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.

As explained in the preface, I have used high-school mathematics to present some of the material in this book. If you want to know what that means, if you want to learn whether you have the background necessary to do the mathematics, then scan through this introductory material. But remember, it is not necessary to follow all the derivations, particularly the ones in the boxes, if you just want to learn what the main ideas in modern gravity and astronomy are. So if you find your mathematics too old or rusty, then see how you get along without it.

High-school mathematics

The mathematics used is basic numeracy, algebra, and a tiny bit of trigonometry (which you can skip).

It is essential to understand scientific notation for numbers, that is how to write numbers in the form 3.2 × 106 and know what the factor 106 means. Scientists use this notation all the time, because otherwise they would be writing out long confusing strings of zeros. The number 3.2 × 106 means 3 200 000, obtained by moving the decimal point in 3.2 six places to the right. Similarly, the number 5.9 × 10−3 is 0.0059, obtained by moving the decimal point three places to the left.

The study of cosmology presents today's physicists with the biggest challenges to their understanding of gravity and of fundamental physics in general. Both on theoretical and on observational grounds, it seems that we will not be able to understand cosmology well until we understand physics better than we do today. But it also seems that cosmology could provide us with the keys to that deeper understanding of physics.

The biggest gap in physics is quantum gravity: we do not yet possess a consistent way of representing gravity as a quantum theory. There is no uncertainty principle in general relativity, no quantization of gravitational effects, no need to use probabilities in making predictions about the outcome of gravitational experiments. This seems inconsistent with the fact that all material systems that create gravity are quantum systems: if we can't say exactly where an electron is, how can we say exactly where its gravitational field is?

The cycle of birth, aging, death, and re-birth of stars dominates the activity of ordinary galaxies like our own Milky Way. The cycle generates the elements of which our own bodies are made, produces spectacular explosions called supernovae, and leaves behind “cinders”: remnants of stars that will usually no longer participate in the cycle. We call these white dwarfs, neutron stars, and black holes.

Governing this cycle is, as everywhere, gravity. An imbalance between gravity and heat in a transparent gas cloud leads to star formation. The long stable life of a star is a robust balance between nuclear energy generation and gravity. This balance is finally lost when the star runs out of nuclear fuel, leading to a quiet death as a white dwarf or to a violent death as a supernova.

We have seen how the Sun's gravity holds the planets in their orbits. The Sun's gravity also holds itself together. Like all stars, the Sun is a seething cauldron, its center a huge continuous hydrogen bomb trying to blow itself apart, restrained only by the immense force of its own gravity. In this chapter, we will see how the Sun has managed to maintain an impressively steady balance for billions of years. In the course of our study, we will learn about how light carries energy and we will build a computer model of the Sun.

Sunburn shows that light comes in packets, called photons

The Sun glows so brightly because it is hot. We can infer just how hot it is from its color. The color and temperature of the Sun are related to each other in just the same way as for hot objects on the Earth.

Black holes. No term evokes the mystery of modern gravity more than this one. The mystery of black holes is more than an invention of popularizers of astronomy and relativity. Black holes were certainly a mystery to Einstein and his contemporaries. Yet today black holes are everywhere: in X-ray binaries, in the centers of galaxies, and of course in books, like this one, on relativity and gravity!

Theorists attacked the problem of understanding black holes, not by using astronomical evidence, but by using lessons they had learned from quantum mechanics. Quantum thinking demanded that physicists ask only questions about things that could be measured, not about what is hidden from experiment. Thus, they can measure that light behaves sometimes as a particle (the photon) and sometimes as a wave, but they find it useless to ask what is a wave–particle.

In the last two chapters we have made a lot of progress in exploring the future and past of the Universe, basically just by using local Newtonian gravity. We argued that the dynamics of an expanding, homogeneous and isotropic cosmology can be calculated from Newtonian gravity, at least if the pressure in the Universe is negligible, because all we need to look at is the local Universe, the part nearest us. The assumption that the Universe is homogeneous guarantees that the rest of the Universe will behave the same as our local region.

1. ▷ The drawing under the text on this page illustrates how complicated three-dimensional solid objects could be. Why is the Universe apparently so simple?

But this line of reasoning has its limitations. Even if we calculate the dynamics of the Universe this way, we don't learn what the distant parts of the Universe will look like in our telescopes. The curvature of space, which is not part of a Newtonian discussion, will affect the paths of photons as they move through the Universe. Moreover, if we want to ask deeper questions about the Universe, such as those we pose in the next chapter, then we should know something more about its the larger-scale structure.

There would be no life as we know it on Earth without the atmosphere. Even life in the oceans would not exist: without the atmosphere's thermal “blanket”, the oceans would freeze. Yet in the beginning, the Earth probably had a very different atmosphere from its present one. The other planets, with their different masses and different distances from the Sun, all have vastly different atmospheres from the Earth's. In the retention of the atmosphere, and in the subsequent evolution of the atmosphere and of life itself, gravity has played a crucial role.

In this chapter, as we look at the role that gravity has played in this story, we shall encounter fundamental ideas about the nature of matter itself: how temperature and pressure can be explained by the random motions of atoms, why there is an absolute zero to the temperature, and even why atoms cannot quite settle down even at absolute zero.

Gravity is the engine that drives the Universe. But it does not work alone, of course. In fact, one of the most satisfying aspects of studying astronomy is that there is a role for essentially every branch of physics when one tries to explain the huge variety of phenomena that the Universe displays. One branch of physics, however, stands out from the rest because of its absolutely central place in helping us to learn about the Universe, and that is the study of the way hot bodies give off light.

Almost all of the information we have from astronomical bodies is carried to us by light, and almost all the light originates as radiation from some sort of hot region. The great breakthrough in physicists' understanding of such thermal radiation was made by the German physicist Max Planck (1858–1947) at the start of the twentieth century. (See Figure 10.2 on page 112.) The story of this breakthrough is the story of physicists' first steps toward quantum theory. It is also the story of the beginnings of a real understanding of the heavens.

Astrophysical spectral lines offer two important insights into the workings of our Universe. First, they are probes of the fundamental (QM) nature of matter because they originate from subatomic, atomic and molecular systems. Second, they provide, via the Doppler effect, critical dynamical information on astrophysical systems ranging in scale from planetary systems to superclusters of galaxies. Examples of major contemporary problems in astrophysics that can be addressed through spectral line studies and the associated quantum mechanics include.

Missing mass and the halos of galaxies The most common element in the Universe is hydrogen and much of it is in a cold state. Given the 10 eV gap between the ground state and the first excited state of the simple Bohr atom, we should have little direct knowledge of this gas, yet it is the best studied gaseous component of the Universe. The reason is the 21 cm line corresponding to the hyperfine splitting of the ground state. The extremely low transition probability of this transition and the consequently narrow width of this line have led to its widespread use in measuring galaxy dynamics and kinematics. Studies of galaxy rotation have shown evidence for missing matter and point to the possibility of dark-matter halos. The nature of the dark matter and the implication on the long-term fate of the Universe remain contentious issues in astrophysics. The nature of this line and its use in these studies is discussed.

We are convinced of a genuine need for a monograph describing the many facets and new developments in numerical relativistic hydrodynamics. Such calculations are crucial to several areas of current research in the physics of stellar collapse, supernovae, and black hole formation, as well as the merging of the final orbits of coalescing binary neutron stars. Both problems are only now entering the level of sophistication where three-dimensional relativistic hydrodynamics simulations are both possible and necessary. In the former problem such calculations are crucial to understand the explosion mechanism. In the latter problem, a great deal of interest in such calculations has recently been inspired by the development of next-generation gravity wave detectors to search for such events, and as a possible explanation of the physics underlying observed astrophysical γ-ray bursts.

The field of numerical relativistic hydrodynamics has developed over the past 30 years, but there has not been written a technical text explaining the many techniques relevant to this discipline, many of which are much different than standard general relativity textbook approaches. This book will present such a review of techniques for numerical general relativistic hydrodynamics developed by one of the pioneers of this field over the past three decades.

We begin by developing the equations and differencing schemes for special relativistic hydrodynamics as an introduction to the metric formulation of the problems. Here, the basic numerical techniques and a number of test problems and applications will be discussed.

Following this, the formalism for matter flows in the curved spacetime of general relativity will be presented in the usual (3+1) formalism.

Progress in computing full general relativistic hydrodynamics in three spatial dimensions has been slow. The problem is not in the hydrodynamics, but in the solution of the field equations. The equation for extrinsic curvature, Kij, is particularly unstable. For example, at the writing of this book no strong field fully relativistic calculation has computed more than two orbits of a neutron star binary without becoming unstable. Currently, there seems to be some promise, however, in a modified version of the ADM equations based upon a conformal decomposition as originally proposed by Shibata and Nakamura and later reinvented by Baumgarte and Shapiro. At the end of this chapter we briefly summarize this method for completeness. First, however, we summarize a useful alternative which the authors have developed for solving strong field systems which avoids the nonlinearities of the full Einstein equations by reducing the problem to an implementation of constraint equations.

The conformally flat approximation

For most gravitating systems studied so far (e.g.), only a relatively small amount of energy is emitted by gravitational waves. Even for the merger of two black holes it is expected that only a few tenths of a percent of the rest mass will be radiated away in gravitation. For the case of two neutron stars we would not expect any more radiation to be emitted during the last few orbits than for a two black hole merger, i.e. during the inspiral, the radiated energy per orbit is a minuscule fraction of the energy in orbital motion. Furthermore, an explicit treatment of the radiation reaction is exceedingly difficult.

As discussed in Chapter 3, the treatment of the spherical collapse of a massive star to produce a supernova is sufficiently complex to warrant a separate chapter. It is believed that a Type II supernova arises from the delicate balance between energy deposited by escaping neutrinos from the core and the gravitational energy of collapsing outer layers. Thus, the relativistic energy and mass transport must be considered with high numerical accuracy to obtain a believable simulation. The model discussed herein includes the experience of about 30 years of development and should be of some guidance to those who wish to understand this fascinating phenomenon.

Collapse supernovae

A brief review of the scenario is as follows. Massive stars (i.e. 10 M/M⊙ 30) evolve until the iron core exceeds ∼1–1.3 M⊙. At this point there can be no more nuclear energy generation in the core. Neutrino emission, electron capture, and photodisintegration cool the inner ∼1 M⊙ and remove pressure support from the core. The central density then rises. When the central density ρc approaches ρc 109 g cm−3, neutrino emission is so large that collapse becomes supersonic, i.e vmax > cs, where cs is the speed of sound. As the core collapses, the inner ∼0.7 M⊙ collapses homologously (e.g.). Once the core density exceeds nuclear density, ρc > 2.5×1014 g cm−3, the pressure rises rapidly and collapse is halted. Matter continues to fall inward, however, so an outward moving shock wave is formed. This is referred to as the core bounce.