I came later to see that, as far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no inconsiderable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonised with ordinary Cartesian work ‐Oliver Heaviside (1850-1925)

A three-dimensional model of spacetime stands up not only to the concepts of special relativity and unaccelerated frames of reference ‐ what physicists call "flat spacetime" ‐ but also to general relativity’s concept of the "curved spacetime" that results in gravity. Admittedly, to do the math, not only are four coordinates required, but also the highly daunting subject of tensors; but Einstein’s description of gravity can be visualized and explained without resorting to a fourth dimension. Neither does one require the Principle of Equivalence with which so many writers introduce the subject of general relativity. As I have discussed elsewhere, this principle is arguably incorrect due to its disregard of tidal forces; furthermore, (as I will show) its significance may be seen as strictly historical, and to begin a theoretical discussion with this principle rather than the one with which Einstein finally completed his theory of gravity is to "bury the lead."

To find our way through general relativity, we will approach its concepts in an order reverse of that in which Einstein worked through them. Einstein started first with a questionable but useful inspiration to regard gravity as indistinguishable from uniform acceleration (his Principle of Equivalence, which he called his "happiest thought"). From there he proceeded to the conclusion that clocks must run more slowly in accelerated frames of reference. He based an early attempt at a theory of gravity on this difference in clock rates, which difference is now called (luckily by only a few) the "curvature of time," an ill phrase if ever there was one (Brockman, p. 79; Schutz, pp. 225- 235). This earlier effort on Einstein’s part resulted in a calculation of the bending of light around heavy bodies that essentially agreed with Newton’s theory of gravity and gave only half of the currently-accepted value. Einstein later added the curvature of space to his formulas, and thus finally arrived at his theory of curved "spacetime." In our study, we may begin at the end; we are presented with Einstein’s brilliant solution of curved spacetime and given the daunting task to comprehend it. We will first find a foothold in the idea of curved space, the concept that Einstein included last. From there we shall be able to work backwards and deduce the so-called "curvature of time," a consequence of curved space that turns out to be nothing more than Newtonian gravity; and as it turns out, we will need neither resort to Einstein’s equivalence principle nor to the concept of time as a physical dimension.

It is relatively rare to see a professor write about the "curvature of space" rather than the curvature of "spacetime," but I find great teaching value in the former approach. It is simpler, easier to visualize, and thus, in my opinion at least, more powerful as a concept. In personal correspondence I received from a qualified professional, the "curvature of space" approach has been criticized as "oversimplified," and I cannot disagree inasmuch as I have yet to see it presented in the fullness of detail I will provide below.

Kenneth Lang writes, "According to Einstein’s theory, space is distorted and curved in the neighborhood of matter. In effect, space has both content and shape. The curved shape is molded into space by its content, the massive objects." And: "Flat Euclidean space ... describes a world which is without matter ... the amount of space curvature is greatest in the regions near the object, while further away the effect is lessened." (Lang, p. 217)

A common method of helping students to visualize the curvature of "spacetime" is to show a heavy object such as a bowling ball resting on a rubber sheet. The bowling ball represents a source of gravitational force. If a smaller ball, such as a marble, is rolled in the vicinity of the larger one, rather than continuing to travel in a straight line it will curve toward the bowling ball, perhaps eventually colliding with it after falling into the dent created by the bowling ball. While not unuseful, this method is unsatisfying for several reasons.

Illustration: A rubber sheet repels water in the event of bed-wetting.

First, it relies on a presumed downward gravity to demonstrate how gravity is created by the distortion of the sheet; it would not work as obviously if the dent in the sheet pointed in any direction other than downward. If we rolled a marble toward a bump rather than toward a depression, we would expect it to curve away rather than toward the bump. Nevertheless, we are supposed to understand that gravity is purely a result of the curvature of the sheet itself, not its orientation in any other gravitational field. Only seldom have I seen the "rubber sheet" model used to better effect, showing for example that light cones embedded in the sheet are bent inward by the curvature of space surrounding a large mass; and this diversion is shown as being representative of the pull of gravity regardless of the orientation of the sheet. (Orzel, pp. 231-32) However, this implies that the upward direction in the model corresponds to the future; this relates to another disadvantage of this model in general.

Second, this model of gravity requires space as we know it to be curved along some additional dimension of space; part of the two-dimensional sheet is pushed along a third dimension to give it curvature. Are we thereby to understand that three-dimensional space curves in the fourth dimension of time? Or does it curve into some other fifth dimension? No, neither of these is the case, and some authors are careful to point this out (Wheeler, p.127), though there are voices to the contrary, speaking of "a three-dimensional space that is curved in the fourth dimension." (Ferris, p. 197) Lawrence Krauss is among those who do not find a fourth dimension necessary to explain the curvature of the other three, but recognizes the difficulties in grasping such a concept:

It is virtually impossible for us, who are confined to live within a curved three-dimensional space, to physically picture what such a curvature implies. We can intuitively grasp a curved two-dimensional object, such as the surface of the earth, because we can embed it in a three-dimensional background for viewing. But the possibility that a curved space can exist in any number of dimensions without being embedded in a higher-dimensional space is so foreign to our intuition that I am frequently asked, "If space is curved, what is it curving into?" (Krauss, p. 56-57)

Before going into the third reason I find the "rubber sheet" model unsatisfying, I will present an alternative. A topographical map shows the elevation at various points on the Earth using contour lines. These contour lines form closed loops and represent a continuous path where the elevation is equal. Where these lines are far apart, the land is relatively flat. Where they are bunched closer together, there is a steep grade. The map also has a grid of north-south and east-west lines that divides the map into squares. Because the boundaries of surfaces shrink when the surface becomes wrinkled, map squares having a greater density of contour lines contain a greater amount of surface area. There is more "area within the area." If you were to draw a circle or square around a steep hill surrounded by flat plain, you might say that the surface area contained by the circle is bigger than the area implied by the circumference of the circle or the perimeter of the square. Extending this concept to three dimensions, you might say that an enclosed curved space is "bigger on the inside than on the outside," a concept already familiar in science-fiction and fantasy which many students will therefore readily seize upon.

Illustration, left: Changes in elevation are represented by contour lines on a flat surface. Right: A topographic map of Stowe, Vermont. The brown contour lines represent the elevation. The contour interval is 20 feet.

This is the idea I prefer for visualizing the curvature of space: that three-dimensional space is denser in the presence of mass, just as the contour lines are denser on steeper areas of a map. Except in the case of curved space, there is no hill rising upward, nor valley sinking downward; nor is there necessarily any fourth dimension into which the three dimensional space is bent. This model requires no preferred orientation to show its curvature, not does it require that the space’s curvature gives it extent in some additional direction. It only requires this small but highly significant amendment to our ideas of geometry as described so far; I will show some of the desirable consequences below, as well as some of the greater implications.

Ironically, though Hans Ohanian’s description of curved space is among the best I have yet seen, he also writes one of the more emphatic denials that such a thing can be visualized:

Curved three-dimensional space ‐ or even worse, curved four-dimensional spacetime ‐ is impossible to visualize. ... Some mathematicians claim they can visualize a curved three-dimensional space, but if so, they are crazy, that is, crazy in the sense of abnormal. The best a normal person can do is to visualize a curved surface, such as the surface of an apple or the surface of the Earth. Such a surface is a two-dimensional curved space which curves into the visualizable third dimension. The curved four-dimensional spacetime of general relativity curves into a fifth, sixth . . . or even a tenth dimension. (Ohanian, p. 192)

Ohanian’s expectations of what curved three-dimensional space should look like are burdened by the idea of additional dimensions ‐ he writes, "If our three-dimensional space is curved, it must be curved into some dimension beyond three dimensions." (Ohanian, p. 192) ‐ and thus he overshoots the mark, not seeing the full implications of the excellent analogy he himself provides for curved space:

[I]n a gravitational field, light propagates more slowly than outside the field. This means that in [this] regard ... the neighborhood of the Sun behaves like a large glass-filled globe encasing the Sun, so this glass slows the propagation of light. The glass is most dense near the sun, less dense further out, and it gradually fades away into empty space at large distances (where the speed of light resumes its standard [vacuum] value). Einstein recognized that such a slowing of the speed of light would bend rays of light in the same way that a glass globe bends [them] ‐ it bends rays of light that strike the right half of the globe toward the left, and it bends rays ... that strike the left half toward the right, that is, it always bends [them] toward the centerline. (Ohanian, pp. 189-90)

If one sees that the density of the globe in this analogy is the density of space itself, then it is not difficult to see how the effects of gravity follow from it, as I will demonstrate. Gravity curves space both positively and negatively at any given point in space, resulting in the tidal forces that cause the parallel paths of freely-falling objects to converge in some directions and diverge in others. At a given point in space, gravity will cause parallel paths near that point to converge in the plane normal to the direction of gravity at that point, due to the positive curvature in that plane; and to diverge near that point in planes having a normal that is perpendicular to the direction of gravity at that point, due to the negative curvature in those planes. These curvatures are illustrated below.

The idea that the same point in three-dimensional space can be a location of both positive and negative curvature ‐ depending on the plane being considered ‐ is analogous to the way a negatively-curved two-dimensional surface has two centers of curvature on opposite sides of that surface, as discussed previously. A two-dimensional surface may have only positive or negative curvature at any point, just as a one dimensional line may at any point on that line only curve toward or away from another point outside that line; but just as a two-dimensional surface may curve both toward and away from an exterior point, three-dimensional space may be both positively and negatively curved.

Wheeler discusses this "double curvature" extensively in his popular-science book on gravity and provides equations quantifying the curvatures described above; the variously-oriented planes are treated by Wheeler as various sides of an imaginary cube having one side facing directly toward a spherical source of gravity. For each side of the cube, the curvature depends on the mass m of the source of gravity and the radius r from the center of that mass. On the facing (toward or away from the mass) sides of the cube, the curvature is proportional to 2m/r^{3}. On the four non-facing sides, the curvature is proportional to ‐m/r^{3}. Curvature is thus positive on two faces and negative on the other four. Summing the curvature of all six sides, these equations neatly balance to zero, and the curvature is not surprisingly in units of density: mass divided by volume. (Wheeler, pp. 88, 141-142) This pairing of sides ‐ one pair of positively-curved sides and two pairs of negatively- curved sides ‐ corresponds directly to the dimensions of space in which gravity acts: one parallel to the force of gravity, or "vertical"; and two perpendicular or "horizontal." The tidal forces tend to increase the vertical separation of freely-falling objects and reduce their horizontal separation in both horizontal dimensions.

The illustrations below show the curvature on Wheeler’s imaginary cube sides, and as far as I am aware, they are an innovation on my part. A single circular contour line, representing a fixed radius from the mass, is drawn through each cube side, though an arbitrary number of lines could be drawn in the same manner as magnetic field lines. On the non-facing sides of the cube, the contour lines tend to cross the side, and do so most clearly at great distances from the mass as the radius increases and the proportion of arc contained in the cube decreases. In these illustrations the radius is small, to show a case in which the center of mass is not far from the edge of the cube.

A triangle is drawn in red between three points on the cube’s side. These points are connected by a representation of the shortest possible path between them. Since space is more dense on the inside of the contour line than on the outside, the paths that must cross this contour line are curved in order to lessen their distance through this denser space. The result is that the interior angles of this triangle are less than 180 degrees. This side of the cube is thus shown to have negative curvature, as do all of the four sides not facing directly toward or away from the mass.

On the facing sides of the cube, the circular contour lines are centered in that side, since the center of that side is nearest the mass. Below, I have chosen three points on this side corresponding to the three points chosen above, and drawn the lines between them in blue, again representing the shortest possible path between the points. And again, since the space inside the contour line is more dense than the space outside, the lines curve to lessen their distance through that denser space. But in this case, that curvature results in the interior angles of the triangle adding to more than 180 degrees, and the resulting curvature is positive.

Thus we see a third failure of the "rubber sheet" model of curved spacetime. The sheet is stretched by the bowling ball into a shape having only a negative curvature in the space surrounding the ball, and a strictly positive curvature where it contacts the ball; whereas the curvature produced by the gravity of such a mass is not so.

All right, then, we have defined the curvature of space. But what is this "curvature of time" that is occasionally spoken of? It means nothing more than the fact that clocks run more slowly near a source of gravity than they do farther away from that source. It may seem a strange idea, but it has been experimentally proven. In fact, if we had not already known Einstein’s theory of gravity by the time satellite-based navigation was invented, someone would have had to come up with the theory in order to synchronize the satellites’ clocks with those on the ground. But why do the clocks run more slowly near sources of gravity? The answer is found in the curvature of space that we have just discussed, and is best illustrated using a "light clock." As the clock nears a massive object, the space it occupies is more dense. The space between the mirrors is therefore greater and the path for one "tick" is longer, though the clock is stationary, and regardless of the direction the mirrors are oriented in. Because this same slowing is applicable to any process by which time may be measured, we say that time passes more slowly in regions of high gravity. The classical theory of gravity touched upon this idea by predicting that the frequency of light would change as a light beam progressed through different altitudes; Einstein extended this much further to connect time itself to the strength of gravity’s pull.

In summary, we see the need for four coordinates of time and space but only three dimensions. To realize the impossibility of building a time machine, we needn’t discuss the causal paradoxes inherent in time travel, such as the possibility of preventing one’s own birth. We need only consider the nature of time. The energies and masses of the past radiate outwards at speeds up to the speed of light. The past of 30 years ago is not located at 30 years’ distance along a single physical time axis; it is scattered in all directions up to thirty light years away. Good luck reassembling all that using the "flux capacitor" in your DeLorean, Doc Brown. I love you but you’re out of your mind.

One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is and will remain the most convenient. ‐ Henri Poincaré (1854-1912)

I came to my "contour map" visualization of gravity rather gradually and naively, beginning with the fanciful supposition that light passed through matter more slowly due to some mysterious distortion of space. I found my first encouragement in a draft manuscript on Edwin Taylor’s website (for *Exploring Black Holes*, a book he was then rewriting with Edmund Bertschinger and John Archibald Wheeler). Space, it said, is stretched in the vicinity of massive objects, even (in the case of black holes) outside of any event horizon. This astonishing idea became central to my understanding of gravity, and eventually the key for understanding its effects. But it was only in the preparation of this essay that I realized that this idea may be considered an alternative to Einstein’s formulation of general relativity rather than a concise explanation of it.

I had naively assumed that my concept of curvature in three dimensions was one of many possible Riemannian "manifolds," or perhaps at least a "pseudo-Riemannian" manifold, as Minkowski space is called. Instead, I found that Riemann had given as a postulate the very principle that I had discarded:

Measure-determinations require that quantity should be independent of position, which may happen in various ways. The hypothesis which first presents itself, and which I shall here develop, is that according to which the length of lines is independent of their position, and consequently every line is measurable by means of every other.

In Riemann’s system, space does not "vary in density." The distance between the mirrors on the light clock remains the same whether the clock is deep within a strong gravitational field or in its weaker outer reaches. Just as Euclid had postulated that parallel lines remain equidistant, Riemann proceeded on the assumption that scales of distance are uniform in a region of n-dimensional space. At first I worried that I had made a misstep in rejecting Riemann’s postulate, but then I realized that Einstein might have done so himself, had he gotten different mathematical advice. Special relativity was a tentative first step in this direction, making measured lengths vary between observers in relative motion. General relativity could have delivered the killing blow in making lengths of measure vary between regions of stronger and weaker gravity. Though he may not have realized it, Einstein’s theory of gravity is one of non-Riemannian geometry if one considers it in three dimensions rather than four.

This non-Riemannian geometry appears to correspond to the "shape dynamics" concept which Lee Smolin describes as an alternative but functionally equivalent theory to general relativity: "General relativity is more or less the opposite. Sizes of objects remain fixed when you move them around, so it’s meaningful to compare the sizes of distant things." In shape dynamics, by contrast, "Size becomes relative and it becomes meaningless to compare the sizes of objects far from one another." (Smolin 2013, pp. 168-170)

Interestingly, one of the earliest attempts to unify gravity with electromagnetism was based on a similar idea. In 1918, "Hermann Weyl suggested that this [unification] could happen through local variations of scale or ‘gauge’ in space." (Wolfram, p. 1028) "Weyl’s new principle of gauge symmetry allowed the use of different distance scales (or gauges) at different points. One could use a different gauge to measure sizes at different points, as long as one had a mathematical entity called a connection that related what one was doing at neighboring points. . . . Einstein quickly objected to Weyl’s gauge principle by noting that the size of a clock would change as it moved through regions containing electromagnetic fields." (Woit, pp. 59-60) Weyl’s was not a bad idea; it was just not the correct application of it. General relativity is a theory showing the scale invariance of special relativity: the freely falling observer measures no change in the speed of light as space becomes more or less "dense," because the rate at which his clock ticks ‐ an indirect expression of the density of space ‐ must decrease accordingly.

Stephen Wolfram also sees a three-dimensional interpretation of Einstein’s equations for gravity: "In their usual formulation, the Einstein equations are thought of as defining constraints on the structure of 4D spacetime. But at some level they can also be viewed as defining how 3D space evolves with time." (Wolfram, p. 1053)

Ohanian, cited above, is far from alone in seeking a higher-dimensional context for the curvature of spacetime; indeed, the conditions of Riemannian geometry seem to imply such. And in the context of Riemannian geometry, he may be justified in asserting the near-impossibility of visualizing the curvature of a three-dimensional space. It is perhaps this difficulty that leads to a confusing use of analogy in higher dimensions. Some have pointed out that a freely-falling trajectory which is curved (or even circular) in three-dimensional space is a straight path in four-dimensional spacetime. Hawking likens this to an airplane in straight-line flight casting a shadow which dances in an abruptly curved path over hilly ground. (Hawking, p. 30) But this analogy seems rather backwards when one considers that according to Riemannian geometry, a straight path in an n-dimensional curved space has curvature that is visible only in a space of n+1 dimensions: for instance, an ant’s straight-ahead march across the two-dimensional surface of the apple results in a curved path in three dimensions. If Hawking’s analogy indeed concerned the behavior of a Riemannian space, one would expect the shadow to appear straight and the flight of the airplane to appear curved. (Ross)

That I arrived independently at a "gauge symmetry" or "shape dynamics" alternative to Minkowski spacetime with nearly no formal instruction in relativity should be an endorsement of its practicality for undergraduate students. Indeed, none of the mathematics presented in this paper have been beyond the level of a common high school student of my generation, or of the middle-school students I have recently tutored. I hope that the methods in this presentation can bring relativity ‐ both the special and general theories ‐ to students of a younger age.

Journalist George Johnson writes of his discouragement with relativity as he considered his college major:

Going through the physics course descriptions in the university catalog, I realized that by my senior year I would be all the way up to the nineteenth century. . . . Only many years later, when I’d earned a Ph.D., would I be taken into a chamber where, like a thirty-third degree Freemason, I’d see the true mysteries revealed ‐the shrinking rulers and slowing clocks . . . and why all this made E equal mc^{2}.

As suggested by the above quotation of Poincaré, a (pseudo-)Riemannian approach to gravity is of course not "wrong;" the math works to the degree of precision in which we have been able to measure experimental results. But inasmuch as it is nearly impossible to visualize, would we not be better served by a geometry ‐ at least in the introductory phase of teaching the subject ‐ which is no less correct and more intuitive? The current paradigm seems to permit the beauty of relativity to be apparent only to those having the considerable will and aptitude necessary to surmount its formidable mathematical barriers. This would seem displeasing to Einstein, who wrote: "Restricting the body of knowledge to a small group deadens the philosophical spirit of a people and leads to spiritual poverty." (Brockman, p. 163)