Fixing up a Flat-Earth cosmology

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Periodically, journalists rediscover that people who believe that Earth is flat actually exist. The most recent incarnation of this curiosity is an article in the Denver Post detailing the lives of so-called "Flat Earthers" in Colorado.

Also with some frequency, skeptics publish lists of evidence that Earth is curved. The evidence is often presented as readily accessible, inexpensive, and easy to understand.

I contend that it is not easy to show that the Earth is a sphere; that, in fact, most of the evidence of Earth's curvature has historically been largely out of reach for most human beings; and that, using modern methods, it is possible to produce a "flat" Earth that is quantitatively consistent with much of the evidence.

The math

For a generic point $P$ in space, define

• $z$, its height above or below Earth's surface;
• $s$, the distance from the North Pole to the point on earth closest to $P$; and
• $\phi$, the longitude of $P$.

Let $\kappa$ ($\kappa^{-1}\approx 6\,371\,\mathrm{km}$) be the curvature of the Earth.

What Flat-Earthers are thinking about when they isn't terribly clear, but it seems that they generally prefer maps that use the azimuthal equidistant projection centered on the North Pole. The range of this projection is a disk: if we extend the projection with respect to height so that it's conformal for constant longitude $\phi$, then our Universe is an infinite cylinder ($-\infty<z<\infty$, $\kappa s<\pi$) whose perpendicular cross sections are parallel to Earth's surface—something like this:

Using differential geometry[^ge], we can show that the physics of this Flat Earth cosmology will be the same as that of a spherical Earth provided that every moving object of velocity $$\boldsymbol{v} =v^{\hat{z}}\boldsymbol{\hat{z}} +v^{\hat{s}}\boldsymbol{\hat{s}} +v^{\hat{\phi}}\boldsymbol{\hat{\phi}}$$ is subject to the specific force[^specificforce] $$\boldsymbol{f}=\kappa\ee^{-\kappa z}\left((v^{\hat{s}})^2+(v^{\hat{\phi}})^2\boldsymbol{\hat{z}} -(g(\kappa s)(v^{\hat{\phi}})^2+v^{\hat{z}}v^{\hat{s}})\boldsymbol{\hat{s}} +(g(\kappa s)v^{\hat{\phi}}v^{\hat{s}} - v^{\hat{\phi}}v^{\hat{z}})\boldsymbol{\hat{\phi}}\right)$$ where $$g(w)\stackrel{\text{def}}{=}\frac{1}{w}-\cot w\text{.}$$

Admittedly, $\boldsymbol{f}$ doesn't seem like much fun to handle directly. Nevertheless, we can still draw qualitative conclusions based on the nature of the terms appearing in $\boldsymbol{f}$.

Key observations

Travelling on Earth

When navigating the sea or the air (and neglecting weather), the optimal path is that given by a great circle on the usual, spherical Earth. On our disk-shaped Earth, the conclusion is the same, but it now follows because such paths are the natural motion of Earthbound bodies when subject to $\boldsymbol{f}$. On the disk, these paths appear to be deflected towards the North Pole.

Light emanating from a point on Earth

Ships appear to move below the horizon after travelling far from the observer. One has a better view from higher vantage points. On a spherical Earth, these facts are consequences of extrinsic curvature: the farther away something is, the more one has to "dip down" to reach it. On a disk-shaped Earth, we can't see distant low-altitude objects because the light they emitted was deflected upward through the action of $\boldsymbol{f}$.

Earth and heavenly bodies

The Sun, Moon, and stars rise and set. Photographs taken by satellites exhibit a round Earth.

For a spherical Earth, these facts are explained by only half of Earth facing a direction at any given moment. But for our disk-shaped Earth, these facts obtain because $\boldsymbol{f}$ acts so as to "defocus" light rays bound for Earth and "concentrate" the light rays that are emitted.

Occam's Razor and Flat-Earth cosmology

I've outlined how introducing fictitious forces can be used to ameliorate the deficiencies of a disk-shaped Earth. Of course, a flat Earth is wrong—but not because it's a priori inconsistent with observations. Instead, in keeping our disk-shaped Earth we are forced to introduce fictitious forces to match the inertial-frame predictions that a sphere-shaped Earth would. From this perspective it is this introduction of needless complexity to preserve a preconceived worldview that makes a Flat Earth unscientific—we have run afoul of Occam's Razor.

transformed Earth compares with that of the usual infinite cylinder in the same orthogonal (necessarily non-holonomic) frame—the difference is the specific force. [^specificforce]: Specific force means force per unit mass. Specific forces carry the same units as acceleration.