The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [186]
5. For the mathematically inclined reader, Einstein’s equations of general relativity in this context reduce to . The variable a(t) is the scale factor of the universe—a number whose value, as the name indicates, sets the distance scale between objects (if the value of a(t) at two different times differs, say, by a factor of 2, then the distance between any two particular galaxies would differ between those times by a factor of 2 as well), G is Newton’s constant, is the density of matter/energy, and k is a parameter whose value can be 1, 0, or -1 according to whether the shape of space is spherical, Euclidean (“flat”), or hyperbolic. The form of this equation is usually credited to Alexander Friedmann and, as such, is called the Friedmann equation.
6. The mathematically inclined reader should note two things. First, in general relativity we typically define coordinates that are themselves dependent on the matter space contains: we use galaxies as the coordinate carriers (acting as if each galaxy has a particular set of coordinates “painted” on it—so-called co-moving coordinates). So, to even identify a specific region of space, we usually make reference to the matter that occupies it. A more precise rephrasing of the text, then, would be: The region of space containing a particular group of N galaxies at time t1 will have a larger volume at a later time t2. Second, the intuitively sensible statement regarding the density of matter and energy changing when space expands or contracts makes an implicit assumption regarding the equation of state for matter and energy. There are situations, and we will encounter one shortly, where space can expand or contract while the density of a particular energy contribution—the energy density of the so-called cosmological constant—remains unchanged. Indeed, there are even more-exotic scenarios in which space can expand while the density of energy increases. This can happen because, in certain circumstances, gravity can provide a source of energy. The important point of the paragraph is that in their original form the equations of general relativity are not compatible with a static universe.
7. Shortly we will see that Einstein abandoned his static universe when confronted by astronomical data showing that the universe is expanding. It is worth noting, though, that his misgivings about the static universe predated the data. The physicist Willem de Sitter pointed out to Einstein that his static universe was unstable: nudge it a bit bigger, and it would grow; nudge it a bit smaller, and it would shrink. Physicists shy away from solutions that require perfect, undisturbed conditions for them to persist.
8. In the big bang model, the outward expansion of space is viewed much like the upward motion of a tossed ball: attractive gravity pulls on the upward-moving ball and so slows its motion; similarly, attractive gravity pulls on the outward-moving galaxies and so slows their motion. In neither case does the ongoing motion require a repulsive force. However, you can still ask: Your arm launched the ball skyward, so what “launched” the spatial universe on its outward expansion? We will return to this question in Chapter 3, where we will see that modern theory posits a short burst of repulsive gravity, operating during the earliest moments of cosmic history. We will also see that more refined data has provided evidence that the expansion of space is not slowing over time, which has resulted in a surprising—and as later chapters will make clear—potentially profound resurrection of the cosmological constant.
The discovery of the spatial expansion was a turning point in modern cosmology. In addition to Hubble’s contributions, the achievement relied on the work and insights of many others, including Vesto Slipher, Harlow Shapley, and Milton Humason.
9. A two-dimensional torus is usually depicted as a hollow doughnut. A two-step process shows that this picture agrees with the description provided in the text. When we declare that crossing the right edge of the screen brings you back to