Chapter 11 Questions

Infinity is a funny number. Remarkably, there are different sized infinities. For example, the infinite number of integers can be shown to be definitely less than the infinite number of real numbers (specifically, it is a countable infinity versus a continuum infinity). However, that isn't the situation here. In the case of geometry, the difference between the sizes of the geometries is defined locally. To see this, choose any point. Go out a radius R and construct a sphere around that point; the sphere consists of the locus of all points exactly at a distance R from the central point, as measured in the geometry in question. The hyperbolic geometry will have move volume within the sphere than will the flat geometry for the same R, while the flat will have more volume than the spherical. As an analogy, consider two hotels, each with an infinite number of rooms. However, the second hotel's rooms are bigger and can hold more guests per room. (Puzzler: if the hotel is full and new guests arrive, how can they be accommodated?)

It is difficult to visualize and requires some practice. Another way to envision negative curvature is the outside of a trumpet bell. The curvature of the bell bends outward (negatively) in contrast to the inward bending of positive curvature (like a sphere). The trumpet bell isn't homogeneous or isotropic since it has a thin end and a thick end, but it provides another illustration of the idea of negative curvature.

Cosmologists used to regard it with the suspicion normally given to a "fudge factor." Most would be extremely reluctant to adopt a cosmological constant unless the alternative were even more distasteful (e.g. complete abandonment of general relativistic cosmology), unless forced to concede by data. Well now it appears that the data have forced cosmologists to accept the cosmological constant. The attitude now is: it seems to be a component of the universe whether we understand it or not.

The cosmological constant has been a feature throughout the historical development of relativistic cosmology. From Einstein's first introduction of it, it was an attempt to make a model consistent with certain expectations; at the time it was thought that the GR equations without it didn't produce a viable alternative. It enjoyed a revival when it was believed that the Hubble time was less than the age of the Earth (better observations eventually showed this belief to be incorrect). It appears that a cosmological constant may have been present in the early universe, creating the period of "inflation." This is discussed in Chapter 16. Now new data indicate the presence of a cosmological constant, which reveals itself as an acceleration, rather than a deceleration, of the rate of expansion of the universe.

It is rationally impossible to accept the literal reading of the Bible as well as the modern cosmological theory. Most modern religions regard the creation stories in the Bible as allegorical. The Pope has declared that there is no conflict between modern cosmological theory and Church teaching. However, humans are not always completely rational. It is entirely possible for a human to believe two mutually exclusive things at the same time, so I wouldn't rule out the possibility of a strict Bible-literal fundamentalist astronomer. After all, one could carry out the mechanics of astronomy, take observations, solve the equations, etc. all without confronting the idea that one is literally seeing billions and billions of distant galaxies with untold numbers of stars and worlds through unfathomable space, and without contemplating the implications for those observations for any narrow geocentric worldview. Most creationist scientists of whom I have heard tend to be engineers, or programmers, or in some such specialty wherein they narrow the focus of their learning sufficiently to avoid the huge conflict between reality and literal creationism.

No. In fact, a big crunch could represent a state of even higher entropy than the "heat death" of the open and flat models.

The confusion here is probably over the situations with and without gravity. (Gas particles are "mass," after all.) First consider the situation when gravity is absent or negligible, such as a gas in a room on the Earth. (Gravity is certainly present here, but it is fixed. The distribution of the gas itself doesn't alter the gravitational field.) In this case, the more clumped state could expand, potentially doing work in the process. This phenomenon happens every day within the cylinders of an internal-combustion engine. Gases expand rapidly and push the piston, doing work. The expanded gas has exhausted much of its ability to do work, and thus its entropy has increased; the original clumped state had lower entropy.

Now consider the case when gravity is important. A self-gravitating aggregation of gas possesses some gravitational potential energy. As it collapses to a more clumped state, it releases gravitational potential energy. This gravitational potential energy could, in principle, do work. (Hydroelectric power is an example of gravitational potential energy doing work on Earth.) The more clumped the gas, the less gravitational potential energy it possesses, and the smaller is its capacity to do work. Thus in a gravitating system, the more clumped state has higher entropy. The ultimate is the black hole, which cannot collapse further and whose gravitational energy cannot be tapped.

If the Hubble constant were determined exactly, would we then know the ultimate fate of the universe?