A misconception conquered!

[Leigh Palmer <palmer@SFU.CA>]

> Since light can't slow down, as it climbs out of the gravity well, it loses
> energy by decreasing frequency. Another way of saying this is that space is
> stretched near the event horizon.

This statement highlights a misconception I held for many years. There
is a gravity well in the abstract space that has gravitational potential
as one of its coordinates, most conveniently the vertical coordinate. It
is, perhaps, possible to depict this well in two spatial dimensions. It
is also possible to imagine that the depiction is equivalent to a
contour map, and in doing so one makes a huge conceptual leap - and
crashes catastrophically in doing so. This picture can be an important
barrier to understanding later on.

To see that this is so, let us follow the analogy a bit farther. If one
imagines the terrain depicted in the contour map to be losslessly rough
(like an ideal billiard table) one will note that a billiard ball placed
on the surface accelerates downhill with a magnitude which increases
monotonically with the slope, or grade, of the terrain. This is analogous
to the gravitational force being proportional to the gradient of the
potential. (Grade and gradient, etymological bedfellows.) One might be
tempted to draw more from this analogy than it will bear, however. It is
quantitatively wrong because it is conceptually wrong at its root. The
topographic grade can tend to infinity, but the acceleration of a billiard
ball rolling on that terrain can never exceed g. pathological contour maps
with overhangs present other conceptual problems, but suffice it to
illustrate that this analogy is just plain RONG [sic]!

That, however, is not the misconception I held for many years. I figured
that one out as an undergrad almost as soon as my thoughtless teacher
foisted it on me. The misconception involves a thing called an "embedding
diagram", an example of which you may see on p.32, Box 1.6, in the phone
book - "Gravitation" by Misner, Thorne and Wheeler. It depicts curved
space as something that looks very much like the science museum exhibit
in which ball bearings or coins are rolled to simulate planetary orbits*.
In the case of the MTW diagram (which is copied in many places, and is
likely not original with them) light is shown to be deflected when
passing the Sun's limb because it skirts what looks like the Sun's
gravitational potential well. It is clear from the text that that is not
what MTW wanted you to think, and the magnitude of the effect is off by
a factor of two if one takes it that way, but nonetheless the door is
left wide open for the casual reader to fall into that conceptual error's
well.

The problem arises because what is depicted as a spatial dimension in
both the potential diagram and the embedding diagram is *not* a spatial
dimension, and making the embedding diagram take the form of a well is
just a gratuitous invitation to the misconception. How thrilled I was
when one of our graduate students gave a seminar (in the course for which
I had responsibility) in which he introduced the embedding diagram and
its correct interpretation. His transparency, however, was made with the
depression transformed into a peak! Why didn't the light rays get
deflected *away* from the Sun in that case, the naive listener might
easily think. But light travels in a locally straight line in the space
in which it is embedded, and the result is the same whether one depicts
the curvature of space upward or downward, or not at all! The emotional
tendency to think of the embedding diagram as a gravitational well is too
great to let it go unrecognized.

There is another important misconception conveyed in the diagram in Box
1.6 of MTW. It is, in fact, just plain RONG, even though I'm sure an
artist made the mistake, and MTW didn't think it sufficiently important
to change it.

Leigh

* This exhibit also gets a low grade in my book for lending aid and
comfort to the aforementioned misconception, and for failing miserably
in its intended purpose. It is fun, however, and I do spend time playing
with it whenever I see one. It can be saved by a sophisticated graphic
which explains how it is like a planetary orbit and how it is unlike a
planetary orbit. Questioning people about what they have learned from an
unqualified exhibit of this sort will reinforce your most pessimistic
guess.