves it under relativity (as we would expect). If you are changing to a relativistic tact, you could come to the conclusion that the arrow was at rest (ignoring gravity) by switching inertial reference frames, but that wouldn't support your discrete time argument.

I will agree that for any object at any given moment there exists an inertial reference frame where it is stationary (the inertial reference frame it occupies at that time) But that is irrelevant to zeno's arrow paradox which assumes an inertial reference frame of the second, stationary arrow and the observer.

There is no need of formula. Virtual particles exist according QM, which is testable theory, isn't it? If they exists, then it is possible to exist groups of virtual particles. The groups of the virtual particles have very low probability to form an invisible pink unicorn, but it's not zero. Each moment an "attempt" to create it is being made. But there are infinite number of moments, according your theory of time continuum. Therefore in any time slice the probability of existence of an invisible pink unicorn is 1, because you have finite probability and infinite number of attempts. Besides, that is valid for any point of space, therefore infinite number of such unicorns exist in any instant. Of course, such structure is too complex, so it would exist for extremely short time.

except you mistakenly assume that the probability of such a structure does not directly scale to the period of time observed. If we make such an assumption, your discrete time can fall in to he same trap. We simply must assume an adequately small slice so as to make the momentary existence of such a structure likely during one of them.

Time is not "just" another dimension. In any other dimension I can turn around and start walking in the opposite direction. I can't do that with time.

An astronaut floating in space has no control over his motion in any of the 3 conventional directions. Like that astronaut, we have no leverage to exert against the time dimension. We can indirectly influence time slightly by motion along the other 3 dimensions, just as approaching the speed of light along the X axis will have a relativistic effect on motion along the Y and Z axis.

You have an arrow moving in the space (no graviy wells around for simplicity). There are two observers. You don't know anything about their reference frames, except they are inertial reference frames. They measure something about the arrow that can be measured and send the information to you. Show me how you can calculate the momentum of the arrow from the data observed.

The two observers (O1 and O2 from here on out)also have measurements about each other as well as the arrow (A). Let's look at length for the calculation of velocity. We'll mark O1's observations with a ' and O2's observations with a ". O1' =rest mass as O1 resides in the same inertial reference frame as himself, likewise, O2" is a resting length. Thus, O1" and O2' serve to calibrate the observations of A' and A"

I'd let you give me a set of observed lengths and actually calculate out what that would mean for the arrow, but my calculator is out of batteries (and still out in my car).

 

[Upisoft]

yup, but you are the one saying that disallows measurement of velocity, not me. The photon would see all matter with a velocity of c and all matter with a complete space/time contraction in the direction of travel.

I will agree with you once you show me how the photon will calculate that everything is moving with speed c.

I'm not sure what you are claiming here.

If you're watching an object, but you have non-inertial reference frame, say you are on accelerating space ship, then you will see the object as it is accelerating in opposite direction. Since momentum p=mv, and v is increasing, then p is also increasing. Therefore, there is no conservation of the momentum in non-inertial reference frame. The same example you can see if you drop an object. Its momentum will start to increase, if you stay firm on the floor of course, until the object hits the floor. Don't try it at your home with an expensive vase.

and both measurements will be accurate. That does not change the inertia the arrow has, it just leaves it under relativity (as we would expect).

I agree that inertia is a property of the object. However, it is not momentum. What you can measure here is the inertial mass of the object: m.

If you are changing to a relativistic tact, you could come to the conclusion that the arrow was at rest (ignoring gravity) by switching inertial reference frames, but that wouldn't support your discrete time argument.

That is Newton's first law, but with relativistic taste. You can always find an inertial reference frame with the object in rest in it, if the forces acting on the object are balanced. Therefore, you can always find an inertial reference frame and the object will have zero velocity and momentum in it. That is the reference frame of the object. I wonder why that is not valid for photons, as you assert? Shouldn't they have zero velocity and momentum in their reference frame?

I will agree that for any object at any given moment there exists an inertial reference frame where it is stationary (the inertial reference frame it occupies at that time) But that is irrelevant to zeno's arrow paradox which assumes an inertial reference frame of the second, stationary arrow and the observer.

I think that we could reach some kind of agreement. At least about that some things are properties of the reference frame. That includes momentum and velocity of an object. Or maybe, it's better to say that they are properties of object/frame interaction.

except you mistakenly assume that the probability of such a structure does not directly scale to the period of time observed.

That will assume that the "attempts" are being made amongst finite number of instants. One possible solution is that there is quantum of time. What is your possible solution?

If we make such an assumption, your discrete time can fall in to he same trap. We simply must assume an adequately small slice so as to make the momentary existence of such a structure likely during one of them.

Nope. You can't achieve probability of 1 this way. The best you can do is 0.9999999, with finite number of digits. So, there will be non-zero probability that they don't exist.

An astronaut floating in space has no control over his motion in any of the 3 conventional directions. Like that astronaut, we have no leverage to exert against the time dimension.

Bad example. The astronaut may have rocket engine and that will not lead to paradox. He can't kill his grandfather this way.

The two observers (O1 and O2 from here on out)also have measurements about each other as well as the arrow (A). Let's look at length for the calculation of velocity. We'll mark O1's observations with a ' and O2's observations with a ". O1' =rest mass as O1 resides in the same inertial reference frame as himself, likewise, O2" is a resting length. Thus, O1" and O2' serve to calibrate the observations of A' and A"

I'd let you give me a set of observed lengths and actually calculate out what that would mean for the arrow, but my calculator is out of batteries (and still out in my car).

That's what I said. They must have measurements about each other. And if you're third party, but you can't directly observe the arrow for some reasons, you must have information about observers to be able to interpret their data in your reference frame. Or, of course, they must have information about your reference frame, then do the calculations and send the prepared data to you.