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Instantly: Thoughts on simultaneous events
By Alan Riddell on Friday, February 12, 1999 - 04:27 pm:

Can anything happen instantly?

This might be more physics than maths. However, when the question first came up in a physics lesson, my first answer was No. There can be no instant changes.

Take a door a door doesn't go from being open to closed instantly. There is a short intermittent stage, where the door is neither open or closed. The same argument can be used with most examples of something changing from one state to another.

However, then later on thinking mathematicaly. Let the door being closed be represented by "0" and the door being open represented by "1".

However, then later on thinking mathematicaly. Let the door being closed be represented by "0" and the door being open represented by "1". Now there must be a stage where the door is inbetween, say, (very) slightly less than 1. I can't see how it jumps from less than 1, to 1 without there being some instantaneous jump. Similarly going from zero to more than zero.

Is there a "reasonable" mathematical answer to this problem? (Without getting into weird physics or philosophy.)

Alan Riddell

By Robert Schulz on Friday, February 12, 1999 - 04:36 pm:

A question which I often pondered as a child was this:

If I hold out my two index fingers next to each other, and then touch a surface with one of my fingers, is it possible for me to then change fingers (remove the first finger from the surface, and touch the surface with the other finger) simultaneously so that at no point are both fingers on the surface, and at no point are there no fingers on the surface?

What do you think?

I suppose the question is, is anything ever truly simultaneous?

Robert Schulz

By James on Friday, February 12, 1999 - 04:39 pm:

Instantaneous speed has always puzzled me. How can you give the speed of
a particle at a fixed place?

In any calculation to do with speed some form of time comes into play.
If not then a distance comes in there.

You could for instance try to use

Power = Driving Force x Velocity

But Remember Power = Work Done / Time (Grrrrrr!!!!)

If anyone can enlighten me on this I'd be grateful. I haven't given this much thought,

James

By Damian Haigh on Friday, February 12, 1999 - 04:41 pm:

I don't think you can give a reasonable answer without at least a little bit of philosophy:

I can see at least two questions here

- when is a door shut, and when is it open?

when does a collection of sand grains become a pile, and when is it a heap? when is something a metre long - when it is 100cm , 100.00000 cm, 100.00000000000cm?

The vast majority of every-day-linguistic concepts have some vagueness built into them, this is how they gain the flexibility that allows language to work.

By 'thinking mathematically' and giving a numerical/logical value to the condition you are producing a model in which you need to tighten up the definition of what it is to be shut. You need to be careful not to conflate the possibility of formulating the situation more rigorously, with there necessarily being a rigorous answer to your 'everyday' question of whether things happen instantly.

In the same way that an eccentric quantity surveyor might demand a definition for a heap of sand, your model needs a working definition of 'shut' - but this doesn't mean there's any significance to what it is to be open or shut in your model, except within your model.

- is time ultimately dividable, or does it fall into discrete steps, with one state of affairs following the next with no intermediate stage?

Is this really the same problem that faced calculus when mathematicians first tried to give it a rigourous foundation - ie how can it be meaningful/true to talk of infinitesimally small periods of time. I don't know the formal justification that was found for this, but I thinkthe key is to remember that things are OK so long as we are only talking of what is happening as eg delta x tends to zero, not when it is actually zero. So things can't happen instantly but they can happen very quickly, and they can happen in lengths of time which tend towards zero. For an example of this you could look at Zeno's paradox (for an interesting source which could be used to extend able Y10 and 11 students who have studied APs and GPs see the Zeno tasksheet SMP 16-19 Foundations booklet).

A related thought: force can be defined as rate of change of momentum. So when a fly hits the cockpit of a jet plane and is instantly squashed, was the force on the fly infinite? Clearly not - as the change of momentum was not actually instantaneous. The fly got squashed and this took some time. How would you account for the force on a cricket ball hitting Brian Lara's bat? Again not an instantaneous change of momentum (deformation of bat and ball takes time) but a very quick one.

Is it possible that quantum mechanics might lead us to conclude that if you keep chopping up time into smaller and smaller bits, you eventually reach a point beyond which you can't chop anymore? I know very little about this. QM does seem to suggest that there are some discrete states of existence eg when electrons change energy levels within an atom. Perhaps what QM really tells us is that the common sense model of the world around us (tangible objects moving in a predictable way etc.), the model on which our language and most of our material concepts are based, is just that - a model - and therefore has limitations which a rigourous examination of language will reveal

If there can be no instant changes, is there a minimum time that anything can take? If the causal connections in nature, such as electrical activity or chemical reactions were really instant at the
'base' level, would everything in life happen instantaneously?

What an interesting question - what did your students think?

Damian Haigh
Rainow

By Damian Haigh on Friday, February 12, 1999 - 04:41 pm:

>Surely another line of reasoning must be to consider the meaning of
>instantaneous. Is the occurrence of change without the passage of a
>time a perceived understanding of the word or the real meaning?

Philosopher here: What is real meaning? Outside formal theories meaning is a question of linguistic convention - ie it depends on the rules of a game where the rules are constantly changing. A hundred years ago a trunk was a piece of luggage you might occasionally strap on the back of a coach, now to millions of americans it is a closed compartment in the back of a car. In science and mathematics we are used to every-day terms having a more strict definition within the academic world, and this is sometimes taken as the 'real' meaning of the term. This does not mean that the meaning of eg the word "consistent" is any more real as used formally in maths, or informally in language. Strict meaning is limited to the confines of a scientific / mathematical model. I suppose you could say that the meaning of 'meaning' depends on what you mean, when you mean it and to whom. This is all getting a bit too Lewis Carroll!

>Does it help to think that a number line moves continuously from 4 to 6.
>But there is one point at which numbers move from being less than 5 to
>being greater than or equal to 5, ie the point 5.

>There is no point which is 'in between' less than 5 and greater than or
>equal to 5.

I think this is an important point.

By James on Friday, February 12, 1999 - 04:42 pm:

This is analogous to finding the gradient of a curve at a fixed point. Indeed, if you plotted a graph of vertical velocity against time for a projectile, the gradient would be the vertical velocity. If a point can have a gradient, in the same way a particle can have a speed at a single instant.

Imagine a fly above a train track. A train comes towards it and whilst they are both moving towards each other they collide. The fly is accelerated in the direction of movement of the train.

The fly has now reversed in direction. Its velocity must have passed through 0 i.e. from -ve to +ve direction. The collision is totally inelastic so they now share the same velocity. So when the fly's velocity is 0 so is the trains! The fly, has stopped the train (instantly)!

James
--
Fishguard High School

By James on Friday, February 12, 1999 - 04:43 pm:

I'm sorry but no. The fly does not stop the train, if the train has stopped it must accelerate back to its previous speed, if it is only stopped for an instant there must be infinite acceleration. This is a feat that can only happen on poorly drawn graphs, never reality, the fly has almost infinite acceleration but not quite, there is an unnoticable duration of change of speed of the fly. There is an unnoticable change in the trains speed, yes, and the fly for an instant does stop. But this is at the point of collision, they only share the same speed after the point of collision. I might seem to be bawling in your face (I suppose it depends on how you read it), but I'd hate to think you'd ruin your chances for a decent grade on the basis of one simple misunderstanding. I hope i've set things straight.

The reason the train does stop is because :

  • The fly hits the train in an instant, during this time period the train does not move, so how can we measure its speed.
  • Flies squash into a nice mess when they hit the train so they don't behave as true particles.
Talking about infinity, has anyone ever thought about the paradoxes that
seem to appear the further into space you look?

Say if the universe is 15,000,000,000,000 years old. If we look back
into space in any direction until we can see the furthest object, we get
older and older the further and further we look. Of course you get so
far and we see just the big bang e.g. a point object. This point object
is visible in every direction!

James
--
Fishguard High School
By Tom Donnellan on Friday, February 12, 1999 - 05:19 pm:

I think Jo Tomalin might have pointed us in a good direction by the "no point on the number line lies between less-than-5 and greater-than-or-equal-to-5" idea; it seems to me that in this case, there is some meaning to the word "instantaneously" - as someone else has suggested, the same will be true of the change from the 20th to the 21st century.

It bothers me, though, when there is something happening which involves matter: take the door-open, door-closed case - - - I am (what is the word I seek?) generously covered, and a door might be closed to me but open to someone less bulky. On the minute scale, then, assuming the door is moving, it becomes progressively closed to the head of a pin, a water molecule, an electron, a neutrino, and whatever else they have found since I last looked seriously at Physics - - there is a bit (vague, I grant you) of meaning of closed here, and it isn't at all instantaneous.

Much work needs doing on this, and if a Foundation will provide the funds to enable me to retire from teaching Maths at Sydney Boys' High School (delightfully situated across the road from Sydney Cricket Ground) I shall be happy to attempt the work. There is - we hope - a generous deadline - - the Day of Judgement, when - of course you remember - we shall all be changed "in a moment, in the twinkling of an eye". I bet that this will happen "instantly".

Hearty greetings to one and all.
Tom Donnellan

By Ed Wallace on Thursday, February 18, 1999 - 02:27 pm:

I've been watching the snooker this week, and it made me think about the "instantly"/discrete values debate. For example, at the end of each shot, it is always possible to say for every ball whether it is in a pocket or not. There is a short moment when a ball is falling into a pocket, but there is no state longer than the fabled "instant" where a ball is neither definitely in a pocket nor on the table. This seems to lead to the conclusion that there is a certain position beyond which the ball goes into the pocket and up to which the ball stays on the table. It would be possible to measure this with arbitrary accuracy, since there is no intermediate position where the ball hangs in the balance, which would be unstable. In this case it is clear that the "in-pocket" state (1) and "not-in-pocket" state are not only different but discrete. This is a closed system but is it relevant to less obvious examples?

Any thoughts?

Ed

By Chris Jefferson (Caj30) on Wednesday, April 14, 1999 - 09:47 am:

The main problem with the door open / door closed is to define EXACTLY what an open door and closed door looks like. If you make sure you define these two things to cover all possibilities and they don't overlap, then the door will be either open or closed. Whenever changing a problem in the English language to one of maths, it is very important to exactly define things exactly.

It is possible to create lots of examples of this with induction

Someone having ONE more hair won't stop them looking bald, so by induction, everyone has no hair.

Eating one chocolate drop won't make you more fat, so you can eat as many as you like!!

By Mark Jordan (P656) on Thursday, April 15, 1999 - 06:32 pm:

If you think of this in deegrees, there is a point where it is closed(0 deegrees) and one where it is open(90 deegrees). However, there is a point where it is halfway(45 degrees).

Bye!
Mark

By Chris Jefferson (Caj30) on Thursday, April 15, 1999 - 08:45 pm:

Ah, but I would say that a door that was at 10 degrees would be called open... :-)

By Mark Jordan (P656) on Wednesday, April 28, 1999 - 04:33 pm:

What if the door would be open at 0.000000000000000000000000000000000000001 mm! :>)

By Mark Jordan (P656) on Wednesday, April 28, 1999 - 04:38 pm:

Also, if you could open it less than the size of an quark (part of a proton) it might be simultaneous..........

8-)

By Graham Lee (P1021) on Sunday, August 8, 1999 - 06:37 pm:

Speaking in physics terms, nothing can be instantaneous because of the passage of information. Information can only travel at the speed of light, so nothing can be proved to have happened between the time it does and the time taken for light to travel from the event to the observer.
It is generally accepted that no two objects can occupy the same point in space, therefore the observer must be some distance away from the event, so the event and its observation cannot be instantaneous.
Heisenberg's uncertainty principle would have us believe that we cannot measure that two events happened at the same time to an accuracy greater than h, Plancks constant.
Also, does simultaneous mean at the same point in time (in which case it depends on proximity of gravitons) or in space AND time (in which case no)?

Think about that lot! I try not to!
GL B-)

By Graham Lee (P1021) on Saturday, August 21, 1999 - 07:37 pm:

And just to follow up, a bit about that big bang theory (that the point that was the singularity of the universe at the time of the big bang is visible in all directions)... no it isn't.
And I speak as an astrophysicist here.
The universe is approx. 8-20GYrs old (about a thousand times younger than the estimate above), according to latest theory. So, at a distance of somewhere over 8GLy (Giga-Light years), we should be able to see what the universe looked like at the time of the Big Bang.
In fact, this distance is so far away that the recession (caused by the expansion of the universe) of anything that does exist would be faster than the speed of light, i.e. if it emits any information we could not see it.
We can "see" back to a few hundred thousand years after the big bang with the COBE sattelite, but that looks in microwave radiation (due to the high redshift). Back then the universe was so uniform that only very small disturbances in the radiation can be measured.
Also, don't forget the the Big Bang was a singularity, and as such anything that did happen during it cannot now be measured.

By Andrew Wyld (Acew2) on Saturday, November 20, 1999 - 11:25 pm:

Whoa! Dude!

An "interesting" possible digression — what is simultaneity? Here's what I mean. This is a thought experiment first suggested by Einstein as a part of special relativity.

Supposing a train travelling along a track is struck by lightning at two points. Let's say, for simplicity, that these points are at the ends of one of the carriages (where the corridor is). There's a guy by the trackside who sees both lightning strikes at exactly the same time. By measuring the distance to the scorch marks on the track, he discovers that the lightning strikes were both at the same distance from him. Because light travels at the same speed always, he concludes, therefore, that the lightning strikes were simultaneous.

Now, at the moment when he calculates the lightning strikes must have taken place, a lady (let's call her Mileva) in the centre of the carriage was passing by. By the time the light from the lightning had travelled to her, however, the train had moved on somewhat. Therefore she saw the light from the front strike before the light from the rear strike.

Now here's the kooky part. Using the same reasoning as the guy by the trackside, she could measure her distance from the scorchmarks and work out how long the light from the lightning strikes would have taken to reach her. She would conclude that each burst of light would have taken the same amount of time to arrive. She would therefore conclude that one burst of lightning had actually happened before the other!

This sounds like messing about, but it's not. The whole point of relativity is that the ground is not more "stationary" than the train is — we are moving around the sun at around 30 kilometres per second, which in turn is orbiting the rough centre of the galaxy, and so on, and so on... There is no reason to prefer the conclusions of the man on the ground (the lightning strikes were simultaneous) to those of the woman in the train (the lightning strikes were not simultaneous). This effect does, genuinely, happen.

It's the bit of relativity they don't tell you about in the Sci-Fi movies!

Sorry for digressing from the question first asked, but I hope it was an interesting digression. Anyone confused? (I'm going to root out the picture from my notes and scan it in so that might make life easier)

have fun

Andrew Wyld

By Andrew Rogers (Adr26) on Sunday, November 21, 1999 - 12:00 am:

Confused, bewildered ? Yep, well that certainly was the case when experimenters first found that they couldn't measure velocity changes of light relative to the 'ether'. Once you're OK with the experimental evidence, or have a feel for Lorentz-Fitzgerald, the above isn't bewildering, but definately odd !

Andrew R