• In modern physics and cosmology, space and time are physical and are part of the universe. The big bang did not happen in a pre-existing space and time.
• The Anthropic Principle has been used to argue that the presence of life constrains the universe or determines why the universe is as it is. Why is this not particularly useful as a scientific concept?
• The Cosmological Principle asserts that the universe is homogeneous and isotropic. Know what those terms mean. Does the universe have a center or an edge? How does the Perfect Cosmological Principle differ from the Cosmological Principle?

Isotropy and homogeneity are important concepts. We observe that on the largest scales the universe appears the same in all directions, hence it is isotropic. Unless we are at the center of the universe, it follows that the universe must also be homogeneous (the same everywhere). Do you understand why? Of course, the universe is not identical everywhere. The Solar System is different from interstellar space; there are galaxies, and clusters of galaxies scattered about. We believe that on the largest scales, those spanning billions of light years, the various objects are statistically indistinguishable. (A somewhat related problem from statistics: How many people must be included in a survey before it becomes representative of humans in general?) Determining the scale upon which the universe really does look the same everywhere is an active field of research.

Isotropy and homogeneity are properties that can be applied to the overall shape (geometry) of the universe. We will see (in Chapter 8) that these properties limit the possible geometries of the universe to a very small subset of all possible geometries. Two of the possibilities are infinite in size, while the third is finite yet still has no center or edge!

• What is an inertial reference frame? What is a noninertial reference frame? How can they be distinguished?
• What is meant by Relativity? What is the difference between quantities that are relative and those that are invariant?
• Why does the theory of electromagnetic waves (Maxwell's equations) run into difficulties with the Galilean rules of relativity? If light were conveyed by the luminiferous ether, it follows that light speeds would be relative to the rest frame of the ether. How did this address the relativity problem? How was the ether sought experimentally?

If you awoke in a closed, circular room, what sort of experiment might you do to determine whether the room was being rotated? In other words, how could you determine if the room was an inertial or a noninertial reference frame?

The Earth rotates and so is not an inertial frame, but it rotates slowly so it is difficult to detect the effects of its rotation. However, the sense of rotation around low and high pressure areas, such as hurricanes and storms visible in photos of the Earth from space, is due to the rotation of the Earth.

Why is "Galilean Relativity" named after Galileo? Galileo worked to come to grips with a requirement of the Copernican cosmology: the Earth is moving through space, and rotating on its axis, but its inhabitants notice no sensation of motion. He concluded that if everything on the Earth shared the same motion, then that motion would be undetectable. From here one proceeds to the idea that there is no absolute frame of rest. (Such an absolute frame of rest had been assigned to the Earth in Aristotlean cosmology.) One inertial frame is completely equivalent to another. Newton's Laws then must be the same in any inertial frame as in another. This is Galilean relativity. Einstein's version of relativity maintains the underlying idea (the equivalence of inertial frames), but replaces Newton's Laws with more general laws of motion.

It is easy to confuse homogeneity with isotropy, or to fail to understand how they are distinct.

In order to illustrate geometries and their properties concretely, we generally must refer to two-dimensional geometries, specifically the surfaces of three-dimensional objects, such as the surface of a sphere. It would be difficult to visualize the geometry of 4-dimensional objects! Take care not to become confused by the third dimension in our 2D analogies. When we say that the surface of a sphere has no center, don't counter by pointing to the center of the 3D sphere. The 2D surface has no center that is part of the 2D surface.

Questions and Answers related to Chapter 6.

On-line biographical information about Albert Michelson and additional links if you would like to know more about the accurate measurment of the speed of light.