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Author Topic: How can photons be localised if the universe is expanding?  (Read 47232 times)

Offline DoctorBeaver

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How can photons be localised if the universe is expanding?
« Reply #25 on: 22/12/2008 12:13:33 »
If, indeed, it's "there" to notice anything.

Well, yes.
 

Offline yor_on

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Re: How can photons be localised if the universe is expanding?
« Reply #26 on: 24/12/2008 16:50:33 »
I can see that you are still insisting that, somehow, the photon (like a bullet) is speeding on its way from a to b. If you look at it that way then I can see there may be a problem. But why does it have to exist between a and b? The only places and times that you can prove it's there are at each end of 'its' journey.
There is a great difference between a photon and all other so-called particles. An electron can make a journey and make its presence felt by many systems on the way - giving a little (in principle, measurable)  nudge to every charge it passes. But a photon only interacts at each end of its 'journey'. As far as I can see, then, it is not necessary to consider what it is getting up to on the way - it gets up to nothing, in fact. I also hold that the 'it', to which we refer , is just the energy, which will behave according to the wave with which it's associated. It can't  / needn't be considered as a particle except where it is interacting.

You seem to be over concerned about the changing distance between source and detector - why should that concern an object which doesn't really exist on the journey? What about the situation where source and detector happen to be approaching each other at the same speed that the space between them is expanding - wouldn't that situation be just as difficult to explain?

My way of looking at it is not to say "Off the photon goes on its journey through the universe without interacting with anything else" because, for a start, you can only say that it hasn't interacted yet. I say it is potentially everywhere at all times between its creation and its absorption. It's journey is only described once it has actually completed the two interactions.

There is, of course, a very finite time involved in the interactions at each end because they are, essentially, resonances which take time to build up. As far as the photon is concerned, the two ends could be one atom's width or the whole universe apart.

The bullet picture makes it extremely difficult to reconcile the effects of diffraction - requiring a message to travel instantly all around the Universe to tell all other atoms that the photon has arrived 'here' and can't be seen anywhere else. But, even in the simplest, two slits, experiment, something has to tell each photon that it is part of a number of others which must, somehow sort themselves out into an interference pattern - which will always occur when there are 'enough of them'. It's a nightmare this way round; stick to the wave interpretation.

SC I read your thoughts with great interest, there are more than you sharing this view.

Still, what does that make the sun?
Isn't that a source of 'light' aka photons?

If there is no 'distance' involved,why do they need a source relative us measured as at a distance?
I'm sorry if I sound 'meta-physical' here, but if they are this in woven' in spacetime, why do 'they' (mostly) get created from 'sources' we measure as being 'distant' in space?

No disrespect meant SC, you are all interesting, I'm just trying to get a 'grip' of how you and others see it:)
 

Offline ukmicky

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How can photons be localised if the universe is expanding?
« Reply #27 on: 24/12/2008 17:15:01 »
As has been explained elsewhere, for a photon travelling at c time has no meaning.
Your basic premise is flawed. Time has no meaning without the observable context of mass. Photons have mass(albeit infinitesimally small mass). This is the reason why black holes can capture light....if photons had no mass they would pass by black holes. Also effects like gravitational lensing would not be observed if photons had no mass.

"It therefore follows that a photon could be everywhere in the universe at the same time."

Time does exist for photons (because they have mass) which is why they cannot accelerate past the speed of light in a vacuum.

Gravitational lensing and the reason photons cant escape blackholes is due to space and the path that photons take through it being bent. Gravity has no direct affect on a photon. 
 

Offline Mr. Scientist

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How can photons be localised if the universe is expanding?
« Reply #28 on: 24/12/2008 19:13:24 »
My little beaver brain is working overtime again. This is what its murky depths have churned out this time.

As has been explained elsewhere, for a photon travelling at c time has no meaning. In fact, time doesn't really exist for it. It therefore follows that a photon could be everywhere in the universe at the same time. But the universe is expanding, so there is now more "everywhere" for it to be in than there was at any time in the past. How can the photon be everywhere at once if everywhere is getting bigger?

 ???

Localization is at a cost though. The Uncertainty inherent in matter, requires it's path be complimentary. Sure, you can localize a particle, if you are willing to give up your time to observe an infinite amount of paths simultaneously.

It would require impossible odds.
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #29 on: 09/01/2009 10:26:37 »
This is one of the many things making photons fascinating.

But we have sources for them.
We have interactions with them at different places in spacetime.
Even though they may change 'entity' in those interactions.
We can still construct a 'particle path' from them.
That will follow everything we know about light.

For us to have that 'shimmering weave' :) of 'light' hiding everywhere.
We don't seem to need any 'sources' at all.

And our ideas of how energy transforms seems childish if so.
As this potential is everywhere, not needing any 'source' at all to my eyes.
But it might be right.

A 'field' of light overlaying a 'field' of time overlaying a 'field' of space all seemingly centered around matter.
And no distance?

It is like we really have two different realities coexisting at all time.
In one there is three dimensions and time creating distance motion space light and matter.
All of them entities by themself.

And then we have ?
 

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How can photons be localised if the universe is expanding?
« Reply #30 on: 09/01/2009 22:37:49 »
yor_on

You say that all those photons 'come from' the Sun. I would agree that photon interactions take place between atoms in the Sun and in your eyes. Between those places, the light behaves like a wave - it is refracted and diffracted on the way. Why can't that be enough?

Why is the idea of Photons being like little bullets such an attractive one, I wonder?
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #31 on: 10/01/2009 12:45:01 »
Yes SC, I see some as coming from the Sun.
Most of them we see in fact:)

And I have no problem with accepting them to be waves, if traveling in spacetime.
What I have problem with is the idea that they only will exist as a 'interaction'.

To me that imply, if that idea is correct of course:) that they would exist independent of any source.
Accepting that view, all our ideas of spacetime seems to collapse.
Then what are what we call sources?

We have built spacetime around concepts as distance motion 'c' etc.
All of those seems questionable, to say the least, if we allow for photons to be at 'all paths'.

Our descriptions 'falters' if you see how I think:)

----------

Then again.
We can send coherent laser light in a very straight path, can't we?
And if we measure that specific wave length outside what we might call its 'path' it won't be there, right?
So light have a path, doesn't it?

Radiation by incoherent light should be able to be treated as the sum of all coherent 'wavelengths' there might be.
And if so, sun light also will have a 'path' in spacetime it seems to me.
Am I right?
« Last Edit: 10/01/2009 16:49:13 by yor_on »
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #32 on: 11/01/2009 13:09:46 »
Let us take the idea of superimposing photons upon each other.
Photons are bosons and can be superimposed as much as you like without taking up any space.

Treated as waves there are experiments proving that.
They can either cancel each other out, or build on each other.
This is called 'Constructive or Destructive Interference'.

If we can do it when they are treated as waves what hinders us from doing it when treating them as particles?
Why would it violate HUP (Heisenberg uncertainty principle) when treated as particles but not as waves?
Am I getting this wrong?

If I assumed that it was possible, as the HUP doesn't forbid it, (as I see it:)?
It only states that we as observers won't be able to prove all parameters, if observing.
That's not the same as stating that under no circumstances, observed or not observed, can all parameters for a given 'system/particle/object' be existing in spacetime at the same time, is it?

Is it possible to superimpose 'photons' on each other?
And will they then have more energy as measured from a point of impact?
« Last Edit: 11/01/2009 14:04:44 by yor_on »
 

Offline lightarrow

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How can photons be localised if the universe is expanding?
« Reply #33 on: 11/01/2009 19:19:08 »
Yes SC, I see some as coming from the Sun.
Most of them we see in fact:)

And I have no problem with accepting them to be waves, if traveling in spacetime.
What I have problem with is the idea that they only will exist as a 'interaction'.

To me that imply, if that idea is correct of course:) that they would exist independent of any source.
Accepting that view, all our ideas of spacetime seems to collapse.
Then what are what we call sources?
Sources of electromagnetic field.

Quote
We have built spacetime around concepts as distance motion 'c' etc.
All of those seems questionable, to say the least, if we allow for photons to be at 'all paths'.
Our descriptions 'falters' if you see how I think:)
Actually, we have built spacetime around tangible concepts as objects with mass.

Quote
Then again.
We can send coherent laser light in a very straight path, can't we?
And if we measure that specific wave length outside what we might call its 'path' it won't be there, right?
So light have a path, doesn't it?
Make this simple experiment: put a coin perpendicular to the light's beam, the coin's area greater than the beam's area. Will the coin stop photons from reaching a point after it? According to your view the answer should be yes, but actually it's not, because of diffraction.
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #34 on: 11/01/2009 20:05:54 »
Are you talking about coherent light or incoherent light here Lightarrow?

But my question wasn't about what would happen if you put matter in its path.
It was if one could prove that the lasers light had a certain path in space.
And from there ask if ordinary light also might have a defined path in spacetime.

As for building spacetime, didn't that start a long time ago:)
With Newton and Einstein, Feynman appearing relatively recently?
It's all a question of definitions of course.

I still have trouble seeing photons as something just appearing as 'interactions'.

---------

But yes, I can see what you mean here.
Still, to me that would be the next question:)
As we first have prove that there are no 'paths' as I see it.

So, do lasers have a 'specific' path in spacetime or not.
« Last Edit: 11/01/2009 20:59:18 by yor_on »
 

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How can photons be localised if the universe is expanding?
« Reply #35 on: 11/01/2009 22:47:01 »
Quote
Are you talking about coherent light or incoherent light here Lightarrow?
No difference at all, in principle.
You get diffraction all the time with all em waves. It's just a matter of degree.

Quote
I still have trouble seeing photons as something just appearing as 'interactions'.
I think you ('one') must try to think through the true consequences of treating light as photons 'on the way'.
I realise that there is a huge photon culture because of the scale of things for visible light but any treatment must encompass all wavelengths of em energy. I can see why, because working out diffraction integrals (even just approximately) is a lot less friendly than drawing lines from a to b and saying that the energy 'goes that way' except when it occasionally gets diffracted.
Do / can you possibly think that a 1500m radio broadcast consists of a shower of little bullets - how big would they be? Would they be the same 'size' as gamma photons?
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #36 on: 12/01/2009 11:10:53 »
Good question SC.
I can give you one question with a twist:)

Do you think you and everything you use/meet today is made of light?
Explain that, no theories.
Just those experiments proving we are light.

What do you think:)

-------------

I haven't said that I don't believe in the result of the two slit experiments?
It's just that I do believe in them :)
And that matter and light, at least looking from where we are, is fundamentally differently expressed.
One of the reasons why I'm not happy with 'many paths' is that it seems to make us all 'probabilities' in a statistical universe.
And considering our self structuring capabilities, as well as entropy and the arrow of time we have?.
To me it seems that we are more than 'probabilities'.

« Last Edit: 12/01/2009 12:16:58 by yor_on »
 

Offline lightarrow

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How can photons be localised if the universe is expanding?
« Reply #37 on: 12/01/2009 12:38:52 »
Are you talking about coherent light or incoherent light here Lightarrow?
Both.

Quote
But my question wasn't about what would happen if you put matter in its path.
It was if one could prove that the lasers light had a certain path in space.
And from there ask if ordinary light also might have a defined path in spacetime.
Yes, but only statistically, in the sense that most of the field can be localized in space, in some cases as this one. If it wasn't so, optical geometry wouldn't have existed...The conditions which the fields must satisfy for the optical geometry approximation to be correct are:
1. Wavelenghts much smaller than all the other physical dimensions involved in the system.
2. Distance between stops/slits and detector screen not much greater than the stop's/slits' dimensions (or you will get a diffraction/interference pattern the same).

Quote
As for building spacetime, didn't that start a long time ago:)
With Newton and Einstein, Feynman appearing relatively recently?
It's all a question of definitions of course.
Remember that spacetime is not simply a mathematical object, but a physical one; to define it you need reference frames, physical objects, physical clocks.

Quote
I still have trouble seeing photons as something just appearing as 'interactions'.
A photon must be a physical object.
1.How would you define the existence of a physical object without ever using the word "measure"?
2.Can you measure a photon between source and detector?
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #38 on: 12/01/2009 14:06:20 »
Hhmmm :)

How about this then:)
If our Sun got quenched.

If now light do have a path, isn't there something called spontaneous particle creation?
That, it is said, the light enjoys itself 'playing' with, even though it goes back to being a wave very quickly?
If that is true, can't then light be seen as a particle having a geodesic path following spacetime.

And if that is true, as well as space containing 'in itself' a huge amount of energy.
And if Feynman is correct in his interpretation of possibilities innumerous for that 'waves/photons' path.
And if I'm right in wanting to allow that wave the properties of a particle?

Will then all be dark here as that sun goes out?
Why?

In my view it's quite simple.
The Sun went out, what do you expect?

From the other view it seems more of a question of probabilities, not discounting the one, wherein the 'light' still will shine with or without a 'source', even if it would be of extremly low probability.

Or am I thinking wrong here?
You could say that even without Feynman's 'many paths', spacetime still would have that possibility.
But to me that concept makes it so much more probable, as it invites us to a spacetime in where all paths are taken at all times, not caring for the 'geometry' we observe versus matter.

Ah the headache:)

"
A photon must be a physical object.
1.How would you define the existence of a physical object without ever using the word "measure"?
2.Can you measure a photon between source and detector? "

How do you mean by 'physical object'?
Only at its moment of 'interaction' or at all times?

1.Yep :)
Lovely question that one.
As well as difficult to answer.

Everything we do or experience comes from some kind of interaction with our surroundings.
So you are definitively right in that this (observing) is the first and last 'proof' of something existing outside ourselves.
Even though it is more of an archetype than a true 'objective evidence', it is still what we build upon.

But we also expect 'cause and effect' (times arrow) to work even without us observing.
For example you filling a bath tube, leaving it for a while and when coming back finding it overflowing.
That is also an archetype, but only at QM level can it be said to be 'broken'.

(Ah, times arrow I meant, not the bath tube:)

So going from those definitions I believe(?:) that this path reasonably could be seen to exist even without any observable interactions.

2. No.
(But see 1.)
(That is if you by measuring, mean observing it 'passively', as an object of its own (a ball)?)

But if you by 'measure' mean.
If it will interact with any 'obstacle' placed in, what I see as, its 'path'?

Then..
Yes, it will.




-----

Another problem I have is with dimensions:)
Take string theory.
Is it eleven dimensions in M theory?

When we start with one dimension.
Nothing special, just trying to see it here in 'spacetime' (in time).
Will we ever see it?
Nope.

Two dimensions then?
Yep, from some angles it will exist, from others it won't.
Have we ever observed anything like that here?
Nope.

What scientists define as 'two dimensional' today is just an 'approximation' of it.
Like a lattice wherein you hinder atoms from moving any other way than back and forth.
Is that two-dimensional?

Use my 'test' in space time and see for your self.
Will you at any angle observe this lattice as 'disappearing?'

And if your answer is no.
Well, then it can't be two-dimensional (as I define it).

What we have here is three dimensions and time.
In this spacetime we find 'matter' and 'time', 'space' and 'light'.
Most of the other 'things' we have defined, as gravity, motion and distance, seems to me to come from them.
Acceleration being a special cause.

We know it is so.
And that it 'works'.
Otherwise I wouldn't be able to write this.

I'll stop here for now:)

Btw: Anyone feeling that they do understand what a 'dimension' is?
You are very welcome to explain how you see it:)

Although DB might want that explanation in a different thread?

-----


(Maybe 'time' and 'space' also fall under matter and light?:)
Could 'time' exist without matter?

As without matter where would 'space' be?
That would be 'time' on its own, and time is a 'relation' right?

And yes, I'm just wondering, nothing more :)

-----------

« Last Edit: 12/01/2009 15:42:36 by yor_on »
 

Offline lightarrow

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How can photons be localised if the universe is expanding?
« Reply #39 on: 12/01/2009 15:38:28 »
Hhmmm :)

How about this then:)
If our Sun got quenched.

If now light do have a path, isn't there something called spontaneous particle creation?
That, it is said, the light enjoys itself 'playing' with, even though it goes back to being a wave very quickly?
If that is true, can't then light be seen as a particle having a geodesic path following spacetime.
Sorry, I don't understand what you want to say.

Quote
And if that is true, as well as space containing 'in itself' a huge amount of energy.
And if Feynman is correct in his interpretation of possibilities innumerous for that 'waves/photons' path.
And if I'm right in wanting to allow that wave the properties of a particle?

Will then all be dark here as that sun goes out?
See up. Could you explain what you want to say in a different way ?

Quote
Why?

In my view it's quite simple.
The Sun went out, what do you expect?

From the other view it seems more of a question of probabilities, not discounting the one, wherein the 'light' still will shine with or without a 'source', even if it would be of extremly low probability.

Or am I thinking wrong here?
You could say that even without Feynman's 'many paths', spacetime still would have that possibility.
But to me that concept makes it so much more probable, as it invites us to a spacetime in where all paths are taken at all times, not caring for the 'geometry' we observe versus matter.
See up.

Quote
Ah the headache:)

"
A photon must be a physical object.
1.How would you define the existence of a physical object without ever using the word "measure"?
2.Can you measure a photon between source and detector? "

How do you mean by 'physical object'?
Only at its moment of 'interaction' or at all times?
Let's make an example. I do the following statement: "everything you see is due to the interaction with your eyes or your camera, ecc. of little angels with blue wings and green faces. They are destroyed at the impact with your eyes or your detectors."
Do you think it's a scientific statement? Let's imagine that you say "No". Then I'll argue that my angels explains very well what you see and the properties of light (tell me a light's property and I'll explain you how the "theory of Angels" describes perfectly that property... ;)).

Quote
But we also expect 'cause and effect' (times arrow) to work even without us observing.
For example you filling a bath tube, leaving it for a while and when coming back finding it overflowing.
Since this is a much more difficult question, for the moment I prefer to say that it's a different situation, because:
1. We are talking about macroscopic objects.
2. We are talking about objects with non-zero mass.
Remember that the fact photons don't have a position operator in QM (as, e.g., electrons, have) is essentially due to the fact they are massless. It is this absence of such an operator that makes it impossible to ascribe them a precise position in space when not detected.

About string theory, M theory and the rest, sorry but I don't know much about them...
« Last Edit: 12/01/2009 15:42:21 by lightarrow »
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #40 on: 12/01/2009 16:08:20 »
Well, :(At least it seemed logic to me:)

Lightarrow, as far as I understand we start everything by archetypes, or do you see it differently?
One of the first archetypes needed to be defined is if 'anything' exist at all outside ones mind.
If your answer is a yes to that one, the rest of what we call 'objectivity' 'observations' 'measurements' will follow and make sense as 'outside phenomena'.
Not hinging on us being here at all.

The second one I used was the arrow of time, and I called that one a archetype too :)
And that is as we without knowing why it does it, or how it does it, observes it as having an arrow.
And that we expect it to continue to do so, at least macroscopically.

And from those two 'postulates' I find it reasonable to expect photons to be able to have a 'path' in spacetime, even without either you nor me observing.

(And no, I don't see light as little angels, but I like the idea, who knows:)
Reminds me a little of that 'postulate' wherein the question was something like.
'How many angels can dance on a needles 'point'':)

As for the beginning of what I wrote, I agree.
It needs to be much clearer.

--------

Give me some 'time' on that one, please:)
« Last Edit: 12/01/2009 16:18:36 by yor_on »
 

Offline Vern

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How can photons be localised if the universe is expanding?
« Reply #41 on: 12/01/2009 16:27:34 »
Quote from: yor_on
Do you think you and everything you use/meet today is made of light?
Explain that, no theories.
Just those experiments proving we are light.
I've been lurking here following this thread with great interest. I've been compiling evidence FOR and AGAINST So far the evidence for that far outweighs the evidence against. It seems that since that is such a restrictive case it should be easy to dispose of.
 

lyner

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How can photons be localised if the universe is expanding?
« Reply #42 on: 12/01/2009 18:19:47 »
Quote
Do you think you and everything you use/meet today is made of light?
Explain that, no theories.
Just those experiments proving we are light.

By 'light' I take it you mean electromagnetic waves(?).
The answer is No, in any case. The waves are only there due to interactions between charge systems, involving mass (thus excluding 'light' which is massless).

How can one explain without theory? "I feel that because it makes sense to me" is no use as an explanation.
 

Offline lightarrow

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How can photons be localised if the universe is expanding?
« Reply #43 on: 12/01/2009 18:30:36 »
Well, :(At least it seemed logic to me:)

Lightarrow, as far as I understand we start everything by archetypes, or do you see it differently?
Archetypes? Do you mean "basic concepts"?
Quote
One of the first archetypes needed to be defined is if 'anything' exist at all outside ones mind.
This is an interesting questions. Harry Palmer (creator of the Avatar course) says: "Let's start to take things away of the universe. Planets, then stars, then all the matter, dust and clouds and atoms or particles. Then even time and space,ecc. What remains? Our awarness. If we took away even our awarness, who could prove there is still something in the universe?"

Anyway this is more phylosophy than physics, at least now (maybe one day we will know something more about awarness).
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #44 on: 13/01/2009 15:53:28 »
As I understands it an 'archetype' is something so 'obvious' that we more or less take it as granted.
Like one and one makes two.
I read somewhere that most animals know the difference between one and two and 'many'.

Sometimes they can be wrong though.
The 'archetypes' I mean, well, the animals too:)

I mean?
Is one and one still two??
It can, as I see it, easily become three:)





 

Offline lightarrow

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How can photons be localised if the universe is expanding?
« Reply #45 on: 13/01/2009 22:57:35 »

Is one and one still two??

I think you will like this (hope to have said it well in english);
<<People are of 10 categories: those who know binary numbers and those who don't>>   :)
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #46 on: 14/01/2009 10:33:07 »
Yep:)
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #47 on: 14/01/2009 19:50:22 »
Apropos nothing at all:)
Considering my inclination for logic..

"If Logic is a systematic method for getting the wrong conclusion... with confidence.
Then surely statistics is a systematic method for getting the wrong conclusion... with 95% confidence."

Or something to that order?
 

Offline LeeE

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How can photons be localised if the universe is expanding?
« Reply #48 on: 14/01/2009 20:52:24 »
My little beaver brain is working overtime again. This is what its murky depths have churned out this time.

As has been explained elsewhere, for a photon travelling at c time has no meaning. In fact, time doesn't really exist for it. It therefore follows that a photon could be everywhere in the universe at the same time. But the universe is expanding, so there is now more "everywhere" for it to be in than there was at any time in the past. How can the photon be everywhere at once if everywhere is getting bigger?

 ???

I've been thinking about this and I think that something I added as a late post in another topic may have a bearing here too, because the premise that time has no meaning for a photon may not be correct.

From that earlier thread:

Quote
Although we can move through three spatial dimensions, at any point in time the movement vectors for all three spatial directions can be summed to a single vector.  Thus movement is essentially in a single direction and can be expressed by a single value, just as when we drive heading North-West we don't say we are driving West at x mph and North at y mph; we just use the summed vector.

With movement induced time dilation, the same thing is happening, except this time the two vectors being summed are the summed spatial movement vector and the temporal movement vector.  The reason we get time dilation is because it is the sum of these two vectors, spatial and temporal, which cannot exceed 'c', so as the spatial movement vector increases, the temporal vector must decrease.  With zero spatial movement then, we move temporally at 'c', which in turn implies we have zero length in that direction

Now, if we are moving at 'c' in the temporal dimension, have no length in that direction, and we are aware of time passing, then it would seem plausible that when we see a photon moving at 'c' though our spatial dimensions we are seeing a two-dimensional object moving along it's temporal dimension; this being our third spatial dimension, in which it has no length.
 

Offline yor_on

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How can photons be localised if the universe is expanding?
« Reply #49 on: 05/02/2009 13:05:13 »
Ok another thought.
If we say that a photon if red shifted, that is, as seen from our frame of observation being of a lesser 'energy' content.
We also say that its wave is longer, right.
So in a BEC where we see photons as frozen/still that wave should be?
Infinite??

And if so then 'expansion' is no obstacle as that 'wave' then will cover 'everything' we can observe.

But if it are able to do this, our ideas of distance seems wrong to me.
They will only work at a macroscopic plane.
 

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How can photons be localised if the universe is expanding?
« Reply #49 on: 05/02/2009 13:05:13 »

 

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