The Elegant Universe - Brian Greene [38]
The Basics of General Relativity
To get a feel for this new view of gravity, let's consider the prototypical situation of a planet, such as the earth, revolving around a star, such as the sun. In Newtonian gravity the sun keeps the earth in orbit with an unidentified gravitational "tether" that somehow instantaneously reaches out across vast distances of space and grabs hold of the earth (and, similarly, the earth reaches out and grabs hold of the sun). Einstein provided a new conception of what actually happens. It will aid in our discussion of Einstein's approach to have a concrete visual model of spacetime that we can conveniently manipulate. To do so, we will simplify things in two ways. First, for the moment, we will ignore time and focus solely on a visual model of space. We will reincorporate time in our discussion shortly. Second, in order to allow us to draw and manipulate visual images on the pages of this book, we will often refer to a two-dimensional analog of three-dimensional space. Most of the insight we gain from thinking in terms of this lower-dimensional model is directly applicable to the physical three-dimensional setting, so the simpler model provides a powerful pedagogical device.
In Figure 3.3, we make use of these simplifications and draw a two-dimensional model of a spatial region of our universe. The grid-like structure indicates a convenient means of specifying positions just as a street grid gives a means of specifying locations in a city. In a city, of course, one gives an address by specifying a location on the two-dimensional street grid and also giving a location in the vertical direction, such as a floor number. It is the latter information, location in the third spatial dimension, that our two-dimensional analogy suppresses for visual clarity.
Figure 3.3 A schematic representation of flat space.
In the absence of any matter or energy, Einstein envisioned that space would be flat. In our two-dimensional model, this means that the "shape" of space should be like the surface of a smooth table, as drawn in Figure 3.3. This is the image of our spatial universe commonly held for thousands of years. But what happens to space if a massive object like the sun is present? Before Einstein the answer was nothing; space (and time) were thought to provide an inert theater, merely setting the stage on which the events of the universe play themselves out. The chain of Einstein's reasoning that we have been following, however, leads to a different conclusion.
A massive body like the sun, and indeed any body, exerts a gravitational force on other objects. In the example of the terrorist bomb, we learned that gravitational forces are indistinguishable from accelerated motion. In the example of the Tornado ride, we learned that a mathematical description of accelerated motion requires the relations of curved space. These links between gravity, accelerated motion, and curved space led Einstein to the remarkable suggestion that the presence of mass, such as the sun, causes the fabric of space around it to warp, as shown in Figure 3.4. A useful, and oft-quoted, analogy is that much like a rubber membrane on which a bowling ball has been placed, the fabric of space becomes distorted due to the presence of a massive object like the sun. According to this radical proposal, space is not merely a passive forum providing the arena for the events of the universe; rather, the shape of space responds to objects in the environment.
Figure 3.4 A massive body like the sun causes the fabric of space to warp, somewhat like the effect of a bowling ball placed on a rubber sheet.
This warping, in turn, affects other objects moving in the vicinity of the sun, as they now must traverse the distorted spatial fabric. Using the rubber membrane–bowling ball analogy, if we place a small ball-bearing on the membrane and set it off with some initial velocity, the path it will follow depends on whether or not the bowling ball is sitting in the center. If the bowling ball is absent,