# Is Krauss looking at this right?

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This is what you said:

I don't think it is that simple.

When the road grows by 2 inches, it means that there is scale factor of 2. When the scale factor is applied to space, it is applied also to any velocity (because velocity is spce/time), and thus it is applied to C also.

That's why we observe galaxies receding at apparent speed multiple of C.

Or:

if the road is 10 inches long, one inch travel is 10% of the road.

When suddenly the road extends to 20 inches long (by scaling), then one "scaled-inch" still is 10% of the length, or 2 inches. The important thing is that for the same time, the distance as measured in the new space system (the scaled-space) remains constant: that's the main property of C.

and it is wrong, when distances increase due to expansion it takes longer time for light to traverse this new distance.

But distance do not increase due to expansion.

The metric is scaled, that is entirely different. And that changes wavelength.

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But distance do not increase due to expansion.

If distance did not increase due to expansion, wouldn't the universe be much smaller than it appears to be?

If distance did not increase due to expansion, why is it that we are just now seeing light from a galaxy that was emitted 12 billion years ago when it was only 1 billion light years away?

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But distance do not increase due to expansion.

The metric is scaled, that is entirely different. And that changes wavelength.

When the metric of space is magnified relative your ruler, then the new distance as measured by your ruler has increased.

The metric expansion of space is the increase of the distance between two distant parts of the universe with time.

http://en.wikipedia.org/wiki/Metric_expansion_of_space

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When the metric of space is magnified relative your ruler, then the new distance as measured by your ruler has increased.

Yes. Agree. That's the distance as measured from the metric inside the "local penny". That is not the distance as measured from the expanded metric.

The metric expansion of space is the increase of the distance between two distant parts of the universe with time.

http://en.wikipedia.org/wiki/Metric_expansion_of_space

Nobody here to slightly correct Wikipedia ?

When the metric changes Light don't travel more distance, only the scale of everything has changed.

Edited by michel123456
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Yes. Agree. That's the distance as measured from the metric inside the "local penny". That is not the distance as measured from the expanded metric.

Nobody here to slightly correct Wikipedia ?

When the metric changes Light don't travel more distance, only the scale of everything has changed.

If light doesn't travel more distance, why is it that we are just now seeing light from a galaxy that was emitted 12 billion years ago when the galaxy was only 1 billion light years away?

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When distant objects gets carried away by expansion of space they are NOT considered traveling through space.

Here is a list of current observational evidence we have of space expanding, as have been said already, there is no theoretical constraint in general relativity for space to expand faster than light. If you want to question relativity or space expansion then I suggest you make a new thread for that.

Observational evidence

Theoretical cosmologists developing models of the universe have drawn upon a small number of reasonable assumptions in their work. These workings have led to models in which the metric expansion of space is a likely feature of the universe. Chief among the underlying principles that result in models including metric expansion as a feature are:

• the Cosmological Principle which demands that the universe looks the same way in all directions (isotropic) and has roughly the same smooth mixture of material (homogeneous).

• the Copernican Principle which demands that no place in the universe is preferred (that is, the universe has no "starting point").

Scientists have tested carefully whether these assumptions are valid and borne out by observation. Observational cosmologists have discovered evidence - very strong in some cases - that supports these assumptions, and as a result, metric expansion of space is considered by cosmologists to be an observed feature on the basis that although we cannot see it directly, scientists have tested the properties of the universe and observation provides compelling confirmation. Sources of this confidence and confirmation include:

• Hubble demonstrated that all galaxies and distant astronomical objects were moving away from us, as predicted by a universal expansion. Using the redshift of their electromagnetic spectra to determine the distance and speed of remote objects in space, he showed that all objects are moving away from us, and that their speed is proportional to their distance, a feature of metric expansion. Further studies have since shown the expansion to be extremely isotropic and homogeneous, that is, it does not seem to have a special point as a "center", but appears universal and independent of any fixed central point.

• In studies of large-scale structure of the cosmos taken from redshift surveys a so-called "End of Greatness" was discovered at the largest scales of the universe. Until these scales were surveyed, the universe appeared "lumpy" with clumps of galaxy clusters and superclusters and filaments which were anything but isotropic and homogeneous. This lumpiness disappears into a smooth distribution of galaxies at the largest scales.

• The isotropic distribution across the sky of distant gamma-ray bursts and supernovae is another confirmation of the Cosmological Principle.

• The Copernican Principle was not truly tested on a cosmological scale until measurements of the effects of the cosmic microwave background radiation on the dynamics of distant astrophysical systems were made. A group of astronomers at the European Southern Observatory noticed, by measuring the temperature of a distant intergalactic cloud in thermal equilibrium with the cosmic microwave background, that the radiation from the Big Bang was demonstrably warmer at earlier times. Uniform cooling of the cosmic microwave background over billions of years is strong and direct observational evidence for metric expansion.

Taken together, these phenomena overwhelmingly support models that rely on space expanding through a change in metric. Interestingly, it was not until the discovery in the year 2000 of direct observational evidence for the changing temperature of the cosmic microwave background that more bizarre constructions could be ruled out. Until that time, it was based purely on an assumption that the universe did not behave as one with the Milky Way sitting at the middle of a fixed-metric with a universal explosion of galaxies in all directions (as seen in, for example, an early model proposed by Milne). Yet before this evidence, many rejected the Milne viewpoint based on the mediocrity principle.

The spatial and temporal universality of physical laws was until very recently taken as a fundamental philosophical assumption that is now tested to the observational limits of time and space.

http://en.wikipedia.org/wiki/Metric_expansion_of_space#Observational_evidence

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This is what you said:

and it is wrong, when distances increase due to expansion it takes longer time for light to traverse this new distance.

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After 9 pages and 175 replies your claim to be in pursuit of logic and knowledge is wearing pretty thin.

When comparing to other large threads in this particular area of the forum, one specific similarity emerges: someone continues to repeatedly make claims against conventional wisdom and dodges any replies that point out flaws or tries to explain why those claims are wrong.

From my experience people searching for knowledge and understanding, ask questions instead of repeating the same inaccuracies.

It is evident that we have failed miserably in our attempts to explain for you, a model in which space can be expanding such that it can bring current visible objects beyond a horizon from where they can't be observed anymore, such that you can understand it and take it to your heart.

The burning question is IF you want to learn and understand or if you simply will reject any explanations that challenges your current belief?

I don't see any point in continue and debate with you if you refuse to try to understand, I gave you the benefit of a doubt but it is fainting fast.

Spyman,

"Beyond a horizon from where they can't be observed anymore"?

This is the crucial consideration that I am questioning, in general. At the moment the galaxy passes over this horizon, and no new photon, has a chance to close the distance, it does not represent a time, here, at the Milky Way, at which the galaxy can't be observed anymore. Remember, we are not seeing the galaxy as it is at the moment. Its present condition is not available to us. There is the lag in time, that it takes light, to travel the emmense distances we are talking about. So, logically, is there not the previous photons, from before the horizon event, that ARE available to us? Are not the photons from before the horizon event, already loaded in the cosmic grid, between us, and the horizon event?

They are yet to get here, and are not wiped from the grid, by the fact that the said Galaxy has now exceeded a C recessional speed. Whatever happens to the space between us, and the horizon event, for the rest of eternity, it, the space between, will ALWAYS contain photons, from prior the horizon event. This requires that the nature of, that is the wavelengths of those, and the frequency which when they arrive at the Milky Way, is what will be apparent to Milky Way observers, as they look in that direction. The photons from that galaxy, that are arriving here, do not just "turn off", at the moment of the horizon event.

I refuse, in this conversation to be lumped in, with other speculation thread type persons, who refuse to accept known science, in favor of their pet theory. I have not offered a pet theory, except for the idea that the space between us, and any distant object, must contain the photons that will, in the future, inform us of events at that distant object. This is already a fact, known by everyone. We do not see distant events, immediately.

I am merely exploring what that might mean.

Regards, TAR2

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Yes. Agree. That's the distance as measured from the metric inside the "local penny". That is not the distance as measured from the expanded metric.

The distance for which you're groping is the 'comoving distance'.

What does this mean?

The speed of a photon must remain constant. and for that to be true the speed must be relative to the size of the grid.

Are you saying that the speed of light is constant over cosmic distance when expressed as comoving distance and proper (unscaled) time? That is how I've understood you.

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So, logically, is there not the previous photons, from before the horizon event, that ARE available to us?

Yes

Are not the photons from before the horizon event, already loaded in the cosmic grid, between us, and the horizon event?

Yes

Whatever happens to the space between us, and the horizon event, for the rest of eternity, it, the space between, will ALWAYS contain photons, from prior the horizon event.

No. Eventually all those photons will crash into us or pass us by.

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The distance for which you're groping is the 'comoving distance'.

What does this mean?

Are you saying that the speed of light is constant over cosmic distance when expressed as comoving distance and proper (unscaled) time? That is how I've understood you.

It may be what I said, it may not, I am not sure.

I am not even sure that light travels a "comoving distance".

From Wiki:

It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to proper length in special relativity) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or spacelike geodesic between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own world lines, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the FLRW metric is set to zero (an empty 'Milne universe'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the flat Minkowski spacetime of special relativity, one where surfaces of constant time-coordinate appear as hyperbolas when drawn in a Minkowski diagram from the perspective of an inertial frame of reference.[4] In this case, for two events which are simultaneous according the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events,(Wright) which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a geodesic in flat Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are simultaneous.

(bolded mine)

I suppose the "distance" I am talking about is a distance along the path of light in expanding spacetime. It is not a distance at a specific time, it is a distance along time.

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Anyway, it is very close to what I ment.

-----------

(edit)

your comment about unscaled time is basically what I ment, but it must be trickier. If one could imagine contracted time, then it could equalize expanded space and produce a result where the photon does not change state of motion at all: it could follow a straight geodesic. IIRC I have seen somewhere such a spacetime diagram.

Edited by michel123456
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No. Eventually all those photons will crash into us or pass us by.

Or, at least require a telescope larger than a planet to see.

this diagram (and accompanying article) is really fantastic for this thread because it shows the event horizon at the infinite future.

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Or, at least require a telescope larger than a planet to see.

this diagram (and accompanying article) is really fantastic for this thread because it shows the event horizon at the infinite future.

Excellent! Thanks. I did a quick read and it is very well done. Now I need to spend some time with it to understand the details. +1

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This diagram is a pure delight to comment.

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"Beyond a horizon from where they can't be observed anymore"?

This is the crucial consideration that I am questioning, in general.

At the moment the galaxy passes over this horizon, and no new photon, has a chance to close the distance, it does not represent a time, here, at the Milky Way, at which the galaxy can't be observed anymore. Remember, we are not seeing the galaxy as it is at the moment. Its present condition is not available to us. There is the lag in time, that it takes light, to travel the emmense distances we are talking about. So, logically, is there not the previous photons, from before the horizon event, that ARE available to us? Are not the photons from before the horizon event, already loaded in the cosmic grid, between us, and the horizon event?

They are yet to get here, and are not wiped from the grid, by the fact that the said Galaxy has now exceeded a C recessional speed. Whatever happens to the space between us, and the horizon event, for the rest of eternity, it, the space between, will ALWAYS contain photons, from prior the horizon event. This requires that the nature of, that is the wavelengths of those, and the frequency which when they arrive at the Milky Way, is what will be apparent to Milky Way observers, as they look in that direction. The photons from that galaxy, that are arriving here, do not just "turn off", at the moment of the horizon event.

Lets say that advanced aliens in that distant galaxy are able to turn off the light, just like you can do in your home. When they do, it does not represent a time, here, at the Milky Way, at which the galaxy can't be observed anymore. There is a lag in time, while the previous photons traverses the distance, but there are only a FINITE amount of photons in route and eventually the last one will arrive and that galaxy will get dark in our sky.

Tonight, when it is dark, I suggest you test this for yourself, turn off the light and watch if there will ALWAYS be photons in the cosmic grid between your eyes and the lamp, for the rest of eternity, or if it will get dark when the time lag catches up. When the distance between the photons and us are increasing faster than what the photons can travel, then they can no longer make progress towards us, therefore photons from that distance will not arrive here. The cosmic horizon will create a gap in the line of photons and that gap will grow and follow the last photon on our side all the way to Earth.

I refuse, in this conversation to be lumped in, with other speculation thread type persons, who refuse to accept known science, in favor of their pet theory. I have not offered a pet theory, except for the idea that the space between us, and any distant object, must contain the photons that will, in the future, inform us of events at that distant object. This is already a fact, known by everyone. We do not see distant events, immediately.

People will judge you for how you behave and not after what you demand. You are currently in conflict with known science and refuse to accept it.

There are photons in the space, between us and a distant object, that will inform us of events in the future, but they will not last forever, if they stop coming there will eventually be a moment when the last one in the line reaches us. This is also already a fact, known by everyone. Whats left for you to speculate about is whether they will stop coming or not, but known science derived from observations agrees with the prediction Krauss made.

I am merely exploring what that might mean.

No, you are merrily exploring a mighty dream.

Edited by Spyman
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There is a lag in time, while the previous photons traverses the distance, but there are only a FINITE amount of photons in route and eventually the last one will arrive and that galaxy will get dark in our sky.

Or in the case of accelerated cosmic expansion, the photons are diluted into the infinite future like Tar says, but they become unobservable because, like Krauss says, the size of telescope needed to detect them is prohibitive.

The explanations and diagrams in this paper are superior in every way to the blog I somewhat haphazardly posted yesterday,

In addition, all galaxies become increasingly redshifted as we watch them approach the cosmological event horizon (z → ∞ as t → ∞). As the end of the universe approaches, all objects that are not gravitationally bound to us will be redshifted out of detectability.

Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the universe, p4

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Or in the case of accelerated cosmic expansion, the photons are diluted into the infinite future like Tar says, but they become unobservable because, like Krauss says, the size of telescope needed to detect them is prohibitive.

The explanations and diagrams in this paper are superior in every way to the blog I somewhat haphazardly posted yesterday,

Iggy,

Yes, I got a bit more from the article, than from the blog.

Interesting in regards to this thread, that the appendix included "Examples of Misconceptions or easily misinterpreted statements in the literature." Number 13 was by some fellow named Krauss.

[13] Krauss, L. M. and Starkman, G. D. 1999, ApJ, 531(1), 22–30, Life, the universe and nothing: Life

and death in an ever-expanding universe, “Equating this recession velocity to the speed of light c, one

finds the physical distance to the so-called de Sitter horizon... This horizon, is a sphere enclosing a

region, outside of which no new information can reach the observer at the center”. This would be true

if only applied to empty universes with a cosmological constant - de Sitter universes. However this

is not its usage: “the universe became -dominated at about 1/2 its present age. The ‘in principle’

observable region of the Universe has been shrinking ever since. ... Objects more distant than the de

Sitter horizon [Hubble Sphere] now will forever remain unobservable.”

Misconception or easily misinterpreted?

Regards, TAR2

Lets say that advanced aliens in that distant galaxy are able to turn off the light, just like you can do in your home. When they do, it does not represent a time, here, at the Milky Way, at which the galaxy can't be observed anymore. There is a lag in time, while the previous photons traverses the distance, but there are only a FINITE amount of photons in route and eventually the last one will arrive and that galaxy will get dark in our sky.

Tonight, when it is dark, I suggest you test this for yourself, turn off the light and watch if there will ALWAYS be photons in the cosmic grid between your eyes and the lamp, for the rest of eternity, or if it will get dark when the time lag catches up. When the distance between the photons and us are increasing faster than what the photons can travel, then they can no longer make progress towards us, therefore photons from that distance will not arrive here. The cosmic horizon will create a gap in the line of photons and that gap will grow and follow the last photon on our side all the way to Earth.

People will judge you for how you behave and not after what you demand. You are currently in conflict with known science and refuse to accept it.

There are photons in the space, between us and a distant object, that will inform us of events in the future, but they will not last forever, if they stop coming there will eventually be a moment when the last one in the line reaches us. This is also already a fact, known by everyone. Whats left for you to speculate about is whether they will stop coming or not, but known science derived from observations agrees with the prediction Krauss made.

No, you are merrily exploring a mighty dream.

Spyman,

According to the article any object that we see above Z=1.6 is already moving away from us, due the expansion of space, at greater than C. Distant galaxies are not like nearby lamps. If I turn the lamp off now, the Mars probe will see it shining for 14 more minutes.

Regards, TAR2

On Pluto, I have not even turned it on yet.

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Or in the case of accelerated cosmic expansion, the photons are diluted into the infinite future like Tar says, but they become unobservable because, like Krauss says, the size of telescope needed to detect them is prohibitive.

Yes, I fully agree, the light beam will get redshifted towards infinity, far beyond possibility of detection.

However on a side note, considering that the beam constitutes of discrete photons, we can conclude that the last photon in the beam must come from a location slightly on our side of the horizon. Just like a light source falling into a black hole, there will be a finite amount of photons emitted in our direction and the last one is from slightly above the horizon. I think that in both cases that last photon will reach us in a finite time.

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tar, I see that you omitted to give me an answer AGAIN, how come you don't give me a straightforward answer?

According to the article any object that we see above Z=1.6 is already moving away from us, due the expansion of space, at greater than C.

Yes, we can currently see objects receding from us faster than light, I have not said anything that is contradicting this. Your point is?

Distant galaxies are not like nearby lamps.

Distant galaxies are like distant lamps, when stars in them dies they stop shining and then eventually we can't see them any more.

If I turn the lamp off now, the Mars probe will see it shining for 14 more minutes.

So you agree that space between us and the Mars probe will NOT contain photons for the rest of eternity, from prior the event when the lamp got dark?

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According to the article any object that we see above Z=1.6 is already moving away from us, due the expansion of space, at greater than C. Distant galaxies are not like nearby lamps. If I turn the lamp off now, the Mars probe will see it shining for 14 more minutes.

On Pluto, I have not even turned it on yet.

You are now forever lumped in with other speculation thread type people. Hand waving, misdirection, heels dug in, refusal to concede any point or to directly confront questons threatening your position.

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The distance for which you're groping is the 'comoving distance'.

What does this mean?

Are you saying that the speed of light is constant over cosmic distance when expressed as comoving distance and proper (unscaled) time? That is how I've understood you.

It may be what I said, it may not, I am not sure.

I am not even sure that light travels a "comoving distance".

From Wiki:

(bolded mine)

I suppose the "distance" I am talking about is a distance along the path of light in expanding spacetime. It is not a distance at a specific time, it is a distance along time.

It is important to the definition of both comoving distance and proper distance in the cosmological sense (as opposed to proper length in special relativity) that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or spacelike geodesic between the two points, observers situated between the two points would have different cosmological ages when the geodesic path crossed their own world lines, so in calculating the distance along this geodesic one would not be correctly measuring comoving distance or cosmological proper distance. Comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a universe empty of mass, where both sorts of distance can be measured. When the density of mass in the FLRW metric is set to zero (an empty 'Milne universe'), then the cosmological coordinate system used to write this metric becomes a non-inertial coordinate system in the flat Minkowski spacetime of special relativity, one where surfaces of constant time-coordinate appear as hyperbolas when drawn in a Minkowski diagram from the perspective of an inertial frame of reference.[4] In this case, for two events which are simultaneous according the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events,(Wright) which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a geodesic in flat Minkowski spacetime), or the coordinate distance between the events in the inertial frame where they are simultaneous.

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Anyway, it is very close to what I ment.

-----------

(edit)

your comment about unscaled time is basically what I ment, but it must be trickier. If one could imagine contracted time, then it could equalize expanded space and produce a result where the photon does not change state of motion at all: it could follow a straight geodesic. IIRC I have seen somewhere such a spacetime diagram.

Am I wrong?

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Yes, I fully agree, the light beam will get redshifted towards infinity, far beyond possibility of detection.

However on a side note, considering that the beam constitutes of discrete photons, we can conclude that the last photon in the beam must come from a location slightly on our side of the horizon. Just like a light source falling into a black hole, there will be a finite amount of photons emitted in our direction and the last one is from slightly above the horizon. I think that in both cases that last photon will reach us in a finite time.

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tar, I see that you omitted to give me an answer AGAIN, how come you don't give me a straightforward answer?

Yes, we can currently see objects receding from us faster than light, I have not said anything that is contradicting this. Your point is?

Distant galaxies are like distant lamps, when stars in them dies they stop shining and then eventually we can't see them any more.

So you agree that space between us and the Mars probe will NOT contain photons for the rest of eternity, from prior the event when the lamp got dark?

Spyman,

Well here are my questions. And yes, I want to learn what we know, but I would like it to make sense, and fit together, from outside in, and inside out.

The photons from my lamp, on for just a few hours, will not forever be in the space between here and Mars, but will they, or will they not, forever be somewhere in the universe, traveling outward from my house, in an ever increasing half shell, in the direction my house was facing during the event of it being on?

Is the event of a galaxy currently exceeding C recessional speed, not a different consideration, from the information we currently are receiving from that galaxy?

If we will never witness the event, but we currently see the galaxy, then the information we will recieve in the future, from that galaxy, must be events that occurred prior the event. Since a finite amount of events occurred prior the event, and after the events we currently are witnessing, should we not expect to see them all, eventually, given an infinite amount of time?

Photons seen now will be of shorter wavelengths, than those we wil see (or fail to notice due to lack of huge telescopes and patience), at the end of time, which will be of very long wavelengths. Why is it not mandatory, that these wavelengths be proportionally, though increasingly lentheningly distributed in the spacetime between here and now, and the C exceeding event?

Is each successive photon, we witness, in the future, from this galaxy, emitted way prior to the exceeding C event, not required to be just that slight bit more time dilated, that is, it will take a longer and longer time, to witness, a second worth of events, happening at that galaxy's location (prior the exceeding C event).

Should these wavelengths eventually grow to the length of the Milky Way, at that time, it will take us 400 thousand years to witness one nanosecond of events that occurred at that galaxy, in the years just prior the exceed C event. Given the wave/particle duality of a photon, it raises the question of whether we would be able to collect the photon at the beginning of the 400 thousand years, the end, or at any time, inbetween. That is, what is the nature of a photon, stretched out over 400 thousand ly. If the wave function would collapse, upon reception of the photon, would that require a physical nullification of the electrical and magnetic fields generated by the photon, instantaneously, over a 400thousand ly distance? Or would the nullification itself travel at C, as well?

Are the photons, that passed us by, from that galaxy, yesterday, still existant, in our galaxy, somewhere behind us, (figuring we are facing the distant galaxy)?

These questions raise a logical question in my mind, of what you might mean, by us recieving the "last" photon, from that galaxy, at which time, there will be no more available, or on their way.

Zapatos,

It is unfortunate that you have labeled me, forever, as a speculation thread ignoramous. It saddens me. I don't feel, from this end, like I earned the label.

Regards, TAR2

Edited by tar
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I have been wanting to make a post in this thread regarding the ant on a rubber rope problem, but I have been extremely busy and have just now found the time.

No. If for every inch of road I traverse, the road grows by two inches, I will never reach my wife. It is that simple. The photons that were emitted prior to the recessional speed exceeding c will reach us. The photons that are emitted after the recessional speed exceeds c will not reach us. After the last photon that was emitted prior to the recessional speed exceeding c reaches us, no other photons from that galaxy will reach us. At that point the galaxy will be lost to us.

Actually, if the road is expanding in accordance to the expansion defined in the "ant on a rubber rope" problem, you will eventually reach your wife. However, you may be dead before you get there due to the amount of time it takes to complete the journey.

This [ant on a rubber rope] problem has a bearing on the question of whether light from distant galaxies can ever reach us if the universe is expanding.[2] If the universe is expanding uniformly, this means that galaxies that are far enough away from us will have an apparent relative motion greater than the speed of light. This does not violate the relativistic constraint of not travelling faster than the speed of light, because the galaxy is not "travelling" as such—it is the space between us and the galaxy which is expanding and making new distance. The question is whether light leaving such a distant galaxy can ever reach us, given that the galaxy appears to be receding at a speed greater than the speed of light.

"Two views of an isometric embedding of part of the visible universe over most of its history, showing how a light ray (red line) can travel an effective distance of 28 billion light years (orange line) in just 13 billion years of cosmological time. Click the images to zoom. (Wikipedia)"

The expansion of space is often illustrated with conceptual models which show only the size of space at a particular time, leaving the dimension of time implicit.

In the "ant on a rubber rope model" one imagines an ant (idealized as pointlike) crawling at a constant speed on a perfectly elastic rope which is constantly stretching. If we stretch the rope in accordance with the ΛCDM scale factor and think of the ant's speed as the speed of light, then this analogy is numerically accurate—the ant's position over time will match the path of the red line on the embedding diagram above.

Now lets move on to the required mathematics for what is being discussed here in order to demonstrate that, according to expanding space equivalent to the "ant on a rubber rope" expansion, photons that are emitted after the recessional speed exceeds $c$ can reach us given enough time. However, this makes no guarantee that the galaxy emitting the photons would actually still be emitting photons, or that the photons will not have redshifted to an undetectable frequency. One can however use the mathematics to show that the distance is traversable.

The ant on a rubber rope

Wikipedia actually provides a decent description of problem.

Ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary,[1][2][3] but a typical form is as follows:

An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). At the same time, the rope starts to stretch by 1 km per second (so that after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc). Will the ant ever reach the end of the rope? At first consideration it seems that the ant will never reach the end of the rope, but in fact it does (although in the form stated above the time taken is colossal). In fact, whatever the length of the rope and the relative speeds of the ant and the stretching, providing the ant's speed and the stretching remain steady the ant will always be able to reach the end given sufficient time.

I modified the variable names used in Wikipedia to make the problem easier to read.

Consider a thin and infinitely stretchable rubber rope held taut along the $x$-axis with a starting-point marked at $x_0$ and a target-point marked at $x_1$.

At time $t=0$ the rope starts to stretch uniformly and smoothly in such a way that the starting-point $x_0$ remains stationary at $x=0$ while the target-point $x_1$ moves away from the starting-point with constant speed $v_r > 0$.

A small ant leaves the starting-point at time $t=0$ and walks steadily and smoothly along the rope towards the target-point at a constant speed $v_a > 0$ relative to the point on the rope where the ant is at each moment.

Will the ant reach the target-point?

According to the definition for the problem, the equation for position of the ant on a rubber rope is a first order linear differential equation:

A key observation is that the speed of the ant at a given time $t>0$ is its speed relative to the rope, i.e. $v_a$, plus the speed of the rope at the point where the ant is. The target-point moves with speed $v_r$, so at time $t$ it is at $x=v_r \, t + x_1$. Other points along the rope move with proportional speed, so at time $t$ the point on the rope at $x$ is moving with speed $(v_r \, x) / (v_r \, t + x_1)$. So if we write the position of the ant at time $t$ as $x(t)$, and the speed of the ant at time $t$ as $x'(t)$, we can write:

$x'(t)=v_a + \frac{v_r \, x(t)}{v_r \, t + x_1}$

We can solve this nonhomogeneous linear differential equation using Integrating Factors. We begin by rewriting the differential equation in the form:

$\frac{dx}{dt} = \alpha (t) \, x + \beta (t)$

where

$\alpha (t) = \frac{v_r}{v_r \, t + x_1}$

$\beta (t) = v_a$

and the integrating factor is

$\mu (t) \, = \, e^{\left(-\int_{} \alpha (t) \, dt\right)} \, = \, \frac{1}{v_r \, t + x_1}$

After working the method, we arrive at:

$x(t) \ = \ \frac{1}{\mu (t)} \int_{} \, \beta (t) \mu (t) \, dt \ = \ \left(v_r \, t + x_1\right) \left(\frac{v_a}{v_r} \, \text{ln} \left(v_r \, t + x_1\right)+C\right)$

Now we have to find the value for $C$ that satisfies our initial condition $x(0) = 0$:

$x(0) \ = \ x_1 \left(\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)\right)+x_1 \, C \ = \ 0$

$x_1 \, C \ = \ - x_1 \left(\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)\right)$

$C \ = \ -\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)$

After substituting $C$ back into our general solution and simplifying the result, we arrive at the following equation for the position of the ant at time $t$:

$x(t) \ = \ \left(v_r \, t + x_1\right) \frac{v_a}{v_r} \, \text{ln} \left(\frac{v_r \, t \, + \, x_1}{x_1}\right)$

However, it is extremely difficult using the above equation to determine the value of $t$ that will tell us how much time it took the ant to complete the journey. If you read the ant on a rubber rope problem in Wikipedia, they offer a different method that will allow us to solve for $t$.

A much simpler approach considers the ant's position as a proportion of the distance from the starting-point to the target-point.[3] Consider coordinates $\psi$ measured along the rope with the starting-point at $\psi = 0$ and the target-point at $\psi = 1$. In these coordinates, all points on the rope remain at a fixed position (in terms of $\psi$) as the rope stretches. At time $t \ge 0$, a point at $x = x_i$ is at $\psi = x_i / (v_r\, t + x_1)$, and a speed $v_a$ relative to the rope in terms of $x$ is equivalent to a speed $v_a/ (v_r\, t + x_1)$ in terms of $\psi$. So if we write the position of the ant in terms of $\psi$ at time $t$ as $\phi (t)$, and the speed of the ant in terms of $\psi$ at time $t$ as $\phi '(t)$, we can write:

$\phi '(t) = \frac{v_a}{v_r\, t + x1}$

$\therefore \phi (t) = \int_{} \frac{v_a}{v_r\, t + x1} \, dt = \frac{v_a}{v_r} \, \text{ln} \left(v_r \, t + x_1\right)+k$ where $k$ is a constant of integration.

Now, $\phi (0) = 0$ which gives

$k = - \frac{v_a}{v_r} \, \text{ln} \left(x_1\right)$, so $\phi (t) = \frac{v_a}{v_r}\, \text{ln} \left(\frac{v_r \, t + x_1}{x_1}\right)$.

If the ant reaches the target-point (which is at $\psi = 1$) at time $t$, we must have $\phi (t) = 1$ which gives us:

$\frac{v_a}{v_r}\, \text{ln} \left(\frac{v_r \, t + x_1}{x_1}\right) = 1$

$\therefore t = \frac{x_1}{v_r} \left(e^{v_r/v_a}-1\right)$

As this gives a finite value $t$ for all finite $x_1, v_r > 0, v_a > 0$, this means that, given sufficient time, the ant will complete the journey to the target-point. This formula can be used to find out how much time is required.

For the problem as originally stated, $x_1 = 1 \text{km}$, $v_r = 1 \text{km}/\text{s}$, and $v_a = 1 \text{cm} / \text{s}$, which gives $t = \left(e^{100000}-1\right) \text{s} \approx 2.8 \times 10^{43429} \, \text{s}$. This is a truly vast timespan, vast even in comparison to the estimated age of the universe, and the length of the rope after such a time is similarly huge, so it is only in a mathematical sense that the ant can ever reach the end of this particular rope.

Just for fun, if we substitute the above equation for the total time into our function $x(t)$ for the position of the ant, we can determine the total distance traveled by the ant.

$d = x\left(\frac{x_1}{v_r} \left(e^{v_r / v_a}-1\right)\right) = x_1 \, e^{v_r / v_a}$

$\therefore d = (1\, \text{km})\, e^{100000} \approx 2.8 \times 10^{43429} \, \text{km}$

Now that we have shown that the distance is traversable in a finite amount of time, we can argue that photons or anything else can reach us from any point in space. However, we would have to use the above equation to calculate the total amount of time it would take for the journey. If we get a time estimate close to that of the ant, then we can be fairly sure that the probability of a photon reaching us is practically nonexistent (much less for anything else) because the Earth will not be around that long and there is a good chance that neither will we : ) In another light, the math does suggests that it is possible for photons to reach us from galaxies that are receding away from Earth at speeds greater than or equal to $c$. Whether or not we can detect them is a different matter altogether.

Edited by Daedalus
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In another light, the math does suggests that it is possible for photons to reach us from galaxies that are receding away from Earth at speeds greater than or equal to $c$. Whether or not we can detect them is a different matter altogether.

Unfortunately when the calculus began my ability to follow ended. I am unsure how to interpret what you are saying given other sources of information that seem to indicate that, assuming current understanding is true, the photons will never reach us. Does your analysis suggest that there is not agreement over whether photons will reach us? Or that under certain conditions or distances the photons can reach us? Or that other sources are being misinterpreted? I am unsure if your example takes into consideration the acceleration of the expansion of space.

The following seems to suggest that any photons emitted from a distance of greater than 16 billion light years will not reach us. Can you help me reconcile what you said and what I read below?

Because the Hubble parameter is decreasing with time, there can actually be cases where a galaxy that is receding from us faster than light does manage to emit a signal which reaches us eventually.[19][20] However, because the expansion of the universe is accelerating, it is projected that most galaxies will eventually cross a type of cosmological event horizon where any light they emit past that point will never be able to reach us at any time in the infinite future,[21] because the light never reaches a point where its "peculiar velocity" towards us exceeds the expansion velocity away from us (these two notions of velocity are also discussed in Comoving distance#Uses of the proper distance). The current distance to this cosmological event horizon is about 16 billion light-years, meaning that a signal from an event happening at present would eventually be able to reach us in the future if the event was less than 16 billion light-years away, but the signal would never reach us if the event was more than 16 billion light-years away.[20]

http://en.wikipedia.org/wiki/Faster-than-light

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Unfortunately when the calculus began my ability to follow ended. I am unsure how to interpret what you are saying given other sources of information that seem to indicate that, assuming current understanding is true, the photons will never reach us. Does your analysis suggest that there is not agreement over whether photons will reach us? Or that under certain conditions or distances the photons can reach us? Or that other sources are being misinterpreted? I am unsure if your example takes into consideration the acceleration of the expansion of space.

The following seems to suggest that any photons emitted from a distance of greater than 16 billion light years will not reach us. Can you help me reconcile what you said and what I read below?

http://en.wikipedia....ster-than-light

No I did not include the acceleration of the expansion of space, which might make it impossible for photons to reach us. However, that is just one interpretation to what accelerated expansion is. If we consider the universe as shaped as a sphere, then the ant on a rope problem still applies to moving across the sphere as it expands. This would result in space expanding at a constant rate, while gravitational bodies are pulling on each other across the sphere (this could apply to a flat universe too). This should impart different velocities onto galaxies which can be interpreted much like the ant moving away from us such that a galaxy would have a velocity $v_a$ with respect to the expansion of space. This will make the galaxy appear to be accelerating away from us while photons that are emitted would only be subjected to the constant expansion of space, and hence could reach us. However, that is just combining certain views on the subject and is not what mainstream science supports, which would mean that such an event horizon as you speak of is possible. I just wanted to use some simple calculus to demonstrate the vast amount of time it would take for the ant to reach the end of the rope : )

Edited by Daedalus
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No I did not include the acceleration of the expansion of space, which might make it impossible for photons to reach us. However, that is just one interpretation to what accelerated expansion is. If we consider the universe as shaped as a sphere, then the ant on a rope problem still applies to moving across the sphere as it expands. This would result in space expanding at a constant rate, while gravitational bodies are pulling on each other across the sphere (this could apply to a flat universe too). This should impart different velocities onto galaxies which can be interpreted much like the ant moving away from us such that a galaxy would have a velocity $v_a$ with respect to the expansion of space. This will make the galaxy appear to be accelerating away from us while photons that are emitted would only be subjected to the constant expansion of space, and hence could reach us. However, that is just combining certain views on the subject and is not what mainstream science supports, which would mean that such an event horizon as you speak of is possible. I just wanted to use some simple calculus to demonstrate the vast amount of time it would take for the ant to reach the end of the rope : )

Got it, thanks. By the way, your signature makes me laugh every time I see it.

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Got it, thanks. By the way, your signature makes me laugh every time I see it.

Yeah I like it too. I got that quote from one of my engineering buddies when I worked at Casino Systems Inc. programming slot machines.

Edited by Daedalus
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Yes, I fully agree, the light beam will get redshifted towards infinity, far beyond possibility of detection.

However on a side note, considering that the beam constitutes of discrete photons, we can conclude that the last photon in the beam must come from a location slightly on our side of the horizon. Just like a light source falling into a black hole, there will be a finite amount of photons emitted in our direction and the last one is from slightly above the horizon. I think that in both cases that last photon will reach us in a finite time.

Yeah, I think the black hole is a perfect analogy. Krauss had a good reading on it,

This means that, the longer we wait. the less we will be able to see. Galaxies that we can now see will one day in the future be receding away from us at faster-than-light speed, which means that they will become invisible to us. The light they emit will not be able to make progress against the expansion of space, and it will never again reach us. These galaxies will have disappeared from our horizon.

The way this works is a little different than you might imagine. The galaxies do not suddenly disappear or twinkle out of existence in the night sky. Rather, as their recession speed approaches the speed of light, the light from these objects gets ever more redshifted. Eventually, all their visible light moves to infrared, microwave, radio wave, and so on, until the wavelength of light they emit ends up becoming larger than the size of the visible universe, at which point they become officially invisible.

I think it might be difficult to say that the last photon reaches us in finite time, just because we could never be sure, no matter how long we waited, that no photon remains.

Am I wrong?

I don't know specifically to what you are referring. You did confuse distance in spacetime with velocity, but that isn't a discussion we should have here.

I have been wanting to make a post in this thread regarding the ant on a rubber rope problem, but I have been extremely busy and have just now found the time.

Actually, if the road is expanding in accordance to the expansion defined in the "ant on a rubber rope" problem, you will eventually reach your wife. However, you may be dead before you get there due to the amount of time it takes to complete the journey.

"Two views of an isometric embedding of part of the visible universe over most of its history, showing how a light ray (red line) can travel an effective distance of 28 billion light years (orange line) in just 13 billion years of cosmological time. Click the images to zoom. (Wikipedia)"

Now lets move on to the required mathematics for what is being discussed here in order to demonstrate that, according to expanding space equivalent to the "ant on a rubber rope" expansion, photons that are emitted after the recessional speed exceeds $c$ can reach us given enough time. However, this makes no guarantee that the galaxy emitting the photons would actually still be emitting photons, or that the photons will not have redshifted to an undetectable frequency. One can however use the mathematics to show that the distance is traversable.

The ant on a rubber rope

Wikipedia actually provides a decent description of problem.

I modified the variable names used in Wikipedia to make the problem easier to read.

According to the definition for the problem, the equation for position of the ant on a rubber rope is a first order linear differential equation:

We can solve this nonhomogeneous linear differential equation using Integrating Factors. We begin by rewriting the differential equation in the form:

$\frac{dx}{dt} = \alpha (t) \, x + \beta (t)$

where

$\alpha (t) = \frac{v_r}{v_r \, t + x_1}$

$\beta (t) = v_a$

and the integrating factor is

$\mu (t) \, = \, e^{\left(-\int_{} \alpha (t) \, dt\right)} \, = \, \frac{1}{v_r \, t + x_1}$

After working the method, we arrive at:

$x(t) \ = \ \frac{1}{\mu (t)} \int_{} \, \beta (t) \mu (t) \, dt \ = \ \left(v_r \, t + x_1\right) \left(\frac{v_a}{v_r} \, \text{ln} \left(v_r \, t + x_1\right)+C\right)$

Now we have to find the value for $C$ that satisfies our initial condition $x(0) = 0$:

$x(0) \ = \ x_1 \left(\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)\right)+x_1 \, C \ = \ 0$

$x_1 \, C \ = \ - x_1 \left(\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)\right)$

$C \ = \ -\frac{v_a}{v_r} \, \text{ln} \left(x_1\right)$

After substituting $C$ back into our general solution and simplifying the result, we arrive at the following equation for the position of the ant at time $t$:

$x(t) \ = \ \left(v_r \, t + x_1\right) \frac{v_a}{v_r} \, \text{ln} \left(\frac{v_r \, t \, + \, x_1}{x_1}\right)$

However, it is extremely difficult using the above equation to determine the value of $t$ that will tell us how much time it took the ant to complete the journey. If you read the ant on a rubber rope problem in Wikipedia, they offer a different method that will allow us to solve for $t$.

Just for fun, if we substitute the above equation for the total time into our function $x(t)$ for the position of the ant, we can determine the total distance traveled by the ant.

$d = x\left(\frac{x_1}{v_r} \left(e^{v_r / v_a}-1\right)\right) = x_1 \, e^{v_r / v_a}$

$\therefore d = (1\, \text{km})\, e^{100000} \approx 2.8 \times 10^{43429} \, \text{km}$

Now that we have shown that the distance is traversable in a finite amount of time, we can argue that photons or anything else can reach us from any point in space. However, we would have to use the above equation to calculate the total amount of time it would take for the journey. If we get a time estimate close to that of the ant, then we can be fairly sure that the probability of a photon reaching us is practically nonexistent (much less for anything else) because the Earth will not be around that long and there is a good chance that neither will we : ) In another light, the math does suggests that it is possible for photons to reach us from galaxies that are receding away from Earth at speeds greater than or equal to $c$. Whether or not we can detect them is a different matter altogether.

my bold.

That analogy works only for a freely coasting universe (one without gravity) where the hubble parameter gets steadily smaller and the scale factor evolves linearly. It does, however, effectively prove that there is no cosmic event horizon at the infinite future of such a universe. Cool

Edited by Iggy

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