Q: If Earth was flat, would there be a horizon? If so, what would it look like? If the Earth was flat and had infinite area, would that change the answer?

Physicist: There’d definitely still be a horizon if the Earth were flat.  It would be in almost exactly the same place, and look essentially identical.  While the Earth isn’t flat, adherents to that theory are correct in that it’s nearly so, and if you’re standing on the surface of a round something the size of the Earth it’s difficult to tell the difference (in fact, mathematically you can make the argument that a flat Earth acts the same as an infinitely big Earth).

Who would even notice?

For someone around 5’6″ tall, if the Earth were perfectly flat the horizon would be about 0.04° higher.  That’s about the width of a (mechanical) pencil lead held at arm’s length.  Unless you have short arms, in which case you’ll need to shave down the lead a little.

Even if the Earth were perfectly flat and went on forever, the horizon would still be exactly level: 180° of sky and 180° of ground (instead of the paltry 179.96° of ground we have).  The only difference between a finite flat Earth and an infinite flat Earth is that no matter how tall you are on an infinite flat Earth, the horizon always stays in the same place.

However, even though the horizon of an infinite and flat Earth might actually be in the same place, it wouldn’t appear to be.  An infinite plane has an extremely simple gravitational field; uniform and exactly the same regardless of distance.  Normally the higher you are the weaker the gravity, but for a flat Earth that isn’t the case.

As such, light, which drops only imperceptibly under Earth’s gravity, has an infinitely great distance over which to do its dropping.  The effect would require tremendous distances (as in; interstellar distances), but if you’ve got an infinite plane, that kind of distance is cheap.

Given enough time and distance light will eventually curve back toward an infinite plane of matter. So if you’re standing somewhere on the surface and you look up you’ll see light that started somewhere else on the surface.  Instead of a horizon, the world would look like it rises up on all sides and encloses you.

When you look up from what should be the horizon you’ll just see more of the infinite-flat-Earth.  The one exception is what you’d see if you looked straight up.  Directly above you you’d find the entire horizon bunched up at that point.


Answer gravy: It’s not interesting enough to include in the post directly, but here’s how the math above was done:

If you’re standing on a sphere with radius R and you’re H tall, then the distance from your head to the middle of the sphere is R+H.  Your line of sight to the horizon is a tangent line to the sphere.  This allows you to draw a right triangle and do a little trigonometry.

θ is the angle between the “true horizon” and the perfectly level “ideal horizon”.  Coincidentally, it’s also the angle between where you’re standing on a sphere and the farthest thing you can see on the surface.

So, \cos{(\theta)} = \frac{R}{R+H} and \theta = \arccos{\left(\frac{R}{R+H}\right)}.  Plug in R = 6,378,100 meters and H = 1.68 meters (which is 5’6″), and you find that θ = 0.04°.

By the end of college, electrical engineers and physicists get sick to death of the example of the infinite plane of anything (be it matter, electrical charge, kittens, whatnot).  The reason an infinite plane is useful is that it has some nice symmetry.  You can argue that, since no direction is special (by being close to an edge for example) gravity always points straight into or out of the plane-o-stuff.  Symmetry is useful when you use a Gaussian surface, because you can ignore buckets of math.

Basically, draw a “bubble” around a lump of matter.  The total gravitational field pointing through that bubble will be proportional to the amount of matter inside.  So, the more matter, the more gravity and the bigger the bubble, the less the strength of the field through any particular part of the surface (by the by, there’s an example of this in action here).  If the Gaussian surface you choose is a rectangular box that punches through the flat-Earth, then you find that it doesn’t matter how tall the box is.

The total amount of gravity pointing into any imaginary box you can draw is always proportional to the amount of matter in that box. In this case, no gravity points through the sides (by symmetry) so it all points through the top. But if you make the box taller that doesn’t change. As a result, for an infinite plane gravity stays the same forever. QED y’all!

Uniform and infinite sheets of matter or charge have gravitational or electric fields that extend, without changing, forever.  Of course, there are barely any infinite planes of stuff out there, so this isn’t a situation ever actually comes up.  However!  If you’re close enough to a surface it can seem to be nearly infinite, so the infinite-plane solutions are often good enough.

And often as not, “good enough” is good enough for physics.

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23 Responses to Q: If Earth was flat, would there be a horizon? If so, what would it look like? If the Earth was flat and had infinite area, would that change the answer?

  1. In a related question, at what altitude does it become visually apparent that the Earth is a sphere?

  2. The Physicist The Physicist says:

    If the Earth is perfectly round (trying this in the middle of the ocean is a good way to approximate that), then you should see a noticeable deviation (a few degrees) when you’re a few kilometers up.

  3. Locutus says:

    Funny how you don’t need a black hole to keep light from escaping…you just need something infinitely big. Of course that would have infinite gravity…which would be a black hole. I think.

    Love Long and Prosper

  4. Will says:

    Black holes don’t have infinite gravity; if they did they’d crush the entire universe. A Black hole actually has slightly less gravity than the star it was created from (due to losing some matter).

    The key is that Black holes are infinitely -small-.

  5. Goodwill says:

    There should be a limit to what we can see(in the infinite flat earth case)? B/c speed of light is finite and it would take an infinite amount of time for light from the edge(of the infinite flat earth) to reach the centre of this flat earth?

  6. The Physicist The Physicist says:

    That’s exactly what you’d have to take into account when you think about what you’d see almost directly above you.

  7. Robert Constant says:

    I don’t believe it is correct to say you need to raise your eye height a few kilometers to see that the Earth is a sphere. Standing on the beach on a clear day, or on the deck of a ship at see, it is plainly apparent that the horizon curves off in all directions.

    I think it is a common misconception that you have to go high to see the curvature of the Earth; I have even heard the Space-X guy, as well as Richard Branson, talk about their “space tourism” vehicles going so high that you can “see the curvature of the Earth…” When I first heard these statements I recall saying (to no one in particular), “Hell, you can see the curvature of the Earth on any clear day standing on the beach!”

    Depending on the size of the hypothetical flat plane on which one was standing, it would appear that you were standing inside a shallow bowl — with the bowl appearing deeper and steeper the larger the plane and the lower the eye height. You can see a similar optical illusion looking down a long runway at the airport.

  8. Daraje Garba says:

    I think there is no change day to night also seasons.

  9. The Cool Dude says:

    I calculated the downward gravity you would experience on this hypothetical infinite plane, for anyone who was curious.
    Since I don’t know the height or density of the infinite plane, only the length and width (infinite), I used infinite sums to add up the downward gravity caused by infinitely smaller segments of a cylinder of a height H, a radius R, and a density D, even taking into consideration the decrease of gravity over distance, as well as the angle of gravity for each point, which would alter the total downward gravity, and after much calculation, my conclusion is that the gravitational acceleration (In m/s^2) is:
    GπD(atan(H/R)R²-atan(R/H)H²+RH)
    Where G is the gravitational constant, 6.67384*(10^-11).
    The limit as R goes to infinity, unless either H or D are zero, is infinity, which means if the plane is infinite in size, which it is, and has any height and density, then it has an infinite downward gravitational force. Further, it has zero horizontal gravitational force, since there is “equal” stuff on all sides.

    Elaborating on the Physicist’s answer to Goodwill’s question, gravity is a function of distance, so the farther out a thing goes from a source of gravity, the less gravity affects it, and the velocity required for an object to completely escape a source of gravity is called escape velocity. Escape velocity is the one reason you would still be able to see the entirety of the infinite plane while looking higher and higher up, because the light doesn’t HAVE to come back down, since the speed of light is definitely greater than the escape velocity of any hypothetical object with the same gravity as Earth.

    Though I suppose if you really want to consider all the mechanics of light, you would have to assume that it would be mostly black, unless the plane its self emits light, since the plane will definitely be almost all you see, except for a near single point directly above the observer, of course, but the real question is how much ambient light that will actually be.

  10. Robert says:

    I am not sure the gravitational bending of light, as Cool Dude calculates, is a substantial phenomenon in such a small scale system.

    Surely, gravity effects light, and we often measure the effect to make measurements in deep space.

    However, the biggest factor bending sunlight as observed from Earth is the density of the atmosphere and not gravity.

    Returning to the main subject of this conversation, IE The optical properties of the horizon if the Earth were flat, I was just re-considering this only yesterday as I sat with my kid on the bluffs overlooking the Pacific ocean at Dockweiler Beach in Los Angeles.

    Looking out at the horizon, it curved off in all directions, a result of the dipping of the earth at a (virtually) uniform rate in all directions. If instead Earth were flat, I don’t see how it is possible for the horizon to appear “essentially identical” as the physicist claims in the original answer to the hypothetical…

  11. The Cool Dude says:

    I am extremely sorry,
    I redact my previous math which calculates for the gravity caused by an infinite plane, due to a slight mathematical error.
    I rechecked my calculus, and through the accidental loss of an exponent, it turns out, as the plane gets increasingly larger (to infinity), the acceleration (m/s^2) is actually equal to 2πDHG, where D is density (kg/m^3), H is the height of the plane (m), and G is the gravitational constant, 6.67384*10^-11 (m^3/(kg*s^2)), meaning it is proportional to the height times the density, assuming the density is uniform.

    This means that the infinite plane does NOT have infinite gravity, despite being infinitely large, unless it is already infinitely dense.

  12. The Cool Dude says:

    Also, Robert,
    The plane in question is infinite, and with a finite gravity, light bouncing from any place on the plane would have the ability to reach you, so long as it was angled correctly. While ignoring air shenanigans, the entire plane would appear to bend upward into the sky, where the infinitely far away edge would be nearly directly above you.
    I don’t really know how air works with respect to light, and over long distances it would definitely increase the effect, but it’s probably more likely that everything would just look really blurry.

  13. The Cool Dude says:

    Additional information:
    Looking at my equation, the properties of the infinite plane allow that your distance from the plane would not actually matter.
    Since if you are Y distance off the surface of the plane height H, then the actual downward acceleration you would experience would be equal to the acceleration you would experience if the plane were height Y+H, minus the acceleration you would experience if the plane were height Y, since all of Y is empty space. Because the function is multiplicative in nature, you could factor out all the multipliers to simplify to get it to the exact same gravitational acceleration as if you were right next to it, which means all things in the universe would be attracted to it at the exact same rate, and more important to the problem we are considering, all light would experience the same acceleration regardless of distance from the plane, and would inherently NOT have an escape velocity, as previously assumed.

    As for how that makes any sense, I do not know, though I have reviewed the Calculus numerous times, and do not see any further errors. I will notify if I do, or at request, can post the math here for the peer review of anyone who understands the properties of this particular situation better than I do.

  14. The Cool Dude says:

    I checked the equation for gravitation caused by standing on top of a cylinder by summing the equation to create the gravity caused by a sphere, (Like a planet), and it resulted in M*G/R^2, which means the equation holds true, and the properties I mentioned in my previous comment should be correct.
    This would imply, however that you would not be able to see the entire plane, as previously suggested, because there would be no escape velocity, and instead you would see that at a certain point, the plane would begin mirroring its self, and so directly upward would in fact be the exact point the observer is standing at, but some time in the past, due to the time it would take light to decelerate back down.

  15. The Cool Dude says:

    Anyway for anyone wondering, here’s a link to the proof Calculus on that, if I am allowed to post links on this site, that is.
    http://tinyurl.com/p6hmy5k

    Though I realize that the math is somewhat irrelevant in proving that the plane has a finite gravity, due to The Physicists Imaginary Box explanation, but I found the problem interesting, and it allows one to find the exact gravitational acceleration based on the density/height of the plane, which leaves room for further consideration, as well as the method of calculating the gravity having many other, perhaps more realistic applications.

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  17. Setuben says:

    So ah – for there to be a finite gravity on a infinite flat plane…My guess there would be an identical “mirrored” plane on the opposite side correct?

  18. The Physicist The Physicist says:

    Nope. The math just works out this way or an infinite plane.

  19. The Cool Dude says:

    I think Setuben is referring to what’s underneath the infinite plane. If the plane has a finite depth, then there’s a mirrored plane on the other side, with an inverted gravity. If it has an infinite depth, it’s the same as being on an infinitely large planet, so the gravity is also infinite. Unless the material has an infinitesimal density, the plane should have a finite depth, but having an infinitesimal density causes a lot of problems with firmness, so you would just fall through it forever, and it would be more like an infinite vacuum plus gravity.

    On the other hand, the density could be shaped in such a way that it gets lesser the deeper in the plane it is, but after a while it would act like a fluid or a gas, so there would still sort of be another surface on the other side, it would just be muddy, or fuzzy.

    In the end though, if you wanted it to be infinitely thin, then it would have to have an infinite density, and in that case although there would be another side to it, the plane would be infinitely hard, so you could never penetrate it, so the other side would be irrelevant.

  20. Mark says:

    I liked the explanation of how a plane would actually appear to be a bowl, due to gravitational bending of light rays. It should have made sense before, but now I understand why the natives of planet Mesklin thought their poles were located at the bottom of a bowl.

  21. Brian says:

    Isn’t that kind of like Gabrael’s horn which has an infinite surface area and a finite volume ?

  22. Happy Forever says:

    Yes for sure. It looks exactly like what you see. Actually we live in a the flat earth and the round earth theory is fake.

  23. Anders says:

    If the Earth were an infinite plane then it’s radios would be infinite and so it would take an infinite velocity to stay in orbit and to escape the planet so it would be a black hole.

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