The Hollow World

This post is derived mainly from this thread at the piazza. Here I put it all together describing the problems of a hollow earth in the real world, the problems I have with how it was represented in the hollow world products published by TSR, and my take on a scientifically sound (mostly) representation of the physics and geology of a hollow earth. 

The Science

The laws of physics as we understand them precludes a planetary body that is hollow from existing, and this is the main reason I have not included the hollow world in my version of Mystara so far. 

Let's start by talking about why hollow planets don't 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 Radioactive 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.

The other big issue has to do with tectonic plates. Given that we know Mystara is the same as earth in the late Jurassic period per this post, we therefore should try to use all that we know about earth's geology to define the geology of Mystara. From this we must concede that continental drift occurs. Continental drift is occurs because a number of tectonic plates forming the earth's crust are 'floating' on a super heated layer of the earth's interior that acts somewhat like caramel. This process is dependent on the heat and pressure derived from the layers of superheated liquids and solids deep inside the earth. Specifically the Asthenosphere (upper Mantle) is what causes the tectonic plates to shift and expand and contract; the other geological layers (the crust, mantle, and core) of the earth also impact other features we take for granted such as the electromagnetic field around the earth, the magnetic north pole, and volcanoes. As it is the hollow models created by Thorf require that the density of the material in the different layers would have to be drastically increased to make up for the hollow center to maintain the same observed force of gravity.

The effects of Gravity cause a whole range of other issues. 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 eventually drift towards the center of gravity.

This would also influence atmospheric pressure. If atmosphere is allowed to flow freely between the outer and inner worlds(such as through the giant polar openings) then the air of the inner world would have a pressure 387000 times that of the outer world (if we assume the outer world still has the equivalent atmospheric pressure as we are familiar with, overall there would be a lot more air in the atmosphere). The solution is fairly simple; there must be barrier of some kind preventing air flow between the two worlds. Essentially this creates two separate atmospheres. The atmosphere on the inside (refer to Solution 2 described below) 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.

The Hollow World Setting

As much as I respect Allston and Heard as authors and creators, there are many things in their works that I don't like. Despite this, there have always been certain things about the idea of a hollow earth that appealed to me; I just really dislike the presentation of most of the official hollow world products. To explain away the scientific problems with magic and Immortals in complete disregard of physics is unacceptable to me.

The world shield and the spell of preservation are my two biggest hang-ups with the published hollow world setting. The spell of preservation just seems unnecessary and heavy handed; magic isn't necessary to maintain a cultural tradition. Societies change and remain stagnant for a variety of reasons and to set arbitrary limits on a certain people is lazy design. For a long time, due to the problems of the physics of the hollow world, I've been thinking about just transplanting the general cultures of the hollow world to a corresponding place in Davania or Skothar (Nithia to Arypt, etc). I don't think it was necessary to put these cultures there in the first place when there was space (and places already specifically named after a historical analog) on the outer world. I see the hollow world as an opportunity to explore some of the concepts from Pellucidar by ERB. The inclusion of weird creatures and societies like the beast-men fits in with my idea of the hollow world, but there is no reason for the Immortals to place a magic spell on the whole world so that people will maintain tradition and not try anything new.

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) Actual 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.

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; yes 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. Even implementing Solution 2 described below, hydrostatic equilibrium would still collapse the openings in the end without some other magical explanation. I think, 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 some material with a high melting point; the tunnels would probably be created by the Annelids and other burrowers like the purple worms.

When I started this project I was pretty firmly in the camp of my Mystara not being hollow; now I consider my Mystara to be hollow because I can create a somewhat scientific model of it, though the peoples inside will probably be different, and the chances of any PCs reaching it are extremely low. Now I'm actually kind of excited to have my version of Mystara be hollow.

Solution 1

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 interior has solidified so it is no longer viscous. This would preclude the presence of continental drift though because the movement of tectonic plates is dependent on the presence of the asthenosphere which is itself dependent on the unique conditions of the upper mantle.

An alternative option 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:

The model that I am using would preclude tectonic plates inside the hollow world because the shell must be deepest layer of the earth. If the model had mirroring layers of mantle and crust on either side of the shell, the interior layers would simply collapse in towards the center the of gravity.

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.

I worked on some preliminary calculations of this model. My results were discouraging to say the least. It seems that this model isn't feasible without eliminating most of the molten elements so that the shell comprises most of the diameter of the planet. The stress that the material in the shell would have to withstand is orders of magnitude greater than any material I could find numbers for. So, again this could be solved by making the shell out of some magical material that can withstand these enormous pressures, but that kinda defeats the purpose of this whole exercise.

I had my brother to check my math and ran more calculations with different variables on the shell model just to confirm my suspicions. After running the numbers it became clear that without some fictional material such as Scrith or Neutronium, which I would rather not use for various reasons, a hollow world with a viscous mantle was not viable. So in the end I decided not to go with it and explored my options using another model I stumbled across while trying to solve the problems of gravity inside the Hollow World.

Solution 2

This second solution occurred to me and my brother while we were trying to figure out a way to reintroduce gravity to the surface of the hollow world without using the mysterious magic of the world shield

The solution we came up with in order 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.

(After more research, what I call anti-gravity here should be referred to as negative gravity or repulsive gravity because anti-gravity is the absence of gravity. In addition, it is generally agreed upon among the scientific community that anti-matter has normal mass the same as normal matter, while it is likely that dark energy has negative gravitational properties. As stated above the actual substance of the red sun is irrelevant here, just the fact that it generates negative gravity, though I do prefer the term anti-gravity, it just makes more sense to me.)

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 is 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 antigravity from an antimatter 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.

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 necessary increase in mass, my brother used the shell theorem 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) inside a hollow earth, and the second the effects of gravity with the anti-matter sun inside the hollow world.

This results in the red sun containing 1.37x10^24 kg of anti-matter or negative mass.

This requires the mass of the earth being increased to 7.34x10^24 kg with an average density of 7.60674 g/cm^3

So overall the mass of the earth increased by 123% and density by 138%, but the surface gravity on both the outer and inner worlds maintains a force of acceleration of 9.8 m/s^2 and center of gravity 1382 miles from outer surface, and 681.55 from inner surface.

So I ended up dividing the planet into 7 layers: the Lithosphere, Asthenosphere, Upper Mantle, Lower Mantle, Outer Core, Inner Core, and the World Shield. I didn't include the transition zones simply because it was too difficult to find the mass/density or volume for all the different transitions. I did decide to include a mundane version of the World Shield just to make the math easier for me, but it doesn't have any special magical properties it is just denser material(osmium) at the center of gravity. The other layers have the same composition and properties as found in the real world.

In figuring the depth of each layer I calculated the percentage of the total radius of the earth that each comprised, and then applied that percentage to the distance between the surface and the center of gravity (another option I explored but didn't go down was to have each layer maintain the same depth as in the real earth but cutting them off at around the center of gravity so portions of the normal core and mantle would be superseded by the world shield). Then I finagled the numbers to get the final mass I needed which was calculated from the real mass of earth plus the effects from the anti-grav sun of the hollow world.

This is a chart with exact measurements for each layer:

Layer	Depth(km)	Thickness(km)	Mass(kg)	Temperature( C )	Density(g/cm3)	Volume(Km3)	Composition
Lithosphere	0	38.91	5.33E+22	200-600	2.7	1.97E+10	granite/basalt
Asthenosphere	38.91	58.37	9.61E+22	1500	3.3	2.91E+10	liquid rock
Upper Mantle	97.28	150.15	2.83E+23	500-900	3.9	7.25E+10	crystalline rocks and hydroxide water in lower transition zone
Lower Mantle (Mesosphere)	247.43	820.57	1.69E+24	2000-4000	5	3.37E+11	semi-solid iron/magnesium/silicon
Outer Core	1068	832.05	2.68E+24	2730-4230	10.7	2.50E+11	liquid nickel/iron
Inner core	1900.05	322.48	1.02E+24	3000-6650	13.5	7.53E+10	solid nickel/iron
World Shield	2222.53	1.97	9.61E+21		22.6	4.26E+08	osmium
center of gravity	2224
Inner core	2224.5	107.6	3.06E+23	3000-6650	13.5	2.26E+10	solid nickel/iron
Outer Core	2332.1	432.2	8.50E+23	2730-4230	10.7	7.94E+10	liquid nickel/iron
Lower Mantle (Mesosphere)	2764.3	426.04	3.09E+23	2000-4000	5	6.17E+10	semi-solid iron/magnesium/silicon
Upper Mantle	3190.34	77.61	3.75E+22	500-900	3.9	9.62E+09	crystalline rocks and hydroxide water in lower transition zone
Asthenosphere	3267.95	31.83	1.26E+22	1500	3.3	3.81E+09	liquid rock
Lithosphere	3299.78	21.22	6.74E+21	200-600	2.7	2.50E+09	granite/basalt
hollow world	3321	1097				1.19E+11
Center	6370

It occurs to me now that I should probably do a similar analysis for the layers of the atmosphere inside the hollow world based on the increase of anti-gravity the closer you get to the red sun. Because of the exponential increases in anti-gravity the atmosphere would essentially give way to vacuum at a much lower altitude than the outer world meaning there is only a thin layer of breathable air near the surface of the hollow world. I will leave that as an exercise for some other time.

Also 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 would have the highest pressures in the center of the gaps where the center of gravity is (maybe this is the deal Urt made with the immortals, it created the polar openings in its shell to facilitate the exchange of atmosphere, see below). 

On Megaliths

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. These continents are rendered virtually uninhabitable by humanoids because of the intense gravity at high altitudes and the thin atmosphere.

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 main mass of the 'planet' forms a protective shell, 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 and assimilating into the different layers of the shell. So during a megalith's active phase they may have grown significantly. After the crust has cooled down and stabilized it will stabilize it's orbit around a stable star and 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 any radioactive matter that has accreted in the shell of the megalith and convert this into energy which fuels the heart of the megalith, thus allowing the megalith to remain active.  A megalith will remain active usually until the all significant sources of radioactive matter in the shell have been consumed.

Towards the end of the active phase a megalith will use the last of its energy to leave its solar system on a trajectory towards a source of fuel, and then become dormant until it reaches that destination. When the megalith becomes dormant the 'heart' cools down to a point that the anti-gravity generated by it is practically zero causing the shell to collapse inward under its own weight. 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 path 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 with no atmosphere whatsoever, 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 that was captured by the gravity well of the sun. 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.