If the world isn't flat and a person is walking around it from east to west. Is he walking downhill or uphill ?
(This a stupid question on many levels, but I'm looking for intelligent answers.)
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If the earth is a perfect sphere, any person walking on it will always be walking downhill (but with caveats).
There are two considerations here; centre of gravity, and gradient. The centre of gravity determines a person's orientation in terms of up or down, and the gradient determines where the next footfall will be in terms of the vertical reference.
Centre of gravity for a perfect sphere of equally distributed mass is the centre of the sphere, meaning that every person's orientation is normal (read as perpendicular) to the plane which is tangental to the point on the sphere's surface on which they stand.
In plain English, what that means is that on your sphere, a person always sees the ground beneath their feet as being straight down from their orientation. BUT, the surface of the sphere is not flat, and actually curves away from them. that means that the gradient is always negative from their current perspective, meaning that their next step is always downhill relative to their current position.
This last point is the important one - At their new position, the centre of gravity reorients their relative concept of down, and every location (even the one they've just arrived from) is now downhill from their current location. In reality, they haven't traveled up or down hill at all relative to the gravity well in which they live, but they have relative to their previous position in that gravity well.
This is exactly what happens in orbital mechanics, by the way. The moon is constantly falling towards the Earth, but it's lateral motion is so fast that it keeps missing it, and the moon keeps turning around the Earth with almost perfect balance. In point of fact, the moon is travelling just a little faster laterally than it is vertically, and will one day escape the Earth's gravity well. That day is a long way off and will likely happen after the Earth is burned by our sun as it expands into a red giant. That said, it's important to note (excluding the elliptical nature of most orbits which is outside the scope of this answer) that because everything falls into a gravity well, on the surface of a sphere one is really just orbiting the centre of gravity.
Indicentally, on a truly flat earth (circle) where the centre of gravity is below the centre of that circle, every step away from the centre would be uphill and every step towards it would be downhill for precisely the same reason.
If you define up and down as a gain or loss in gravitational potential energy then during a trip around a perfect (in density and shape) sphere you would go neither up or down. That's because the surface would exist as an equipotential. This definition of up or down runs into problems when your surface has variable density, (i.e. you could step onto a lead surface from a wooden surface and you could have said to gone up) or influences by outside forces, (i.e. the moon being overhead). The benefit of this method is it would be easier to measure, one could feasible measure the gravitational force on two objects (assuming they're not in orbit) and know which one is more "up".
Another definition would be to use your distance from the center of gravity of Earth, This is hard to measure (as we cant really put a ruler all the way through the Earth) but is not affected by changes in density or the moon.
The answer depends on several factors. Let's assume that the surface of the Earth doesn't have any hills where you are. In other words you are walking across a flat and level plain. For a short walk we can ignore the curvature of the Earth - it is effectively zero.
(1) The positions of the Moon and the Sun. They exert a gravitational pull depending where they are in the sky relative to you and each other. If you are walking the same way that a nearby tide is moving then you are walking downhill just as the water is running downhill. Note: When the tide appears to be climbing up the beach it obviously can't do so because water can't run uphill. This seems contradictory until you think about it enough.
(2) If you neglect the tides, then you are walking uphill. This is because of the rotation of the Earth. If you walk with the rotation, you are slightly being thrown off the surface. You say East to West which is counter to the Earth's rotation. You will feel the effects of gravity slightly more so you are effectively walking uphill if you turn that way. You will experience level walking by travelling at right-angles to the direction of rotation.
(3) If you are near a mascon then you are walking downhill as you approach it and uphill as you leave it. https://en.wikipedia.org/wiki/Mass_concentration_(astronomy)
You have to combine all of these forces together with any local slope to determine whether you are walking up- or down-hill
"up" and "down" (hills, stairs, ladders, air) refer to doing more work against gravity, or less work against gravity, respectively.
You are mistakenly thinking of the gravity being produced from outside the earth (like an ant walking on a basketball you hold aloft), but it is (mathematically equivalent to being) produced from the center of the sphere, and pulling (accelerating) objects toward the center.
So no matter where you stand on the sphere, the gravitation pull on you is exactly the same. Walking on the sphere -- North, South, East, West, any direction -- does not require you to work against gravity, and is not assisted by gravity. The only work you do is remaining upright, the same as you would think about walking on a flat surface.
You aren't moving "up" or "down" because the strength of gravity pulling on you isn't changing.
I would argue that, on a perfectly flat, perfectly round, perfectly uniform sphere (which the earth emphatically is not), you cannot walk either uphill or downhill - all of your walking will be perfectly level.
Obviously this depends on the definition of "up" and "down". For the purposes of this answer, I'm going to define these in terms of potential energy. One position is "uphill" of another if an object at that position has more potential energy from gravity than the same object at the other position. Therefore, by definition, moving uphill requires an input of energy, while moving downhill allows a release of energy. In layman's terms, it's harder to move uphill than it is to move level, and easier to move downhill.
This definition is useful in a couple of ways. First, as mentioned above, it fits our intuition that moving downhill should be easier, and that unsupported objects should fall, slide, roll, etc. downhill. It also makes sense in terms of pressure - assuming a perfectly uniform atmosphere at a constant temperature, higher pressure will always be found "downhill" of lower pressure. It's also universal (ignoring, once again, the confounding effects of gravity from things that aren't earth) in that it gives the same result for the same two points regardless of whether we're looking from one, from the other, or from some other third point.
Using this uniform definition of up and down, we find that every point on a perfect sphere has an equal amount of potential energy, and thus are level. We think of level surfaces as planes, but in this metric, a level surface is actually a spherical shell. A given object has the same potential energy at every point on the shell, and so is going neither uphill or downhill by traversing the shell.
Incidentally, this is why the earth is round in the first place: because gravity pulling all the material in the earth "downhill" pulls it into a sphere. Obviously the real earth is more complicated than our idealized model: it's made up of different materials at different temperatures, it's influenced by the gravity of the sun and moon, and it rotates, all of which have their own effects on the surface.