How are "down" and "up" defined?

I disagree with the answer given in the mailbag: http://www.straightdope.com/mailbag/mdownup.html

The question was:

I agree with the first paragraph of the answer, when they define “down” as the direction that an object goes when dropped. But, the second paragraph says that down is therefore directly towards the center of mass of the earth. That is incorrect. The same effect that causes the Earth to be oblate also causes objects to fall away from the center of the earth.

In the third paragraph, they claim that defining down as as the direction perpendicular to the tangent to the earth won’t work. This is also incorrect. When treating the Earth as an oblate spheroid, down is the perpendicular (or normal) to the tangent. If it were otherwise, a flat puddle of water at your feet would flow.

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I don’t think you read the article very carefully, since Karen very specifically addresses WHY the normal (perpendicular to the tangent) is NOT the same as the “down” that you get from a plumb line in all situations.

RM, try reading this thread. It pretty much addresses the points you make (and agrees with you).
http://www.straightdope.com/ubb/Forum6/HTML/000125.html

And just to be thorough, try
http://www.straightdope.com/ubb/Forum6/HTML/000130.html

CK, RM did understand the article, and Karen’s answer didn’t fully explain his question. The linked thread above covers those details.

Essentially, what makes the earth oblate also pulls the plumb bob.

CKDextHavn

There is a technical term called “deviation of the vertical” that seems to be similar to what is described in the article, but the description in the article is wrong. Here are a couple links:
http://164.214.2.59/GandG/geolay/80003006.GIF http://164.214.2.59/GandG/geolay/TR80003A.HTM

Notice, in the illustration, that neither line goes through the center of the earth.
Irishman

Thanks for the links! When I tried to search for ‘plumb bob’ a few days ago, I had some problems with the search engine, apparently.

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Yeesh, you’d think a person with a PhD in physics would get a little more respect.

I did neglect the effects of the rotation of the earth in my discussion of perpendicular-to-the-tangent and direction-of-gravity. I also neglected the fact that the earth is NOT a perfect oblate spheroid. I also neglected mountains, basements, etc. My main point was that the direction that gravity acts locally is most relevant, and plumb bobs measure that, whereas it is impossible to measure the perpendicular to the tangent (there being no “tangent” to a bumpy, not-quite oblate spheroid). The direction a plumb bob takes is NOT the perpendicular-to-the tangent.

So, what this boils down to is: if you neglect rotation, and assume the earth is a perfectly smooth oblate spheroid, my original answer stands. If you wish to add the addition effects of rotation only, and assume that the earth is perfectly smooth oblate spheroid, made of an elastic liquid, at equilibrium with its rotation, then the plumb bob measures the perpendicular-to-the-tangent. However if you continue to add effects, such as the non-oblateness and the mountains and basements, then the plumb bob again does NOT measure the perpendicular-to-the-tangent.

As for the “flat” puddle of water at your feet, you need to be careful of circular definitions. I would define “flat” as the perpendicular to gravity, so a “flat” puddle, by definition, does not flow on any shape planet.


Karen Too

I’m struggling with your last answer. What do you mean by the effects of “non-oblateness” and “basements”?

Thanks

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You know, all this time, all this thought, and I can’t believe you don’t understand by now that, to define up, you take your right hand and, curling the fingers… :wink:

Karen Too

The amount that a plumb bob will miss the center of the Earth is the same value that you calculated in your sixth paragraph, except you said that it was the normal (perpendicular to the tangent) that would miss the center by that far. The tops of those two hypothetical poles would actually be separated by zero feet.

Because the Earth is a not a perfect sphere, the original questioner asked: is down a) the perpendicular to the tangent, or b) the most direct line to the core?

Your answer was b), as you explained in your second paragraph. The actual answer is a). One doesn’t have to calculate the normal, though, because the plumb bob does it for you.

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I guess I’m being dense, because I’m still not seeing the objections to Karen’s original comments.

The tangent to a 2-dimensional surface (for the sake of this discussion, the surface of the earth) is a thing that happens at a point. Thus, if you are standing on the side of a hill, the tangent plane is nowhere near “flat” to an observer who is not standing on that hill; and hence perpendicular to the tangent is not the direction that objects fall or roll.

Indeed, if you are on the side of a straight-edged incline, and drop an object, it will fall “down” in the direction of the tangent (not orthogonally to the tangent) – that is, it will roll along the hill – rather than fall towards the center of the earth.

OK, I’m speculating here because I’m not able to do the math while I’m at work, but… assuming that the earth is a smooth surface (ellipsoid, say), it it NOT THE CASE that the perpendicular to the tangent points to the gravitational center of the ellipsoid. Draw an ellipse for yourself, draw a tangent, see where the orthogonal goes.

Or am I missing something obvious here?

I didn’t need to be so waffly in my prior post. The perpendicular to the tangent in an ellipse does NOT got through the center of the ellipse.

A further example: we have assumed that mass is evenly spread. But suppose that mass is concentrated – suppose that the center of gravity is NOT the center of the earth, but a different inner point. Then the shape of the earth’s surface (that is, the slope of the tangent plane at a point and hence the orthogonal to the tangent plane) is irrelevant to the center of gravity. If “down” is towards the center of gravity, then Karen’s plumb line method does the trick… and the tangents are irrelevant.

Dex

That’s right, but the plumb bob follows that perpendicular. Karen Too admits as much in the second paragraph of her post there on 1/4/2000, and I’m pretty sure that her claim in the last sentence of that post is wrong–the the effect of basements, for instance, do deform the shape of the earth’s surface, but the net effect is that the plumb bob follows the normal.

The plumb line method certainly does do the trick, but it is the same thing as the perpendicular to the tangent, contrary to the mailbag answer.
There are many definitions of the “surface” of the earth. Is it the top of the atmosphere, or the bottom of the oceans? The usual definition is what is known as the geoid–or “sea-level”. The geoid is not a perfect oblate spheroid, as Karen Too points out. It is deformed by mass concentrations such as mid-ocean ridges–in fact, satellite images of the ocean surface can actually detect subsurface topography by noticing where the ocean water is “piled up”. But, a plumb bob follows the normal to the geoid even there.

If you use the rock surface as your definition, you will run into the problems you describe. That is also contrary to the intent of the original question, and also to the way it was treated in the original mailbag answer.

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Non-oblateness: the deviation of the earth’s shape from a true oblate spheroid; e.g., the north pole is farther from the center of the earth than is the south pole. There are also additional bulges and flattened spots on the earth which cause some people to say the earth is pear-shaped [1]. I believe these non-oblateness features are on the order of 20 meters, tiny
compared to the 21 km oblateness, which is tiny compared to the 6000 km radius of the earth.

mountains and basements: a mountain is a natural formation of extra mass at larger radius. Basements are artificial formations of removed mass at smaller radius. My “mountains and basements” phrase is short for “mountains and valleys and buildings and basements” by which I mean the natural and artificial bumpiness of the earth’s surface. I was thinking of an anecdote from a “Fifth Force” seminar I heard years ago [2].

Dex: I think the objections to my original answer are better stated in the other thread: the same “forces” that fling the earth into oblateness also fling the plumb bob, such that the plumb bob is flung into line with the normal. This may be true for an elastic liquid perfectly oblate spheroid earth, but I am not going to answer that question (we particle physicists hate non-inertial reference frames.) I prefer to evaluate the assumptions by rephrasing the question: are the “forces” that fling the plumb bob the same ones that cause the shape of the earth? The answer is an unequivocal NO. If the rotational and self-gravitational forces are the only effects determining the shape of the earth, then the earth WOULD be a perfect oblate spheroid. We KNOW the earth is not a perfect oblate spheroid; people have measured the difference between the north pole radius and the south pole radius; people have measured the bulges and flattened spots; I myself have seen a mountain or two. Thus, we are forced to conclude that there are additional effects at
work, presumably the rigidity of the earth’s crust, plate tectonics, tidal effects from the sun and the moon, and maybe the Chandler wobble for all I know. (There is also considerable earth-shaping caused by those bulldozers that wake me every morning with their bleepity bleep beeping.) The shape of
the earth is in fact a hot topic of current research; I’m sure the GPS folks tear their hair out in frustration everyday trying to model it.

Therefore the earth’s shape is much more complex than a simple oblate spheroid, and as such we cannot expect that a plumb bob would ever line up with the normal. I will admit that the plumb bob is probably flung a bit by the earth’s rotation, as well as the local mass distribution. Nonetheless,
the plumb bob measures what you typically want to know – the direction
stuff will fall, aka “down”.
If you insist that the normal of an oblate spheroid goes through its center, then I can’t help you.
Footnotes:

[1] I believe it is more correct to say that the average radius at a specific latitude (i.e., integrated over longitude) resembles the cross section of half a pear.

[2] As you know, gravitational and electromagnetic forces are very long range, while the weak and strong forces are very short range. This irks some people and about 15 years ago, scientists were all hot to find a medium range “fifth” force. To measure this hypothetical force, you have to
account for very subtle effects of gravity, and the scientists therefore had to know the distribution of mass in their location, so they went down the street knocking on the doors of serious corporations, in truly Cecilian fashion, asking the dimensions of their basements. As you can imagine, the
corporations were suspicious of such inquiries, and the scientists had to do a lot of explaining before the corporations would give them their basement blueprints.

I never said that the plumb bob follows the normal to the surface of the earth. I strongly disagree. I have a couple of remaining points:

  1. The actual shape of the earth is not the same as the geoid. Nor is the actual shape of the earth the same as the idealized oblate sphereoid.

  2. Unless the geoid takes into account basements, then a (very sensitive) plumb bob is not going to be perpendicular to the geoid either. There are plenty of instances where the gravity from local mass distributions affect scientific measurements. For instance, the Fifth Force experiment I mentioned in my earlier post. Also, the particle accelerator at CERN is quite sensitive to the rainfall, and thus the level of water in Lake Geneva, and the resulting changes in the gravitational pull on the particles in the accelerator.

  3. If the geoid takes into account the basements and water level in Lake Geneva, that is, the geoid is defined to be the perpendicular to the plumb bob, then we have another circular definition.


Karen Too

I didn’t.

We both agree that the plumb bob should define down. The original question asked whether down was the direct line to the center of the earth, or it was the normal. Your answer was the line to the center, whereas the normal is what the plumb bob follows.

In other words, I’m saying that neither the normal, nor the plumb bob, point to the center. They both point in the same direction, away from the center.

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Karen Too:

I didn’t say you did. I was replying to Dex’s remarks about a smooth ellipsoid, and was referring to your post where you said (1/4/2000 10:37 PM):

Even if you use a smooth ellipsoid to approximate the surface of the earth, rather than the actual geoid, the difference between that ellipsoid’s normal and the plumb bob is an order of magnitude smaller than the difference between them and the line to the center of the earth. So, the answer to the original question should not be the line to the center of the earth, it should be the perpendicular to the tangent.

The geoid[1] is what is measured by earth satellites (and earth-based gravity measurements), and is not exactly an oblate spheroid. One of the first such deviations to be measured, by satellites in the early sixties, was the so-called “pear-shape.” Also, the equator is “pinched”: one equatorial radius is longer than another at right angles to it. These deviations are a result of deep mass heterogeneities, probably related to mantle convection and plate tectonics. The deep mass actually changes the shape of the earth: ocean water “piles” up higher in some places (180 meters higher) than in others.

The pinching of the equator is a degree two effect. Imagine a sine wave wrapped around the equator, following the geoid surface. It would have two highs, and two lows–so, two periods, or degree two. The pear-shape is degree three: follow the outline of the pear from pole to pole and back. There are three highs, and three lows–so, degree three. Over the decades, models of the geoid have been published out to degree 360, which requires over 13,000 coefficients[2], but they are not exactly the geoid. The geoid is perpendicular to the plumb bob–that is not just a circular definition, because the geoid conforms to sea-level also. It is an equi-potential surface. Geodesists don’t tear their hair out in frustration anymore, but they might if they had any left.

An interesting consequence of all of this is that the center of mass of a body is identical to its center of figure of its geoid. So, Dex’s example (01-05-2000 03:25 PM) of shifting centers of gravity is…impossible. Consider a circle shifted from the center of reference. As you moved around the circle, you’d have one high then one low, and back to the start. So, it is called “degree one.” Degree one shifts are called “forbidden” by geodesists, in imitation of the forbidden transitions of physical chemistry. This is more than just flattery–the functions that are used to model the geoid are the same spherical harmonics that are used in physical chemistry.

[1] http://164.214.2.59/GandG/geolay/TR80003A.HTM#ZZ5
[2] Each degree n has 2n+1 spherical harmonics, so in a degree 360 approximation, you’d have to use 361^2 different spherical harmonics.

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RM says: << Your answer was the line to the center, whereas the normal is what the plumb bob follows >>

Here’s the confusion in the nutshell. If by “normal” you mean the orthogonal to the tangent plane at a point, then I strenuously disagree. The plumb bob does NOT follow the orthogonal to the tangent. The counterexamle of your assertion that it does is the side of the hill situation.

Now, if you means something different by “normal”, that’s a different story altogether.

Dex

I don’t think that is the confusion at all. Do you think that Andrew Mattison, the person who posed the original question to the mailbag, was talking about perpendiculars to the tangents of the side of the hill?

I don’t think he was. And I’m not.

I think he was asking about perpendiculars to the tangent of the spheroid. And that’s where the calculations about the poles as tall as the Empire State building come from for the example in the mailbag answer, so apparently the mailbag thought so also.

If you take any plumb bob anywhere in the United States, it is closer to that perpendicular to the tangent than it is to the line to the center of the earth by [em]at least[/em] a factor of ten. If you adjust the geoid for mountains and basements, they line up even better.

Let me be even more simplistic. If you want to determine which direction is “down”, the simplest thing to do is to drop something. Whichever way it falls, that’s “down”.

Ergo, the plumb line.

And, RM, I really don’t understand what it is you’re trying to say. On the one hand, you say that the center of gravity (center of mass) must be the same as the center of the geoid. (That’s physics, not math, so I don’t pretend to understand it, I’ll accept it.) On the other hand, you say that the plumb line (which presumably points to the center of the mass) is not as accurate as the orthogonal to the tangent (which does NOT point to the center of the geoid).

Well, the fact is, that these are all wrong. “Down” is actually towards the sun, that’s where all the planets and stuff would fall if they didn’t have orbital velocity.

I’m goin’ for another beer, and I’ll drop a plumb line in it, and see how long I can stay orthogonal.

Orthogonal to what? If Karen Too is there, I’m thinking I want to get parallel. Or is that coincident? :wink:

Dex

Thanks for sticking with this. (Not you Irishman! I’ll see you at the pub.) I think I see the light at the end of the tunnel.

Your presumption is wrong, the plumb line does [em]not[/em] point to the center of mass. It points along the “orthogonal to the tangent.”

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