Historical background

In 1915 Einstein put the finishing touches to the general theory of relativity. The Schwarzschild solution described in Chapter 9 was the first physically significant solution of the field equations of general relativity. It showed how spacetime is curved around a spherically symmetric distribution of matter. The problem solved by Schwarzschild was basically a local problem, in the sense that the deviations of spacetime geometry from the Minkowski geometry of special relativity gradually diminish to zero as we move further and further away from the gravitating sphere. This result can be easily verified from the Schwarzschild line element by letting the radial coordinate go to infinity. In technical jargon a spacetime satisfying this property is called asymptotically flat. In general any spacetime geometry generated by a local distribution of matter is expected to have this property. Even from Newtonian gravity we expect an analogous result: that the gravitational field of a local distribution of matter will die away at a large distance from the distribution. Can the Universe be approximated by a local distribution of matter?

Einstein rightly felt that the answer to the above question would be in the negative. Rather, he expected the Universe to be filled with matter, howsoever far we are able to probe it. A Schwarzschild-type solution cannot therefore provide the correct spacetime geometry of such a distribution of matter. Since we can never get away from gravitating matter, the concept of asymptotic flatness must break down.

In this chapter we review the challenges of, and opportunities for, 3D spectroscopy and how these have led to new and different approaches to sampling astronomical information. We describe and categorize existing instruments on 4 m and 10 m telescopes. Our primary focus is on grating-dispersed spectrographs. We discuss how to optimize dispersive elements, such as VPH gratings, to achieve adequate spectral resolution, high throughput, and efficient data packing to maximize spatial sampling for 3D spectroscopy. We review and compare the various coupling methods that make these spectrographs ‘3D’, including fibres, lenslets, slicers, and filtered multi slits. We also describe Fabry–Perot (FP) and spatial-heterodyne interferometers, pointing out their advantages as field-widened systems relative to conventional, grating-dispersed spectrographs. We explore the parameter space all these instruments sample, highlighting regimes open for exploitation. Present instruments provide a foil for future development. We give an overview of plans for such future instruments on today's large telescopes, in space and in the coming era of extremely large telescopes. Currently-planned instruments open new domains but also leave significant areas of parameter space vacant, beckoning further development.

Fundamental challenges and considerations

The detector limit I: six into two dimensions

Astronomical data exist within a six-dimensional hypercube sampling two spatial dimensions, one spectral dimension, one temporal dimension, and two polarizations. In contrast, high-efficiency, panoramic digital detectors today are only two-dimensional (with some limited exceptions).

Introduction

The description of gravity based on Einstein's general theory of relativity is quite satisfactory in most respects. It has been repeatedly verified experimentally as regards those features which could be directly tested while other parts of it are conceptually very elegant and beautiful. Nevertheless, it is obvious that this theory is fundamentally flawed or – at the least – incomplete.

Such a conclusion emerges from the fact that there exist well defined situations in which the theory is incapable of predicting the future evolution of the dynamical variables owing to the development of singularities. To see this concretely, consider the example of a collapsing sphere of dust described in Chapter 8. An observer comoving along with the dust particle will find that the trajectory of the dust particle hits a singularity (at which the curvature and density diverge) within finite proper time τ as shown by the observer's clock. In other words, the observer can not ascertain beforehand her future evolution for arbitrarily large values of τ using Einstein's theory of gravity. As another example, consider the standard description of our universe in terms of a Friedmann model discussed in Chapter 10. For reasonable values of the parameters of the model at the present moment – which are determined observationally – the theory is incapable of describing the state of the universe, say, 20 billion years ago for any equation of state for high density matter having positive pressure and energy density.

In 1978 I wrote an introductory textbook on general relativity and cosmology, based on my lectures delivered to university audiences. The book was well received and had been in use for about 15–20 years until it went out of print. The present book has been written in response to requests from students as well as teachers of relativity who have missed the earlier text.

An Introduction to Relativity is therefore a fresh rewrite of the 1978 text, updated and perhaps a little enlarged. As I did for the earlier text, I have adopted a simple style, keeping in view a mathematics or physics undergraduate as the prospective reader. The topics covered are what I consider as essential features of the theory of relativity that a beginner ought to know. A more advanced text would be more exhaustive. I have come across texts whose formal and rigorous style or enormous size have been off-putting to a student wishing to know the A, B, C of the subject.

Thus I offer no apology to a critic who may find the book lacking in some of his/her favourite topics. I am sure the readers of this book will be in a position to read and appreciate those topics after they have completed this preliminary introduction.

Cambridge University Press published my book An Introduction to Cosmology, which was written with a similar view and has been well received.

Introduction

The work covered in Chapter 14 did not tell us two important items of information about the Universe: (1) the rate at which it expands as given by the function S(t); and (2) whether its spatial sections t = constant are open or closed as indicated by the parameter k. To find answers to these questions, it is necessary to go beyond the Weyl postulate and the cosmological principle. We require a dynamical theory that tells us how the scale factor and curvature are determined by the matter/radiation contents of the universe.

A comparison of Newton's law of gravitation with the general theory of relativity shows the latter as enjoying advantages both on the theoretical and on the observational front. General relativity gets round the criticism of Newtonian gravity of violating the light-speed limit. It allows for the permanence of gravitation by identifying its effect with the curvature of spacetime. Observationally it performs better vis-à-vis the Solar-System tests and explains the shrinking of binaries through gravitational radiation. It therefore generates greater confidence than Newton's approach does, especially for use in cosmology, where strong gravitational fields are likely to be involved and where distances are so large that the assumption of instantaneous action at a distance would be misleading. Hence we will adopt general relativity as the underlying theory for constructing models of the Universe.

We will now undertake that exercise by constructing the models which Friedmann in 1922–4 and Lemaître in 1927 came up with before Hubble's results became known.

Introduction

This chapter develops the ideas of classical field theory in the context of special relativity. We use a scalar field and the electromagnetic field as examples of classical fields. The discussion of scalar field theory will allow us to understand concepts that are unique to field theory in a somewhat simpler context than electromagnetism; it will also be useful later on in the study of topics such as inflation, quantum field theory in curved spacetime, etc. As regards electromagnetism, we concentrate on those topics that will have direct relevance in the development of similar ideas in gravity (gauge invariance, Hamilton–Jacobi theory for particle motion, radiation and radiation reaction, etc.).

The ideas developed here will be used in the next chapter to understand why a field theory of gravity – developed along similar lines – runs into difficulties. The concept of an action principle for a field will be extensively used in Chapter 6 in the context of gravity. Other topics will prove to be valuable in studying the effect of gravity on different physical systems.

External fields of force

In non-relativistic mechanics, the effect of an external force field on a particle can be incorporated by adding to the Lagrangian the term −V(t, x), thereby adding to the action the integral of −V dt. Such a modification is, however, not Lorentz invariant and hence cannot be used in a relativistic theory.

Parallel propagation around finite curves

Figure 5.1 repeats the previous example of non-Euclidean geometry on the surface of a sphere which we discussed in Section 2.2 of Chapter 2. We have the triangle ABC of Figure 2.3 whose three angles are each 90°. Consider what happens to a vector (shown by a dotted arrow) as it is parallely transported along the three sides of this triangle. As shown in Figure 5.1, this vector is originally perpendicular to AB when it starts its journey at A. When it reaches B it lies along CB; it keeps pointing along this line as it moves from B to C. At C it is again perpendicular to AC. So, as it moves along CA from C to A, it maintains this perpendicularity, with the result that when it arrives at A it is pointing along AB. In other words, one circuit around this triangle has resulted in a change of direction of the vector by 90°, although at each stage it was being moved parallel to itself!

A similar experiment with a triangle drawn on a flat piece of paper will tell us that there is no resulting change in the direction of the vector when it moves parallel to itself around the triangle. So our spherical triangle behaves differently from the flat Euclidean triangle.

The phenomenon illustrated in Figure 5.1 can also be described as follows.

Introduction

We obtained Einstein's equations in Chapter 6 from an action principle in which we varied the four-dimensional metric tensor gab. The resulting equation, Gab = kTab, is generally covariant in the spacetime. We also described in Section 6.3 several peculiar features of Einstein's equations. In particular, we noticed the following. (i) The time derivatives of g00 and g0α do not occur in any of the equations. (ii) No second time derivatives of gαβ occur in the time–time or space–time components of Einstein's equations. These equations contain only the first time derivatives of gαβ. (iii) Only the space–space part of Einstein's equations involves the second time derivatives of gαβ.

These peculiarities introduce several complications when we attempt to study Einstein's theory as describing the evolution of some well defined dynamical variables. It is clear from the above properties that one cannot treat all the components of the metric tensor on an equal footing; the real dynamics is essentially contained in the evolution of gαβ. At the same time, the generally covariant description treats the metric as a single entity which allows for a nice geometrical interpretation of Einstein's theory. The question arises as to whether one can maintain the geometrical structure of the theory and yet perform a split of Einstein's equations, along with the dynamical variables, into space and time. We shall now describe how this can be achieved.

Introduction

Once we venture into spacetime dimensions other than four, the possible theoretical models increase enormously and it is impossible (and, to a certain extent, unnecessary) to do justice to all of them in a comprehensive manner. Hence, in this chapter, we will confine our attention to a few selected topics which deal with the description of gravitational field in dimensions other than four.

The motivation to study gravity in D < 4 is quite different from the motivation to study gravity in D > 4. The interest in two and three spacetime dimensions arises from the hope that the simplified models in these lower dimensions will provide a better understanding of gravity in four dimensions as well as help us to appreciate those features of gravity which are closely tied to the fact that the spacetime has four dimensions. While these models are interesting, it is probably fair to say that the study of lower dimensional models has not significantly enhanced our understanding of general relativity in four dimensions. Therefore, our discussion of these models will be quite brief.

The motivation to study models in D > 4 comes from two key factors. First, there is a very natural class of gravitational theories (called Lanczos–Lovelock models) which exist in higher dimensions and share several properties of Einstein's theory in four dimensions.

Introduction

This chapter describes some interesting results that arise when one studies standard quantum field theory in a background spacetime with a nontrivial metric. It turns out that the quantum field theory in a curved spacetime (or a non-inertial coordinate system) with a horizon exhibits some peculiar and universal properties. In particular, the study of the quantum field suggests that the horizon is endowed with a temperature T = k/2π (in natural units with ħ = c = kB = 1), where k is the surface gravity of the horizon. This result can be viewed from very different perspectives, not all of which can be proved to be rigorously equivalent to one another. In view of the importance of this result, most of this chapter will concentrate on obtaining this result using different techniques and interpreting it physically. The latter part of the chapter will develop quantum field theory in an external Friedmann universe and will apply that formalism to study the generation of perturbations during the inflationary phase of the universe.

Review of some key results in quantum field theory

Fortunately, most of the important results we are interested in can be obtained with a minimum amount of background knowledge in quantum field theory. In order to set the stage, we shall rapidly review the necessary concepts in this section.

Quantum field theory attempts to describe particles as excitations of an underlying field.

In this set of lectures, I review recent observational progress on extragalactic studies using integral field spectroscopy (IFS) techniques, highlighting the importance of IFS for the study of the nuclear regions of nearby galaxies, of low-z active galactic nuclei (AGN) and massive star-forming galaxies, and of high-z galaxies, including lensed quasars, lensing galaxies and bright submillimetre galaxies. Emphasis is given to the study of (ultra)luminous infrared galaxies as examples of low-z systems where the physical processes relevant to the formation and evolution of galaxies can be investigated in more detail. Research projects involving future ground-based facilities and satellites are also briefly presented.

Introduction

The use of IFS for extragalactic studies has burgeoned over the past 10 years and is already becoming a standard observational technique used by several groups in many different areas. Most IFS systems (INTEGRAL, GMOS, PMAS, SAURON, SINFONI, VIMOS, etc.) allow us to simultaneously obtain spectra covering a wide spectral range over a wide field of view (up to 1 arcmin square for VIMOS). These instruments in their standard configurations provide low–intermediate spectral resolution (R of 1000 to 4000) with a relatively low angular resolution (0.5″ to 3.0″). In addition, a few IFS systems, such as OASIS on the William Herschel Telescope and SINFONI on the Very Large Telescope (VLT), can provide very high angular resolution (i.e. 0.1″) in the optical (OASIS) and near-infrared (SINFONI) when combined with adaptive optics (AO) systems.

There is a need for a comprehensive, advanced level, textbook dealing with all aspects of gravity, written for the physicist in a contemporary style. The italicized adjectives in the above sentence are the key: most of the existing books on the market are either outdated in emphasis, too mathematical for a physicist, not comprehensive or written at an elementary level. (For example, the two unique books – L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, and C. W. Misner, K. S. Thorne and J. A. Wheeler (MTW), Gravitation – which I consider to be masterpieces in this subject are more than three decades old and are out of date in their emphasis.) The current book is expected to fill this niche and I hope it becomes a standard reference in this field. Some of the features of this book, including the summary of chapters, are given below.

As the title implies, this book covers both Foundations (Chapters 1–10) and Frontiers (Chapters 11–16) of general relativity so as to cater for the needs of different segments of readership. The Foundations acquaint the readers with the basics of general relativity while the topics in Frontiers allow one to ‘mix-and match’, depending on interest and inclination. This modular structure of the book will allow it to be adapted for different types of course work.

For a specialist researcher or a student of gravity, this book provides a comprehensive coverage of all the contemporary topics, some of which are discussed in a textbook for the first time, as far as I know.

3D spectroscopy has a relatively short history. Most of the present instrument concepts were developed in the 1980s and early 1990s. During those pioneering years a great deal of work was done in optical labs in an attempt to understand how the optical fibres, microlenses and image slicers behave. Only a few groups (often formed by one or two people) worked on this topic. Communications were not very good, which explains why virtually all the groups decided to refer to this technique by a different name. So we ended up with ‘spectral imaging’, ‘bidimensional spectroscopy’, ‘integral field spectroscopy’, ‘two-dimensional spectroscopy’, ‘3D spectroscopy’, etc.

During those years it was more than doubtful whether this technique was going to be useful at all. In fact, it looked like a kind of curiosity of limited practical interest to astronomy. However, in the 1990s the first scientific results were obtained and they immediately produced a change of perception.

In the last few years investment in this type of instrumentation has been enormous. Large telescopes all around the world are now equipped with integral field units. Two instruments of the future James Webb Space Telescope will also have integral field spectroscopic capabilities, etc. Instead of being based in the optical lab trying to characterize optical fibres or micro-lenses, more effort is dedicated nowadays to refining techniques for reducing, analysing and interpreting the data obtained with a new generation of 3D spectrographs.

Introduction

The general theory of relativity, like any other physical theory, must submit itself for experimental verification. It started with a disadvantage in that it was competing with a well-established paradigm, viz. the Newtonian laws of motion and gravitation. Any test that could be designed for testing general relativity had at the same time to show ways of distinguishing its predictions from those of the Newtonian framework. Here the situation has been different from the case of special relativity. Many laboratory tests have been designed (see some in the opening chapter) to study the dynamics of fast-moving particles. For, in this case, the crucial factor γ, distinguishing relativity from Newtonian dynamics, is significantly different from unity. For really testing general relativity we need situations of strong gravitational fields that cannot be arranged in a terrestrial laboratory. The differences from Newtonian predictions can and do exist in relatively weak fields, however, provided that we look at astronomical situations. Therefore astronomical tests have figured prominently in establishing the general theory.

In the early days Einstein proposed three tests, which are known as the classical tests of general relativity. More tests emerged in later years, although their number is still small. In this chapter we will disuss both classes of tests. All except one require an astronomical setting.

To place matters in proper perspective, let us see how ‘strong’ or ‘weak’ the Earth's gravitational field is at its maximum, i.e., on the surface of the Earth.

Introduction

One of the key new phenomena that arises in general relativity is the existence of solutions to Einstein's equations which represent disturbances in the spacetime that propagate at the speed of light. Such solutions are called gravitational waves and this chapter will explore several features of them.

Propagating modes of gravity

Within the context of special relativity, it is easy to identify a wave solution. For example, a propagating, monochromatic spherical wave will be described by an amplitude that varies in space and time as f(t, r) ∞ r−1 exp[−iω(t − r)]. This disturbance clearly propagates from the origin with the speed of light (which is unity in our notation) with an amplitude that decreases as (1/r). Since the energy flux of a wave varies as the square of the amplitude, this wave transports a constant amount of energy across every spherical surface. Such a description can be easily made Lorentz covariant in terms of an appropriate wave vector, etc., and has an unambiguous meaning.

The situation is somewhat more complicated in the case of gravity for two (closely related) reasons. First, not all the components of the metric gab enjoy equal status in the dynamics of gravity. We saw in Section 6.3 that the g00 and g0α components do not propagate in general relativity. The equations governing them are constraint equations involving and and are analogous to the equation governing the gauge dependent mode in electrodynamics.

Introduction

In this chapter, I give an introduction to observing with integral field units and performing basic reduction of the resulting data, prior to scientific analysis. After briefly considering the context of the lectures, I begin by discussing strategies for observing. This is followed by a short tutorial on sampling theory and its application to integral field unit (IFU) data, before continuing with an overview of the requirements for each stage of data reduction. I finish by considering the data reduction process as a whole, along with associated issues such as error propagation and file formats.

Background

Techniques for integral field spectroscopy (IFS) have been in development for at least two decades (Vanderriest, 1980). During the 1980s–1990s, numerous prototype IFUs and even a few public instruments were deployed at observatories and used for scientific work. Nevertheless, IFS has only become widely available at major telescopes during the past five years or so, following two centuries of slit spectroscopy. Experience in observing with IFUs and processing the data is just starting to become commonplace within the community, but will be spread more widely by the current generation of postdoctoral and student astronomers.

In terms of data reduction and analysis, IFS poses some non-trivial new requirements. The most obvious factor is the introduction of 3D datasets to mainstream optical and near-infrared (NIR) (as opposed to radio) astronomy. Although older scanning methods such as Fabry–Perot interferometry produce higher-dimensional datasets, these techniques are relatively specialized by comparison.