.......The Elegant Universe
THE ELEGANT UNIVERSE, Brian Greene, 1999, 2003
```(annotated and with added bold highlights by Epsilon=One)
Chapter 3 - Of Warps and Ripples
A Few Caveats
The rubber membrane—bowling ball analogy is valuable because it gives us a visual image with which we can grasp tangibly what we mean by a warp in the spatial fabric of the universe. Physicists often use this and similar analogies to guide their own intuition regarding gravitation and curvature. However, its usefulness notwithstanding, the rubber membrane—bowling ball analogy is not perfect and for clarity we call attention to a few of its shortcomings.

First, when the sun causes the fabric of space around it to warp this is not due to its "being pulled downward" by gravity as in the case of the bowling ball, which warps the rubber membrane because it is pulled earthward by gravity. In the case of the sun, there is no other object to "do the pulling." Instead, Einstein has taught us that the warping of space is gravity. The mere presence of an object with mass causes space to respond by warping. Similarly, the earth is not kept in orbit because the gravitational pull of some other external object guides it along the valleys in the warped spatial environment, as occurs for a ball bearing on the warped rubber membrane. Instead, Einstein showed that objects move through space (spacetime, more precisely) along the shortest possible paths—the "easiest possible paths" or the "paths of least resistance." If the space is warped, such paths will be curved. And so, although the rubber membrane—bowling ball model provides a good visual analogy of how an object such as the sun warps the space around it and thereby influences the motion of other bodies, the physical mechanism by which these distortions occur is totally different. The former appeals to our intuition about gravity in the traditional Newtonian framework, whereas the latter expresses a reformulation of gravity in terms of curved space.

A second shortcoming of the analogy stems from the rubber membrane's being two-dimensional. In reality, although harder to visualize, the sun (and all other massive objects) actually warps the three-dimensional space surrounding it. Figure 3.6 is a rough attempt to depict this; all of the space surrounding the sun—"below," "on the sides," on "top"—suffers the same kind of distortion, and Figure 3.6 schematically shows a partial sampling. A body, like the earth, travels through the three-dimensional warped spatial environment caused by the sun's presence. You may find this figure troubling—why doesn't the earth slam into the "vertical part" of curved space in the image? Bear in mind, though, that space, unlike the rubber membrane, is not a solid harrier. Instead, the warped grids in the image are but a couple of thin slices through the full three-dimensional warped space in which you, the earth, and everything else are immersed fully and move freely. Perhaps you find that this only makes the problem seem worse: Why don't we feel space if we are immersed within its fabric? But we do. We feel gravity, and space is the medium by which the gravitational force is communicated. As the eminent physicist John Wheeler has often said in describing gravity, "mass grips space by telling it how to curve, space grips mass by telling it how to move." 8

Figure 3.6 A sampling of the warped three-dimensional space surrounding the sun.
A third, related shortcoming of the analogy is that we have suppressed the time dimension. We have done this for visual clarity because, notwithstanding the declaration of special relativity that we should think of the time dimension on par with the three familiar spatial dimensions, it is significantly harder to "see" time. But, as illustrated by the example of the Tornado ride, acceleration—and hence gravity—warps both space and time. (In fact, the mathematics of general relativity shows that in the case of a relatively slow-moving body like the earth revolving around a typical star like the sun, the warping of time actually has a far more significant impact on the earth's motion than does the warping of space.) We will return to a discussion of the warping of time after the next section.

Important as these three caveats are, so long as you hold them in the back of your mind, it is perfectly acceptable to invoke the warped-space image provided by the bowling ball on the rubber membrane as an intuitive summary of Einstein's new view of gravity.