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.

Given the minimal assumptions of the standard models, the matter-energy content of the universe and its geometry are strictly tied together and do not change as the universe evolves. Less restrictive models (incorporating additional complications) are required if the geometry of the universe were to change. An example is the inflationary model (Chapter 15) wherein the early universe with arbitrary geometry evolves rapidly to become flat. After this brief epoch the universe evolves onward as a standard flat model.

It depends on what you mean by "universe." If you mean "everything" then by definition there is only one universe. If you mean that volume of stuff that is consistent with what we observe (i.e. everything consistent with our understanding of the cosmological principle) then you could admit the idea of other universes that have far different properties. Our cosmological principle would then be true only for our universe within this greater "metauniverse." It is intriguing to speculate along these lines, and the point about the Copernican Principle is interesting. However, we have no real way to know.

They 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. If you like the inflationary model of the universe then you must postulate a cosmological constant during the earliest moments of creation. For the times afterward, however, most theorists prefer zero, since explaining the existence of a nonzero, but small, cosmological model is difficult within our current understanding. So, maintaining a healthy skepticism, cosmologists await further data.

To a large extent the cosmological constant is a relic of the historical development of 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). For the purpose of this course, the cosmological constant models provide instructive alternatives. It is possible that a cosmological constant may have been present in the early universe, creating the period of "inflation." There are reasons why this alternative model is interesting. These reasons are discussed in Chapter 15. Furthermore, some new data are tending to 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.

*
Would a big crunch be contrary to the 2nd law of thermodynamics?
*

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

*
Evenly distributed gas particles have higher entropy than particles
clumped in one spot, yet evenly distributed mass has lower entropy
compared to mass clumped in one spot. Why is this?
*

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.

The early universe was incredibly hot, incredibly dense, and filled with high-energy matter and antimatter. Stars were not produced for approximately a billion years after the big bang, since the contents of the universe had to cool sufficiently for atoms to exist, and gravity had to have enough time to draw overdense regions closer together.

The Hubble constant is not enough. We would need to know something else like the mass density, or the true age of the universe. And then there is that pesky cosmological constant.... But knowing the Hubble constant accurately would be an excellent start to determining the fate of the universe.

We don't know. At the moment the only answer we can give is that expansion was part of the initial conditions.

Fire or ice. We don't know which, but most cosmologists think ice is nice.

You don't even need a steady state universe. If the cosmological principle holds, and the universe is spatially infinite, there are an infinite number of planets out there with an infinite number of people, doing an infinite number of things....But we are ourselves by definition; if there are an infinite number of versions of ourselves they certainly are not interacting with us, and hence we are distinguishable.

Basically, this occurs when all the stars that are capable of supporting life burn out. A star like the Sun lasts for approximately 10 billion years, but there are still sun-like stars being formed today. Smaller stars last even longer. So it might be several tens to hundreds of billions of years before life as we know it would not be possible.

It depends on what you mean by the "density." If you mean matter density, then it doesn't necessarily affect it (except insofar as the matter density helps to determine the overall gravity). But if you mean a sort of "effective density" determining the evolution of the universe, the curvature of space ("gravity") does play a role in how the universe evolves. Matter isn't needed for an evolving spacetime (e.g., any of the various empty models).

If we knew the Hubble constant for certain, we would still need to know one additional parameter to know the fate of the universe (age, density, deceleration). And even then we would have to know one more to rule in or out a nonzero Lambda force.

Plenty of such cosmologies are possible, although they are more difficult to study because they are more complicated. Bianchi models are a type that is homogeneous but not isotropic (universe expands at different rates in different directions). The Goedel model is neither isotropic nor homogeneous. It is generally not possible to display their evolution through an R-t plot. The scale factor R determines everything in homogeneous and isotropic universe - R depends neither on location nor on direction so a simple plot showing its time dependence is sufficient. More complicated universes would require more complicated descriptions.

Nope. Everything in the universe will get caught in the big crunch. It is analogous to falling into the singularity of a black hole. All futures lie in the singularity.

Nope. Everything in the universe will get caught in the big crunch. It is analogous to falling into the singularity of a black hole. All futures lie in the singularity.

It's possible perhaps. People have wondered about that and the idea has been taken seriously in the past. There doesn't seem to be any compelling reason for the universe to do so, and several reasons why it might not. But since we don't know really what goes on in the singularities we can't say.

It has a broader interpretation as a vacuum energy density, or an exotic form of matter-energy. It is taken seriously, and if you believe the latest Hubble expansion measurements, it may even be necessary to explain their observation that the universe is accelerating (q < 0).

See above. Whether or not any of the current hypotheses are "valid" remains to be seen.

Space getting smaller is a good way to put it. In the spherical model space is finite in volume, and as the collapse proceeds the volume goes to zero.

Nothing. The Lambda factor is completely valid as a term in Einstein's equations. We just don't know whether the Lambda term is equal to zero or some other value.

It takes as long to collapse as it takes to expand. The specific number would depend on the model.

I like to think that we will have a pretty certain answer within the next few decades. We have made a lot of progress on this question in the twentieth century and telescopes are getting better and better.

The linear model doesn't have any matter in it, so there is nothing to slow down the rate of expansion. With matter the expansion rate slows with time. But it doesn't slow enough ever to stop and recollapse.

My frank view is that they are like somebody trying to keep Ptolemy's model alive after Kepler made his discoveries. I am perfectly willing to entertain the idea that our current best big bang model isn't the final word on cosmological models, but we certainly have progressed beyond the philosophical leanings that motivate the attempts to maintain the steady state model.

Also see the well-written "rebuttal" in the following article.

Return to Chapter 11 | Table of Contents

Copyright © 1998 John F. Hawley