Our purpose in this chapter is to develop the mathematical techniques required in the description of congruences, the term designating an entire system of nonintersecting geodesics. We will consider separately the cases of timelike geodesics and null geodesics. (The case of spacelike geodesics does not require a separate treatment, as it is virtually identical to the timelike case; it is also less interesting from a physical point of view.) We will introduce the expansion scalar, as well as the shear and rotation tensors, as a means of describing the congruence's behaviour. We will derive a useful evolution equation for the expansion, known as Raychaudhuri's equation. On the basis of this equation we will show that gravity tends to focus geodesics, in the sense that an initially diverging congruence (geodesics flying apart) will be found to diverge less rapidly in the future, and that an initially converging congruence (geodesics coming together) will converge more rapidly in the future. And we will present Frobenius' theorem, which states that a congruence is hypersurface orthogonal – the geodesics are everywhere orthogonal to a family of hypersurfaces – if and only if its rotation tensor vanishes.

The chapter begins (in Section 2.1) with a review of the standard energy conditions of general relativity, because some of these are required in the proof of the focusing theorem. It continues (in Section 2.2) with a pedagogical introduction to the expansion scalar, shear tensor, and rotation tensor, based on the kinematics of a deformable medium. Congruences of timelike geodesics are then presented in Section 2.3, and the case of null geodesics is treated in Section 2.4.

This chapter covers three main topics that can all be grouped under the rubric of hypersurfaces, the term designating a three-dimensional submanifold in a fourdimensional spacetime.

The first part of the chapter (Sections 3.1 to 3.3) is concerned with the intrinsic geometry of a hypersurface, and it examines the following questions: Given that the spacetime is endowed with a metric tensor gαβ, how does one define an induced, three-dimensional metric hab on a specified hypersurface? And once this three-metric has been introduced, how does one define a vectorial surface element that allows vector fields to be integrated over the hypersurface? While these questions admit straightforward answers when the hypersurface is either timelike or spacelike, we will see that the null case requires special care.

The second part of the chapter (Sections 3.4 to 3.6) is concerned with the extrinsic geometry of a hypersurface, or how the hypersurface is embedded in the enveloping spacetime manifold. We will see how the spacetime curvature tensor can be decomposed into a purely intrinsic part – the curvature tensor of the hypersurface – and an extrinsic part that measures the bending of the hypersurface in spacetime; this bending is described by a three-dimensional tensor Kab known as the extrinsic curvature. We will see what constraints the Einstein field equations place on the induced metric and extrinsic curvature of a hypersurface.

The third part of the chapter (Sections 3.7 to 3.11) is concerned with possible discontinuities of the metric and its derivatives across a hypersurface.

This first chapter is devoted to a brisk review of the fundamentals of differential geometry. The collection of topics presented here is fairly standard, and most of these topics should have been encountered in a previous introductory course on general relativity. Some, however, may be new, or may be treated here from a different point of view, or with an increased degree of completeness.

We begin in Section 1.1 by providing definitions for tensors on a differentiable manifold. The point of view adopted here, and throughout the text, is entirely unsophisticated: We do without the abstract formulation of differential geometry and define tensors in the old-fashioned way, in terms of how their components transform under a coordinate transformation. While the abstract formulation (in which tensors are defined as multilinear mappings of vectors and dual vectors into real numbers) is decidedly more elegant and beautiful, and should be an integral part of an education in general relativity, the old approach has the advantage of economy, and this motivated its adoption here. Also, the old-fashioned way of defining tensors produces an immediate distinction between tensor fields in spacetime (four-tensors) and tensor fields on a hypersurface (three-tensors); this distinction will be important in later chapters of this book.

Covariant differentiation is reviewed in Section 1.2, Lie differentiation in Section 1.4, and Killing vectors are introduced in Section 1.5. In Section 1.3 we develop the mathematical theory of geodesics. The theory is based on a variational principle and employs an arbitrary parameterization of the curve.

Origins

Science is a process

Astronomy is the study of the Universe and everything in it. Astronomers use the tools and language of many different disciplines. Sometimes they are physicists, at other times chemists, biologists, or geologists. They follow the systematic scientific process that has developed over the last 400 years to ask questions about nature and to answer them convincingly. Astronomers also design (and sometimes build) telescopes to help collect the data needed to discover answers to the questions they ask.

In many ways, doing science is like putting together a jigsaw puzzle. Scientists start by observing some aspect of the world (or the Universe), then try to fit that piece into the bigger picture. Occasionally the answer to a simple question may be profound, as when Einstein asked himself what the world would look like if he were riding on a beam of light. The answer was the theory of special relativity.

Space is vast

The scale of space can be both confusing and daunting. The Earth seems like a very big place to us, but it is really a very tiny place in the Universe. For example, a jet airliner flying at 500 miles per hour (800 kilometers per hour) would take about 50 hours to fly around the Earth — a distance of about 25 000 miles.

Administration

Kitt Peak National Observatory was dedicated on March 15, 1960. Its founding organization, the Association for Universities for Research in Astronomy, Inc. (AURA), has evolved in its mission since that time, including the establishment of the National Optical Astronomy Observatories (NOAO) which manages Kitt Peak National Observatory; the Cerro Tololo Inter-American Observatory in Chile; and until recently the National Solar Observatory with facilities at Kitt Peak and Sacramento Peak, New Mexico. NOAO is headquartered in Tucson, Arizona.

Tohono O'odham Nation

Kitt Peak is located on the Schuk Toak District of the Tohono O'odham Reservation, and is under perpetual lease to AURA for “as long as the property is used for astronomical study and research and related scientific purposes.”

Situated in the Quinlan Mountains, Kitt Peak is one of several nearby mountains sacred to the Tohono O'odham people. The Tribe's consent was necessary prior to the construction of telescopes and support facilities. Preliminary talks between astronomers and tribal elders led to a demonstration of what was being proposed for construction on Kitt Peak. After the Tohonos' viewed the moon and stars from the 36-inch telescope at Steward Observatory in Tucson, they dubbed the astronomers “people with the long eyes” and gave consent for the mountain's use.

This edition of The Clementine Atlas of the Moon represents a significant enhancement over the first edition. The primary improvement is that we use new mission data for the annotated pages of the atlas. Specifically we use a new global topography product generated using data acquired by LROC, the imaging camera on board NASA's Lunar Reconnaissance Orbiter. For allowing us to use these wonderful data we wish to thank Dr Mark Robinson, the LROC team, and Frank Scholten and colleagues at the DLR Institute of Planetary Research in Berlin, Germany. We'd like to acknowledge the LOLA team whose high-resolution topography of the lunar polar regions was used for LACs 1 and 144.

Also since the initial release of the atlas, the Astrogeology branch of the United States Geological Survey has released a comprehensive set of annotated LACs that use Lunar Orbiter data as their basemap. We thank Jenny Blue and her colleagues for this invaluable resource for lunar scientists and enthusiasts. Finally we thank Danny Caes of Ghent, Belgium, whose meticulous and careful reading of the atlas uncovered several typographical errors in the first edition.