With this chapter, we let gravity lead us out of the familiar territory of the Solar System and into the arena of the stars. This is a tremendous leap: the furthest planet, Pluto, is never more than 50 AU away from the Earth, while the nearest stars to the Sun – the αCentauri system – are 270 000 AU away! In between is almost nothing. Yet, just as gravity determines the structure of the Sun, so also it governs the stars.

Stars are the workplaces of the Universe. Stars made the rich variety of chemical elements of which we are made; they created the conditions from which our Solar System and life itself evolved; our local star – the Sun – sustains life and, as we shall see, will ultimately extinguish it from the Earth.

1. ▷ The biggest stars are called giants, and the smallest are neutron stars.

Leaping out of the Solar System

The huge variety of kinds of stars gives a clue to why they can do so many different things. There are stars that are 20 times larger than the whole Solar System, and others that are smaller than New York City. Big stars can blow up in huge supernova explosions; small ones can convert mass into energy more efficiently than a nuclear reactor.

Our introduction to special relativity in the last chapter covered the basics, but it may have raised more questions for you than it answered. Before reading the chapter, you may have been very happy with the simple idea that everyone would agree on the length of a car, or the time it takes for the hands on a clock to go around once. If so, you have now learned to question these assumptions, that Nature does not really behave like that. If you want to fit these ideas together into a more logical framework, and if you want to learn something about why scientists are so sure that Nature really follows the principles of special relativity, then this chapter is for you. Read on.

1. ▷ The image under the text on this page illustrates length contraction. The top figure is after Leonardo da Vinci's famous drawing. The bottom figure has the dimensions that an experimenter would measure if the experimenter were flying across the original drawing at a speed of 0.9c.

In the previous chapter I listed some important effects of special relativity and gave a brief description of each, such as time dilation and the equivalence of mass and energy.

Cosmology is the study of the Universe as a whole. A century ago, scientists had only a vague idea about what even the Milky Way galaxy was like, and they were only able to make guesses about the Universe beyond. Most educated people believed that the nature and history of the Universe were simply matters for religious belief. The word “cosmology” referred to the set of beliefs one had about the whole world: Earth, God, Universe, Creation.

1. ▷ The image under the text on this page reminds us that creation myths and cosmologies were central parts of the belief systems of ancient peoples. It is remarkable how many cultures believed in a beginning of time, a moment of creation. The ancient Egyptians had several creation myths. The Hebrew creation story even orders the events in much the same way that modern science would, although on a vastly different time-scale. More than any other branch of physics, the scientific study of cosmology raises religious sensitivities and addresses questions that have long been regarded the domain of philosophy and belief.

As children of our age, we find it natural to think of the planets as cousins of the Earth: remote and taciturn, perhaps, but cousins nevertheless. To visit them is not a trip lightly undertaken, but we and our robots have done it. Men have walked on the Moon; live television pictures from Mars, Jupiter, Saturn, Uranus, and Neptune have graced millions of television screens around the world; and we know now that there are no little green men on Mars (although little green bacteria are not completely ruled out).

1. ▷ This name is pronounced “Tolemy”. His full name was Claudius Ptolomæus, and he lived in Alexandria during the second century AD. Little else is known of him.

Among all the exotic discoveries have been some very familiar sights: ice, dust storms, weather, lightning, erosion, rift valleys, even volcanos. Against this background, it may be hard for us to understand how special and mysterious the planets were to the ancients.

As we have progressed through the story set out in this book, we have met and begun to understand many of the objects that astronomers regularly photograph: planets, stars, galaxies, supernovae. Astronomical photographs show, in fact, the astonishing variety of objects that make up our Universe. But, to my eye, the most spectacular and entertaining astronomical photographs are fashioned by the objects we will study in this chapter: gravitational lenses. Let's start this chapter with two, shown in Figure 23.1 on the following page. Gravitational lenses are a spectacular illustration of the working of general relativity in the Universe. And besides entertaining us with pictures of eerie beauty, they have become an important tool of astronomy, a way of probing the distribution of mass (and in particular the dark matter) in galaxies and clusters of galaxies.

There are about 1080 particles in the Universe. Most of them are concentrated in stars though some can be found in interstellar and intergalactic space. At the centers of stars, particle densities are sufficiently high to allow nuclear reactions to take place. The energy liberated by these reactions heats the gas that makes up the star to the point where the gas pressure balances the gravitational pressure leading to hydrostatic equilibrium and long-term stability. Stars like the Sun are stable for about 1010 years, a large fraction of the age of the Universe.

The radiation produced by the nuclear reactions achieves a near equilibrium with the gas particles in the stellar interior, leading to local (but not global) thermodynamic equilibrium. The presence of local thermodynamic equilibrium (LTE) means that radiation inside the star is close to that of a black body at the local temperature. The absence of global equilibrium allows the radiation to leak out through the surface of the star and into space. Thus, stars are objects which attain hydrostatic equilibrium and radiate as near-black bodies for most of their lives.

The early Universe, just after the Big Bang, was much like the interior of the star. The radiation from that epoch has the characteristics of black-body radiation and we see it nowadays as the 2.7 K cosmic background radiation. The early Universe also experienced a period of thermonuclear reactions when most of the hydrogen and helium was produced.

In previous chapters, we have seen how the new ideas in Einstein's gravity make small but striking corrections to the predictions of Newton's gravity, bending light more strongly as it passes the Sun and causing the orbits of planets to precess. Working out these corrections helped to ease us into the theory, to see that relativistic gravity is a natural development from Newtonian gravity. But the real excitement in modern astronomy and theoretical physics is in situations where Newtonian gravity doesn't even come close to being right. The Universe demands that astronomers use general relativity to explain what they see, and the deepest questions of fundamental physics demand that physicists even go beyond general relativity to find their answers. In this chapter we open the door on the richness of modern gravity by studying our first example of really strong gravitational fields: neutron stars.

We are now ready to go to the heart of general relativity, to learn how matter generates gravity. This subject is usually left out of discussions of general relativity below the level of an advanced university course. The reason is mathematics, not physics: Einstein formulated his field equations, his gravity-generating equations, using the language of differential geometry. This is the mathematical discipline that deals with curvature, and it is far from elementary. The physical ideas that Einstein expressed in this mathematical language are simply too important, however, to pass over. In this chapter we whittle down the mathematics to a form that is as close as possible to the algebra we used in our earlier chapters on Newton's gravity. This allows us to share in Einstein's thinking, to see what general relativity really predicts about the world we live in.

We have allowed gravity to take us on a tour of the Universe in the first half of this book. It has taken us from the planet Earth to the rest of the Solar System, then to other stars, and from there to galaxies. Gravity wants to lead us further, because we have not yet come to understand its most profound consequences. These include black holes, which we met briefly in Chapter 4, and the Big Bang, which is the beginning of time itself.

These matters require strong gravity: gravity that is strong enough to trap light in a black hole or to arrest the expansion of the entire Universe. Studying strong gravity takes us beyond the limits where we can trust Newton's theory of gravity and his laws of motion.

The visible Universe contains hundreds of billions of galaxies, each consisting of billions of stars. Recent discoveries of extrasolar planets lead us to believe that a typical galaxy may contain billions of planets (and presumably, asteroids and comets). The planets, stars and galaxies interact on a hierarchy of scales ranging from AU to parsec to megaparsec, experiencing forces arising from gravity, on all scales, and cosmic expansion on the larger scales. The combination of gravitational attraction and cosmic expansion has shaped the visible matter in the Universe into a hierarchy of structures leading to clusters and superclusters of galaxies.

A full description of the interactions that define the large-scale structure of the Universe and its constituent parts requires the application of general relativity on all scales and the introduction of a new force, as embodied in the recently proposed cosmological constant, on the largest scales. In this part, however, we limit ourselves largely to the application of classical (Newtonian) mechanics which is sufficiently accurate to describe the topics covered in this part and has the advantage of being more intuitive and accessible to the reader.

This part begins with a review of the basic elements of classical mechanics, subsequently used to derive Kepler's laws, the Virial theorem and various aspects of orbital motion. The resulting derivations are applied to specific astrophysical problems such as planetary motion, extrasolar planets, binary stars, galaxy rotation curves, dark matter, the large scale structure of the Universe and cosmic expansion.

Geometry is at the heart of Einstein's picture of gravity. The best place to see how gravity as curvature works is in the Solar System, where the predictions must be very close to the description given by Newton. In this familiar arena, we can compare the old and new ways of looking at gravity. In this arena, too, general relativity meets and passes its first two crucial tests: explaining the anomalous advance in the perihelion of the planet Mercury (which we puzzled over in Chapter 5), and predicting that light should be deflected as it passes the Sun by twice the amount that would be calculated from Newtonian gravity (see Chapter 4).

We saw in the last chapter that the equivalence principle tells us that it is not possible to represent gravity just by the curvature of space; the curvature of spacetime must include the curvature of time as well.

Astrophysics strives to describe the Universe through the application of fundamental physics. The Cosmos manifests phenomena in which the physics can appear in its most extreme, and therefore more insightful, forms. Consequently, developing astrophysical concepts from fundamental physics has the potential to achieve two goals: to derive a better understanding of astrophysical phenomena from first principles, and to illuminate the physics from which the astrophysics is developed. To that end, astrophysical topics are grouped, in this book, according to the relevant areas of physics. For example, the derivation of the laws of orbital motion, used in the detection of extrasolar planets, takes place in the classical mechanics part of the book while the derivation of transition rates for the 21 cm neutral hydrogen line, used to probe galaxy kinematics, is performed in the quantum mechanics part. The book could serve as a text for graduate students and as a reference for established researchers.

The content of this book is based on the material used by the author in support of advanced astrophysics courses taught at the University of New Mexico. The intended audience consists of graduate students and senior undergraduates pursuing degrees in physics and/or astrophysics. Perhaps the most directly relevant demographic is the combined Physics and Astronomy departments. These departments tend to emphasize the fundamental physics regardless of the research track pursued by the student. In many cases a separate astrophysics degree is not an option.

One of the most radical changes in the behavior of gravity in going from Newton's theory to Einstein's is that Einstein's gravity has waves. When two stars orbit one another in a binary system, the gravitational field they create is constantly changing, responding to the changes in the positions of the stars. In any theory of gravity that respects special relativity, the information about these changes cannot reach distant experimenters faster than light. In general relativity, these changes in gravity ripple outwards at exactly the speed of light.

These gravitational waves offer a new way of observing astronomical systems whose gravity is changing. They are an attractive form of radiation to observe, because they are not scattered or absorbed by dust or plasma between the radiating system and the Earth: as we saw in Chapter 1, gravity always gets through. Unfortunately, the weakness of gravity, which we also noted in Chapter 1, poses a severe problem. Gravitational waves affect laboratory equipment so little that only recently has it become possible to build instruments sensitive enough to register them.

Gravity is everywhere. No matter where you go, you can't seem to escape it. Pick up a stone and feel its weight. Then carry it inside a building and feel its weight again: there won't be any difference. Take the stone into a car and speed along at 100 miles per hour on a smooth road: again there won't be any noticeable change in the stone's weight. Take the stone into the gondola of a hot-air balloon that is hovering above the Earth. The balloon may be lighter than air, but the stone weighs just as much as before.

1. ▷ Remember, terms in boldface are in the glossary.

This inescapability of gravity makes it different from all other forces of nature. Try taking a portable radio into a metal enclosure, like a car, and see what happens to its ability to pick up radio stations: it gets seriously worse. Radio waves are one aspect of the electromagnetic force, which in other guises gives us static electricity and magnetic fields.