1. Internal Forces of a Thick Infinite Plan
Suppose you have a flat disc with infinite radius, constant thickness 2H, and uniform density. Let height h be the distance from the central plane of the disc, with positive and negative values of h denoting the two different sides of the disc.
Let −G be the gravitational acceleration experienced by a particle of mass m on the positive surface of the disc, at h = H. It is a fact that the gravitational force on a particle near an infinite plane is independent of the particle’s distance to the plane. As such, if the particle is outside of the disc, then the gravitational force on the particle is simply F = ±Gm. And if the particle is inside the disc, then the gravitational force is proportional to the distance from the central plane.
The gravitational potential energy of the particle is found by integrating force over distance:
2. Traveling Through Aerb
Now suppose an entad is used to render the particle intangible, so that only gravity exerts a force upon it, and the particle is free to fall through the mass of the disc. Acceleration is the second derivative of h with respect to time, and combining this with the previous formula yields
This resembles the properties of a sinusoidal function. To verify, let c(t) be the cosine of t, s(t) the sine of t, and define y = Ac(Bt). Then:
Substitute h for y to get
This equation holds if
Thus the motion of the intangible particle can be described as a cosine with period and amplitude H. If the particle starts at t = 0 on the positive surface of the disc, then the position at time t is given by the equation
The time needed to cross all the way through the disc and reach the negative surface is
Applying these result to Aerb, the parameters are G = 32 feet per second², and H = 4000mi = 21120000ft.
Thus it would take an incorporeal particle ≈ 2552 seconds, or about 43 minutes, to pass through Aerb.
The Council of Arches has access to entads which can provide both the incorporeal effect and the supply of oxygen needed to last through the travel, and so disc-slipping should be considered as an option if it ever becomes strategically or economically important to travel to the Other Side.
3. Practical Complications
Firstly, it is unknown how the currently available incorporeality entad interacts with heat and pressure. It may be necessary to take additional precautions to mitigate these effects.
Secondly, the surfaces of Aerb do not exhibit uniform altitude. If the exit point has higher altitude than the entry point, then the traveler will come to a stop underground, eventually reversing direction and returning to their entry point after passing through Aerb twice. If the exit point has lower altitude than the entry point, then the traveler will exit at extremely high speeds. Incorporeality protects the traveler in both of these situations, and attempts with large altitude differences can still be used to gain information about the terrain of the other side.
It is also likely that Aerb has increasing density towards the central plane, but this is not an issue by itself as such concentrations will only decrease travel time.
A more worrying issue is if the gravity equations above don’t model Aerbian forces at all. They are based on Earth data, which indicates that every particle pulls on every other particle with a force inversely proportional to the square of the distance between them. But suppose that the interior of Aerb exhibits inconsistent gravitational effects. For example, suppose that below 1600 feet, the acceleration entirely cuts off. Then the traveler will continue through Aerb at a velocity of 320 feet per second, and it will take over 36 hours for them to reach the other side.
Even worse, there could be unforeseeable anomalies deep underground. Hidden exclusion zones, regions that
nullify entad magic, exotic radioactive ores, etc.
Taking all of these factors into consideration, it would be best to convince the renacim to carry out any experiments.