One of the greatest intellectual achievements of the twentieth century is surely the realization that space and time should be considered as a single whole – a four-dimensional manifold called spacetime – rather than two separate, independent entities. This resolved at one stroke the apparent incompatibility between the physical equivalence of inertial observers and the constancy of the speed of light, and brought within its wake a whole new way of looking at the physical world where time and space are no longer absolute – a fixed, god-given background for all physical processes – but are themselves physical constructs whose properties and geometry are dependent on the state of the universe. I am, of course, referring to the special and general theory of relativity.

This book is about this revolutionary idea and, in particular, the impact that it has had on our view of the universe as a whole. From the very beginning the emphasis will be on spacetime as a single, undifferentiated four-dimensional manifold, and its physical geometry. But what do we actually mean by spacetime and what do we mean by its physical geometry?

A point of spacetime represents an event: an instantaneous, pointlike occurrence, for example lightning striking a tree. This should be contrasted with the notion of a point in space, which essentially represents the position of a pointlike particle with respect to some frame of reference. In the spacetime picture, a particle is represented by a curve, its world line, which represents the sequence of events that it “occupies” during its lifetime. The life span of a person is, for example, a sequence of events, starting with birth, ending with death, and punctuated by many happy and sad events.

Introduction

More than their large sizes, the key defining property of Giant HII regions (GHIIRs), as a distinct class of objects, is the supersonic velocity widths of their integrated emissionline profiles (Smith & Weedman 1972; Melnick 1977; Melnick et al. 1987 and references therein). Since supersonic gas motions will rapidly decay due to the formation of strong radiative shocks, the detection of Mach numbers greater than 1 in the nebular gas poses an astrophysically challenging problem.

Melnick (1977) suggested that the ionized gas is made of dense clumps moving in an empty or very tenuous medium, so that the integrated profiles reflect the velocity dispersion of discrete clouds rather than hydrodynamical turbulence. In this model, the relevant time scale for radiative decay of the kinetic energy is the crossing-time of the HII regions which turns out to be comparable to the ages of the ionizing clusters.

In this chapter we shall consider the intrinsic geometric structure of flat spacetime M. In particular, we shall use the results of the previous chapter to show that M has a natural physically defined affine structure (i.e., the notion of a displacement vector makes sense) and, most importantly, that the space of displacement vectors possesses a natural, physically defined metric.

Spacetime Vectors

A spacetime displacement is simply an ordered pair of spacetime points. We write OP = - PO and OP + PQ = OQ. If P lies inside N(O), the null cone of O, we say that OP is timelike, in which case O and P lie on the world line of an observer or a massive particle. There are two types of timelike displacements: if P lies in the future of O (according to an observer whose world line passes through O and P), we say that OP is future-pointing otherwise, of course, we say it is past-pointing. If P lies on N(O), we say that OP is null, in which case O and P lie on a null ray or the world line of a massless particle. Again, there are two types of null vectors, future-pointing and past-pointing, depending on whether they lie on N+(O) or N−(O). If OP is neither timelike nor null, that is, if P lies outside N(O), we say that OP is spacelike (Fig. 4.1).

For a timelike displacement OP, the points O and P lie on a world line l with proper time t. If Q also lies on l and t (Q) – t (O) = a[t (P) – t (O)], we write OQ = α OP.

The isothermal gravitational collapse and fragmentation of a molecular cloud region and the subsequent formation of a protostellar cluster is investigated numerically. The clump mass spectrum which forms during the fragmentation phase can be well approximated by a power law distribution dN/dM ∝ M−1.5. In contrast, the mass spectrum of protostellar cores that form in the centers of Jeans unstable clumps and evolve through accretion and N-body interaction is best described by a log-normal distribution. Assuming a star formation efficiency of ∼ 10%, it is in excellent agreement with the IMF of multiple stellar systems.

Introduction

Understanding the processes leading to the formation of stars is one of the fundamental challenges in astronomy and astrophysics. However, theoretical models considerably lag behind the recent observational progress. The analytical description of the star formation process is restricted to the collapse of isolated, idealized objects (Whitworth & Summers 1985). Much the same applies to numerical studies (e.g. Boss 1997; Burkert et al. 1997 and reference therein). Previous numerical models that treated cloud fragmentation on scales larger than single, isolated clumps were strongly constrained by numerical resolution. Larson (1978), for example, used just 150 particles in an SPH-like simulation. Whitworth et al. (1995) were the first who addressed star formation in an entire cloud region using high-resolution numerical models. However, they studied a different problem: fragmentation and star formation in the shocked interface of colliding molecular clumps.

Introduction

Spiral galaxies, quasars, active galactic nuclei, X-ray binaries, cataclysmic variables, and young stars: these are a few of the astronomical objects that contain disks. Disks are common in astrophysics because it is usually difficult to change the specific angular momentum of gas, but easy to radiate away its thermal energy. Gas injected into in a spherically symmetric potential thus naturally shocks, radiates, and settles down into a plane normal to its mean angular momentum.

Because they are so common, disks occupy a lot of the astronomical community's time and energy (that would otherwise be entirely dissipated in attempting to measure Ω0). Although there are enormous differences between individual disk systems in global structure and observational appearance, there are a number of fluid dynamical processes common to all disks. These processes are worth understanding in detail.

The most fundamental process in disks, analogous to nuclear reactions in stars, is angular momentum transport. The disk cannot evolve unless gas in the disk can be persuaded to give up some of its angular momentum and spiral down the gravitational potential.

Introduction

Cosmic rays are very fast charged particles which are accelerated to high energies by plasma processes, principally collisionless shock waves, occuring in astrophysical plasmas. The acceleration at collisionless shock waves relies on the interaction of the charged particles with turbulence, which causes spatial diffusion both along and perpendicular to the magnetic field. This allows some of the particles to cross the the shock many times, to gain many times their original energy.

Physical properties of the atomic gas in spiral galaxies are briefly considered. Although both Warm (WNM, 104 K) and Cool (CNM, ∼ 100 K) atomic phases coexist in many environments, the dominant mass contribution within a galaxy's star-forming disk (R25) is that of the CNM. Mass fractions of 60 to 90% are found within R25. The CNM is concentrated within moderately opaque filaments with a face-on surface covering factor of about 15%. The kinetic temperature of the CNM increases systematically with galactocentric radius, from some 50 to 200 K, as expected for a radially declining thermal pressure in the galaxy mid-plane. Galaxies of different Hubble type form a nested distribution in TK(R), apparently due to the mean differences in pressure which result from the different stellar and gas surface densities. The opaque CNM disappears abruptly near R25, where the low thermal pressure can no longer support the condensed atomic phase. The CNM is apparently a prerequisite for star formation. Although difficult to prove, all indications are that at least the outer disk and possibly the inter-arm atomic gas are in the form of WNM, which accounts for about 50% of the global total. Median line profiles of the CNM display an extremely narrow line core (FWHM ∼ 6 km s−1) together with broad Lorentzian wings (FWHM ∼ 30 km s−1). The line core is consistent with only opacity broadening of a thermal profile.

A synopsis of results stemming from the analysis of the radial velocity fluctuation fields of 6 HII regions (Sh 142, M 17, Sh 158, Sh 170, Orion and Sh 212) are presented. In addition new data from the DENSITY fluctuation fields of the HII region Sh 269 will also be shown. The analysis was done using the well known two-point correlation functions. However I innovated by using the higher order structure functions on the Sh 269 data. PDF increment calculations were also done hinting at the presence of intermittency in Sh 269.

Introduction

HII regions were the first interstellar objects where scale dependent brightness and velocity fluctuations were identified and attributed to turbulent motions (von Hoerner 1951; Courtes 1955; Münch 1958). The study of turbulent motions in HII regions was then forgotten for many years until the work of Roy & Joncas (1985) and of O'Dell and collaborators later on. The discovery of such motions in HII regions should not come as a surprise. These objects possess large scale velocity gradients that are explained by ionized gas flows produced by the erosion of the parent molecular cloud. The newly born massive stars produce the necessary UV photon flux. Turbulence becomes a natural companion of the kinematics of HII regions since the ionized gas flows can reach twice the speed of sound enabling the Reynolds number to reach high values (ℜ > 105).

An isolated physical body such as a star may be described in terms of a spacetime picture by an asymptotically flat spacetime, (M, g), containing a world tube representing the region of M occupied by the body. Inside the world tube, the energy–momentum tensor, Tab, will be nonzero and should describe some physically reasonable matter distribution. Outside, in the vacuum, matter-free region, Tab=0, and hence, by Einstein's equation, Rab = 0. The curvature, and hence the gravitational field, in the vacuum region is thus completely given by the Weyl tensor, that is, Rabcd = Cabcd.

While the situation described above serves as a framework for the discussion of isolated bodies in general relativity, it is too general to provide much real information; what we need is some simplification to make it more tractable to calculation. Fortunately, most stars are more or less spherically symmetric and stationary. We can thus obtain a useful, but considerably simpler, model by imposing spherical symmetry and stationarity. The problem of finding the most general spacetime satisfying these conditions was, in fact, essentially solved by Schwarzschild in 1916, not long after Einstein first proposed his general theory of relativity. In particular, he showed that the vacuum region outside a spherically symmetric and stationary body has a simple geometric structure, which we shall refer to as a Schwarzschild geometry, that depends on only one number, m, which may be identified with the total mass of the body. This result is similar to the Newtonian case, where the external gravitational field of any symmetric body is given simply by the inverse square law.

We come now to the physical interpretation of the curvature tensor Rabcd. As we have seen, Rabcd arises from the spacetime metric, which in turn arises from the properties of inertial world lines and null rays. If there are no gravitational tidal effects, then spacetime is flat and hence Rabcd = 0. Conversely, if Rabcd = 0, then, apart from the possibility of topologies other than ℝ4, spacetime is flat and there are no gravitational tidal effects. A nonvanishing curvature tensor is thus an indication and a measure of gravity. In this chapter we shall show that a gravitational field is completely described by Rabcd. But first we need the concept of a geodesic.

Geodesics

The world line of an inertial particle has the remarkable property of being uniquely determined by its initial four-velocity vector. If we take proper time to be the parameter along the world line, then its tangent vector at any point will be a four-velocity vector. Similarly, a null ray is uniquely determined by specifying a null vector at some point, O say, and the set of all null rays through O forms a null cone N(O). We recall that it was precisely these properties of inertial world lines and null rays that allowed us to define a unique spacetime metric. Furthermore, the metric led to a unique connection and, finally, to the curvature tensor.

We present two-dimensional MHD numerical simulations for the interaction of high-velocity clouds (HVC) with a magnetized gaseous disk. The initial magnetic field is oriented parallel to the disk. The impinging clouds move in oblique trajectories and fall toward the disk with different initial velocities. The B-field lines are distorted and compressed during the collision, increasing the field tension and preventing the cloud material from penetrating into the disk. The perturbation, however, creates a complex, turbulent, pattern of MHD waves that are able to traverse the galactic disk and, for unstable disks, can trigger the Parker instability.

Introduction

High velocity clouds (HVC) are atomic H I clouds located at high latitudes in our Galaxy, and moving at velocities ∣VLSR∣≥ 90 km/s (see Bajaja et al. 1985, and Wakker & van Woerden 1997). Their distance is unknown, but limits to the locations of some particular clouds indicate z-heigths of a few kiloparsecs, setting a possible mass range of 105-106 M⊙. Thus, a HVC complex moving with a speed of 100 km/s has a kinetic energy of about 1052−53 erg. These values indicate that the bulk motion of the HVC system could represent a rich source of energy and momentum for the interstellar medium (equivalent to that from several generations of superbubbles).

There is evidence for possible collisions between HVCs and gaseous disks, both in our Galaxy and in external galaxies.

Introduction

Molecular clouds (MCs) are recognized to be the sites of present day star formation in our galaxy. The description of their dynamics is an essential ingredient for the theory of star formation.

A lot of work has been devoted to understand i) how super-sonic random motions in MCs can persist for at least a few dynamical times and ii) why MCs do not collapse, or fragment gravitationally into stars, on a free–fall time–scale. The magnetic field has been advocated as the solution for both problems. Magneto–hydrodynamic (MHD) waves were believed to dissipate at a significantly lower rate then super–Alfvénic and super–sonic random motions.

Introduction

Electromagnetic radiation from astronomical objects encounters different turbulent zones in its way to Earth-based telescopes. Interstellar turbulence, for instance, provokes phase fluctuations of radio waves which are exploited to study the interstellar medium, as presented by several authors in this volume. In the optical domain, wave perturbations occur in the Earth's atmosphere, due to turbulent fluctuations of the refractive-index of air (which is frequently referred to as optical turbulence). This has severe negative effects on astronomical observations as the commonly known “seeing” that strongly limits the achievable angular resolution (Roddier 1981). A number of high angular resolution techniques, like adaptive optics and interferometry, are being developed to overcome this limitation. They owe their good results yet obtained to the knowledge of the optical effects of atmospheric turbulence, and their optimization demands an increasingly high precision of that knowledge.

Introduction

Interstellar turbulence may affect the chemistry of translucent and dense molecular clouds in various ways. Turbulent mixing of material from dense cores to the surface of molecular clouds, and vice versa, may alter the abundances inferred from chemical networks. In particular, turbulent transport and diffusion was invoked to explain the large abundance of atomic carbon and that of complex organic molecules which is observed in dense molecular clouds (Boland & de Jong 1982; Chièze, Pineau des Forêts & Herbst 1991; Xie, Allen & Langer 1995).

The dissipation of turbulence in translucent molecular clouds is another physical process which has recently been considered to alter chemical abundances.

As anyone who has paid an electricity bill knows, energy is a very real physical quantity but unlike other expensive commodities such as books, records, and bottles of wine, it has a universal and all-pervading character. But what actually is energy? We shall attempt to answer this question by simply listing its defining properties. We must, however, be careful to list enough properties so as to capture the notion of energy as is actually used in fundamental physics, and also to distinguish it from other conserved quantities such as electric charge. We must, of course, not list too many properties, or properties that are too rigid, as this may lead to a trivial or nonexistent quantity. We do not want, for example, to end up with a quantity that turns out to be identically zero.

Energy is a measurable quantity possessed by all physical objects, which is always found to be strictly positive. Energy comes, of course, in many forms, ranging from pure radiative energy, which tends to be its most useful but most dangerous form, to its most benign form, which is safely locked away, like a genie in a lamp, in all inert matter. Even the paperweight in front of me contains energy (quite a considerable amount) but, fortunately for me, this is in its most benign form. I could, in principle, if I so wished, convert this energy into its radiative form by bringing my paper weight into contact with an otherwise identical one made of antimatter, and this would result in a spectacular firework display of pure radiative energy. (Such experiments are, however, not recommended, as they can cause quite a mess.)