Hollow World Physics and Structure
by Lance DuncanThis got me inspired to go deep researching the last couple days. As I stated above physics precludes a planetary body that is hollow from normally existing, and this is the reason I have not included the hollow world in my version of Mystara so far. There has always been certain things about it that have appealed to me though, I just really dislike the presentation of most of the offical hollow world products. To explain away the problems with magic and Immortals in complete disregard of physics is unacceptable to me.
So after doing some digging into the whys and hows of planet formation and speaking with my brother(currently attending university to become an engineer) for help with some equations I think I've come up with a solution for both the structural integrity of the hollow world and the reversal of gravity in the interior.
Let's start by talking about why hollow planets dont exist. Within Astrophysics there is a principle called Hydrostatic Equilibrium. Essentially, this is a balance between gravity and pressure. Gravity pushes all the mass of a planetary (or star) body inward toward a single point, while the substance of the body resists compression and builds up pressure pushing outward away from the center of gravity. When these two forces are equal, there is an equilibrium. If a body is not in equilibrium it will either expand or contract depending on whether the force exerted by gravity or pressure is greater. Hydrostatic Equilibrium is what causes all planets and Stars to be spherical in shape.
The reason a hollow world is unstable is because we have removed the material in the center exerting outward pressure, so now there is vacant space into which the rest of the planetary mass will collapse. Now, you may be asking why a solid object would change shape and collapse. The answer to this requires a different way of thinking about solids and liquids. Most of the mantle and core of the earth would normally be called a solid. But we're talking about millions of tons of material here constantly being acted upon by both pressure and gravity and heat(a significant portion of the heat in the Core comes from Radiocative materials). On a geological timescale the mantle and core behave viscously, basically the main body of the earth is a fluid. So when any significant hole is opened up inside the earth, the mass surrounding it would flow into the empty space until a new equilibrium was reached.
Now for the solutions.
I'm going to explain the physical structure of a viable hollow planet first. We understand at this point that a hollow earth can't exist because the interior of the earth is viscous. In order to support a cavity inside the center of the planet a solid non viscous material is required. If the mass of the earth didn't behave as a fluid, then Hydrostatic Equilibrium would be irrelevant because these principles only apply to fluids. The extremely easy solution is just to make the hollow world be a dead planet, meaning the core and mantle are completely cooled down and the intrior has solidified so it is no longer viscous. This would preclude the presence of continental drift though because the movement of tectonic plates is dependant on the presence of the asthenosphere which is itself dependant on the unique conditions of the upper mantle.
An alternative option which I prefer is to have an inner shell of solid non viscous material that would support all the viscous layers of the rest of the earth so that it wouldn't collapse in on itself. I think it's better explained through an image:
My next step with this is to determine what material the inner shell should be made of and how thick it needs to be. The shell must be able to act as a replacement of the pressure that would normally be in the center of the earth. There are a series of equations that I must iterate to determine exactly what type of material for the shell is best.
First I pick a substance and decide on how thick I want the shell to be. From there I calculate Hydrostatic Equilibrium
dP=-D(h)G(h)dh where P is pressure, D is density, G is gravity, h is height
Then I calculate stress exerted on the shell
Rs=P((R^3-r^3)/(r^3-R^3)) where Rs is radial stress, R is outerradius of the shell, r is inner radius of the shell
Cs=(P/2)(2(R^3+r^3)/(r^3-R^3)) where Cs is Circumferential stress
Equivalent Total Stress=Rs-Cs which must be less than the yield strength of the material comprising this inner shell. This will be a process of trial and error, picking a material doing calculations and figuring out what works out the best.
It's getting late and I have work in the morning. I will talk about the gravity of the inner world tomorrow and some ideas about Urt and Megaliths in general
OK, Gravity.
There are all kinds of issues resulting from this. To summarize, even if the hollow world structure is viable, anyone or object at any point inside this hollow space would feel weightless essentially and maybe eventually drift towards the center of gravity. Also the floating continents wouldn't maintain their orbits; my brother and I haven't been able to figure out any solutions for the floating continents, so we'll have to attribute that to magic. However we were able to figure out a way to reintroduce gravity to the surface of the hollow world without using the mysterious magic of the world shield (that and the spell of preservation are my 2 biggest hangups with the published hollow world).
The solution we came up with to allow gravity in the Hollow world is in the realm of pseudo science, yet the principles used should still apply if the details are off. We decided the best way to generate the necessary gravitational force is to replace the portal to the plane of fire(the red sun) with an anti-matter core. This anti-matter would generate anti-gravity counteracting the gravity generated by the mass of the Earth itself. This also requires us to increase the mass of the earth so that the equivalent gravity on the outer world remains the same. This solution comes with a big caveat; we don't really know what anti-matter is, how it interacts with matter, or whether it even has an anti-gravity force associated with it. Everything about it are hypotheses and there is no evidence so far to really support one hypothesis over another. Regardless of this, if it turns out anti-matter doesn't behave in this way, I would simply reason that the nature of the red sun inside the hollow world, whatever it's scientific or magical properties, is a source of anti-gravity such that our equations still hold true.
Now just for some basic statistics to get an idea of the changes that occur from this implementation.
The real Earth has a mass of 5.97x10^24 kg with an average density of 5.51 g/cm^3.
My model of the Hollow World has an outer diameter of 7917.5 miles(the same as earth's) and an inner diameter of 3790 miles(based on the canon circumference of 11908 miles) giving us a shell 2063.5 miles thick. To maintain the same gravity on the outer world as exists on earth we must compress the same mass of the earth into a smaller volume giving us an average density of 6.18695105 g/cm^3 for the hollow earth.
To calculate the gravitational effect of the anti-matter and the neccessary increase in mass, my brother used the shell theorm and put the results in a spreadsheet. What follows are a couple of graphs derived from that spreadsheet; the first shows the normal force of gravity(assuming a uniform density for the sake of simplicity), and the second the effects of gravity with the anti-matter sun inside the hollow world.
This results in the red sun containing 1.62x10^24 kg of anti-matter.
The mass of the earth being increased to 7.59x10^24 kg with an average density of 7.86582219 g/cm^3
So overall the mass of the earth increased by 127% and density by 143%, but the surface gravity on both the outer and inner worlds maintains a force of acceleration of 9.8 m/s^2.
The atmosphere has problems of it's own. If atmosphere is allowed to flow freely between the outer and inner worlds(such as through the giant polar a openings) then the air of the inner world would have a pressure 387000 times that of the outer world. The solution is fairly simple; there must be barrier of some kind preventing air flow between the two worlds. In my model I plan to not have the curving polar openings(I will probably keep an entrance at the poles, but they will be smaller and harder to find akin to Journey to the Center of the Earth) creating a physical barrier, but if you want to maintain the polar openings as per canon then the air should probably be blocked by some sort of magical force field or something. Essentially this creates two separate atmospheres. The atmosphere on the inside would settle into a layered strata as described in the canon sources, with the space around the red sun being similar to the vacuum of space.
I also have some idea of how all this can tie into the nature and life cycle of the megalith called Urt. This requires some revision from the Immortals set, but I think it still maintains the central idea presented there.
A megalith is different from a standard planet in a structural nature as described above. A naturally formed planet follows all the forms and patterns that we currently know about planetary bodies, while a megalith is hollow on the inside but appears to be a normal planet from the outside. The red antimatter sun in the center of the hollow is the heart of the megalith. The floating continents inside can be equated to different organs and they are held aloft and in their orbits through unknown forces which the megalith controls.
There are a few option for defining the the role of the planetary mass of the megalith. This could be the standard explanation offered in the Immortal set, but I'm leaning toward something else. The inner shell which supports the main mass of the 'planet' is like the skin of the megalith, and the mantle and crust, etc are simply matter that has accreted on the outside of the shell during the active phase of the megaliths lifecycle (by the way, the active/dormant phases should be measured in the millions of years at a minimum). During the early stages of a megalith's active phase they seek out(yes they are capable of voluntary movement while active) matter which heats up from impact and pressure and gradually cools down forming a crust. At this point they begin cultivating life on their outer surface.
Burrowers(annelids, purple worms, etc) are part of the biological systems of the megalith. They have a symbiotic relationship with the megalith and can be compared to the bacteria in the human gastro-intestinal system. They consume whatever matter has accreted above the shell/skin of the megalith and convert this into the material which the megaliths shell is made of, thus allowing the megalith to grow. so at the end of a megalith's active phase they may have grown significantly. A megalith will remain active usually until the accreted matter above the shell cools down to a similar temperature as the shell itself.
Towards the end of the active phase a megalith will become hyperactive. The 'heart' will increase in anti-matter, causing the forces of anti-gravity to overcome the gravity of the accreted mass, thus causing the matter to be ejected into space away from the megalith. At this point the megalith will use the last of its energy to stabilize it's orbit around a stable star, and then become dormant. When the megalith becomes dormant the 'heart' cools down to a point that the anti-gravity generated by it is practically zero. While Dormant the megalith will not act in any way(no involuntary volcano eruptions or earthquakes), though it may be awakened by drastic changes in its environment such as disruptions to it's orbit or significant impacts on it's shell. After lying dormant for a period of time, it's energy reserves will be replenished, the heart will become active again and the megalith repeats the cycle.
Urt is a special case, normally the interior of the megalith is uninhabitable, but the Immortals in the hollow world council have persuaded Urt to allow life to grow within the shell as well as on it's crust. Urt carefully regulates the temperature and gravity and atmosphere of the inner world to best support life. Also some Immortals suspect Mars of being a dormant megalith. Venus is thought to be a normal planet because of the chaotic and hostile nature of it's atmosphere. Megalith's tend to carefully manicure their surface to encourage the evolution of life, and all evidence points to an uncontrolled surface environment on venus.
All of these are just some preliminary ideas, I'm sure I've forgotten or ignored some facet of the lifecycle of a megalith. Let me know what you guys think of these ideas, or contribute your own ideas. I will also be posting my calculations later for the material of the inner shell.
Am I correct in thinking you're going to discard the polar openings altogether? I would imagine they'd be the hardest bits to explain with science, since they break the tension between the inner and outer layers.
Yes, in my version of mystara, no matter what physical model I adopt for the hollow world, I don't plan to have polar openings. This is for various reasons, 1) I'd rather not have to deal the mapping issues that would entail, ye it is possible to accommodate the polar openings, i just don't want to do it. 2) If I include the Hollow World I want it to be more difficult to enter/find than just a trip to the north pole(not that that is an easy task either). I think the hollow world should be nigh impossible to reach for an outworlder. It should be a completely unknown and mysterious place. 3) polar openings would probably disrupt the structural integrity. I haven't done the math so I can't say for sure, but it's a likely possibility.
Given that, I may still have entrances at the poles, maybe a mile wide shaft directly over the pole. However, for purposes of separating the two atmospheres the shaft would not go directly to the hollow world; it would probably lead to a system of tunnels with doors or cave-ins through which someone could eventually reach the hollow world, though it wouldn't be a guarantee. I might also place other smaller entrances in other places. To counteract the heat of the mantle which these tunnels would pass though I think they would have to be lined by the same material that makes up the inner shell; the tunnels would probably be created by the Annelids.
Your solution is essentially the same as Aaron Allston's: a world shield. The only significant difference seems to be the location close to the Hollow World's surface.The model that I am using would preclude tectonic plates inside the hollow world. If someone wanted to include them they could place the supporting shell in the center instead of the inner edge, but then the gravity equations would get really complicated.
Wouldn't it be easier to go with this solution, since after all it's the one already laid out?
For me, the thing about the Hollow World is that it's such a crazy idea that it's beyond explanation through science alone. I think you can see from Aaron Allston's work that he probably went through the same kind of thought experiments, reaching broadly the same conclusions. His solution was the world shield — and it's a magical solution, by definition, because there's only so far you can go with science for this.
A further consideration is that Aaron Allston seems to have based his design on a comic panel showing a hollow earth — see the picture here: https://tvtropes.org/pmwiki/pmwiki.php/Main/HollowWorld . If this is indeed the case, the chances are high that he started with the design and went about explaining it as best he could — just as you are doing — rather than designing it all by himself.
As much as I respect Allston and Heard as authors and creators, there a many things in their works that I don't like. The World Shield has no explanation; from what I've been able to gather it is treated as a source of gravity, not the center of gravity. It is just explained away as a magical plane, in fact very similar to Spelljammer's gravity plane, which make no sense either. In spelljammer "[gravity] is there at full strength or it is not there at all." (Concordance of Arcane Space, pg 13) Gravity is measured as acceleration and is completely dependent on distance. I understand this is fantasy, but real world scientific terms refer to very specific things; if something else is meant a new/different term should be used(maybe bodily attraction instead of gravity in spelljammer). My issue with the World Shield is that it solves the problem of gravity by simply saying the problem is solved without explaining where this new nonsensical gravity comes from, and as I said earlier 'Magic' is not acceptable for me.
When I said the supporting material could be placed in the center to allow tectonic plates on both surfaces, I didn't really think it through. If we leave gravity as normal acting as we observe it here on earth, the supporting inner shell must be on the inside or else the rest of the viscous material of the earth would collapse in on itself.
Introducing the Anti-Matter sun opens up some more options however. Because the forces of gravity and anti gravity are working against each other, they balance out somewhere inside the shell of the earth. A new center of gravity appears as sphere at this point of balance. Based on our latest equations, Gravity achieves 0.00 m/s^2 at 1382 miles below the outer surface of the earth. This center of gravity similar in some ways , and different in other to the world shield. it is the same in that is the center of gravity, that is where all mass accelerates towards, yet it is different in that the change of gravity is gradual. If a person were 1382 miles below the surface of the planet they would not experience a sudden shift in gravity. They would feel weightless, and within a few miles of this point they probably wouldn't notice the effects of gravity because they are so minuscule. If the shadow elves truly had a Cavern this deep and had lived there their whole lives, once they returned to the surface they would be unable to function; they just wouldn't have the bone density or the muscle mass to exert the kind of forces necessary to even move their limbs.
Another consideration for layers of the earth and tectonic plates, is that with this antimatter sun, the inner solid shell would be unnecessary for structural support. This is because the center of gravity is now within the earth itself, and not at a single point at the center of the hollow world. Therefore,there are two possible models to make the hollow earth structurally sound; one with an inner supporting shell, and another one with no solid shell, but with a completely viscous interior with a center of gravity inside the thickness of the earth which is generated by the competing forces of anitgravity from an anitmatter interior sun and gravity from the matter which comprises the earth. If one desires the movement of tectonic plates on both the inner and outer surfaces, it is possible to have mirror layers of core, mantle, and crust on either side of of this center of gravity.
Personally I am going to stick with the inner solid shell model because I like the idea of a megalith with an inner skin, feeding on other sources of matter to grow both in thickness and diameter. Also the idea of the interior sun of the hollow world being the heart of the megalith has been a pet idea of mine for a long time.
updated statistics:
red sun has 1.37x10^24 kg of antimatter
mass of earth is 7.34x10^24 kg with an average density of 7.60674 g/cm^3
mass of earth increased by 123% and density by 138% with equal gravity on inner and outer surfaces and center of gravity 1382 miles from outer surface, and 681.55 from inner surface.