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It has been determined by observation of the stars that the gravity on Flat Earth is due to it accelerating upwards at 1G. This is what holds people and objects to the surface. It has also been determined that items do not experience any mutual attraction due to 'gravity'.

The problem is that the energy to do this (e.g by rockets or by being pushed by an ever accelerating giant turtle) seems impossibly large. Scientists want to find an alternative explanation for the acceleration.

One hypothesis put forward is that the Flat Earth is largely made of iron and that we are being pulled towards a giant magnet that is immovably fixed in the firmament.

A counterargument is that there is no way that the Earth has existed for the 6000 years since its creation without having already crashed into the magnet.

Question

Given the above conditions with no gravitational force existing other than produced by acceleration but magnetism obeying the standard laws of physics:

(a) Is it possible to sustain a constant level of acceleration by magnetism? (perhaps even adjustable electromagnetism)

(b) What would have to apply for it to have taken at least 6000 years without a collision occurring?

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ Your world is already magic. Why are you looking for a non-magical explanation? $\endgroup$ – Tim B Dec 14 '18 at 12:10
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    $\begingroup$ What causes stars to form/not explode? What keeps things in orbit around them? If not in orbit then how do you stay warm? Or are you accelerating in a circular orbit but using acceleration to do it? $\endgroup$ – Tim B Dec 14 '18 at 12:16
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    $\begingroup$ If there is no gravity what clumped all that matter and holds it together? Stars? At 1 G it would take just below 1 year to get to light-speed, that makes things complicated to work for ages.. $\endgroup$ – Artemijs Danilovs Dec 14 '18 at 13:29
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    $\begingroup$ -1 for using "flat Earth" un-ironically in a question tagged hard-science. There is no way to answer this question with "equations, empirical evidence, scientific papers, other citations, etc." $\endgroup$ – T.J.L. Dec 14 '18 at 13:34
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    $\begingroup$ @T.J.L. - Irony has nothing to do with the validity of the question. This question can be answered with maths and equations. If you take the trouble to read it, the actual question is purely about magnetism - which I define to be the same as in our universe. Just look at (a) and (b) in my question - anything non-scientific about that? The system I describe in the question could actually occur in interstellar space. The rest is just background for the story. $\endgroup$ – chasly from UK Dec 14 '18 at 13:40
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After thinking about this problem a bit, I'm fairly sure that it's impossible. Strap in, this is a long one.

You see, the reason a magnet attracts a piece of iron is that the magnetic field induces a bunch of little dipoles in the iron (more precisely, the magnet coerces each atom's intrinsic dipole moment to line up in the same direction, but the net result looks the same). But here lies the problem: the force exerted on a dipole by a magnetic field is not due to the strength of the field, but rather on its gradient (ie rate of change). Specifically,

$$\mathbf{F = \nabla (m \cdot B)}$$

where $\mathbf{F}$ is the force on the dipole, $\mathbf{m}$ is the dipole moment, and $\mathbf{B}$ is the magnetic field.

The force that the entire chunk of iron feels is just the sum of all the forces that each tiny constituent dipole feels. Therefore, when we see magnets attract iron, the fact that the field of the magnet fringes at the ends and becomes weaker is crucial, so that the gradient is non-zero. So while Dubukay is correct that an infinite sheet of dipoles creates a constant magnetic field throughout all space, this field won't accelerate a piece of iron at all, let alone uniformly!

In case the analogy to electrostatics is still confusing you, the key difference here is that magnetic sources only come in dipoles (as far as we've encountered, at least). So, the problem is fundamentally different from the case of a charged particle in an area with a uniform electric field, because electric charges are monopoles.

So, now knowing that a uniform magnetic field won't serve our purpose, the question becomes: can we come up with one that can? The answer appears to be no. To see how, we will take the simplest model for magnetization of iron, which posits that it will be directly proportional to the magnetic field strength $\mathbf{H}$ (we use $\mathbf{H}$ instead of $\mathbf{B}$ because it doesn't change within the iron):

$$\mathbf{M} = \chi_{m} \mathbf{H}$$

while

$$\mathbf{B} = \mu \mathbf{H},$$

where $\mu$ is the permeability of iron.

Putting it into our formula above, for a small chunk of our iron we have we have $$\mathbf{F} = \mu \chi_{m} \nabla (H^2) dV$$ Since we want the force to be constant in a single direction, we end up with: $$\frac{\partial}{\partial z}(H^2) = constant$$ which implies $$ H = \sqrt{Cz + f(x,y)}$$

But we have one further hurdle to clear. Maxwell's equations require that $\nabla \cdot \mathbf{H} = 0$, so we must have $f$ chosen so that

$\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + C = 0$,

where C is a constant relating to the acceleration desired and the permeability of iron. There are many such functions that exist, so so far our task is possible. But we have one more hangup: unless you want this field caused by currents permeating all of space (which is rather unphysical), Ampere's Law requires that the curl of $\mathbf{H}$ also be zero in the vacuum through which the Earth is traveling. It turns out that this requires $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$, something incompatible with our previous constraints on f.

Now, I must point out that all of this hinges on the assumption that our magnetic field only points in the z direction for all space, and that's mostly because I'm lazy and this answer is already very long. I suspect that a similar answer would turn up even if I did take into account a more general case, or at least it would lead to problems with stability of the Earth's motion.

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    $\begingroup$ Damn. I knew I should've sat down and worked out the math. I'll bounty you the rep and make some edits to my answer to direct people here. Nicely answered! $\endgroup$ – Dubukay Dec 16 '18 at 5:26
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    $\begingroup$ Well done! "...this field won't accelerate a piece of iron at all, let alone uniformly." The pedantic individual in me wants to point out that, actually Dubukay's answer does indeed produce a uniform acceleration.... just an acceleration of zero! $\endgroup$ – Cort Ammon Dec 17 '18 at 4:36
  • $\begingroup$ Good answer. At least this does not disprove that Earth is flat. We can be safe for now. $\endgroup$ – Artemijs Danilovs Dec 17 '18 at 13:10
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EDIT: THIS WILL NOT ACTUALLY WORK

Check out el duderino's answer below for a mathy explanation of why an infinite magnetic sheet won't exert any forces on a chunk of iron, despite my intuition.

The magic of an infinite sheet of magnetic monopoles:

One topic I truly struggled with in physics was the formula for the electric field produced by an idealized, infinite sheet of charge. The formula is below:

$$E = \frac{\sigma}{2\epsilon_0}$$

What's wild about this is that there's no distance (r) term - the field is felt equally strongly at every distance from the sheet. If you're a charged particle one light year away from the sheet, the force you feel as a result of it is the same as the force you'd feel when right next to the sheet.

This holds true for an infinite magnetic sheet. Although I mention monopoles in the title, one could simply have an infinite wall of bar magnets and obtain the same effects, as long as they're all aligned in the same direction.

Implications for worldbuilding:

Now, there are a few ways to handle this in a Worldbuilding context. One is to simply accept that there can exist an infinite field of magnets, and place your iron-bottomed planet a very large distance away. Even though this planet will eventually approach the speed of light, remember that the distance doesn't matter and that you can choose the initial distance for whatever works best for your plot. Of course, this requires handwaving what exactly the field of magnets is and how it remains stationary - is it a crease in the galactic plane? Is it a megastructure left by aliens? Is it a grey-goo style scenario as a result of magnetically-minded nanobots who have been collecting all the iron in the universe to use to expand their magnet wall? On the plus side, throwing out gravity means that you could reasonably have a magnetic wall made of mass and not worry about its eventual collapse under its own weight.

The second option, if you're unwilling to accept an infinite magnetic sheet, is a little bit trickier. Without an infinitely large sheet, we run into edge effects at some distance away, and if we get far enough the force exerted on our planet decays as the very large sheet nonetheless appears as a point as the distance away approaches infinity. The best way to handle this as far as I can imagine is with portals or traversable wormholes. Two wormholes are placed close enough to the magnetic sheet to avoid edge effects with your iron planet in between, and the planet "falls" toward the wormhole closer to the sheet at a constant acceleration. It passes through the wormhole there and warps back to the more distant wormhole. Because the magnetic field doesn't fall away as we move away from the sheet, acceleration is maintained and the consequences for the planet would make GLaDOS proud.

Image of falling through portals indefinitely, courtesy of Valve

Further considerations:

Light problems

The sun and moon will requre additional explanation - possibly iron cores in each of them. Then they’ll “fall” at the same speed as the planet itself.

What might end up being highly problematic are the stars. If these stars are fixed to something external like the magnetic sheet, then as we approach them at close-to-light-speed, they’ll turn into gamma-ray beams of death due to extreme blue shifting. This would hold true under both scenarios outlined above, so I hope the “firmament” that you describe them being fixed in is also moving along with the Earth!

Starting distance

Although I'm not comfortable with the special relativity required, there's a question on the Space Exploration stack and another note on Wikipedia that can give us some idea of how far away we'll need to start.

Wikipedia mentions that under 1*g* constant acceleration, the a ship would get close to light speed in about a year and will have traveled half a light year in doing so. The Space Exploration question sums this up by saying we should add a little more than a year per light year. Thus, we should start a little less than 6,000 light years away, which is about the dimension of the Orion Arm of the Milky Way.

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    $\begingroup$ I love it! I'm always happy when infinities make the problem easier rather than harder! That being said, I doubt many who profess to Flat Earth would accept an argument that demands an infinity unless they get to choose the precise shape of the infinity to wed cleanly with their religion of choice. $\endgroup$ – Cort Ammon Dec 14 '18 at 18:04
  • $\begingroup$ @CortAmmon On the other hand, the Bible never says that there's not an infinite magnetic sheet, so I think believers have some freedom to decide. I wonder why Jesus never got around to mentioning it though... $\endgroup$ – Dubukay Dec 14 '18 at 18:06
  • $\begingroup$ Or maybe it's just beeing kept as a surprise. Just because the magnets could be arbitrarily far away doesn't mean we can't have a planned dinner-date with them in the next years! $\endgroup$ – Cort Ammon Dec 14 '18 at 18:13
  • $\begingroup$ This is great! It does raise other questions but I'd better save them for another thread. $\endgroup$ – chasly from UK Dec 14 '18 at 18:38
  • $\begingroup$ It occurs to me that I should (according to the hard-science tag) ask for equations! Suppose the mass of all the free iron in the flat earth is 5.0 × 10^24 kg, what would the magnetic field of the infinite magnetic sheet need to be to cause an acceleration of 1G and what distance would we have had to start at in order to not collide with it in 6000 years? P.S. I'm still accepting this answer and I don't know how difficult those questions are so I won't insist on them. (I suspect the latter one requires relativity) $\endgroup$ – chasly from UK Dec 14 '18 at 19:36
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If you had an extremely strong monopole magnet you could have the flat world orbit that monopole and get constant acceleration that way.

Note that you would get some interesting side effects here, for example magnetic items would be essentially weightless (accelerating up at the same speed as the base of the planet) but still have full mass.

The equation for magnetic force says that the force is inversely proportional to the distance squared (i.e. it obeys an inverse square law with distance).

$$F∝{1 \over r^2}$$

This is the exact same property as gravity holds, so with a sufficiently powerful magnetic monopole in a vacuum you would see magnetically active materials orbit it in the exact same way that in our universe planets orbit larger masses.

So long as the flat earth is tidally locked (which could be achieved and stabilized by having a large anchor hanging from it) then all magnetically active materials would orbit it, and anything not magnetically active would experience simulated gravity.

By making the monopole weaker or stronger and varying the radius and radial velocity of the orbit you can tweak it to give whatever length of "year" and level of simulated gravity, etc as you desire.

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  • $\begingroup$ I'm concerned that such a magnetic orbit would be very sensitive to the slightest radial displacement and thus become unstable. Should I be? $\endgroup$ – chasly from UK Dec 14 '18 at 12:41
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    $\begingroup$ No, it would act exactly the same way that gravity does. Note that monopoles while theoretically possible are unknown in our universe. $\endgroup$ – Tim B Dec 14 '18 at 12:43
  • $\begingroup$ So what we'd have is essentially a giant coin tidally-locked to its star and orbiting around with its face always pointing toward the magnet? I like the way you think! $\endgroup$ – Dubukay Dec 14 '18 at 18:04
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The magnet idea feels like a good idea on paper, until you consider that magnetic field strength is exponential based on distance. There are other factors but I'm going to keep it simple for time.

The pulling force between two magnets can be (at least roughly,) calculated by the equation known as the Gilbert model. I'm going to simplify as best I can and something is going to be lost in translation, but the biggest thing is the last part, four times pi times the distance between the two objects squared.

Squaring causes exponential changes in field strength, every time you half the distance you make the pull four times stronger. The same applies but for weakening pull strength when you move the two objects farther apart. If two magnets are one meter apart, they have a pull that is four times stronger than if they were two meters apart.

Even if the planet itself lacks a magnetic charge, it will still be increasing its acceleration rate constantly as the distance between your planet (If you can call a flat body a planet.) nears the magnet. There is a solution to this in that both your disc and the magnet pulling it are moving, and there is a system that keeps the distance between the two constant by altering the strength of the magnetic field(s).

Only problem with this is it requires the planet being artificial, although giving it's flat this is not that much of a logical stretch anyway. It could be super advanced ancient tech, magic or God/gods.

This could deal with the problem of time, and the planets creation given its unnatural shape and is more or less believable in a fiction setting.

TL; DR, the magnet would have to be moving and your magnets would need to have some way to change strength or the planet would get too close or far from the magnet.

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The Doppler Effect

Essentially any wave has its wavelength shorten as you approach it, and lengthen as you depart from it. This has the rather unsettling effect, but accounts for why sounds are louder when driving toward them, and quieter when driving away. It also applies to light, making it bluer when moving toward it, and redder when moving away from it.

Accelerating at a constant pace toward a light source will mean that it shifts up in frequency, becoming bluer. This bluer light is more energetic and progressively becomes deadlier as higher energy light is capable of ionising mater. Now Ionising radiation is not a new thing. The sun produces a lot of lower energy ionising light that is dealt with by the magnetosphere and the atmosphere of Earth.

The magnetosphere removes a lot of this by deflecting most away, and channeling the rest toward the poles. The atmosphere backs it up with molecules such as ozone that absorb the high frequency energy. Now on Earth as we know it, there is a constant acceleration as we orbit in circle (actually an ellipse but lets be simple here). The nice thing about circles (ellipses) is that the speed traveled is rather constant with a little variation between a maximum and minimum speed. This means that the atmosphere and the magnetosphere are deformed by the planets motion rather predictably and constantly.

A Pizza planet is also undergoing constant acceleration, but it does not have a constant (or near constant) speed. As this speed notches up the magnetosphere and the atmosphere will deform more and more. The deformation will be toward the planets surface as the atmosphere is compressed, and the magnetosphere cannot project out as easily. At the same time the radiation (light) will notch up higher and higher becoming steadily more deadly and ionising.

At first the magnetosphere and atmosphere will hold their own, but shortly they will be overwhelmed. Atmosphere will be bleed off into the great beyond as the magnetosphere shrinks. More ionising radiation will come into contact with the atmosphere as it was not magnetically deflected accelerating the process. Eventually, actually in pretty short order, perhaps a few hundred years (being very generous here) the planet becomes uninhabitable.

At this point the atmosphere is now comparable to Mars in thickness, and the average surface temperature is comparable to Venus. The world is slowly being eroded away and any surviving life will be living deep within the rock beds, trying to protect their delicate organic molecules from the cruel gamma rays scorching the planets surface. You still have over 5 millennia to go.

Nova

To worsen this problem, any nova (not just super nova, but also the less overwhelming cousins) would effectively behave like a hyper-nova. This is not good news. A Hypernova within a few thousand light-years will sanitise a planet right down to its bedrock. Better hope none of those stars explode, or they are accelerating at or faster than the planet in the same direction.

Relativity

This is without considering the shear stresses created by relative time-dialation that will have already ripped this world into pieces. This problem is caused by even slight differences in time between adjoining parts of the surface which only increase as the planet approaches light-speed. Even if the speed of light could be exceeded, the time issue will only worsen.

Rewriting Physics

You will actually need to rewrite a lot of physics. Dropping gravity simply isn't enough.

Photons (light), magnetism, electricity, and the fundamental properties of electrons/protons cannot exist as currently understood. You would have to rewrite them to not obey the Doppler effect. This will be hard. It ties up time, and motion.

Consequentially Relativity has to be dropped. Trying to fix Time and motion to avoid Doppler shifts implies that there can be no single distinct set of consistent physics. This is because two observers cannot agree on the number of waves seen. A faster observer would have to see fewer waves of light when approaching, and more waves of light when departing than someone stood still so as to preserve the relative energy of the light. Thus there will either be rules that change, or fundamental constants that change based on location, direction, velocity, or some other property.

This will also make the notion of time hard. A person further away from a source may actually see more than a moving person next to the source of an event. This will make causality near impossible to handle.

I can't offer much of a solution here, but not as we know it is pretty accurate.

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Option #1

A downward acceleration of $1 g$ could be a linear acceleration. This means that it is constantly increasing in kinetic energy in its own reference frame. This means mechanical energy needs to be supplied to it at a rate of

$$ \frac{\partial E}{\partial t} = \vec{F} \cdot \vec{u} = m \vec{a} \cdot \vec{u} \text{,}$$

in the non-relativistic limit, where $\vec{F}$ is the net force on Flat Earth, $\vec{u}$ is Flat Earth’s velocity, $m$ is its mass, and $\vec{a}$ is its upward acceleration.

This would require a truly unimaginable power source that would have to be infinite in extent or always traveling (and accelerating) with Flat Earth. It is possible that Flat Earth could be in the chamber of a gigantic rail gun. Problems exist: do people see the rails? 2) Do the incredibly strong electromagnetic fields affect the physics anywhere else on Flat Earth, or just a superconducting base? Such a setup would produce a ton of heat!

I deem that this approach is just physically untenable because of the enormous amount of energy it would require. (Let me know if you want me to push beyond this point!)

Note: In terms of special relativity, it is possible for Flat Earth to experience an acceleration of 1 g in its own reference frame indefinitely (it asymptotically approaches the speed of light in other reference frames).

Option #2

Accelerating in the direction of motion requires a lot of input energy, but accelerating perpendicular to the motion requires zero energy, as can be seen from the above equation (we’re here ignoring the electromagnetic radiation of an accelerating charged particle!).

Many spaceship designs include centrifuges for creating artificial gravity. Flat Earth could be constrained to move in a circular path, where “up” (from the perspective of the inhabitants of Flat Earth) is actually radially inward.

But what would cause the circular motion?

  • Gravity? Possibly, but that would require Flat Earth to have sufficient mass, but having sufficient mass would also mean Flat Earth’s self-gravity would be significant. Unless “Flat Earth” were sufficiently spherical, its own self-gravity would cause it to break apart.
  • Electromagnetic forces? My first thought was that the Flat Earth has a net charge and orbits a body with an equal and opposite charge. This would create an electric field. The charges and orbital radius and velocity could be determined such that the acceleration was indeed 1 g. The problem is that the inhabitants of Flat Earth would be in a very strong electric field! The other problem is how to make a body or Flat Earth have such a sufficiently strong net charge but be stable, since a high charge density makes material unstable, since like charges repel each other.

At any rate, although such a setup would require a ton of energy to set up, it would not require a constant influx of energy to maintain the acceleration (again, ignoring radiation due to accelerating charges).

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