Hyperventilating over hyperspheres

[Stegyre]

This is a solicitation for crunchy comments. If that is not to your taste, you may want to pass this thread by.

I am working on a hypersphere as a pocket universe for a sci-fi setting and seeking crunchy input to “get it right.”

Spoiler:  

Before anyone writes, “Don’t worry about it; hand wave it,” let me say that getting it right interests me. I am not seeking narrative shortcuts. I like this idea, and I want to implement it accurately.

Here are the basic parameters, so far. The radius of the hsphere is 6 units. (What a unit is remains to be decided; could be as large as a parsec; could be less than a light year.) It contains 29 stars distributed among 23 systems (four binary; one trinary).

First quandary: the volume of the hypersphere. (I suspect this is important for calculating the background “heat” of the universe, discussed further, below.) I get a formula that I’m actually able to handle from wiki. If I did it right, my 4-dimensional hsphere has a volume of 6,395 units^4.

As I think about it, though, I probably don’t care about the 4-dimensional “volume,” do I? That’s the space where we 3-dimensional beings cannot travel. Isn’t my 3-dimensional “surface area” just the volume of a sphere, so about 905 units^3?

Second quandary: coordinates. I’ve plotted my little universe using Cartesian coordinates (x,y,z). (Heavenly help me if the only accurate way of doing it is using the hyperspherical coordinates from the wiki. Trig was not my strong point.)

My possibly-mistaken-understanding is that I can recalibrate my little universe on any point that I want by simply making that the 0 point (0,0,0). To get the new coordinates of all the other systems, I then adjust their coordinates by the same amount. For example, if the original coordinates for my 0 point were (1,2,3), I would adjust coordinates for all systems by (-1,-2,-3). Anytime this resulted in a number with an absolute value greater than 6, it would “loop around,” so a -7 would become a 5 (-6 is equal to 6). Is that right, or does a hypersphere not work quite like that?

Spoiler:  

I’m wondering if I am not inadvertently creating a hypertoroid instead.

Third quandary: I wonder what the night sky would look like. The universe is small enough for starlight to circle it multiple times in every direction, so the same star would be visible in the sky in several different places. Is that right?

Fourth quandary: where there’s light, there’s heat. Because of its small volume, over the 3 or so billion years minimum for planetary evolution, isn’t my universe going to be a bit warm? This could be good or bad. On the one hand, I may be able to justify habitable zones around my red dwarfs; on the other hand, maybe it’s so hot that everything cooks, and I need to come up with some sort of heat sink?

[jeff_wilson]

Originally Posted by Stegyre

Because of its small volume, over the 3 or so billion years minimum for planetary evolution, isn’t my universe going to be a bit warm?

If you want to do everthing by the book, you'll need a bigger space at least twice as old to allow for enough supernovae to form metals, disperse them into clouds, then compact the cloud nodes into 2nd generation solar systems - then the 3 billion years of terrestrial planetary evolution can begin. Before that, you'll have essentially all Hydrogen-Helium systems, with only a tiny contamination of heavier nuclei like Lithium and Berylium.

[Fred Brackin]

Originally Posted by Stegyre

Fourth quandary: where there’s light, there’s heat. Because of its small volume, over the 3 or so billion years minimum for planetary evolution, isn’t my universe going to be a bit warm? This could be good or bad. On the one hand, I may be able to justify habitable zones around my red dwarfs; on the other hand, maybe it’s so hot that everything cooks, and I need to come up with some sort of heat sink?

I beleive that with continued output of 29 stars you have to have an expanding space to keep heat down to a maneageable level.

Your closed space is far less than 3 billion years across. Every photon visible, IR, X-ray, etc has gone round and round multiple times except if it's hit something. I suspect this may be a serious problem but I can't crunch the nimbers myself.

[Stegyre]

Originally Posted by jeff_wilson

If you want to do everthing by the book, you'll need a bigger space at least twice as old to allow for enough supernovae to form metals, disperse them into clouds, then compact the cloud nodes into 2nd generation solar systems - then the 3 billion years of terrestrial planetary evolution can begin. Before that, you'll have essentially all Hydrogen-Helium systems, with only a tiny contamination of heavier nuclei like Lithium and Berylium.

My thought is I can have one (or maybe two) large 3rd generation stars that supernova in a few hundred million years. (I have not worked out the complete stellar evolution, but I at least did some looking into it.)

[Anthony]

In an actual hyperspherical space, you have a problem that light will pass around the sphere and return to its starting point; thus, in effect, a star will focus its light back on itself. Unless your space is extremely large, this will cause a star to overheat in a fairly short time.

If you can get rid of the focusing problem, I believe a space several parsecs across can avoid overheating, though the sky will be extremely bright (billions of visible stars).

[Grouchy Chris]

Originally Posted by Stegyre

The radius of the hsphere is 6 units. (What a unit is remains to be decided; could be as large as a parsec; could be less than a light year.) It contains 29 stars distributed among 23 systems (four binary; one trinary).

First quandary: the volume of the hypersphere. (I suspect this is important for calculating the background “heat” of the universe, discussed further, below.) I get a formula that I’m actually able to handle from wiki. If I did it right, my 4-dimensional hsphere has a volume of 6,395 units^4.

As I think about it, though, I probably don’t care about the 4-dimensional “volume,” do I? That’s the space where we 3-dimensional beings cannot travel. Isn’t my 3-dimensional “surface area” just the volume of a sphere, so about 905 units^3?

You do indeed want the "surface area," but it's not just the area of a sphere of the same radius. (The surface area of an ordinary sphere is not the same as the area of a circle of the same radius, either.) From Wikipedia and also from MathWorld, the hyper-surface-area of a 4-sphere is 2 * pi^2 * r^3. For r=6, that's about 4264 units.

Second quandary: coordinates. I’ve plotted my little universe using Cartesian coordinates (x,y,z). (Heavenly help me if the only accurate way of doing it is using the hyperspherical coordinates from the wiki. Trig was not my strong point.)

My possibly-mistaken-understanding is that I can recalibrate my little universe on any point that I want by simply making that the 0 point (0,0,0). To get the new coordinates of all the other systems, I then adjust their coordinates by the same amount. For example, if the original coordinates for my 0 point were (1,2,3), I would adjust coordinates for all systems by (-1,-2,-3). Anytime this resulted in a number with an absolute value greater than 6, it would “loop around,” so a -7 would become a 5 (-6 is equal to 6). Is that right, or does a hypersphere not work quite like that?

Spoiler:  

I’m wondering if I am not inadvertently creating a hypertoroid instead.

You have indeed described a hypertorus. Cartesian coordinates aren't going to work very well here for the same reason that they don't work on the surface of the earth: lack of flatness. Your two fixes are a) shrug and say, "Okay, hypertorus it is," or b) use a different and more complicated coordinate system. One possible coordinate system is mentioned in the Wikipedia page I linked to above.

Third quandary: I wonder what the night sky would look like. The universe is small enough for starlight to circle it multiple times in every direction, so the same star would be visible in the sky in several different places. Is that right?

Fourth quandary: where there’s light, there’s heat. Because of its small volume, over the 3 or so billion years minimum for planetary evolution, isn’t my universe going to be a bit warm? This could be good or bad. On the one hand, I may be able to justify habitable zones around my red dwarfs; on the other hand, maybe it’s so hot that everything cooks, and I need to come up with some sort of heat sink?

The night sky is going to look like your eyes burnt to a plasma as soon as you entered this universe. I haven't done any calculations, but with a universe that old and that small, I'm pretty sure nothing human could survive in it. You're going to have to either make it bigger, make it younger, or find a way to vent the heat somewhere else. Perhaps another universe?

[Extrarius]

How well do black holes "eat" heat? I know they emit energy via hawking radiation, but I'm not sure how the output from that relates to possible intake. From the wiki article on hawking radiation, I think I'm understanding that the output can be miniscule and is inversely proportional to the gravity and mass of the black hole such that a huge black hole could absorb significant heat while emitting an insignificant amount via hawking radiation.

Interestingly, under this model (ignoring probably tons of other factors), the black hole might eventually consume all matter in the universe only to 'evaporate' via hawking radiation and repopulate the universe with matter.

Though if your universe 'wraps' in any way (such as it would for a hypersphere surface, etc), it might be difficult to work out orbits etc such that stars and planets last long enough to be interesting while excess heat is kept low enough for comfort.

[teviet]

As long as the sphere is large in size and finite in age, it will be a while before you have to worry about runaway heating. Say you have 29 "average" stars (where an average star has about 0.05 Solar luminosities), and they've been burning for 5 billion years (our Sun's age). That's a total radiant energy of about 1e44 joules in a 3-volume (hypersurface) of 4264 cubic "units". Taking a unit to be a parsec, that's an energy density of about a nanojoule per cubic metre, which is the equivalent of a 25K blackbody.

If your stars are brighter and more Sunlike, the radiant energy will increase accordingly. But the effective temperature only goes up as the 1/4 power of the energy density, so you've got some safety margin.

The focusing problem that Anthony mentioned can be avoided by making the universe a quadraxial hyperellipsoid, rather than a perfect hypersphere.

But your simple coordinate system only works for a hypertorus, not a hypersphere. You can always make your pocket a hypertorus (why not?), and the volume formulae get much simpler.

TeV

[teviet]

Oh, and as for the night sky, it will look like you said: multiple images of each star at greater and greater distances, with the more distant images appearing younger and younger. If the stars have been burning for a finite time, there will be a horizon beyond which there appear to be no more stars. As time goes by, the horizon will expand, with images of newborn stars appearing in the distance (just additional time-delayed images of the existing stars). The overall brightness of the sky will increase steadily over time, but, as I said, should still be manageable even after a few billion years.

TeV

[Grouchy Chris]

Originally Posted by Anthony

In an actual hyperspherical space, you have a problem that light will pass around the sphere and return to its starting point; thus, in effect, a star will focus its light back on itself. Unless your space is extremely large, this will cause a star to overheat in a fairly short time.

This should not be a problem is the stars have any motion at all. If the unit is a parsec, then it's about 40 years before the light comes back to where the star was, and the star has moved on.

Originally Posted by teviet

As long as the sphere is large in size and finite in age, it will be a while before you have to worry about runaway heating. Say you have 29 "average" stars (where an average star has about 0.05 Solar luminosities), and they've been burning for 5 billion years (our Sun's age). That's a total radiant energy of about 1e44 joules in a 3-volume (hypersurface) of 4264 cubic "units". Taking a unit to be a parsec, that's an energy density of about a nanojoule per cubic metre, which is the equivalent of a 25K blackbody.

Wow, I thought it would be much hotter. That's what I get for not figuring.