What happens to light when it reaches the edge of the universe?

  • Context: Undergrad 
  • Thread starter Thread starter fbsthreads
  • Start date Start date
  • Tags Tags
    Edge Light Universe
Click For Summary
SUMMARY

The discussion centers on the behavior of light at the edge of the universe, concluding that the universe does not possess an edge in any credible cosmological model. Participants assert that light cannot reach an "edge" because the universe is either closed, open, or flat, and its three-dimensional space is curved back on itself. The initial expansion of the universe during the Big Bang occurred at a speed exceeding that of light, leading to the understanding that light emitted from the Big Bang could not reach a hypothetical edge. Furthermore, the expansion of the universe is accelerating, which means that any edge would be receding faster than light, making it unreachable.

PREREQUISITES
  • Understanding of cosmological models (closed, open, flat)
  • Familiarity with the concept of light speed and recession velocity
  • Knowledge of the Big Bang theory and its implications
  • Basic grasp of the curvature of space-time
NEXT STEPS
  • Research the implications of the Big Bang on the observable universe
  • Study the curvature of space-time and its effects on light propagation
  • Explore the concept of recession velocity in cosmology
  • Learn about Hubble's Law and its relation to the expansion of the universe
USEFUL FOR

Astronomers, cosmologists, physics students, and anyone interested in the fundamental nature of the universe and the behavior of light in cosmological contexts.

  • #31
Space curvature is nice, but to really understand how things work you need to know what's going on at the quantum level, which is anyones guess.

meemoe_uk's law is a simple idea which deals with the edges of space with QM.
- every traveling quantum must have a start and end point
so no light can be sent off the edge on an infinite journey.

Quasars are the most distant objects we can detect, so they are our current best bet for 'objects on the edge'. By meemoe_uk's law, all their radiation must be transmitted back into the universe. This of course makes them seem to radiate much more intensely then we'd expect if we thought their radiation was free to fly off in any direction. Also this may explain why the universe appears to expand. Galaxys and quasars effectively have radiation thrusters which accelarate them into the least dense space, in the case of quasars this is off the edge of the universe! lol
 
Astronomy news on Phys.org
  • #32
Flatland said:
What do you not understand? The universe has no edge, regardless if it's closed or not.
Some people tend to overlook the fact that the very premise of their argument is flawed. Be patient.
 
  • #33
there is no 'edge' of the universe. only spatially bound objects have edges and the universe in not an 'object'.. it is all of spacetime. to think of it as an object requires the existence of non-object regions [i.e. spacetime continuums that originated independent of our universe]. it also requires them to interact [have observable effects] with this universe to be of any theoretical consequence. we have not yet observed any phenomenon in this universe that require the existence of other universes to be explained.

there is however a theoretical limit to the observable universe. hubbles constant predicts objects at a distance around 15 billion light years would be receding at the speed of light. we could therefore say the observable universe is a sphere 30 billion light years in diameter. of course, you would arrive at this same conclusion no matter where you were in the universe. this imaginary edge only exists in your local reference frame. under general general relativity, no matter where you are or what speed you travel, the 'edge' of the universe will always appear to be 15 billion light years [according to whatever clock you happen to have along] away from your current position. the long and the short of it is you can't get there from here, or from anywhere else for that matter.
 
  • #34
Chronos said:
...a theoretical limit to the observable universe. hubbles constant predicts objects at a distance around 15 billion light years would be receding at the speed of light. we could therefore say the observable universe is a sphere 30 billion light years in diameter...

Chronos you might enjoy reading "Expanding Confusion" by Tamara Davis. I will put a link, in case you want to.
http://arxiv.org./abs/astro-ph/0310808
It addresses some misconceptions about the expansion of the universe and the dimensions of what is observable.

You are right that there is a bound to the observable universe. But I don't think it is 15 billion LY.
Hubble's parameter has not been constant over time.
One cannot go by the present value of it.

Indeed, it turns out that
the light reaching us from many of the galaxies we see now was emitted at a time when that galaxy was receding from us at faster than the speed of light---and yet the light managed to reach us, curiously enough.
The Davis and Lineweaver article explains how this can be.

For example, galaxies are routinely observed at redshifts greater than 3.

In fact, one was recently detected to have z = 10 (by Roser Pello's group).

A galaxy observed at z = 3 must have been receding from us, at the time it emitted the light we are now receiving from it, at a speed greater than light.

using the standard "Sky and Telescope" calculator at S. Morgan's website
http://www.earth.uni.edu/~morgan/ajjar/Cosmology/cosmos.html

you can see that a galaxy observed with z = 3 would have emitted the light
when it was receding at 1.6 times c
and it would now be at a distance of 21 billion LY
and currently receding at a speed of 1.5 times c.

If you want to use the online calculator, put in 0.73 for dark energy (lambda)
and 0.27 for matter (omega), and z = 3 or whatever you want the redshift to be.

BTW it looks like you, flatland and Thor are all saying no edge to the U
and I can only say amen to that! I'm not certain myself there is even a back fence in the time direction :smile:, but for sure nobody I know thinks there's a spatial boundary
 
Last edited by a moderator:
  • #35
i've always wondered about "expanding" space
are the objects (protons, photons, whatever) embedded? do they expand along with space? or do little distance markers move away from each other? Does matter expand along with space? isn't it part of spacetime or is it totally separate? Does space "expanding" simply mean that MORE space got put in between two objects? If 2 objects are attracted to each other gravitationally (sitting in little spacetime wells) then wouldn't the "expanding" space between them affect their gravitational pull on one another?
 
  • #36
marcus: 15 billion light years is the average of what is currently measured as the distance the hubble's constant approaches the speed of light [between 10 and 20 billion light years]. this approximation is supported by stellar evolution models that predict it would take no less than 10 billion years [by local inertial reference frame clocks] and no more than 20 billion years to result in globular clusters composed mainly of white dwarf stars. the fact these clusters are only observed at the extreme fringes of galaxies strongly suggests they are the most ancient gravitationally condensed collections of matter in any galaxy.

regarding recessional velocities. you are right. hubble's constant is not constant over time. the fact the red shift increase with distance, and therefore over time, insists the universe expanded more rapidly in the past than it does now. that is no surprise. the early universe had to expand with more force than the attractive force of gravitation. it would otherwise have collapsed upon itself before we had the opportunity to observe and ask the question 'what happened?'. we know gravity is attractive, hence, the existence of a repulsive [anti-gravity] force is virtually assured. the main question is which force will prevail? i would guess neither. the universe, as we perceive, will eventually reach a state of equilibrium. when the matter density exactly balances the energy density, the universe will acquire a state of perfect equilibrium. this implies other consequences, but, we don't have to deal with that for at least another 6 billion years.
 
  • #37
shrumeo: i think the real problem here [and with quantum gravity] is that gravity ceases to be a 'player' at the subatomic level. number one, the other 3 forces [strong, weak, and electromagnetic] are vastly more powerful than gravity at short distances. i think gravity is nonexistent at such short distances. quantum field theory requires forces to act at integer distances. from what i have seen, the wave length of gravity must be a recipocal of c^2. this is not consistent with the theoretical distance between an electron and proton in atomic hydrogen.
 
  • #38
Doesn't a flat and infinite universe imply there's a center ?

Assume the universe is flat and infinite.
Is there an infinite amount of matter covering this infinite universe ? Not according to big bang, right ?
So is this limited amount of matter spread across the entire universe ? No, cause then there'd be infinite distance between these "pieces" of matter, right ?

The only alternative is that there is a limited amount of matter localised somewhere in the universe. So I guess it's a matter of definition, right ? Do we define "the center" as the center of all space (i.e. there is none), or do we define it as the gravity center of all matter/energy ? Or perhaps the point where the sum of all coordinates of matter equals zero ?
.. or what ?

If there is to be no center (of any kind) to the universe, isn't a closed borderless one the only option ?
 
  • #39
Hydr0matic said:
Doesn't a flat and infinite universe imply there's a center ?

Assume the universe is flat and infinite.
Is there an infinite amount of matter covering this infinite universe ? Not according to big bang, right ?
So is this limited amount of matter spread across the entire universe ? No, cause then there'd be infinite distance between these "pieces" of matter, right ?

The only alternative is that there is a limited amount of matter localised somewhere in the universe. So I guess it's a matter of definition, right ? Do we define "the center" as the center of all space (i.e. there is none), or do we define it as the gravity center of all matter/energy ? Or perhaps the point where the sum of all coordinates of matter equals zero ?
.. or what ?

If there is to be no center (of any kind) to the universe, isn't a closed borderless one the only option ?

No a falt nfinite unievrse doe not imply a centre, the big bang means that if the unievrse is infinite in space it is infinite in energy.
 
  • #40
hmm.. ok ?
So how did this energy get from being localised in a single point to being distributed in an infinite space with finite density ? Did this happen in an inifinitesimal timeframe ? Was there ever a moment where the energy was somewhere in between the singularity and the infinite distribution ?

Feels like I've got some reading to do ...
 
  • #41
Hydr0matic said:
hmm.. ok ?
So how did this energy get from being localised in a single point to being distributed in an infinite space ...

Hydr0 there is actually a linguistic (not physics) problem here
that confuses people again and again

In ordinary non-technical English, "singularity" means peculiarity, or oddness, or abnormality

(it has no connotation of happening at a single point!)

a theoretical model can experience a singularity at an infinite set of points

it simply means that there is a boundary or limit to its applicability

it means you cannot push the model past a certain limit, because (say) it blows up and fails to compute, or it computes infinities or meaningless numbers----then there is a singularity

this limit could be pictured as a 3D hypersurface bounding a 4D region where the model works well----a little bit like the 2D crust on a 3D loaf of bread is a boundary of the bread (no, I cannot think of a good image, all the images seem to make it more confusing)

But because "singularity" sounds like the word "single"
it suggests to many people that there is a single point
where the singularity happens!
so they imagine a single isolated point
this is a wrong image and leads to much confusion

the BB singularity may have been confined in a small point-like region, but this is not the prevailing view. It could also have occurred at every point of an infinite 3D hypersurface---this is nowadays a very common view
 
  • Like
Likes   Reactions: 1 person
  • #42
Ok, I see. So the difference between now and the beginning is simply a matter of energy density, where "the critical density" marks the point to which the theory is limited.

right ?
 
  • #43
not right yet

the "critical density" is a very nice density of about 0.83 joules per cubic kilometer

which is either exactly (or else very near) the actual density
of the universe right at this moment!


at the bigbang time the energy density was probably "off the chart"
infinities are usually a sign that the model is breaking down
and in the usual model
things like density and curvature go off to infinity as one approaches
the instant that expansion began
and so one really must admit that if one wants to approach
that instant and continue computing them one needs a new model!


perhaps you are wondering what is the "critical density"
it is the density the U would need to have (in the simplified Friedmann picture) in order to be exactly spatially flat, at this moment

too much mass/energy and she will be positively curved
too little and she will be negatively curved
and critical is just right
and since the U is observed to be indistinguishable from flat, spatially, it is usually assumed that the actual density, smoothed out to a uniform average level, would be equal to or very near the critical
 
  • #44
Ok :smile: .. "critical" sounded more dramatic so I assumed it was the point when the model brakes down.

thnx for clearing things up :smile:
 
  • #45
Chronos said:
shrumeo: i think the real problem here [and with quantum gravity] is that gravity ceases to be a 'player' at the subatomic level. number one, the other 3 forces [strong, weak, and electromagnetic] are vastly more powerful than gravity at short distances. i think gravity is nonexistent at such short distances. quantum field theory requires forces to act at integer distances. from what i have seen, the wave length of gravity must be a recipocal of c^2. this is not consistent with the theoretical distance between an electron and proton in atomic hydrogen.

hmm, maybe I'm missing something...(total non-physicist here trying to understand these things in plain english)

i didn't really mean to imply anything about gravity among subatomic particles. it was just a string of nonsequential questions. But still, let's say we are at the subatomic level and we are "looking" (forgive me Heisenberg) at an electron surrounding a proton. Now, space is expanding even at this scale, right? Now, are the electron and proton something that is not part of the fabric of spacetime? In other words is there more "vacuum" being added to the existing "vacuum" that spearates them? is the electromagnetic force constantly overcoming the expansion OR are they wrinkles in spacetime that also expand along with the vacuum?

On the other scale, say a planet and a moon. Is the space between them expanding to where gravity is constantly overcoming all the new space between them?

Also, there is amount of space between the atoms and molecules that make up the planet and moon (forgot tht scale). Is the electromagnetic force constantly overcoming the expansion of space between them?

If objects and matter remain intact (which they appear to do), do any equations that, say, deal with their, say, group velocity, for example, automatically have built in them, from experiment or derivation, a term that overcomes the expansion of space? :confused:
 
Last edited:
  • #46
http://www.space.com/scienceastronomy/mystery_monday_040524.html

Universe Measured: We're 156 Billion Light-years Wide!
By Robert Roy Britt
Senior Science Writer

...

Stretching reality

The universe is about 13.7 billion years old. Light reaching us from the earliest known galaxies has been travelling, therefore, for more than 13 billion years. So one might assume that the radius of the universe is 13.7 billion light-years and that the whole shebang is double that, or 27.4 billion light-years wide.

But the universe has been expanding ever since the beginning of time, when theorists believe it all sprang forth from an infinitely dense point in a Big Bang.

"All the distance covered by the light in the early universe gets increased by the expansion of the universe,"

The scientists studied the cosmic microwave background (CMB), radiation unleashed about 380,000 years after the Big Bang, when the universe had first expanded enough to cool and allow atoms to form...

...findings have shown "no sign that the universe is finite, but that doesn't prove that it is infinite."

The results do render impossible a "soccer ball" shape for the universe...



"If the universe was finite, and had a size of about 4 billion to 5 billion light-years, then light would be able to wrap around the universe, and with a big enough telescope we could view the Earth just after it solidified and when the first life formed," Cornish said. "Unfortunately, our results rule out this tantalizing possibility."


"The problem is that funny things happen in general relativity which appear to violate special relativity (nothing traveling faster than the speed of light and all that).

=====
might help-might not
 
  • #47
how did that guy earn a doctorate? i agree GR has potential flaws, but, not on the basis of such patently flawed arguments. you cannot look into a telescope and see the back of your head. that violates causality. i can play along with any kind of universe anyone predicts, but, i absolutely reject any model that violates causality. even GR forbids that. god may play dice, but a universe without causality is impossible to observe.
 
  • #48
Chronos said:
how did that guy earn a doctorate? i agree GR has potential flaws, but, not on the basis of such patently flawed arguments. you cannot look into a telescope and see the back of your head. that violates causality. i can play along with any kind of universe anyone predicts, but, i absolutely reject any model that violates causality. even GR forbids that. god may play dice, but a universe without causality is impossible to observe.

i think you are agreeing with the article(?)

:confused:
 
  • #49
According to the most accurate measurements of the position of "planet earth" in relation to the rest of the universe, we are in the Dead center. And so one could assume 2 things about the universe: 1) we are extremely lucky. or 2) the universe is closed. The reason being, if we do indeed live in a closed universe, then it would appear that we we're in the center. For example, Imagine yourself standing on planet Earth (pretty hard huh? XD) But without all of the mountains, valleys, trees, grass etc. Just you and a nice two dimensional surface. If you assumed that the Earth was flat (and if it was), then you would observe that you were in the very center (just as we observe). However, it is very unlikely that we are at the very center of the universe. Therefore, there is only one other possibility:We live in a closed universe. However, this is only hypothetical. And I don't think that we will know for sure until we are able to venture out into the universe and make more observations. So the answer to your question in my opinion is that Light propagating from stars and galaxies eventually comes back around to the point of origin.
 
  • #50
jcsd said:
The rate of expansion of the universe is measure by Hubble's constant, which is equal to vr/d, so you should be able to see that even if Hubbles constant is very small, as long as the distance between the two objects (d)is large enough the recession velocity (vr) will exceed c. In an infinite expanding universe there must be a distance where vr exceeds c.

I though the speed of light was the speed at which "Stuff" (Pardon my ignorance here) of 0 mass can travel, ie photons. So how can there be ANYTHING that travels faster than this?
 
  • #51
This is what lead me to post this, something Marcus wrote on a post somewhere, i thought it would take infinite energy to move objects of mass at c ?


(For example, galaxies are routinely observed at redshifts greater than 3.

In fact, one was recently detected to have z = 10 (by Roser Pello's group).

A galaxy observed at z = 3 must have been receding from us, at the time it emitted the light we are now receiving from it, at a speed greater than light).
 
  • #52
Disclaimer on all of this: I am a mathematician, not a physicist; this is all just a hobby for me and I have a lot to learn myself.

EMFsmith said:
I though the speed of light was the speed at which "Stuff" (Pardon my ignorance here) of 0 mass can travel, ie photons. So how can there be ANYTHING that travels faster than this?

There can't. According to special relativity, you are correct. Objects with 0 mass must move at exactly c and objects with more than 0 mass must have velocities less than c. However, this rule can be bent in general relativity. The objects themselves must have velocities less than c, but the distance between them can grow at a rate faster than c.

Imagine the universe is a typical piece of graph paper, complete with the little grid lines and everything. On this graph paper are two marbles that represent galaxies. To keep this simple, we will use regular Newtonian mechanics, but impose a speed limit on the marbles. So they can move around, but are limited to a maximum speed of, say, 1 grid box per second. You initially place one at (-1, 0) and one at (1, 0). The first one moves due left and the latter one moves due right. So they are moving apart, and it follows that the fastest the distance between them can grow is 2 boxes / sec.

Now imagine that after the first second, Superman comes by and quickly rips your graph paper cleanly in half, right down the x=0 line. He then separates the halves and pastes a whole new 20x20 sheet of graph paper in between the two halves, glues everything together, and speeds away back to comic book land. The distance between your marbles just increased by 22 boxes in 1 second, but each one has only moved 1 box relative to the box it was just in. This is how galaxies can recede from each other faster than the speed of light; locally they are not moving very fast (or even at all) but space is being inserted in between them.


shrumeo said:
i didn't really mean to imply anything about gravity among subatomic particles. it was just a string of nonsequential questions. But still, let's say we are at the subatomic level and we are "looking" (forgive me Heisenberg) at an electron surrounding a proton. Now, space is expanding even at this scale, right? Now, are the electron and proton something that is not part of the fabric of spacetime? In other words is there more "vacuum" being added to the existing "vacuum" that spearates them? is the electromagnetic force constantly overcoming the expansion OR are they wrinkles in spacetime that also expand along with the vacuum?

On the other scale, say a planet and a moon. Is the space between them expanding to where gravity is constantly overcoming all the new space between them?

Also, there is amount of space between the atoms and molecules that make up the planet and moon (forgot tht scale). Is the electromagnetic force constantly overcoming the expansion of space between them?

If objects and matter remain intact (which they appear to do), do any equations that, say, deal with their, say, group velocity, for example, automatically have built in them, from experiment or derivation, a term that overcomes the expansion of space? :confused:

I don't know much about QM so I can't really say what happens on the atomic/subatomic level. As others have noted QM may require integer distances or some other such sillyness that may cause the effects of gravity and expansion to be exactly zero at those scales (as opposed to just being really really small effects).

In most models, according to the Hubble data, the expansion "force" between two objects should be proportional to the amount of space between them. The effect is only really significant at intergalactic scales. For the earth-moon scale, it is pretty negligible and will not overcome gravity. As long as the expansion constant does not change, it will just sort of slightly decrease the strength of the Earth's gravity. The earth-moon system will not gradually grow any larger from this effect.
 
  • #53
marcus said:
Chronos you might enjoy reading "Expanding Confusion" by Tamara Davis. I will put a link, in case you want to.
http://arxiv.org./abs/astro-ph/0310808
It addresses some misconceptions about the expansion of the universe and the dimensions of what is observable.

Hi marcus.. I've read the first few pages of Tamara's report, but being a layman, I started getting bogged down in all the items being referenced. The major premise seems based on the graphs (Page 3, Figure 1). I can understand proper distance, comoving distance and time, and I think I understand conformal time (time as it conforms to the location of the observer?) but beyond that, the paper depends a lot on the reader comprehending a bunch of things.

I Googled ACDM and found it refers to the cold dark matter model (which raises other questions). I tried to understand the ACDM concordance model (0.3,0.7 etc), and got lost with the last part of "the event horizon is the distance light can travel from a given time t to t = ∞".

But in reading page 4, para 2, I got the idea that she was saying, the light we're seeing from superluminal galaxies is like an artifact that the expanding Hubble sphere has given us access to. Did I read that right? If I did, I still don't understand how light which is effectively receding from us (observer) can end up heading towards us when the Hubble sphere reaches that distance from us.

Maybe in simple math terms, it could be like this. Say the superluminal galaxy is moving away from us at c + 100kph. If we are moving in the direction of that superluminal galaxy at say 200kph, then that light is effectively traveling towards us at 100kph. It won't reach us as fast, but eventually it will reach us. But I get the idea I'm not on the right track.
 
  • #54
Shovel said:
locally they are not moving very fast (or even at all) but space is being inserted in between them.

Where does this inserted space come from? Does it relate to curvature?

I'm trying to imagine this in my mind. The big gaps between masses (the space) is where time moves slower (slower than time in the neighborhood of a mass). So... umm it's like space is being inserted (relative to our time). Is that it?
 
  • #55
GodAdoresU said:
According to the most accurate measurements of the position of "planet earth" in relation to the rest of the universe, we are in the Dead center.

We are in (or near) the centre of what we can observe. Light from beyond that bubble hasn't reached us yet. We are not in the centre of anything else, let alone the universe.
 
Last edited:
  • #56
narrator said:
Where does this inserted space come from? Does it relate to curvature?

I'm trying to imagine this in my mind. The big gaps between masses (the space) is where time moves slower (slower than time in the neighborhood of a mass). So... umm it's like space is being inserted (relative to our time). Is that it?

I think what shovel is trying to say (Or here's how I see it) is that the space is expanding, the universe is expanding in all directions from every point within it, so there's no "More" space, its just the amount we have has stretched so to speak.

The thing I don't get is the amounts of Redshift, why are some objects moving faster? Is it the further away they are the faster there moving? I know further things are Redshifted more because the universe has expanded more in the time its taken the light to reach us, I am just confused why some thing move faster.
 
  • #57
narrator said:
We are in (or near) the centre of what we can observe. Light from beyond that bubble hasn't reached us yet. We are not in the centre of anything else, let alone the universe.

Right. Doesn't that imply that the universe is closed, because whatever is on the other side of the sphere is out of view. Just like we can't see around the world.
 
  • #58
GodAdoresU said:
Right. Doesn't that imply that the universe is closed, because whatever is on the other side of the sphere is out of view. Just like we can't see around the world.

Only closed from view, not closed in actuality. Have you ever watched a distant jet in the sky? For some time the jet is silent, then eventually the sound of it reaches you. The jet is there all the time. You just haven't heard it yet. The same is true of light traveling from very distant space. The stars and galaxies are there, just like the jet, but the light (like the sound from the jet) hasn't reached us yet. Hence, using the word closed suggests the wrong conclusion. To use your analogy, the rest of the world is not closed, it's just out of view.

As a lay person myself, one thing I understand clearly is that we must make sure our concepts are unambiguously clear, as best as possible. The idea of a closed universe (no more stars/galaxies existing beyond some boundary), linked to the idea that we are in the center is suggestive of design, and while it would be nice to think that, the concept itself is incorrect.
 
  • #59
narrator said:
Only closed from view, not closed in actuality. Have you ever watched a distant jet in the sky? For some time the jet is silent, then eventually the sound of it reaches you. The jet is there all the time. You just haven't heard it yet. The same is true of light traveling from very distant space. The stars and galaxies are there, just like the jet, but the light (like the sound from the jet) hasn't reached us yet. Hence, using the word closed suggests the wrong conclusion. To use your analogy, the rest of the world is not closed, it's just out of view.

As a lay person myself, one thing I understand clearly is that we must make sure our concepts are unambiguously clear, as best as possible. The idea of a closed universe (no more stars/galaxies existing beyond some boundary), linked to the idea that we are in the center is suggestive of design, and while it would be nice to think that, the concept itself is incorrect.

The poster here had a point - although I don't think he was intentionally making it.

To take the jet anaology further - there are jets that exist that we will never see nor never hear. These stars/galaxies are just too distant and receeding>c. So in effect our OU is closed causally, anything outside our OU is currently causally disconnected from our current OU - at least this is my current understanding. :)
 
  • #60
Understand that the topology of the universe [closed, flat, open] mainly depends on the matter-energy content of the universe. The illusion we reside at the center of the observable universe is unrelated to its topology. I would argue it makes more sense to assert we reside at the temporal edge of the universe [i.e., the most ancient observable region of the universe]. No matter which direction we look from earth, we see the universe as it was in our past. The center of the universe appears to be 13.7 billion light years distant in every direction [apparent distance to the surface of last scattering]. This merely implies the universe has no center, not that it is closed, flat or open..
 

Similar threads

  • · Replies 17 ·
Replies
17
Views
3K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 49 ·
2
Replies
49
Views
8K
  • · Replies 17 ·
Replies
17
Views
2K
High School The M paradox
  • · Replies 4 ·
Replies
4
Views
526
  • · Replies 10 ·
Replies
10
Views
5K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 13 ·
Replies
13
Views
6K
  • · Replies 4 ·
Replies
4
Views
2K