Playing with 4D Planetary Scales

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Playing with 4D Planetary Scales

Postby loganrk » Tue May 03, 2022 9:10 pm

Lets assume that, to maintain life, a 4D planet needs active geology just like Earth, which means retaining heat to drive tectonics and volcanism. If primordial heat is the same between a 4D planet and Earth (very debatable, but I'll use it as a first approximation because I don't know of anything better), and assuming that thermal conductivities and emissivity and so on are all equal, then we want the the surface-to-volume ratio of Earth to match the surface-volume-to-bulk ratio of the 4D planet. That ratio will always be equal to the radius of the planet scaled by the ratio of dimension-specific coefficients. Going from 3D to 4D, it turns out that the radius must increase by a factor of 4/3rds to maintain the surface-to-volume ratio.

In order for figure out how much material that corresponds to, we'll measure radius in units of 2 angstroms (i.e., one average atomic diameter). Calculating the volumes of Earth and the 4D planet in terms of cubic atomic volumes and quartic atomic bulks, and then dividing, we find that this 4D world with equal heat retention capacity as Earth must contain 1.18515e17 times as many atoms! Earth's mass in kilograms in about 6e24kg. Multiplying it out and assuming atomic masses are comparable between universes, this 4D planet would have a mass of about 7e+41kg--which is somewhere between 1/3 and 1/4 the mass of our entire galaxy! Meanwhile, its surface comes out to 2e28 cubic kilometers--equal to the volume of a sphere 23 AUs in diameter! (That's a couple of AUs larger than Saturn's orbit.)

So, uh... dang. That is a lot of space, and all crunched down in four dimensions so its all within reasonable Earthy travel distances....

Of course, if you fiddle with physics in other ways, planets could retain heat at smaller sizes. Or maybe life doesn't need geological recycling, and then they don't need to retain heat at all. But hey--the volume of a solar system to play with all accessible on the surface of a planet? That's pretty neat.
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Re: Playing with 4D Planetary Scales

Postby PatrickPowers » Mon Jun 27, 2022 2:46 pm

My view is that you can't take this kind of thing seriously. If you try to use 3D physics in the 4D world then that ultramassive planet would collapse into a black hole. I don't worry about it and consider it a pedogogical exercise to imagine an Earthlike planet in 4D.
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Re: Playing with 4D Planetary Scales

Postby quickfur » Mon Jun 27, 2022 5:03 pm

Yeah, before one can build a serious model of 4D physics, one first needs to answer difficult fundamental questions as, why is 3D physics the way it is, and which part of that derives from the dimensionality of 3D space, and which part is dimension-independent. You can't just arbitrarily change the dimension of space and expect physics to remain consistent or produce the same macroscopic effects. For example, if you subscribe to string theory, which postulates that much of subatomic physics arises from rolled-up extra dimensions, then "unrolling" another of those dimensions into a macroscopic 4th dimension is sure to have some pretty radical consequences to the resulting physics. It may not look anything like the 3D physics we know anymore. Also, parameters such as the Planck length, the mass of an electron, the speed of light, etc., "work" in a 3D universe, but who's to say that in a hypothetical 4D universe they aren't going to be different? And besides, why are these parameters the way they are in the 3D universe anyway? Without understanding why they are that way, who's to say that they aren't dependent on the dimensionality of space in some way?

If you go down this route, pretty soon you find yourself questioning some really fundamental things about why 3D physics is the way it is, and how it might translate to a 4D universe, and pretty soon you find yourself staring at a modified physics native to 4D whose macroscopic consequences are completely alien and difficult for our poor 3D brains to even begin to comprehend. The resulting 4D universe will likely look nothing like anything we're familiar with, and will have completely foreign phenomena that we'd have a hard time wrapping our brains around. And more importantly, it will have nothing to do with any of the familiar things from our 3D universe, and thus it wouldn't be very interesting; it'd just be a collection of bizarre phenomena with no real analogies to 3D, and we wouldn't be interested in it because we wouldn't be able to relate to it at all. But it'd be consistent.

So eventually, we realize that the whole thing has missed the mark to begin with: the whole point of this exercise wasn't to invent a completely self-consistent 4D universe from ground up; the whole point was to use dimensional analogy from 3D to 4D as a way for us 3D beings to come to grips with 4D geometry. By extrapolating familiar 3D things to 4D, we can gain insights into how having an extra dimension of space might change the way things work -- but the starting point is always some familiar object or phenomenon in 3D. Once we get beyond that starting point of familiar 3D objects or phenomena, we start losing track of the original purpose of the whole exercise.
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