In this lesson we'll solve the FRW equation (one of the EFE's) for the scaling factor \(a(t)\) to determine the expansion rate of the universe in two different idealized scenarios: a universe filled with only radiation and a universe filled with only matter. These two different scenarios are called a radiation dominated universe and a matter dominated universe, respectively.

Shortly after Isaac Newton published his law of gravitation, the philosopher Richard Bentley and the astronomer Heinrich Olbers pointed out two paradoxes that arise from this law. The first, which is called Bentley's paradox, points out that if the universe is finite in size then, since the force of gravity is always attractive, all of the stars and galaxies in the universe should collapse in on themselves. The second, called Olbers' paradox, states that if the universe is infinite and if the distribution of stars in the universe is uniform then the night sky should be filled with infinitely many stars and the night sky should therefore be blindingly bright.

Using Newton's law of gravity and shell theorem, we can derive the FRW equation. The FRW equation is a differential equation whose solution is the scaling factor \(a(t)\). The FRW equation essentially give us the relationship between the distribution of mass/energy and the scaling factor; we can use the FRW equation to find out what \(a(t)\) for various different kinds of mass/energy distributions including a universe dominated by matter, one dominated by radiation, or one dominated by vacuum energy. We'll see that the FRW equation predicts scenarios where the universe could be expanding. One might point out, and correctly so, that this contradicts Newton's law of gravity since, according to this law, gravity cannot act as a repulsive force. To reconcile this (and, we'll do this in a separate lesson), we could derive the FRW equation from Einstein's general theory of relativity by plugging the FRW metric into the Einstein Field Equations; plugging in the FRW metric, one could simplify the Einstein Field Equations to the FRW equation. This essentially would resolve the contradiction of the FRW equation predicting that, under the right circumstances, the universe could be expanding because general relativity (unlike Newton's law of gravity) allows for things like expanding universes.

According to the Big Bang theory, for hundreds of thousands of years the entire universe was hotter than a star. The universe was so hot that electrons could not bind with atomic nuclei; this meant that the universe was opaque to radiation. If you were living in such a universe, you wouldn't be able to see very far; in fact, you would only be able to "see things" that are microscopic distances away. But as the universe expanded, everything cooled down. What we'll do in this lesson is calculate that after about 300,000 years since the origin of the universe, the universe cooled enough (due to expansion) for electrons to bind with atomic nuclei. After this happened, light was no longer constantly scattering off of nucleons in a plasma soup; once the universe cooled enough for all of the plasma (which was everywhere in the universe) in the universe to become a gas, light could travel through space freely without constantly "bumping into things."

According to Planck's relation each element in the periodic table emits only discrete frequencies of light, not a continuous spectrum. Also, each element has its own unique signature and emits particular frequencies and wavelengths of light. An instrument called a spectroscope allows us to measure and record the particular frequencies of light emitted by a substance which, in turn, allows us to determine what particular kinds of elements that substance is comprised of. Using this technique, astronomers were able to determine the compositions of the atmospheres of other planets, of the Sun and other stars, and of entire galaxies.

The Milky Way galaxy—our home galaxy—is a grand assemblage of over one-hundred billion stars that spans one-hundred thousand light-years across space. But that isn't all that there is in our galaxy. Enormous clouds of gad and dust float in between the stars; this is called the interstellar medium. Mapping the Milky Way and understanding its size and composition was made possible by advances in astronomy. Techniques such as stellar parallax and the use of a class of stars known as Cepheid variable stars made it possible for astronomers to measure vast distances across space. Despite this it was still very difficult to make some long-range distance measurements across the galaxy because the interstellar medium blocks out so much visible light; the advent of infrared astronomy, however, circumvented this issue and made it possible for 20th century astronomers to determine the size of the Milky Way.

Since most of the mass in our home Milky Way galaxy is at its center we would expect, from Newton's law of gravity and second law, that the tangential velocity of each star in our galaxy would increase with increasing distance away from the Milky Way's center. But observations showed that, actually, the tangential velocity of each star is roughly constant. There were many hypotheses put forward around the time of this discovery which attempted to reconcile Newton's laws with experiment. But today, the prevailing explanation of why the tangential velocities of the stars in our galaxy is roughly constant is the theory that an invisible substance known as dark matter pervades the galaxy. Researchers speculate that this substance is comprised of particles which do not interact with light or radiation and are therefore invisible.

The mapping of the Cosmic Microwave Background Radiation (CMBR) was hailed by Stephen Hawking as one of the greatest achievements in 20th-century science because it gave us an image of our "baby universe" when it was very young. The CMBR proved, beyond a shadow of a doubt, that the Big Bang theory—a prediction of Einstein's general theory of relativity—is correct. Before the CMBR, many did not think of cosmology as a serious science since it lacked high-precision experiments and measurements like astronomy; many viewed cosmology as merely theoretical speculation that could never be confirmed by experiment. Many people view the mapping of the CMBR as the turning point when cosmology transitioned from being purely theoretical to a truly experimentally rigorous science.

In this lesson we'll use the FRW equation to solve for the scaling factor \(a(t)\) in a universe dominated by vacuum energy or dark energy. We'll see that the scaling factor grows exponentially with time; this means that not only is the size of the universe getting bigger and bigger with time, but the rate at which it is doing so is increasing. In other words, the expansion of the universe is accelerating with time.

In this lesson, we'll prove that the capacitance of a parallel-plate capacitor does not depend on the charge or voltage of the capacitor but, rather, on only the size and geometry of the capacitor.