Does the Universe Cast a Shadow?

[Rudy Toody]

Imagine a two-dimensional room whose walls form a simple closed figure, and that these walls are completely covered with mirrors. If the room were a convex body every point in it would be illuminable by a single lamp or candle placed anywhere in the room, with or without the mirrors. But suppose it is not convex. Is there a shape for the curved walls such that from some point in the room a light source would fail to illuminate the whole room even with the aid of mirrors? The answer is yes; and more surprisingly perhaps, we can exhibit a room (Figure) not fully illuminable from any point in it. The area marked green is bounded by a semi-ellipse and its major axis through foci p,p'. The blue region is another such semi-ellipse area, foci at q,q'. The other curves are tangent to the major axes at the foci.

Now it is a property of the ellipse that any light ray crossing the major axis between the foci is reflected along another path that crosses between the foci; likewise a ray crossing the major axis beyond one focus is reflected beyond the other focus. This is because a ray eminating from a focus is reflected to the other focus, the reason for the name. Now if R stands for red points and O for orange points, we see that any light ray starting from a red point can never reach a yellow, blue, or orange point, and one from an orange point can never reach a yellow, green, or red point. A ray starting from a green point misses all the orange points, from a blue point misses all the red points, and from a yellow point misses the orange and red points. Thus the whole region, bounded by smooth curves, is not illuminable from any of its points.

The Figure and text is from "Tomorrow's Math" by C. Stanley Ogilvy 2nd Edition 1972 pp.59-61 Oxford University Press.

My question is: Could a three-dimensional shape with these characteristics exist in space and perhaps account for dark matter and dark energy? In other words, does the universe cast a shadow?

 

[Anteaus]

My question is: Could a three-dimensional shape with these characteristics exist in space and perhaps account for dark matter and dark energy? In other words, does the universe cast a shadow?

Given that dark matter and dark energy are purely hypothetical and that your example leaves out the all important fourth dimension (time), I don't think there is enough data to suggest anything.

In addition, your treating light and matter as if they behave equally when in fact there are completely different. The truth is dark matter and dark energy can't be accounted for at all because there is currently nothing to account for. Some people believe they might exist, and that's about as far as it goes.

 

[Rudy Toody]

Given that dark matter and dark energy are purely hypothetical and that your example leaves out the all important fourth dimension (time), I don't think there is enough data to suggest anything.

In addition, your treating light and matter as if they behave equally when in fact there are completely different. The truth is dark matter and dark energy can't be accounted for at all because there is currently nothing to account for. Some people believe they might exist, and that's about as far as it goes.

I think I should have said that something like this shape could hide regular matter and energy. Because we cannot see it from our point, it would seem to be dark.

 

[Anteaus]

I think I should have said that something like this shape could hide regular matter and energy. Because we cannot see it from our point, it would seem to be dark.

I guess then my question would be, how can this shape hide it?

Shadow in and of itself is restricted to use in regards to light, and light is not transmitted from the walls of the universe. Any star in existance can prove that. Matter is not transmitted in the same way as light, therefore it can't be "blocked" in the same way as light, thus no shadow can exist in the first place.

I'm not telling you that it isn't possible that something could potentially hide matter and energy, but that your arguement for it is based on the assumption that matter behaves like light, which isn't true. Relativity theory is quite specific about that.

 

[Farmer]

You ask a question that is probably worth a Ph.D. thesis or at least a paper in physics or math.

You presented a fairly complicated geometrical demonstration of the "shadow" thing using first order/ray optics, and that is fascinating! I would argue that if you consider light a wave then in the enormous superposition of waves that composes the universe, intuitively, it would certainly possible to find regions where they interfere to zero. If we speak of waves then we can speak in generality, since both matter and "energy" can certainly be interpreted as waves. I'm not a mathematician, so I don't know if I'm even close to right, or how I would approach proving that. Since I am not a physicist either, in a naive model where the universe is a closed sphere with a dielectric wall (the "boundary of the universe," whatever that might be), then the infinite number of eigenmodes (number of photons, if you will) of the EM field are automatically configurations where there definitely are locations of no field.

But the structure of the field (I speak of the EM field since you are speaking of light) itself is determined by the geometry of the universe. I.e., the fact that certain fields are eigenfields is because Planet A is here, or Mass B is there, etc., when solving what essentially is a boundary value problem. Then, the location of "dark spots" would betray the configuration of the universe entirely.

Anteaus:

I'm not sure I understand what you're saying at all. By the de Broglie hypothesis, matter has wave like behavior. By Einstein's photo electric effect, light can be quantized into photons, hence the wave-particle duality. Light interacts with matter via scattering processes. What does relativity say on this subject that is of consequence?

 

[PsiStar]

I will have to think about your postulate quite a bit more.

Your reference to a 1972 text may be a clue ... in that "theories" can become obsolete, outdated, and sometimes rapidly. This is assuming that there was any consensus originally at the time the text was written. Forty years in theoretical physics is ancient history ... ages almost as fast as PCs.

One of my fav texts is "Mr Tompkins". The reprints of that little book are not necessarily about correcting English and mis-statements, but more to update the story lines with the current understanding and theories of Modern Physics. For what it is worth, it is cheap, entertaining, and humorous.

 

[silverpig]

Given that dark matter and dark energy are purely hypothetical and that your example leaves out the all important fourth dimension (time), I don't think there is enough data to suggest anything.

In addition, your treating light and matter as if they behave equally when in fact there are completely different. The truth is dark matter and dark energy can't be accounted for at all because there is currently nothing to account for. Some people believe they might exist, and that's about as far as it goes.

Uh, no.

 

[SMOGZINN]

I think the problem with this is that dark matter/energy is needed to explain not just the total mass of the universe, but the local effects of the shape and movement of galaxies. There would need to be a shadow zone around every (?) galaxy. If that were the case, I think we would notice large dark zones, and areas of reflection, when looking out from our own galaxy.

 

[Rudy Toody]

Your reference to a 1972 text may be a clue ... in that "theories" can become obsolete, outdated, and sometimes rapidly...

The shape is Ogilvy's and the theory is mine based on his explanation.

I think the problem with this is that dark matter/energy is needed to explain not just the total mass of the universe, but the local effects of the shape and movement of galaxies. There would need to be a shadow zone around every (?) galaxy. If that were the case, I think we would notice large dark zones, and areas of reflection, when looking out from our own galaxy.

If a galaxy was at point R and we were in blue, we would not see it at all. If something else was at point R'. it would affect that galaxy, but we would not know that either. If we were in green we would be aware of both objects, but we would not know about objects at the orange points.

Edit: If one R was a black hole and we were in blue, we would not see it. But, if we look at an object in green, we would notice that it is being affected by something we cannot see.

So, what we know depends on our point of view.

 

[wuliheron]

All the evidence collected to date shows the universe is about as flat as we can measure. If there is some larger shape it appears to be completely outside the visible universe and, therefore, completely speculative.

 

[Sunny129]

My question is: Could a three-dimensional shape with these characteristics exist in space and perhaps account for dark matter and dark energy? In other words, does the universe cast a shadow?

without a doubt the 3-dimensional analogue of this 2-dimensional space with these properties could exist. whether it accounts for dark matter/energy, and whether the universe casts a shadow, i don't know.

regarding the first part of the question, simply start by projecting the existing image into 3 dimensions. that is, rotate the existing 2D image about a vertical axis running through its center (an axis that is perpendicular to the major axes of both half-ellipses). see below:

if you can visualize this, you'll see that the major axes of the half-ellipses that separate Blue from Yellow & Green from Yellow actually become planes separating these zones when rotated about the vertical axis and projected into 3 dimensions. since every possible 2D image can be treated the same as the original 2D image (so long as it is rotated symmetrically around the vertical axis) - that is, since all possible 2D images projected about the vertical axis have the same properties as the original 2D image regarding the motion and reflection of light - this 2D image's 3-dimensional analogue will also have these properties.

you can double-check this by allowing light rays to originate from different points in different zones (red, green, yellow, blue, etc.), and in specific directions so that they do things like cross zone barriers both inside and outside the foci. you'll see that any ray originating in the yellow zone that crosses into either the blue or green zones must get reflected back into the yellow zone, thus never reaching the orange or red zones. the orientation of the light ray about the vertical axis does not matter, since everything around/about it is symmetrical. thus the path of any light ray will be confined to a plane, despite the fact that it may traversing space inside a 3-dimensional figure.

what the implications for dark matter, dark energy, or the universe in general are, i don't know.

 

[Rudy Toody]

Great explanation, Sunny129.

I imagine that if we created a huge circle whose circumference is made up of billions of half-ellipses and we rotated them to get the 3D version. we would have points that still could not see all the other points.

The circumference would be alternating blue and green half-ellipses with their corresponding orange and red points. The inner part would be the yellow.

If we had an even number of blues and greens, we could rotate about the axis for each same color pair. this would have a lot of overlaps. So, we could be in some shade of blue and see some shade of orange? Or not?

Edit: perhaps the red and oranges rings are Hopf Fibrations.

This sphere would also explain why we could look one direction for a long time and eventually see the back of our head. And, if we could look the opposite direction at the same instant, we would see our face. (This might also help explain entanglement---one of two opposites can be viewed depending on our point of view.)

Edit to answer 4D concern: In 4D the half-ellipses would not be boundaries, however the behavior would be the same. This is a 3D explanation of a 4D structure.

 

[Biftheunderstudy]

There are a couple of problems that I can pick out right off the bat. First, Dark Matter is not inferred by taking the total visible mass and comparing it to the total gravitating mass in the universe. Instead, its more of a local effect, if you can call galactic scales local.

The first piece of evidence is the rotation of galaxies, you would expect that if you could see all the mass that you could just use newton's laws to predict the distribution of rotational velocities against the radial distance. The resulting prediction should be a curve which peaks close to zero and falls off quickly with distance. Instead, we observe "flat" rotation curves which peak close to zero and remain roughly constant out to very far distances. This suggests that the bulk of the material in a galaxy is in fact in the halo or in the galactic suburbs...where we don't see much mass.

There are several other lines of evidence, such as gravitational lensing and x-ray emitting intra-cluster gas.

The effect you describe, sounds more like missing mass on a universal scale (i.e. hidden galaxies, clusters, gas etc.). Unless I'm mistaken and you mean that these spatial structures are higher dimensional spaces embedded on small scales in our universe -- much like the high dimensional spaces of string theory....then I have no idea what kind of effect that would have....

Now to address the Dark Energy part, missing mass is not what Dark Energy is.....rather, its like a pressure that is expanding the universe. Its effects are so insignificant on scales that we can observe, only at the largest scales does it have an effect. That said, it is a couple hundred orders of magnitude larger than QFT predicts....

 

[JackSpadesSI]

Right, in order to account for local (indirect) observation of dark matter the analog would have to be a higher-than-3 dimensional shape where the matter isn't merely tucked around a corner of the universe but actually tucked "behind" each atom we see.

 

[FTL_Diesel]

I think this is an intersting question, but the problem with thinking of DM as a portion of the universe that is "not illuminated" is that we *can* see it gravitionally. Whatever higher-dimensional sight line that the gravity-mediating bosons follow should also be available to photons.

So DM is still probably a WIMP with just an extremely low photon interaction cross-section.

 

[pandemonium]

Dark matter and universe shadowing are one and the same and are explained by the 4th dimension. Until we can truly understand how the upper dimension(s) provide us with such information we're just barely starting to grasp, it's "form" will remain algebraic; e.g. E=mc^2. Also, the shape will be inconsequential once we understand that the "plane" doesn't adhere to any perceivable form. At that point the shape can be whatever it wants to be; the results will be the same regardless.

 

[looper]

"Who's on first?"
"I don't know"
"Third base..."