Hyperspace (56 page)

1
. Heinz Pagels,
Perfect Symmetry: The Search for the Beginning of Time
(New York: Bantam, 1985), 324.

2
. Peter Freund, interview with author, 1990.

3
. Quoted in Abraham Pais,
Subtle Is the Lord: The Science and the Life of Albert Einstein
(Oxford: Oxford University Press, 1982), 235.

4
. This incredibly small distance will continually reappear throughout this book. It is the fundamental length scale that typifies any quantum theory of gravity. The reason for this is quite simple. In any theory of gravity, the strength of the gravitational force is measured by Newton’s constant. However, physicists use a simplified set of units where the speed of light
c
is set equal to one. This means that 1 second is equivalent to 186,000 miles. Also, Planck’s constant divided by 2π is also set equal to one, which sets a numerical relationship between seconds and ergs of energy. In these strange but convenient units, everything, including Newton’s constant, can be reduced to centimeters. When we calculate the length associated with Newton’s constant, it is precisely the Planck length, or 10−
³³
centimeter, or 10
¹⁹
billion electron volts. Thus all quantum gravitational
effects are measured in terms of this tiny distance. In particular, the size of these unseen higher dimensions is the Planck length.

5
. Linda Dalrymple Henderson,
The Fourth Dimension and Non-Euclidean Geometry in Modern Art
(Princeton, N.J.: Princeton University Press, 1983), xix.

1
. E. T. Bell,
Men of Mathematics
(New York: Simon and Schuster, 1937), 484.

2
. Ibid., 487. This incident most likely sparked Riemann’s early interest in number theory. Years later, he would make a famous speculation about a certain formula involving the zeta function in number theory. After 100 years of grappling with “Riemann’s hypothesis,” the world’s greatest mathematicians have failed to offer any proof. Our most advanced computers have failed to give us a clue, and Riemann’s hypothesis has now gone down in history as one of the most famous unproven theorems in number theory, perhaps in all of mathematics. Bell notes, “Whoever proves or disproves it will cover himself with glory” (ibid., 488).

3.
John Wallis,
Der Barycentrische Calcul
(Leipzig, 1827), 184.

4
. Although Riemann is credited as having been the driving creative force who finally shattered the confines of Euclidean geometry, by rights, the man who should have discovered the geometry of higher dimensions was Riemann’s aging mentor, Gauss himself.

In 1817, almost a decade before Riemann’s birth, Gauss privately expressed his deep frustration with Euclidean geometry. In a prophetic letter to his friend the astronomer Heinrich Olbers, he clearly stated that Euclidean geometry is mathematically incomplete.

In 1869, mathematician James J. Sylvester recorded that Gauss had seriously considered the possibility of higher-dimensional spaces. Gauss imagined the properties of beings, which he called “bookworms,” that could live entirely on two-dimensional sheets of paper. He then generalized this concept to include “beings capable of realizing space of four or a greater number of dimensions” (quoted in Linda Dalrymple Henderson,
The Fourth Dimension and Non-Euclidean Geometry in Modern Art
[Princeton, N.J.: Princeton University Press, 1983], 19).

But if Gauss was 40 years ahead of anyone else in formulating the theory of higher dimensions, then why did he miss this historic opportunity to shatter the bonds of three-dimensional Euclidean geometry? Historians have noted Gauss’s tendency to be conservative in his work, his politics, and his personal life. In fact, he never once left Germany, and spent almost his entire life in one city. This also affected his professional life.

In a revealing letter written in 1829, Gauss confessed to his friend Friedrich Bessel that he would never publish his work on non-Euclidean geometry for fear of the controversy it would raise among the “Boeotians.” Mathematician Morris Kline wrote, “[Gauss] said in a letter to Bessel of January 27, 1829, that he
probably would never publish his findings in this subject because he feared ridicule, or as he put it, he feared the clamor of the Boeotians, a figurative reference to a dull-witted Greek tribe” (
Mathematics and the Physical World
[New York: Crowell, 1959], 449). Gauss was so intimidated by the old guard, the narrow-minded “Boeotians” who believed in the sacred nature of three dimensions, that he kept secret some of his finest work.

In 1869, Sylvester, in an interview with Gauss’s biographer Sartorious von Waltershausen, wrote that “this great man used to say that he had laid aside several questions which he had treated analytically, and hoped to apply to them geometrical methods in a future state of existence, when his conceptions of space should have become amplified and extended; for as we can conceive beings (like infinitely attenuated book-worms in an infinitely thin sheet of paper) which possess only the notion of space of two dimensions, so we may imagine beings capable of realizing space of four or a greater number of dimensions” (quoted in Henderson,
Fourth Dimension and Non-Euclidean Geometry in Modern Art
, 19).

Gauss wrote to Olbers, “I am becoming more and more convinced that the (physical) necessity of our (Euclidean) geometry cannot be proved, at least not by human reason nor for human reason. Perhaps in another life we will be able to obtain insight into the nature of space, which is now unattainable. Until then, we must place geometry not in the same class with arithmetic, which is purely a priori, but with mechanics” (quoted in Morris Kline,
Mathematical Thought from Ancient to Modern Times
[New York: Oxford University Press, 1972], 872).

In fact, Gauss was so suspicious of Euclidean geometry that he even conducted an ingenious experiment to test it. He and his assistants scaled three mountain peaks: Rocken, Hohehagen, and Inselsberg. From each mountain peak, the other two peaks were clearly visible. By drawing a triangle between the three peaks, Gauss was able to experimentally measure the interior angles. If Euclidean geometry is correct, then the angle should have summed to 180 degrees. To his disappointment, he found that the sum was exactly 180 degrees (plus or minus 15 minutes). The crudeness of his measuring equipment did not allow him to conclusively show that Euclid was wrong. (Today, we realize that this experiment would have to be performed between three different star systems to detect a sizable deviation from Euclid’s result.)

We should also point out that the mathematicians Nikolaus I. Lobachevski and János Bolyai independently discovered the non-Euclidean mathematics defined on curved surfaces. However, their construction was limited to the usual lower dimensions.

5.
Quoted in Bell,
Men of Mathematics
, 497.

6
. The British mathematician William Clifford, who translated Riemann’s famous speech for
Nature
in 1873, amplified many of Riemann’s seminal ideas and was perhaps the first to expand on Riemann’s idea that the bending of space is responsible for the force of electricity and magnetism, thus crystallizing Riemann’s work. Clifford speculated that the two mysterious discoveries in mathematics (higher-dimensional spaces) and physics (electricity and magnetism) are
really the same thing, that the force of electricity and magnetism is caused by the bending of higher-dimensional space.

This is the first time that anyone had speculated that a “force” is nothing but the bending of space itself, preceding Einstein by 50 years. Clifford’s idea that electromagnetism was caused by vibrations in the fourth dimension also preceded the work of Theodr Kaluza, who would also attempt to explain electromagnetism with a higher dimension. Clifford and Riemann thus anticipated the discoveries of the pioneers of the twentieth century, that the meaning of higher-dimensional space is in its ability to give a simple and elegant description of forces. For the first time, someone correctly isolated the true physical meaning of higher dimensions, that a theory about
space
actually gives us a unifying picture of
forces
.

These prophetic views were recorded by mathematician James Sylvester, who wrote in 1869, “Mr. W. K. Clifford has indulged in some remarkable speculations as to the possibility of our being able to infer, from certain unexplained phenomena of light and magnetism, the fact of our level space of three dimensions being in the act of undergoing in space of four dimensions … a distortion analogous to the rumpling of a page” (quoted in Henderson,
Fourth Dimension and Non-Euclidean Geometry in Modern Art
, 19).

In 1870, in a paper with the intriguing title “On the Space-Theory of Matter,” he says explicitly that “this variation of the curvature of space is what really happens in that phenomenon which we call the
motion of matter
, whether ponderable or ethereal” (William Clifford, “On the Space-Theory of Matter,”
Proceedings of the Cambridge Philosophical Society 2
[1876]: 157-158).

7
. More precisely, in
N
dimensions the Riemann metric tensor
g
_μυ
is an
N
×
N
matrix, which determines the distance between two points, such that the infinitesimal distance between two points is given by
ds
²
= Σdx
^μ
g
_μυ
dx
^υ
. In the limit of flat space, the Riemann metric tensor becomes diagonal, that is,
g
_μυ
= δ
_μυ
, and hence the formalism reduces back to the Pythagorean Theorem in
N
dimensions. The deviation of the metric tensor from
δ
_μυ
, roughly speaking, measures the deviation of the space from flat space. From the metric tensor, we can construct the Riemann curvature tensor, represented by R
^β
_μυα
.

The curvature of space at any given point can be measured by drawing a circle at that point and measuring the area inside that circle. In flat two-dimensional space, the area inside the circle is
πr
²
. However, if the curvature is positive, as in a sphere, the area is less than
πr
²
. If the curvature is negative, as in a saddle or trumpet, the area is greater than
πr
²
.

Strictly speaking, by this convention, the curvature of a crumpled sheet of paper is zero. This is because the areas of circles drawn on this crumpled sheet of paper still equal
πr
²
. In Riemann’s example of force created by the crumpling of a sheet of paper, we implicitly assume that the paper is distorted and stretched as well as folded, so that the curvature is nonzero.

8.
Quoted in Bell,
Men of Mathematics
, 501.

9.
Ibid., 14.

10.
Ibid.

11
. In 1917, physicist Paul Ehrenfest, a friend of Einstein, wrote a paper entitled “In What Way Does It Become Manifest in the Fundamental Laws of Physics that Space has Three Dimensions?” Ehrenfest asked himself whether the stars and planets are possible in higher dimensions. For example, the light of a candle gets dimmer as we move farther away from it. Similarly, the gravitational pull of a star gets weaker as we go farther away. According to Newton, gravity gets weaker by an inverse square law. If we double the distance away from a candle or star, the light or gravitational pull gets four times weaker. If we triple the distance, it gets nine times weaker.

If space were four dimensional, then candlelight or gravity would get weaker much more rapidly, as the inverse cube. Doubling the distance from a candle or star would weaken the candlelight or gravity by a factor of eight.

Can solar systems exist in such a four-dimensional world? In principle, yes, but the planets’ orbits would not be stable. The slightest vibration would collapse the orbits of the planets. Over time, all the planets would wobble away from their usual orbits and plunge into the sun.

Similarly, the sun would not be able to exist in higher dimensions. The force of gravity tends to crush the sun. It balances out the force of fusion, which tends to blow the sun apart. Thus the sun is a delicate balancing act between nuclear forces that would cause it to explode and gravitational forces that would condense it down to a point. In a higher-dimensional universe, this delicate balance would be disrupted, and stars might spontaneously collapse.

12.
Henderson,
Fourth Dimension and Non-Euclidean Geometry in Modern Art
, 22.

13
. Zollner had been converted to spiritualism in 1875 when he visited the laboratory of Crookes, the discoverer of the element thalium, inventor of the cathode ray tube, and editor of the learned
Quarterly Journal of Science
. Crookes’s cathode ray tube revolutionized science; anyone who watches television, uses a computer monitor, plays a video game, or has been x-rayed owes a debt to Crookes’s famous invention.

Crookes, in turn, was no crank. In fact, he was a lion of British scientific society, with a wall full of professional honors. He was knighted in 1897 and received the Order of Merit in 1910. His deep interest in spiritualism was sparked by the tragic death of his brother Philip of yellow fever in 1867. He became a prominent member (and later president) of the Society for Psychical Research, which included an astonishing number of important scientists in the late nineteenth century.

14
. Quoted in Rudy Rucker,
The Fourth Dimension
(Boston: Houghton Mifflin, 1984), 54.

15
. To imagine how knots can be unraveled in dimensions beyond three, imagine two rings that are intertwined. Now take a two-dimensional cross section of this configuration, such that one ring lies on this plane while the other ring becomes a point (because it lies perpendicular to the plane). We now have a point inside a circle. In higher dimensions, we have the freedom of moving this
dot completely outside the circle without cutting any of the rings. This means that the two rings have now completely separated, as desired. This means that knots in dimensions higher than three can always be untied because there is “enough room.” But also notice that we cannot remove the dot from the ring if we are in three-dimensional space, which is the reason why knots stay knotted only in the third dimension.