Introduction

This paper will deal with turbulence in the ionized portion of the interstellar medium. The theoretical ideas invoked will therefore be from the field of plasma turbulence, which in some respects differs from hydrodynamic turbulence. My primary interest, in keeping with the title of this paper, will be in the fluctuations which occur on spatial scales much less than the outer scales of the turbulence, scales which may be termed part of the inertial subrange.

There is no purely internal physical experiment which will demonstrate the relative motion of a Galilean reference system – this is the principle of relativity. It follows that all physical laws must necessarily have the same form in all inertial systems (see Chapter 2). To express this property in a simple analytic way, and to avoid the need to show explicitly in each case that the principle of relativity is satisfied, it is extremely useful to write the equations of physics in a form which is manifestly covariant under Lorentz transformations. Not only do we avoid the need for such proofs (often very delicate, and subject to error), but this also allows us to set physics in the context of space-time furnished with a pseudo-metric. Further, the need for manifest covariance of the analytic formulation of a physical phenomenon restricts its possible forms. For this reason we shall use various representations of the Lorentz group: spinors, tensors, etc.

Tensor formalism

(1) Unless otherwise stated, we shall only use cartesian coordinates which are orthogonal in the sense of the metric on M. They thus have a timelike axis Ox0 and three spacelike axes Oxi (i = 1, 2, 3) orthogonal in pairs, in the sense of the metric of R3; these three spatial axes are orthogonal to the axis Ox0, the time axis, in the sense of the metric of Minkowski space (Fig. 3.1).

We shall see at the end of this chapter that relativistic gravitation will require the introduction of curved space–time. We should ask what “curved space” actually is. Our intuition, based on surfaces in R3, can be extended to spaces of dimension larger than two. We can deduce the essentials from simple examples [like the sphere] of surfaces in R3. We shall do this, first by studying some geometric properties of known surfaces, and then comparing them with corresponding properties of the plane R2. We then define the Riemann curvature and finally give arguments leading to curved space–time.

Some manifestations of curvature

Here we consider only a sphere of radius R embedded in R3: clearly this is a curved surface. We shall try to construct elementary geometrical figures whose properties we compare with the analogous plane figure.

(1) Geodesic triangle (Fig. 6.1): In the plane, a triangle is formed by the intersection of three non-parallel straight lines. On a sphere, arcs of great circles play the role of straight lines: a straight line in the plane R2, is the shortest path (geodesic) between two points, while for the sphere S2 the geodesies are arcs of great circles. We thus can construct a triangle on the sphere S2; between two points A and B there is an arc of a great circle (exactly one, if the distance AB is to be a minimum and A, B are not at poles).

The usual equations of motion of a particle or a system of particles can be deduced from a variational principle [principle of least action leading either to Lagrange's or Hamilton's equations, depending on the variables used, see H. Goldstein (1980) or L. Landau and E. Lifschitz (1960)], in both the Newtonian and relativistic cases [see A.O. Barut (1965) or J.L. Anderson (1967)] with some subtleties concerning the constraint uµuµ = 1 in the latter case [G. Kalman (1961); A. Peres, N. Rosen (1960)].

In the same way, the equations satisfied by the fields (continuous systems with an infinite number of degrees of freedom), whatever tensor nature they may have, can often be deduced from variational principles. This is true of the equations of electromagnetism, for example, but not the equation for heat transfer.

There are many analytic procedures for describing the motion of a particle or the evolution of a field. There is no a priori reason to confine oneself to differential equations or second order partial differential equations.

The main advantage of a variational formalism is that it allows one to find the conserved quantities in the motion (i.e. the first integrals) and directly to exploit the symmetries of the physical problem considered; these two aspects are connected, as we shall see. We shall introduce such a formalism here only because it allows us to define the energy and momentum of a field very simply.

For classical physics, space and time provide the arena in which the phenomena of nature unfold. These phenomena do not change the space–time frame, which is inert and absolutely fixed for all time. Moreover, space and time are regarded as completely distinct and having no connection with each other. Relativity theory links space and time, and reaches its culmination in General Relativity, which connects the space–time properties with the dynamical processes occurring there.

Newtonian space–time

Physical space possesses the usual properties of continuity, homogeneity and isotropy which we attribute to the space R3 when equipped with its affine structure (parallelism, existence of straight lines) and its usual metric structure (Pythagoras' “theorem”). However, we must understand the physical significance of the mathematical concepts connected with R3. Thus, the existence of physical phenomena which can be represented by straight lines (mathematics) leads to the (experimental) notion of alignment: three points are (physically) aligned if we can find a viewing point from which they appear to coincide. From this it follows that light constitutes our standard of straightness; it is only by a further step (which may prove to be incorrect) that we can identify the trajectory of a light ray with a straight line in R3. Similarly, the mathematical concept of parallelism in R3 is directly related to the (physical) notion of rigid transport and of distance. Finally, we must recognise that the (mathematical) properties of homogeneity and isotropy of physical space only express our experience of mechanical systems: that these remain unaltered when placed in any position or place.

We have seen in the preceding chapters that the changes in the theory of gravity introduced by relativity, chiefly as a consequence of the famous relation E = mc2 (Chapter 4), amount to the near-necessity of introducing curved space-time, also a consequence of the Equivalence Principle (Chapters 6 and 7). However, this principle does not specify the equations determining the ten components of the metric tensor gµv. The relation E = mc2 suggests that these equations must be non-linear (Chapter 4). In this chapter we shall study the simplest equations compatible with observation, and show how they arise. These are Einstein's equations, and the theory they define is called general relativity. We shall derive some elementary consequences which are astrophysically important.

The curvature of space–time expresses the effects of gravitation on physical phenomena through a metric tensor gµv, which cannot be reduced to ηµv everywhere. However, this does not rule out the existence of other long-range fields which might also be important. The theory of C. Brans and R.H. Dicke (1961) is the best-known example; here a scalar field coexists with the metric tensor. The existence of such fields must ultimately be decided by experiment or observation [see C. Will (1981)]. Currently it appears that the metric tensor alone appears capable of ensuring agreement with observation, and that general relativity is the correct relativistic theory of gravity.

We note again that simple arguments rule out relativistic theories of gravity based solely on a scalar or vector field.

This book is devoted to general relativity, i.e. to the synthesis of special relativity and gravitation. This Relativistic Gravitation, as it is sometimes called, appears to be of uppermost importance in all those astronomical phenomena that involve velocities close to that of light or intense gravitational fields. The study of the latter constitutes a new subject, Relativistic Astrophysics, an expression due to Alfred Schild (1967).

The content of this book is the minimum minimorum needed to approach this relatively recent domain.

It may be interesting at this point to recall how and why this new subject started. Once the classical tests of general relativity were performed (bending of light rays by the Sun (1919), gravitational redshift (in white dwarfs) [W.S. Adams (1925)]; perihelion advance of Mercury), the subject became very formal, as current technology did not provide contact with experiment or astronomical observation. Although much research had great conceptual interest (unified theories of gravity and electromagnetism, for example), general relativity became rather arid [see J. Eisenstaedt (1986)] because of the lack of laboratory experiments or observations of relativistic objects, which were in any case unknown to theory before the 1930s, and even then ignored in the 1940s and 1950s. Thus, cosmology was regarded more as a “free area for thinking about relativity” [J. Eisenstaedt (1989)] than a field for astronomical verifications of general relativity, or even the “Science of the Universe” [E.R. Harrison (1981)].