Introduction

This chapter introduces the special theory of relativity from a perspective that is appropriate for proceeding to the general theory of relativity later on, from Chapter 4 onwards. Several topics such as the manipulation of tensorial quantities, description of physical systems using action principles, the use of distribution function to describe a collection of particles, etc., are introduced in this chapter in order to develop familiarity with these concepts within the context of special relativity itself. Virtually all the topics developed in this chapter will be generalized to curved spacetime in Chapter 4. The discussion of Lorentz group in Section 1.3.3 and in Section 1.10 is somewhat outside the main theme; the rest of the topics will be used extensively in the future chapters.

The principles of special relativity

To describe any physical process we need to specify the spatial and temporal coordinates of the relevant event. It is convenient to combine these four real numbers – one to denote the time of occurrence of the event and the other three to denote the location in space – into a single entity denoted by the four-component object xi = (x0, x1, x2, x3) ≡ (t, x) ≡ (t, xα). More usefully, we can think of an event P as a point in a four-dimensional space with coordinates xi. We will call the collection of all events as spacetime.

This book can be adapted by readers with varying backgrounds and requirements as well as by teachers handling different courses. The material is presented in a fairly modular fashion and I describe below different sub-units that can be combined for possible courses or for self-study.

1. 1 Advanced special relativity

2. Chapter 1 along with parts of Chapter 2 (especially Sections 2.2, 2.5, 2.6, 2.10) can form a course in advanced special relativity. No previous familiarity with four-vector notation (in the description of relativistic mechanics or electrodynamics) is required.

3. 2 Classical field theory

4. Parts of Chapter 1 along with Chapter 2 and Sections 3.2, 3.3 will give a comprehensive exposure to classical field theory. This will require familiarity with special relativity using four-vector notation which can be acquired from specific sections of Chapter 1.

5. 3 Introductory general relativity

6. Assuming familiarity with special relativity, a basic course in general relativity (GR) can be structured using the following material: Sections 3.5, Chapter 4 (except Sections 4.8, 4.9), Chapter 5 (except Sections 5.2.3, 5.3.3, 5.4.4, 5.5, 5.6), Sections 6.2.5, 6.4.1, 7.2.1, 7.4.1, 7.4.2, 7.5. This can be supplemented with selected topics in Chapters 8 and 9.

7. 4 Relativistic cosmology

8. Chapter 10 (except Sections 10.6, 10.7) along with Chapter 13 and parts of Sections 14.7 and 14.8 will constitute a course in relativistic cosmology and perturbation theory from a contemporary point of view.

9. 5 Quantum field theory in curved spacetime

10. Parts of Chapter 8 (especially Sections 8.2, 8.3, 8.7) and Chapter 14 will constitute a first course in this subject.

11. […]

Introduction

In this chapter, we will develop further the mathematical formalism required to understand different aspects of curved spacetime, focusing on the description of spacetime curvature. It uses extensively the concepts developed in Chapter 4, especially the idea of parallel transport. Most of the topics described here will be used in the subsequent chapters, except for the ideas discussed in Section 5.6 related to the classification of spacetime curvature which fall somewhat outside the main theme of development.

Three perspectives on the spacetime curvature

The discussion in the previous chapter used only the fact that the metric tensor depended on the coordinates. This dependence can arise either due to the use of curvilinear coordinates in flat spacetime or due to genuine curvature of the spacetime. The gravitational field generated by matter manifests itself as the curvature of the spacetime and hence, to study the gravitational effects, we need to develop the mathematical machinery capable of describing and analysing the curvature of spacetime. This will be the aim of the current chapter and we shall begin by introducing the concept of spacetime curvature from three different – but closely related – perspectives. At a fundamental level, these three perspectives stem from the same source, viz. behaviour of vectors under parallel transport; however, we will discuss them separately in the next three subsections for greater clarity.

Parallel transport around a closed curve

The first perspective on curvature originates from the changes induced in a vector when it is parallel transported around a small, closed, curve in the spacetime.

Introduction

Having acquainted ourselves with the trials and tribulations of working in non-Euclidean spacetimes we are now prepared for the next step, that of describing physics in such curved spacetimes. For we recall from Chapter 2 that the Einstein programme for general relativity consists of replacing the Newtonian perception of gravitation as a force by the notion that its effect makes the geometry of spacetime ‘suitably non-Euclidean’. What we mean by ‘suitably’ will be clear in the next two chapters. But given that the geometry is non-Euclidean we first need to know how the rest of physics is described in it.

For example, how do we describe the motion of a particle under a non-gravitational force? How do we write Maxwell's equations? What is the role of energy-momentum tensors? Can a dynamical action principle be written in curved spacetime? Such questions need our attention before we turn to the basic issue of how gravity actually leads to curved spacetime.

To this end we will introduce a concept that Einstein took as a basic principle in formulating general relativity. It is known as the principle of equivalence.

The principle of equivalence

Let us go back to the purely mathematical result embodied in the relations shown in Section 4.6 and attempt to describe their physical meaning. These relations tell us that special (locally inertial) coordinates that behave like the coordinates (t, x, y, z) of special relativity exist in the neighbourhood of any point P in spacetime.

Introduction

We begin this chapter by introducing the action functional for gravity and obtaining Einstein's equations. We then describe the general properties of Einstein's equations and discuss their weak field limit. The action functional and its properties will play a crucial role in Chapters 12, 15 and 16, while the linearized field equations will form the basis of our discussion of gravitational waves in Chapter 9.

Action and gravitational field equations

Let us recall that we studied the scalar and electromagnetic fields in Chapter 2 in two steps. First, in Sections 2.3.1 and 2.4.1, we considered the effect of the field (scalar or electromagnetic) on other physical systems (like material particles). In the second step, in Sections 2.3.2 and 2.7, we studied the dynamics of the field itself, by adding a new term to the action principle. This new term depended on the field and on its first derivatives; by varying the field in the total action we could obtain the dynamical equations governing the field.

In the case of gravity, we have already completed the corresponding first step in the previous two chapters. We have seen that the effect of gravity on any other physical system (particles, fluids, electromagnetic field, …) can be incorporated by modifying the action functional for the physical system by changing d4x to, partial derivatives by covariant derivatives and replacing ηab by gab.

Introduction

This chapter applies the general theory of relativity to the study of cosmology and the evolution of the universe. Our emphasis will be mostly on the geometrical aspects of the universe rather than on physical cosmology. However, in order to provide a complete picture and to appreciate the interplay between theory and observation, it is necessary to discuss certain aspects of the evolutionary history of the universe. We shall do this in Section 10.6 even though it falls somewhat outside the main theme of development.

The Friedmann spacetime

Observations show that, at sufficiently large scales, the universe is homogeneous and isotropic; that is, the geometrical properties of the three-dimensional space: (i) are the same at all spatial locations and (ii) do not single out any special direction in space.

The geometrical properties of the space are determined by the distribution of matter through Einstein's equations. It follows, therefore, that the matter distribution should also be homogeneous and isotropic. This is certainly not true at small scales in the observed universe, where a significant degree of inhomogeneity exits in the form of galaxies, clusters, etc. We assume that these inhomogeneities can be ignored and the matter distribution may be described by a smoothed out average density in studying the large scale dynamics of the universe.

Space, time and gravitation

The special theory of relativity reviewed in the last chapter marked a major advance in physics. The basic assumption that the fundamental laws of physics are invariant for all inertial observers looks at first sight a reasonable premise. However, as we saw in Chapter 1, its application to Maxwell's equations of electromagnetic theory led to a drastic revision of how such observers make and relate their measurements of space and time. One consequence was that the Newtonian notions of absolute space and absolute time had to be abandoned and replaced by a unified entity of spacetime. The Galilean transformation relating the space and time measurements of two inertial observers had to be replaced by the Lorentz transformation. Strange and non-intuitive though the consequences of this transformation were, as we saw in Chapter 1, several experiments confirmed them.

In spite of these successes, Einstein felt that the special theory addressed limited issues. For example, what was the nature of physical laws when viewed not in the inertial frames of reference, but in an accelerated one? Was there some more general principle that, when applied to these laws, preserved their form? Intuitively Einstein felt that some such situation must prevail. But that required a formalism more general than that provided by the Lorentz transformation.

On another matter, of the two classical theories of physics known in the first decade of the twentieth century, the electromagnetic theory had played a major role in the genesis of special relativity.

This lecture introduces the scientific motivation for integral field spectroscopy (IFS) and describes the results from this novel technique in Galactic studies as of 2005. The following chapter by Luis Colina then picks up on extragalactic studies and rounds out the picture giving an outlook to future integral field spectroscopic studies. Following the five one-hour lectures, this chapter is broken down into sections on:

• science motivation;

• the Galactic Centre black hole;

• the Galactic Centre stellar population;

• star formation; and

• the Solar System.

The publications discussed in this lecture are selected on the basis of their scientific impact as measured by the citation index and/or because they specifically qualify for a lecture from a didactic point of view.

Science motivation

Motivation for the development and application of integral field spectroscopy is manifold and is closely tied to personal interest and background. This is most obvious in the history of instrument development, outlined in the beginning of this section. The ‘ideal’ objects for integral field observations can be well characterized by their angular size, substructure and spectral characteristics. These qualifiers are used to identify the ideal objects on the distance ladder. The section ends comparing the apparent interest of the Galactic and extragalactic astronomical communities in integral field spectroscopy.

Motivation for integral field spectroscopy

The development of the first-generation integral field spectrographs was largely driven by an ‘experimental’ motivation.

Introduction

The evolution of the homogeneous universe was described in Chapter 10. Following up on that, we shall now turn to the study of the formation of structures in the universe. The key idea is that if there were small fluctuations in the energy density in the early universe, then the gravitational instability could amplify them leading – eventually – to structures like galaxies, clusters, etc. The most popular model for generating the initial fluctuations is based on the paradigm that, if the very early universe went through a phase of accelerated expansion (called the inflationary phase), then the quantum fluctuations of the field driving the inflation could lead to fluctuations in the energy density. (We will discuss this idea in detail in Chapter 14.) When the perturbations are small, one can use the linear perturbation theory to study its growth. The observations of cosmic microwave background radiation (CMBR) at z ≃ 103 show that the fractional perturbations in the energy density were quite small (about 10−4–10−5) at z ≈ 1000 when the matter and radiation decoupled. Hence, linear perturbation theory can be used to make clear predictions about the state of the universe at z ≈ 1000 and to study the anisotropies in the CMBR. This will be one key application of the formalism developed in this chapter.

Structure formation and linear perturbation theory

The basic idea behind linear perturbation theory in cosmology is well defined and simple.

The concept of general covariance

We begin this chapter by introducing the idea of a field in physics (to be distinguished from the ‘field’ in algebra that mathematicians talk about). The idea was popularized by Michael Faraday in the context of the electric and magnetic fields. Figure 4.1 shows what happens when iron filings are sprinkled in the vicinity of a bar magnet. The filings get distributed in a pattern somewhat like that in this figure. Faraday called these curves lines of force. If we imagine a magnetic pole placed anywhere on one of these lines, it will move along that line, being guided by the magnetic force on it. The lines of force therefore represent the ‘magnetic field’ B both in strength and direction at any point in the vicinity of the magnet. In short the magnet generates a ‘field’ of B vectors all around it, representing the force exerted by it on another magnetic pole.

We generalize the concept of a vector field by defining a vector function of spacetime variables, so that at each point a vector is defined. This idea may further be generalized by having tensor fields as functions of spacetime coordinates. Thus we could argue that equations of physics involve fields related by partial differential equations. If we additionally require that these equations do not change their form under changes of coordinates (thereby being the same for all observers), then they should be represented by tensor fields.

Introduction

Do Einstein's equations permit the existence of gravitational waves? As in the case of electromagnetism, where Maxwell's field equations led to the important deduction of electromagnetic waves carrying energy and momentum with the speed of light, one expects the relativistic equations to imply the existence of gravitational waves that do the same. However, several issues intervene to make the answer to our question nontrivial.

The first problem is posed by the non-linearity of the Einstein field equations. In the wave motion discussed in electromagnetic theory, acoustics, elastic media, etc. the basic equations are linear and a superposition principle holds. There is no corresponding situation in general relativity. Secondly, there is no corresponding vector or tensor in relativity that plays the role of the Poynting vector in the transport of electromagnetic energy.

A third difficulty arises from the general covariance of the field equations. With the facility available to use any coordinate system as per convenience, it is not clear whether a particular ‘wavelike’ solution is a real physical effect or a pure coordinate effect. Thus one has to be on guard against solutions that describe coordinate waves that may travel ‘with the speed of thought’.

Even during Einstein's lifetime, the above question did not receive an unequivocal answer.

Historical background

1905 is often described as Einstein's annus mirabilis: a wonderful year in which he came up with three remarkable ideas. These were the Brownian motion in fluids, the photoelectric effect and the special theory of relativity. Each of these was of a basic nature and also had a wide impact on physics. In this chapter we will be concerned with special relativity, which was arguably the most fundamental of the above three ideas.

It is perhaps a remarkable circumstance that, ever since the initiation of modern science with the works of Galileo, Kepler and Newton, there has emerged a feeling towards the end of each century that the end of physics is near: that is, most in-depth fundamental discoveries have been made and only detailed ‘scratching at the surface’ remains. This feeling emerged towards the end of the eighteenth century, when Newtonian laws of motion and gravitation, the studies in optics and acoustics, etc. had provided explanations of most observed phenomena. The nineteenth century saw the development of thermodynamics, the growth in understanding of electrodynamics, wave motion, etc., none of which had been expected in the previous century. So the feeling again grew that the end of physics was nigh. As we know, the twentieth century saw the emergence of two theories, fundamental but totally unexpected by the stalwarts of the nineteenth century, viz., relativity and quantum theory.

Introduction

In this chapter we will obtain the simplest of the exact solutions to Einstein's equations, which are the ones with spherical symmetry. The chapter also discusses the orbits of particles and photons in these spacetimes and the tests of general relativity. All of this will be used in the study of black holes in the next chapter.

Metric of a spherically symmetric spacetime

One of the simplest – but fortunately very useful – class of solutions to Einstein's equations is obtained when the source Tik and the resulting metric possess spherical symmetry. We shall first obtain the general form of the metric in the spherically symmetric context and then use Einstein's equations to relate the metric to the source. While the form of the spherically symmetric metric (given in Eq. (7.12) below) can be obtained almost ‘by inspection’, we shall perform a rather formal analysis in order to illustrate a useful technique.

If the spacetime exhibits a particular symmetry which can be characterized by the action of an element of a group, then the functional change in the form of the metric under the action of this element of the group should vanish. In the case of spherical symmetry, the relevant group is the group of rotations under which the Cartesian coordinates will change by xa → xa + ξa, where ξa has the components

Here εαβ = −εβα is a set of arbitrary infinitesimal constants (with only three elements being independent due to the antisymmetry) which describe infinitesimal rotations.