What is down, exactly? What’s up?

SDStaff Karen replies:

There’s an easy answer to this question, Andrew. However, your talk of tangents and perpendicular lines suggests you hanker for higher knowledge on the up vs. down front. If so, my friend, get ready. You’re about to learn about the geoid.

First the easy answer. Down is the direction an object goes when dropped. To put it another way, down is the direction a carpenter’s plumb bob points, and up is the opposite direction.

But that raises another question: Where exactly is the plumb bob pointing? From gravitational physics, we know that two objects gravitationally attract each other according to their centers of mass. So it seems reasonable to suppose that down is directly towards the center of mass of the earth, which for our purposes is its geometrical center.

But that’s wrong. For a variety of reasons, one of which is the earth’s rotation, objects don’t fall directly toward the center of our planet, but rather at an angle to it.

So let’s try again. Is down perpendicular to the tangent to the earth? Sorta, but what does that mean — tangent to the spot where you happen to be? Obviously not; you could be standing on a hillside. Do we instead mean tangent to the idealized earth, which is a type of ellipsoid known as an oblate spheroid, or slightly flattened ball?

No, we don’t mean that either. Here’s your answer: Down is perpendicular to the geoid.

The geoid is an imaginary surface defined as mean sea level — that is, the hypothetical surface that would result if the earth were covered with water. By definition, this surface is perpendicular to a plumb bob everywhere (water seeks its own level).

The geoid takes into account the gravitational pull of the Earth’s mass, the centrifugal effects of the Earth’s rotation, the flattening of the poles due to the Earth’s rotation, the semi-random bumps and bulges in the Earth’s shape, inhomogeneities in the Earth’s density, and mountains and valleys and tides and buildings and whatnot. The geoid is only approximately the same shape as an oblate spheroid, and has almost no correlation with the local terrain.

You can see an illustration of the geoid here. Note that the geoid (the solid line) is somewhat irregular compared to an ellipsoid. You may ask: if the geoid is mean sea level, how can there be high and low spots in it? Won’t the water flow downhill? Answer: no. The geoid is an “equipotential” surface, permanently deformed by the many forces acting on and within the Earth. It may look bumpy, but you experience it — water experiences it — as perfectly level. Think about it. Elegant concept, really.

So let’s review:

(1) Down is the direction a plumb bob points — but a plumb bob doesn’t necessarily point to the center of the earth.

(2) Down isn’t perpendicular to an ellipsoid, because that’s not what shape the earth is (it’s close though).

(3) Down is perpendicular to the geoid, or mean sea level.

It may take a while to grasp all this, Andrew. But you’ll be a better person when you do.

Bonus info: When people hear the earth is an ellipsoid or oblate spheroid, they tend to think “hamburger bun” when they should be thinking “billiard ball.” The earth shrunk to the size of a billiard ball would be out-of-round by ±0.004″, whereas the tolerance for billiard balls according to the Billiard Congress of America is ±0.005″ — see here. I think you’ll agree the press has really blown the flattening of the earth’s poles all out of proportion.

SDStaff Karen, Straight Dope Science Advisory Board

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