I love a good time travel story. Why wouldn't I? I loved the Harry Potter series too, and didn't waste a single moment trying to figure out the theoretical basis of the characters’ ability to transform things with the wave of a wand. But the problem with time travel stories, by contrast, is that they are too often regarded as science fiction rather than fantasy. And as such, they can get in the way of real learning about the relationship of space and time. Time travel offers the imaginative mind unique possibilities for adventure, narrative complexity, and righting of wrongs; it is no wonder that it is a very popular plot device and subject of speculation. As a consequence, by the time students are introduced to modern physics ‐ namely the theory of relativity ‐ they are likely to have several false preconceptions of time, accumulated through a childhood abounding in time machines, which make for great fiction but bad science. This problem can be exacerbated by educators who use terminology and analogies that reinforce these preconceptions rather than help to dispel them.

I am not an expert on any matter of science and am not writing as an educator *per se*; rather my area of "expertise" is in being a lifelong student. It is as a student that I write, my purpose being to point out what has been the greatest stumbling-block in my education and to advocate an approach to teaching relativity ‐ and the relationship between space and time ‐ that would have saved me a great deal of trouble. I will make some effort to make this article comprehensible to a general audience, but this is primarily intended for those who are already familiar with the concepts, terminology, common metaphors, and widely-adopted diagrams associated with the teaching of relativity, whether as a student or instructor. For the experienced student, I hope to provide a remedy for the confusion that may have resulted from less-effective teaching; to the instructor, I hope to offer more effective methods. Those readers who are less familiar with relativity may find it useful to skip ahead to the explanation of the " light clock " below before proceeding with the lengthy discussion of the spacetime interval that precedes it.

As I survey the current literature on relativity, it is clear to me that physics is pleading for a more adequate definition of time. In his essay " The Gift of Time, " Richard A. Muller laments at length that though " physics uses time, it is our dirty little secret that we don’t understand it. . . . Physicists love confusion, mystery, and being surrounded by things they don’t understand. " (Brockman, pp. 213-222) Lee Smolin wrote: " More and more, I have the feeling that quantum theory and general relativity are both deeply wrong about the nature of time. It is not enough to combine them. There is a deeper problem, perhaps going back to the origin of physics. " (Smolin 2006, p. 256) Smolin’s *Time Reborn *begins with over a hundred pages criticizing the current philosophy of time, denouncing its implications for a predetermined future and calling our failure to understand time the " single most important problem facing physics. " (Smolin 2013, p. xi, xxii) Einstein himself struggled with the concept of time, particularly with the definition of " the present. " Once in a letter of consolation to the widow of a departed friend, he wrote that " the distinction between past, present, and future is only a stubbornly persistent illusion. " (Smolin 2013, pp. 88, 91)

In this multipart essay I will show that a great deal of confusion arises from the treatment of time as a fourth dimension, and offer an alternative that resolves many of the issues pointed out by Muller and Smolin. When discussing the problem faced by one presenting a difficult subject to a general audience, Einstein wrote:

Either he succeeds in being intelligible by concealing the core of the problem and by offering to the reader only superficial aspects or vague allusions, thus deceiving the reader by arousing in him the deceptive illusion of comprehension; or else he gives an expert account of the problem, but in such a fashion that the untrained reader is unable to follow the exposition and becomes discouraged from reading any further.

If these two categories are omitted from today’s popular scientific literature, surprisingly little remains. But the little that is left is very valuable indeed.(Barnett, p. 9)

I have noticed this dilemma in much of the materials I have read during my career as a student of relativity, having myself suffered under the " illusion of comprehension. " I hope in the following discussion to demonstrate the " valuable " combination of both clarity and validity Einstein writes of.

Time is said to have only one dimension, and space to have three dimensions. [. . . ] The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be. ‐William Rowan Hamilton (1805-1865)

H.G. Wells gave us the idea of the time machine in his 1895 book of the same name, and countless stories since then have featured similar devices. In a story that - interestingly ‐ predates even the automobile, Wells' Time Traveller makes a clever argument that time is a fourth dimension in addition to the familiar three dimensions of space, and demonstrates a machine for traversing this fourth dimension. This particular four-dimensional understanding of space and time, which I shall call *Wells spacetime*, still has a great hold on the popular mind. Though neither Wells nor his fictional time traveler invoke the name of the ancient Greek mathematician Euclid in their description of this space-time relationship, it is a Euclidean geometry that governs this spacetime; namely, each of the four dimensions is at right angles to the others, and the familiar rules of flat-surface geometry apply.

"Any real body must have extension in *four* directions," the Time Traveller argues. "It must have Length, Breadth, Thickness, and Duration." He makes the claim that "There is no difference between Time and any of the three dimensions of space except that our consciousness moves along it." He then goes on to cite the efforts of "some philosophical people" to "construct a Four-Dimension geometry" and points in particular to a recent lecture on the subject by Simon Newcomb, a real-life mathematician and astronomer who was at the forefront of the efforts that would eventually give birth to relativity. The Fourth Dimension was indeed a hot topic at the time *The Time Machine* was written. William Rowan Hamilton, quoted above, also spoke of time as a fourth dimension, and his now-*passé* mathematical innovation the quaternion makes a fitting symbol in this essay for the idea of time as a dimension. I argue, as Heaviside did regarding the quaternion (quoted below), that in some cases we are better off without it.

Perhaps there would have been no *Time Machine *had it not been for the mathematician Karl Gauss and his student Bernhard Riemann, who more than anyone else was responsible for introducing the "fourth dimension" to both academia and the popular imagination. As a surveyor, Gauss had noted that over great distances, the surface of the Earth did not follow the rules of geometry established millennia before by Euclid. Specifically, "he found that the sum of the angles in his largest survey triangle was less than 180 degrees. ... The deviation observed by Gauss ‐ almost 15 seconds of arc (a quarter minute of arc, or 1/240^{th} of a degree) ‐ was both inescapable evidence for and a measure of the curvature of the Earth." (Wheeler, pp. 4-5) Though it would have been tantamount to heresy in an earlier time, Gauss explored possibilities beyond the ancient paradigm and in 1854 asked Riemann to prepare a presentation on "the foundation of geometry" as "Gauss was keenly interested in seeing if his student could develop an alternative to Euclidean geometry." (Kaku, p. 34) Riemann "hit it out of the park," announcing 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, even (gasp!) more than three. "Riemann’s famous lecture was popularized to a wide audience via mystics, philosophers and artists" and "the ideas originated by Gauss and Riemann permeated literary circles, the avant garde, and the thoughts of the general public, affecting trends in art, literature, and philosophy." (Kaku pp. 62, 79)

When the common person thinks of curvature, he or she is most likely to think of curves that could be described in two dimensions, such as the tendency of a one-dimensional " line " to curve toward or away from a point. At any point on a line, that line may curve either toward or away from another point not on the line. In this sense, a circle is curved around its center. When solid objects have this kind of curvature, they can easily be made flat. A cylinder and a cone are curved around their axis but one simple cut will allow them to be unrolled into a flat surface. This " extrinsic " curvature is not what is meant by " curvature " in the context of Gauss, Riemann and Einstein.

The degree of curvature of a line at any point is related to the radius of the circle with which it is congruent.

A more complex curvature than that of the one-dimensional line is the curvature of two-dimensional surfaces. A surface may curve at one point on that surface both toward *and* away from a point not on that surface. Such a surface is said to have " negative curvature. " Take the donut-shaped *torus* as an example. 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 *toward* its center in one direction, forming a circle with the center of the torus as its own center and enclosing the " hole " of the donut shape. Notice also that in a direction perpendicular to the first, the surface bends *away* from the center of the torus, forming a different circle that has a center enclosed by the torus itself.

A torus.

Other examples of negative curvature include the shapes of saddles, trumpets, and ‐ my favorite ‐ Pringles potato chips. In contrast to negative curvature, there is the " positive curvature " of spherical 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.

From left to right, surfaces having negative, zero, and positive curvature.

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. 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. Even smaller areas of the Earth have measurable curvature, as Gauss discovered.

*Left*: A negatively-curved surface in which parallel lines diverge and a triangle has interior angles that add up to less than 180 degrees. *Center*: A positively-curved surface in which parallel lines converge and a triangle has interior angles which add up to more than 180 degrees. The Earth’s surface (*right*) is such a positively-curved surface. Longitude lines that are parallel at the equator meet at the poles.

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 " action at a distance, " 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)