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<!DOCTYPE html>
<html lang="en">
<head>
<link rel="apple-touch-icon" href="apple-touch-57.png"/>
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<title>Einstein's Cat</title>
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<hr>
<h2>Section Two<br>
<span class="section-subhead">The End of the World As We Knew It</span></h2>
<p>Science has overturned our concept of the world many times. We used to
think the world was flat and that the sun revolved around it (some people
still do think this). The next several chapters will talk about the
surprising experiments and revolutionary theories that have most
fundamentally changed our view of the physical world over the past 120
years or so, overthrowing the dynasty established by Euclid and ruled over
most gloriously by Isaac Newton. Modern physics, which emerged in the
beginning of the twentieth century, held two great surprises for mankind:
quantum theory and relativity, and these will be the focus of Section Two.</p>
<!--
**********************************************************************************************************
**********************************************************************************************************
******************************* CHAPTER 17 ******* **************************************************
**********************************************************************************************************
**********************************************************************************************************
-->
<hr>
<div id="Chapter17">
<h3 id="ChapterSeventeen">Chapter Seventeen<br>
Mapping The Earth (II)</h3>
<p>In the previous section, we looked at how the curvature of the earth
means that systems of measurement that work over small distances begin to
fall apart or can't be applied over large distances. We talked a little
bit about what curvature means, and now it's time to revisit that. The
point at which humanity looked beyond Euclidean (flat) geometry to
describe the surface of the Earth might be considered the first phase in
the history of what I am calling in this section <q>The End of the World As We Knew It;</q> and non-Euclidean geometry is the foundation of something
important later in that history, namely general relativity.</p>
<p>We will take a brief look at curvature in one dimension, which in a
two-dimensional context might be considered clockwise or counterclockwise,
up or down. Some types of curvature are more easily understood by looking
at them in the context of one more dimension. In this chapter we will
mostly look at two-dimensional curvature, which also comes in two types,
positive or negative, but has features that you simply can't have in one
dimension; a two-dimensional surface can curve into what we see in three
dimensions as sphere, egg, donut or saddle shapes. We will look at the
effects of this kind of curvature and how it and its effects can be
measured. This is all interesting enough in itself, but it will also
prepare us for a discussion of three-dimensional curvature (Chapter
Thirty), which is as conceptually different from two-dimensional curvature
as two-dimensional curvature is from one-dimensional curvature.
Three-dimensional curvature is both positive and negative at the same
time, but in different aspects. This sounds really abstract, but we will
get around to visualizing all of it, despite the people who say it can't
be done because we can't conceive a four-dimensional context to place it
in.</p>
<div class="figdiv fright nofloat_600 float_300_66 float_800_50 g2-1500" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Osculating_circle.svg.png" style="max-height: 180px;">
<figcaption>
<b class="figlabel">Figure 17-1</b>.
<span class="ficaption_description">
The degree of curvature of a line at any point is related to the radius of the circle with which it is congruent. Source: <a href="https://commons.wikimedia.org/wiki/File:Osculating_circle.svg">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/2022_11_24_11_40_50.png" style="max-height: 180px;">
<figcaption>
<b class="figlabel">Figure 17-2</b>.
<span class="ficaption_description">
The graph of the function 1+ cos (<i>x</i>+π).
</span>
</figcaption>
</figure>
</div>
<p>When we talk about one-dimensional things, we usually talk about <q>lines,</q>
but not always in the same sense. There is the sense of a <q>line</q> that
means something that always goes in the same direction and follows the
shortest distance between any two points on that line, and then there is
the sense of <q>line</q> as we use it in <q>line drawing:</q> something very thin,
perhaps infinitesimally thin, that may change direction or <em>curve</em>.
So in this chapter I will refer to both kinds of lines, but write
<q>straight lines</q> when I mean the former type.
</p>
<p>At any point on a line, that line may curve either toward or away from
another point not on the line (Figure 17-1). The shorter the distance is
to that point from the line, the more tightly the line must curve to
maintain a constant distance from it. In this sense, a circle is curved
around its center.</p>
<p>On a Cartesian graph, we might say that a function has positive curvature
in ranges where its slope increases from left to right (where it points
more upward), and negative curvature where its slope decreases in that
direction (where it points more downward). In Figure 17-2, we see the
graph of a function with positive curvature on both sides and negative
curvature in the middle. From points A to C, the slope is upward;
from A to B it gets steeper, but from B to C, that slope decreases until
it becomes zero and finally goes back downward. B is an inflection point
where the curvature goes from positive to negative. In calculus terms, we
would say that the <em>second derivative</em> of this function — what
might be called slope of the function's slope — changes from positive to
negative at point B. In the terms we used to describe Figure 17-1, the
curving line in Figure 17-2 that represents the function is curving around
a point <em>above</em> it from A to B (and from D to E), and curving
around a point <em>below</em> it from B to D.</p>
<p>Two-dimensional surfaces can have a curvature similar to this, if they
curve radially around a line that serves as an axis. If that radius
changes only in a linear fashion (or not at all), these surfaces can
easily be made flat. A cylinder and a cone are the two best examples of
this: a cylinder has a constant radius around an axis, and though that
radius changes on a cone, the sides of the cone are straight lines when
viewed in profile. One simple cut will allow either of these to shapes to
be unrolled into a flat surface (Figures 17-3 and 17-4). This kind of
curvature of two-dimensional surfaces is called <q>extrinsic.</q>
Two-dimensional surfaces may also have an <q>intrinsic</q> curvature; this is
what is more interesting to us, and what takes us beyond the limits of
Euclidean geometry.
</p>
<div class="figdiv fleft nofloat_600 float_300_66 float_800_50 g2-900" style="clear: both; background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Circular_Cylinder_Quadric.png" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-3</b>.
<span class="ficaption_description">
A cylinder can be cut along any of the vertical lines to allow it to lie flat. Source: <a href="https://commons.wikimedia.org/wiki/File:Circular_Cylinder_Quadric.png">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/cone3.png" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-4</b>.
<span class="ficaption_description">
A cone can be cut along any line from the center to the edge to allow it to lie flat. Conversely, a circle can be cut into a cone shape by making two radial cuts from the center; the angle between the two cuts determines the shape of the cone.
</span>
</figcaption>
</figure>
</div>
<p style="clear: both;">
A two-dimensional surface may curve at any point on that surface both
toward <i>and</i> away from a point not on that surface. Such a surface
is said to have <q>negative curvature.</q> Take the donut-shaped <i>torus</i>
as an example (Figure 17-5). Part of its surface is negatively curved.
Choose a point on the torus nearest the center of the torus, and notice
that the surface of the torus bends <i>toward</i> its center in one
direction, forming a circle with the center of the torus as its own center
and enclosing the <q>hole</q> of the donut shape (Figure 17-6). Notice also
that in a direction perpendicular to the first, the surface bends <i>away</i>
from the center of the torus, forming a different circle that has a center
enclosed by the torus itself (Figure 17-7).
</p>
<div class="figdiv fleft nofloat_600 float_300_66 g3-1200" style="clear: both; ">
<figure>
<img class="Figure" loading="lazy" src="images/Torus.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-5</b>.
<span class="ficaption_description">
A torus. Source: <a href="https://commons.wikimedia.org/wiki/File:Torus.png">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/Torus2.png" style="max-height: 220px;">
<figcaption>
<b class="figlabel">Figure 17-6</b>.
<span class="ficaption_description">
A circle in the torus centered on its <q>hole.</q>
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/Torus3.png" style="max-height: 220px;">
<figcaption>
<b class="figlabel">Figure 17-7</b>.
<span class="ficaption_description">
A circle in the torus having a center enclosed by the torus.
</span>
</figcaption>
</figure>
</div>
<p style="clear: both;">
Other examples of negative curvature include the shapes of saddles,
trumpets, and — my favorite — Pringles potato chips. In Figure 17-8 we see
a two-dimensional surface being transformed between two types of flat
surfaces: a cylinder and a pair of cones; in between these shapes, the
surface is negatively-curved in a shape called a <dfn>hyperboloid</dfn>.
The sides form a hyperbola (Chapter Thirteen), and the geometry of
negatively-curved surfaces is sometimes called <dfn>hyperbolic</dfn> geometry
to distinguish it from flat Euclidean geometry.</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 g2-900" style="clear: both; background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Cylinder_-_hyperboloid_-_cone.gif" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-8</b>.
<span class="ficaption_description">
Animation of a cylinder transforming into a hyperboloid (see also Figure 17-10) and cones. Source: <a href="https://commons.wikimedia.org/wiki/File:Cylinder_-_hyperboloid_-_cone.gif">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/1600px-Finger_trap_toys.jpg" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-9</b>.
<span class="ficaption_description">
The toys called <q>Chinese finger traps</q> are built on a principle similar to what we see in Figure 17-8. As the ends are pulled, the radius constricts. Source: <a href="https://commons.wikimedia.org/wiki/File:Finger_trap_toys.jpg">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>In contrast to negative curvature, there is positive curvature, the <em>elliptic</em>
geometry of objects like the Earth, tennis balls, and so on. At
every point on a perfectly spherical object, the surface curves only
toward the object's center.
</p>
<div class="figdiv nofloat_600 float_300_66 float_1200_50 float_1500_33">
<figure>
<img class="Figure" loading="lazy" src="images/Gaussian_curvature.PNG" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-10</b>.
<span class="ficaption_description">
From left to right, surfaces having negative, zero, and positive curvature (a hyperboloid, cylinder, and sphere, respectively). A vertical cut through the left and right surfaces will still result in surfaces that will not lie flat. Source: <a href="https://commons.wikimedia.org/wiki/File:Gaussian_curvature.PNG">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>What these two types of curvature, positive and negative, have in common
is that the rules of Euclidean geometry do not apply. On such a surface,
the interior angles of a triangle do not add up to 180 degrees. The
circumference of a circle is not equal to pi times its diameter. Parallel
lines do not remain equidistant. Furthermore, this intrinsic curvature
cannot be eliminated by cutting the surface. If you cut a sphere into
pieces, none of those pieces will lie flat. This is the reason world maps
are drawn with distortions or discontinuities; large regions of a curved
surface cannot be faithfully represented on a flat surface (see Chapter
Four). Using a globe and a felt pen, a triangle can be drawn with one
corner at the North Pole and two other corners on the equator, and with
each of the three interior angles being 90 degrees, for a total of 270
degrees rather than the 180 degrees demanded by Euclid (Figure 4-5,
Chapter Four). Even smaller areas of the Earth have curvature which can be
measured in this manner.
</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 g2-900" style="clear: both; background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/1200px-Hyperbolic_triangle.svg.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-11</b>.
<span class="ficaption_description">
A negatively-curved surface in which parallel lines diverge and a triangle has interior angles that add up to less than 180 degrees. Source: <a href="https://commons.wikimedia.org/wiki/File:Hyperbolic_triangle.svg">Wikimedia
Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/Earth_geo.png" style="max-height: 280px;">
<figcaption>
<b class="figlabel">Figure 17-12</b>.
<span class="ficaption_description">
A positively-curved surface in which parallel lines converge and a triangle has interior angles that add up to more than 180 degrees. The Earth's surface is such a positively-curved surface; longitude lines that are parallel at the equator meet at the poles. Source: <a href="https://commons.wikimedia.org/wiki/File:Earth_geo.png">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>We tend to think of curved two-dimensional surfaces as being <q>embedded</q>
in a three-dimensional space. It's easier to visualize them this way, but
that third dimension is not an intrinsic property of the space, and need
not necessarily exist. By the same token, it's one thing to look at
two-dimensional curvature from the outside, like examining a globe; it's
quite another thing to notice it from the inside. For century after
century, it was known that the earth was round, and yet Euclid's geometry
worked just fine, because no one was taking highly accurate measurements
over very large distances.</p>
<p>It is thought that on a geographic survey commissioned in 1827, the
mathematician Karl Gauss noted that over great distances, the surface of
the Earth did not follow the rules of geometry established millennia
before by Euclid. Specifically, John Wheeler writes, <q>[Gauss] found that the sum of the angles in his largest survey triangle was less than 180 degrees;</q> this was
<q>both inescapable evidence for and a measure of the curvature of the Earth.</q> Even then, the difference observed by Gauss was
only 1/240<sup>th</sup> of a degree, so the fact that he even took notice
might also be considered a measure of his own confidence and competence.
<!-- (Wheeler, pp. 4-5)-->James Hartle, in <cite>Gravity</cite>,
<!--p. 15-->gives more details regarding Gauss' legendary survey while
cautioning the reader that the <q>historical evidence is not conclusive</q>
that it took place. Though dramatizing this development would make for a
great story, like Newton's supposed epiphany while observing the fall of
an apple, Gauss would have been well prepared for such a discovery.</p>
<p>For many years, Gauss had begun working on non-Euclidean geometry without
having published anything on it. Others who had similar thoughts began
publishing them soon afterward, not only beating Gauss to the press but
also writing about hyperbolic geometries like that which Gauss himself was
considering, and which were likewise quite different from what anyone
would notice on the spherical surface of the earth. Russian mathematician
Nikolai Lobachevsky was the first to publish in 1830, followed closely by
the Hungarian János Bolyai in 1832. In 1853, Gauss asked his student
Bernhard Riemann to prepare a presentation on <q>the foundation of geometry</q>
in hopes he could further develop alternatives to it.<!--(Kaku, p. 34)-->
Riemann succeeded beyond his mentor's hopes, announcing the next year that
not only was there a system for mathematically describing the curvature of
a non-Euclidean space at any point, but that this system could be extended
to an arbitrary number of dimensions. A handful of numbers could describe
the curvature of three-dimensional space and not many more would describe
four.
<!-- (Wheeler 6)--></p>
<p>Anticipating Einstein's work by more than half a century, Riemann
explored non-Euclidean geometry as the explanation for the invisible
forces of gravity, electricity, and magnetism. Rather than being satisfied
with Newton's <q>action at a distance,</q> Riemann saw that these forces could
be explained as properties of space itself. But lacking the field
equations later developed by Einstein and Maxwell, this ambition was
unrealized in his lifetime.
<!--(Kaku, pp. 36-37, 42-43)--></p>
<p>Riemann's 1854 lecture <q>On the Hypotheses Which Underlie Geometry</q>
proposed a <q>family</q> of non-Euclidean geometries, of which spherical
geometry is just one. It was published posthumously twelve years later;
Riemann died in 1863. Arthur Cayley is credited with initiating the study
of elliptic (or spherical) geometry in his 1859 writing <q>On the definition of distance.</q><!--https://en.wikipedia.org/wiki/Elliptic_geometry--> It
seems strange that scholarship on the geometry of spheres, like the Earth,
actually lagged behind the much less intuitive geometry of negative
curvature, on which papers had been published decades earlier.</p>
<div class="figdiv nofloat_600 float_300_66 float_1200_50 float_1500_33">
<figure>
<img class="Figure" loading="lazy" src="images/1599px-Dent_de_Vaulion_-_360_degree_panorama.jpg" style="max-height: 300px;">
<figcaption>
<b class="figlabel">Figure 17-13</b>.
<span class="ficaption_description">
A panoramic photo that shows all directions of the horizon in a circle. Source: <a href="https://commons.wikimedia.org/wiki/File:Dent_de_Vaulion_-_360_degree_panorama.jpg" target="_blank">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>I hope you will permit me a few more paragraphs on hyperbolic geometry
before we return to the more familiar geometry of spheres and an
overview of how the math works. I promise to make it fun. We saw in
Chapter Four that there are challenges in representing a positively-curved
surface on a flat one, and that there are different projection methods for
doing so. Similar challenges exist in representing a negatively-curved
surface, and I'd like to show two different projection methods here.
Whereas spheres tend to come back and meet themselves after curving around
a single point, this isn't true of all curved surfaces; they aren't
necessarily <em>finite</em>. So the methods of projection tend to take
this into account, and represent all directions of infinity as a fixed
distance from any given point, in much the same way that the horizon line
and its vanishing points represent an infinite distance in panoramic
photos and perspective drawings (See Figure 17-13 and Chapter Three).
</p>
<div class="figdiv nofloat_600 float_300_66 float_900_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Uniform_tiling_73-t1_klein.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-14</b>.
<span class="ficaption_description">
A tiled Beltrami-Klein projection model of a negatively-curved space in which all tile edges represent straight lines. Although this one looks a little like a garishly-colored ball, the space it represents is neither spherical nor flat. Source: <a href="https://commons.wikimedia.org/wiki/File:Uniform_tiling_73-t1_klein.png" target="_blank">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>In Figure 17-14, we see what is called the Beltrami-Klein projection
model of a negatively-curved space. Like the Mercator projection of the
spherical surface of the Earth, this projection also has its unavoidable
distortions. Lines are shown as lines, but angles are not accurate. We see
that lines which are not parallel do not necessarily meet, even at the
infinite distances represented at the circular boundary of the model; this
would be impossible in a Euclidean space, but it is one of the defining
characteristics of hyperbolic geometry.
</p>
<div class="figdiv nofloat_600 float_300_66 float_900_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/1188px-Droites_disquePoincare.svg.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-15</b>.
<span class="ficaption_description">
Three straight lines in negatively-curved space represented on a Poincaré disk. Source: <a href="https://commons.wikimedia.org/wiki/File:Droites_disquePoincare.svg" target="_blank">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>An alternative to the Beltrami-Klein model is the Poincaré disk (Figures
17-15 and 17-16), which has some amusing properties. In this model,
straight lines appear curved, but angles and circles are preserved. All
lines must meet the boundary, which represents an infinite distance, at
right angles; this has the result that any line not passing through the
center will be curved.
</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 g2-900" style="clear: both; background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/Hyperbolic_domains_642.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-16</b>.
<span class="ficaption_description">
A tiled Poincaré disk in which all tile edges represent straight lines. Click to enlarge. Source: <a href="https://en.wikipedia.org/wiki/File:Hyperbolic_domains_642.png" target="_blank">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/Escher_Circle_Limit_III.jpg" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-17</b>.
<span class="ficaption_description">
M. C. Escher's 1959 woodcut <i>Circle Limit III</i> , inspired by a Poincaré disk like the one in Figure 17-16. Source: <a href="https://en.wikipedia.org/wiki/File:Escher_Circle_Limit_III.jpg" target="_blank">Wikimedia Commons</a>.
</span>
</figcaption>
</figure>
</div>
<p>Henri Poincaré (1854-1912) was on the leading edge of scientific thought
during his time and under slightly different circumstances might well have
arrived at a theory of relativity before Einstein. In 1905, he described a
world much like the one represented in the Poincaré disk; this description
was both reminiscent of Riemann's efforts to explain the universe's
fundamental forces in terms of geometry and anticipatory of Einstein's
completion of what Riemann had begun:</p>
<blockquote>
Suppose, for example, a world enclosed in a large sphere and
subject to the following laws: The temperature is not uniform; it is
greatest at their centre, and gradually decreases as we move towards the
circumference of the sphere, where it is absolute zero. The law of this
temperature is as follows: If <i>R</i> be the radius of the sphere, and
<i>r</i> the distance of the point considered from the centre, the
absolute temperature will be proportional to <i>R</i><sup>2</sup> - <i>r</i><sup>2</sup>.
Further, I shall suppose that in this world all bodies have the same
co-efficient of dilatation, so that the linear dilatation of any body is
proportional to its absolute temperature. Finally, I shall assume that a
body transported from one point to another of different temperature is
instantaneously in thermal equilibrium with its new environment. … If
they construct a geometry, it will not be like ours, which is the study of
the movements of our invariable solids; it will be the study of the
changes of position which they will have thus distinguished, and will be
<q>non-Euclidean displacements,</q> and this will be non-Euclidean geometry. So
that beings like ourselves, educated in such a world, will not have the
same geometry as ours.
</blockquote>
<p>So let's talk now about how we make measurements and detect curvature in
our world. We have mentioned a few measurable effects of curvature
already: whether parallel lines converge or diverge; how many degrees are
in the sum of the angles of a triangle; and the ratio of a circle's
circumference to its radius. There is also the <q>parallel transport</q> issue
mentioned in Chapter Four, before we had really talked about vectors. As
we saw in Chapter Eight (Figure 8-9) with the moving billiard ball
vectors, to add vectors at different locations, you have to move them
together, taking care to keep their components the same, or in other
words, keeping them parallel throughout the move. That is why we call it
<q>parallel transport.</q> All of these effects are measurable from <em>within
</em> the space; one doesn't need, for example, to look at the globe from
the outside to measure the curvature of the Earth, nor does one need to
have a direct measurement of the Earth's radius.</p>
<p>For a practical example of how this works, let's imagine ourselves at the
north pole with Alice and Zeno. By the way, in elliptic geometry, a <em>pole
</em> has a meaning closely related to the one we know. Every straight
line has at least one pole where all lines perpendicular to that line will
meet, so any pole is defined in relation to a line. In the case of the
north and south poles, that line is the equator. All lines of longitude
cross the equator at right angles, meeting at both of these poles. Lines
of latitude parallel to the equator do not meet, not because they are
straight but because they curve around the nearest pole.</p>
<p>With that said, let's imagine that Alice has driven a stake into the ice
at the north pole, around which she has tied a rope; and that there is
sufficient ice all around her to walk a great distance in any direction
from the pole. She remains at the pole with a radio while Zeno uncoils the
rope and pulls it tight. Together, they make measurements. To make the
differences in these measurements easier to notice, let's also suppose
that the Earth is much smaller than it really is, and that it curves much
more tightly around its center. The rope is marked at equal intervals to
help Zeno measure its length. With one length of rope uncoiled, which is
quite long, he walks in a direction perpendicular to the rope and measures
how far he walks until he hears from Alice on his radio that he has walked
one tenth of the circumference of a circle, which Alice has measured by
watching the changing angle of the rope on the stake. Zeno radios back
with the distance he has walked, which happens to be sixty one hundredths
of a rope length. They each write down <q>1</q> and
then <q>0.61</q> Zeno uncoils
another length of the rope and heads back in the opposite direction,
measuring the distance he walks until Alice radios again and tells him
that he has walked the same angular distance. Zeno reports that this walk
was equal to 1.19 rope lengths. They each write down <q>2</q>
under the <q>1</q> and
then <q>1.19</q> under
the <q>0.61</q> On the third pass, with three rope lengths,
the distance of his walk is 1.74 rope lengths.</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 g2-900" style="clear: both; background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/polar-arc.png" style="max-height: 200px;">
<figcaption>
<b class="figlabel">Figure 17-18</b>.
<span class="ficaption_description">
Thirty-six degrees of arc (one tenth of a circle) marked off at radii of one, two, and three units.
</span>
</figcaption>
</figure>
<figure>
<table class="bordered_table">
<tbody>
<tr>
<th>Radius</th>
<th>Arc length</th>
</tr>
<tr>
<td>1</td>
<td>0.61</td>
</tr>
<tr>
<td>2</td>
<td>1.19</td>
</tr>
<tr>
<td>3</td>
<td>1.74</td>
</tr>
</tbody>
</table>
<figcaption>
<b class="figlabel">Table 17-1</b>.
<span class="ficaption_description">
Measurements of arc at radii of one, two, and three units.
</span>
</figcaption>
</figure>
</div>
<p>Alice looks at her table of measurements, and notices two things. First,
Zeno's walks get longer for every length of rope that he is away from the
pole. That makes sense, because the circumference of a circle grows in
direct proportion to its radius, and Zeno has been walking a constant
tenth of a full circle around the pole. But the second thing that she will
notice, if their measurements are accurate enough, and their rope is long
enough in relation to the curvature of the Earth, is that for each
successive rope length between Zeno and the pole, his walks gain a little
less distance each time. At one rope length of radius, Zeno's walking
distance was 0.61 lengths. At two lengths, the distance was only an
additional 0.58 lengths. At a radius of three lengths, Zeno only gained an
additional 0.55 lengths in his walk. This decreasing incremental distance
tells Alice that the surface they are measuring has positive curvature.</p>
<p>If Alice and Zeno were at liberty to fly high above the earth, they would
be able to see the reason why their measurements of walking distance are
increasing more slowly at greater distances. If Zeno were to have walked
to the equator, his walking distance would have reached its maximum; and
at greater radial distances after that, his walking distance would have
actually grown shorter. If Zeno were able to make it all the way to the
opposite side of the earth, still holding the rope tight and connected to
the north pole, he would find that he had been walking around the south
pole as well as (or rather than?) the north, until he reached the south
pole and his walking distance became nearly zero.</p>
<p>If we establish a (naturally) polar coordinate system with the north pole
as its origin, calling the length of the rope the radius <i>r</i> and
the direction of the rope the angle <i>θ</i>, we can generalize Alice
and Zeno's table of measurements into something called a <q>metric</q> of the
Earth's surface. At every point on that surface, there is not only an <i>r</i>
for its radial distance from the pole and a <i>θ</i> for its
direction, these being the <em>coordinates</em> of the point; but there
is also a distance <i>s</i> associated with every successive linear
distance <i>r</i>, and another distance <i>s</i> associated with
every successive angular distance <i>θ</i>. These last two numbers are
the <em>metric</em> for that point. At the coordinates <span class="row_vector">[0,0]</span>, that
metric is <span class="row_vector">[1.00, 0]</span>; in other words, there is only one direction from the
pole, and that direction is <em>outward</em>. The numbers mean this: if
Alice or Zeno goes one rope length in the <i> r</i> direction (in this
case, any direction, since we are at the origin), he or she will have gone
one rope length (of course). If they confine themselves to moving only in
the <i>θ</i> direction, they will only be rotating in place, so they
will have gone no distance at all. Polar coordinates aren't really the
best system to use <em>at</em> the pole, but they work very well <em>near
</em> it. At any other point, the numbers are different. At the
coordinates <span class="row_vector">[1,0]</span>,
the metric becomes <span class="row_vector">[1.00, 0.017]</span>. The number 0.017 is
the 0.61 lengths of walking distance divided by the 36 degrees of angle
that Zeno's walk traversed at a radius of one length. In other words, for
every rope length Zeno travels in the <i> r</i> direction from that
point, he will have gone (… wait for it …) one rope length. But for
every degree he travels in the <i>θ</i> direction, he will have gone a
distance <i>s</i> of 0.017 rope lengths. Alice and Zeno will find that
the metric does not change for increasing or decreasing <i>θ</i> coordinates,
but only for increasing or decreasing <i>r</i> coordinates. So at the
coordinates <span class="row_vector">[1,0]</span>
or at <span class="row_vector">[1, <i>θ</i>]</span> for any value of <i>θ</i>, the
metric will be <span class="row_vector">[1.00, 0.017]</span>.
At the coordinates <span class="row_vector">[2,0]</span>
(or at <span class="row_vector">[2, <i>θ</i>]</span>
for any value of <i>θ</i>), the metric will be <span class="row_vector">[1.00, 0.033]</span>. The <i>r</i>
metric is of course unchanged, because it is a linear measurement. But at
<span class="row_vector">[2, 0]</span> the <i>θ</i> metric is a little less than twice what it was at
the point <span class="row_vector">[1, 0]</span> (Table 17-2).
</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 float_1050_33 float_1050_15" style="background-color: white;">
<figure>
<table class="bordered_table">
<tbody>
<tr>
<th>Coordinate</th>
<th>Metric</th>
</tr>
<tr>
<td><span class="row_vector">[1,0]</span></td>
<td><span class="row_vector">[1.00, 0.017]</span></td>
</tr>
<tr>
<td><span class="row_vector">[2,0]</span></td>
<td><span class="row_vector">[1.00, 0.033]</span></td>
</tr>
<tr>
<td><span class="row_vector">[3,0]</span></td>
<td>?</td>
</tr>
</tbody>
</table>
<figcaption>
<b class="figlabel">Table 17-2</b>.
<span class="ficaption_description">
The metric values at several coordinates. Can you calculate the metric at <span class="row_vector">[3,0]</span>?
</span>
</figcaption>
</figure>
</div>
<p>If we assign a metric to every point on this surface (or any other) in
this way, we can do more precise quantitative analyses on this metric and
use calculus to talk about the ratios of tiny differential changes <i> dr
</i> and <i>dθ</i> in coordinates <i>r</i> and <i>θ</i> to tiny
differential changes <i>ds</i> in the metric. More importantly, we can
talk about how those ratios <i>ds</i>/<i>dr</i> and <i>ds</i>/<i>dθ</i>
themselves change over tiny increases <i>dr</i> in the distance <i>r</i>
(or in any other coordinate). If these ratios change, then we know
that the surface is curved. In calculus, these tiny changes in ratio are
called the <em>second derivatives</em> (<i>d</i><i><sup>2</sup>s</i>/<i>dr</i><sup>2</sup>
and <i>d</i><i><sup>2</sup>s</i>/<i>dθ</i><sup>2</sup>)
of the metric with respect to the coordinate system. They are more precise
and smaller-scale descriptions of the same changes that we noticed. For
more on calculus, see Chapter Eleven if you skipped over it.</p>
<p>Speaking of which, if you did skip over Chapter Eleven, you may prefer to
skip over the remainder of this chapter, and that is perfectly all right.
We're going to go over some more technical details that really aren't
necessary for enjoying the chapters to follow, but if you really do want
to learn the math, then the rest of this chapter will be really helpful.
This will be the material I wish I would have had when I was struggling to
make sense of what is called <dfn>differential geometry</dfn>.</p>
<h4>The metric tensor in two-dimensional polar coordinate systems</h4>
<p>Wow, that chapter sub-heading is a mouthful, isn't it? Have I scared you
off yet? No? Good.</p>
<p>I should point out that from here on, when I talk about angles, I will
mean measurements in radians rather than degrees, unless I write
<q>degrees.</q> Quantities like arc length are simpler to calculate in radians;
the length is simply the radius <i>r</i> times the angle <i>θ</i>
rather than <i>r</i> times <i>θ</i> times π/180. I will also on
occasion use the <q>*</q> character to indicate
multiplication when the <q>x</q>
character might be mistaken to mean the wrong thing, such as a vector
cross-product (Chapter Eleven).</p>
<p>In a two-dimensional polar coordinate system, Alice and Zeno were able
to demonstrate curvature using only two numbers for the metric because the
lines of constant <i>r</i> and <i>θ</i> are perpendicular (or <em>orthogonal</em>)
at every point on the surface. This <em>orthogonality</em> is what makes
the variables <i>r</i> and <i>θ</i> independent. There could be
other coordinate systems in which this isn't so, and in such cases more
numbers are needed for the metric, to describe how the variables depend on
each other. But the polar coordinate system tends to break down at the
pole itself, where the lines of constant <i>θ</i> all cross one
another, because they are not parallel at most points on the surface.
Later in this chapter we will discuss a Cartesian coordinate system on a
spherical surface that has the surprising property of its <i>x</i> and
<i>y</i> coordinates not being entirely independent; it will have lines
of constant <i>x</i> and <i>y</i> that eventually cross one another.
The fact that coordinate systems such as these have to be accounted for is
why we have something called the <dfn>metric tensor</dfn>. Let's hold off
for a moment on the question of what a tensor is in general. The metric
tensor is an invention of Riemann's which, like the metric, tells us about
distances surrounding any point in a coordinate system. John Wheeler calls
it a machine for calculating the real distance between any to points in a
given system. You feed it pairs of coordinates, and it gives you a
distance.<!--(Ohanian 194) -->The difference between the metric tensor and
the metric as we have described it so far is that the metric tensor is a
two-dimensional array of numbers rather than just a single pair. This is,
again, because we will need a generalized system that can handle
non-orthogonal coordinate axes. So at the point <span class="row_vector">[1, 0]</span>, where our metric
is <span class="row_vector">[1.00, 0.017]</span>. the metric tensor looks like this:
</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 float_1050_33 float_1050_15" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/tensor1.png" style="max-height: 100px;">
<figcaption>
<b class="figlabel">Figure 17-19</b>.
<span class="ficaption_description">
A metric tensor named G describing the metric of a two-dimensional surface at a given point on that surface.
</span>
</figcaption>
</figure>
</div>
<p>
The zero elements (upper right and lower left) in Figure 17-19 are where
we would see nonzero values in cases where our two coordinate axes were
not orthogonal and independent. We will have such an example later on.
</p>
<p>Now let's talk about what <dfn>tensor</dfn> means. A tensor is an idea
that includes, and expands on, the ideas of scalars and vectors. A scalar,
you will recall, is a single quantity having no direction associated with
it; we call this a tensor of <q>rank zero.</q> A vector is a quantity with an
associated direction, so a vector will have as many numbers as there are
dimensions in the space it is associated with. A two-dimensional vector
will typically have <i>x</i> and <i>y</i> components in a Cartesian
system and <i>r</i> and <i>θ</i> components in a polar system. We
can call a vector a tensor of <dfn>rank one</dfn>. To reiterate: no
directions, rank zero; one direction, rank one. The metric tensor is a
tensor of <dfn>rank two</dfn>, and as such it can represent the
relationships between multiple <em>pairs</em> of directions: for
example, between the components of the coordinate system.</p>
<p>The elements of a tensor are indexed by which components they represent.
A rank one tensor, what we more commonly call a vector, has one index, and
one element for each of the coordinates in the coordinate system. Let's
recall Figure 8-10 for some concrete examples.
</p>
<div class="figdiv fleft nofloat_600 float_300_66 float_800_50 float_1050_33 ">
<figure style="min-width: 250px;">
<p><b>a</b> = <span class="row_vector">[1,-1]</span></p>
<p><b>b</b> = <span class="row_vector">[1, 1]</span></p>
<p><b>c</b> = <span class="row_vector">[2, 0]</span></p>
<figcaption>
<b class="figlabel">Figure 8-10</b> (repeated).
<span class="ficaption_description">
The components <span class="row_vector">[<i>x</i>, <i>y</i>]</span> of the vectors <b>a</b>, <b>b</b>, and <b>c</b>. As discussed in the text, these components might also be named <span class="row_vector">[<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>]</span>.
</span>
</figcaption>
</figure>
</div>
<p>In Figure 8-10 we have some vectors with <i>x</i> and <i>y</i>
components. As we consider vectors more generally as examples of tensors,
it sometimes becomes more convenient to <em>number</em> the components
of the vector than to assign them various <em>letters</em>. So in place
of <i>x</i> and <i>y</i>, we might have <i>x</i><sub>1</sub> and
<i>x</i><sub>2</sub>, or even ε<sub>1</sub> and ε<sub>2</sub>. The
subscripted number in either of these cases is the component <em>index</em>.
We could say of the vector <b>a</b> in Figure 8-10 that its <i>x</i><sub>1</sub>
component is 1 and <i>x</i><sub>2</sub> component is -1.</p>
<p>A rank two tensor has two indices, one for a row in the tensor, and
another index for a column; and a metric tensor has one element for every
<em>combination</em> of the coordinates in the coordinate system. Skip
ahead a little bit and look at the metric tensor <i>G</i> in Figure
17-22. It has four elements, G<sub>11</sub>, G<sub>12</sub>, G<sub>21</sub>,
and G<sub>22</sub>. G<sub>11</sub> and G<sub>12</sub> are the top row (1
and 0), the topmost elements in each column, as specified by the first
index having the value of one. G<sub>21</sub> and G<sub>22</sub> are
the bottom row (0 and 1), as specified by the first index having the value
of two.</p>
<p style="clear: both;">In general, we can read the elements of a metric tensor using a sort of key
like one of the following:</p>
<div class="figdiv nofloat_600 float_300_66 g2-900" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/t2neg.jpg" style="max-height: 150px;">
<figcaption>
<b class="figlabel">Figure 17-20</b>.
<span class="ficaption_description">
A key for the indices of metric tensors in terms of <i>r</i> and <i>θ</i>.
</span>
</figcaption>
</figure>
<figure>
<img class="Figure" loading="lazy" src="images/t3neg.jpg" style="max-height: 150px;">
<figcaption>
<b class="figlabel">Figure 17-21</b>.
<span class="ficaption_description">
A key for the indices of metric tensors in terms of <i>x</i> and <i>y</i>.
</span>
</figcaption>
</figure>
</div>
<p>The first index in each tensor will specify a row in that tensor. In
Figure 17-20 these are the <i>r</i> and <i>θ</i> rows; in Figure
17-21 these are the <i>x</i> and <i>y</i> rows. The second index in
each tensor will specify a column in that tensor. In Figure 17-20 these
are the <i>r</i> and <i>θ</i> columns; in Figure 17-21 these are
the <i>x</i> and <i>y</i> columns. Bear with me a moment longer,
and we'll start to see how this all comes together with what we have been
talking about.</p>
<p>Metric tensors help us in calculating distance in a way that closely
relates to the Pythagorean theorem. We are most familiar with this theorem
in a form something like <i>c</i><sup>2 </sup>= <i>a</i><sup>2
</sup>+ <i>b</i><sup>2</sup>, which describes the relationship
between the sides of a right triangle, but we have also seen it in the
form <i>s</i><sup>2 </sup>= <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup>,
which describes the sum of two orthogonal vectors in a Cartesian
coordinate system and in flat Euclidean space. We could rewrite this as <i>s</i><sup>2
</sup>= <i>x*x</i><sup> </sup>+ <i>y*y</i>, and then more
generally as <i>s</i><sup>2 </sup>= 1(<i>x*x</i>)<sup> </sup>+ 0(<i>x</i>*<i>y</i>)
+ 0(<i>y*x</i>)+ 1(<i>y*y</i>) to show every possible pairing of <i>x</i>
and <i>y</i>. Then, using the index key in Figure 17-21, we could place
each of the zero and one coefficients (multipliers) into a metric tensor:
</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/tensor4.png" style="max-height: 100px;">
<figcaption>
<b class="figlabel">Figure 17-22</b>.
<span class="ficaption_description">
The metric tensor for flat Euclidean space in Cartesian coordinates (also known in linear algebra as the identity matrix).
</span>
</figcaption>
</figure>
</div>
<p>
So now, courtesy of Bernhard Riemann, we have a very robust system for
describing distances in any coordinate system, flat or curved, orthogonal
or not. We could take the metric tensor in Figure 17-19 and the index key
in Figure 17-20 and use them to calculate distances near the coordinates
it is associated with. For every small <i>dr</i> and <i>d</i><i>θ</i>,
we would calculate the distance <i>ds</i> as follows:
</p>
<blockquote class="equation">
<p><i>ds</i><sup>2 </sup>= 1.00(<i>dr*dr</i>)<sup> </sup>+ 0(<i>dr*d</i><i>θ</i>)
+ 0(<i>d</i><i>θ</i><i>*dr</i>)+ 0.017(<i><i>d</i><i>θ</i>*</i><i>d</i><i>θ</i>)</p>
<b>Equation 17-1.</b> <span class="equation_description">Calculating a small distance using the metric tensor in Figure 17-19.</span>
</blockquote>
<p>Of course, over distances which are large enough to be accompanied by a
significant change in the metric, we will have to take those changes in
the metric into account as well, and that involves integral calculus which
will tell us how the distance <i>s</i> accumulates over all the
incremental changes. This takes us into a subject area called differential
geometry.</p>
<p>Though I think it's a good idea to show specific examples, it isn't often
that you will find specific metric tensor values for specific coordinate
values in introductory texts. More often, you will see tensors populated
by variables that apply over an entire surface. For example, the metric
tensor of a flat surface in polar coordinates can be written this way:</p>
<div class="figdiv nofloat_600 float_300_66 float_800_50 float_1200_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/tensor5.png" style="max-height: 100px;">
<figcaption>
<b class="figlabel">Figure 17-23</b>.
<span class="ficaption_description">
The metric tensor for flat Euclidean space in polar coordinates.
</span>
</figcaption>
</figure>
</div>
<p>Using this metric, we calculate that <i>ds</i><sup>2</sup> (the square
of the distance <i>ds</i>) is <b><span style="color: #3515f4;">1</span></b> (<i>dr</i><sup>2</sup>)
+ <b style="color: #7b7778;">0</b> (<i>dr*dθ</i>) + <b
style="color: #7b7778;">0</b> (<i>dθ*dr</i>) +<b><span
style="color: #3515f4;"> <i>r</i><sup>2</sup></span></b> (<i>dθ</i><sup>2</sup>)
= <i>dr</i><sup>2</sup> + <i>r</i><sup>2</sup><i>dθ</i><sup>2</sup>.
In other words, for very small values of <i>θ</i>, the distance <i>s</i>
approximates the vector addition of the distance <i>dr</i> and the
arc length <i>r*dθ</i>. For larger values of <i>θ</i>, it gets
more complicated and involves calculus again, because (for example) the
direction of <i>dθ</i> changes as <i>θ</i> itself changes. If you
dive deep enough in the math of it all, you will hear this aspect of it
referred to as <em>changes in the basis vectors</em>. See Chapter
Nineteen for more on this.</p>
<p>For example, imagine we are standing at the north pole and that our
coordinate system is such that <i>θ</i> is zero degrees longitude at
our right hand, 90 degrees in front of us, 180 degrees at our left hand,
and so on. In terms of places these longitude lines would pass near, we
would have London far off to our right, Krasnoyarsk in front of us, and
the Bering Strait and Aleutian Islands to our left (Figure 17-24). So, at
a radius of one meter, and at <i>θ</i>=0, to our right, imagine there
is a vector (red) pointing counterclockwise, in the direction of
increasing <i>θ</i>. If that vector were moved (or <dfn>parallel
transported</dfn>) to be right next to us, it would point the same way
that we are facing, towards Krasnoyarsk. In fact, right where it sits it
is still pointing in the general direction of Krasnoyarsk. Also at a
radius of one meter, directly in front of us, there is another vector
(orange), also pointing in the direction of increasing <i>θ</i>; this
one points in the direction of the Bering Strait. There is a third vector
(purple) that we can add to the first one to get the second one, and that
third vector points in the direction of minus <i>r</i> (the three
vectors are brought together at lower left to show that the red and
purple vectors add to make the orange one). In other words, as we move in
the direction of increasing <i>θ</i>, the vector <i>dθ</i> changes,
and this change is in the direction of decreasing <i>r</i>. If you
repeat this exercise, you will find that as we move in the direction of
increasing <i>θ</i>, the vector <i>dr</i> also changes, and this
change is in the direction of increasing <i>θ</i>; as
we move in the direction of increasing <i>r</i>, the vector <i>dθ</i> changes, and this change is in the direction of increasing <i>θ</i>.</p>
<div class="figdiv nofloat_600 float_300_66 float_900_50 float_1500_33" style="background-color: white;">
<figure>
<img class="Figure" loading="lazy" src="images/polar-vectors.png" style="max-height: 350px;">
<figcaption>
<b class="figlabel">Figure 17-24</b>.
<span class="ficaption_description">
The vector <i>dθ</i> changes direction for increasing values of <i>θ</i>, and its change is in the minus <i>r</i> direction.
</span>
</figcaption>
</figure>
</div>
<p>If you think this is all terribly complicated, you're absolutely right.
And the math is pretty ugly, which is why our exploration of it will be
somewhat superficial here. But here's a table of what we have covered so
far for two-dimensional polar coordinate systems:
</p>
<div class="figdiv nofloat_600 float_300_66 float_1500_50" style="background-color: white;">
<figure>
<table id="Table17-3" style="border: 1px solid black;
border-collapse: collapse;">
<tbody>
<tr>
<th>As you move in the direction of increasing ___</th>
<th>this vector or value</th>
<th>does this</th>
<th>when the space is</th>
</tr>
<tr>
<td><i>θ</i></td>
<td><i>dθ</i></td>
<td>changes in the -<i>r</i> direction<br>
</td>
<td><br>
</td>
</tr>
<tr>
<td><i>θ</i></td>
<td style="height: 19px;"><i>dr</i> </td>
<td>changes in the +<i>θ</i> direction </td>
<td><br>
</td>
</tr>
<tr>
<td><i>r</i><br>
</td>
<td><i>dθ</i></td>
<td>changes in the +<i>θ</i> direction </td>
<td><br>
</td>
</tr>
<tr>
<td><i>r</i> </td>
<td><i>ds</i>/<i>dθ</i></td>
<td>increases<br>
</td>
<td><br>
</td>
</tr>
<tr>
<td><i>r</i> </td>
<td><i>d<sup>2</sup>s</i>/<i>dθ</i><sup>2</sup> </td>
<td>stays the same<br>
</td>
<td>flat<br>
</td>
</tr>
<tr>
<td><i>r</i> </td>
<td><i>d<sup>2</sup>s</i>/<i>dθ</i><sup>2</sup> </td>
<td>increases<br>
</td>
<td>positively curved<br>
</td>
</tr>
</tbody>
</table>
<figcaption>
<b class="figlabel">Table 17-3</b>.
<span class="ficaption_description">
Changes in the metric and in the basis vectors in a two-dimensional polar coordinate system.