Lectures on General Relativity

Chapter 1

First overview

1.1 General relativity as a dynamical theory of
space-time and gravitation

Physical space-time can be defined as the collection of all events or, rather, equivalence classes of events with unrestricted causal connections. It is thus the stage on which physical events take place, deprived of the characteristics of specific events. Experiments within high energy physics have revealed no sign of discontinuity at least down to distances of 10-18 m. Because of this, together with the local validity of special relativity, we use as mathematical model a four-dimensional C8 manifold M with Lorentz metric g, i e a metric with signature +2 [81].1

Physical events in space-time are described through various fields on defined in general by tensor equations on These can be found by assuming that the equations of special relativity hold in the tangent space at every point of (the principle of equivalence). The metric g will then enter the equations though covariant derivatives and becomes a physical field of its own. From a formal point of view g is the gauge field of local Poincaré invariance. The equations of physical phenomena are, thus, covariant with respect to arbitrary coordinate transformations on as is also evident since they express relations between tensors on

The gravitational field, unlike other physical fields, does not fit this description since light rays, whose world-lines are null curves, are deflected by gravitation. This means that the description of gravitation requires curved space-time and has no meaning in the tangent space. However, as is realized eg by means of the equivalence principle, a complete description of the gravitational field is furnished by the metric tensor g alone and does not require the introduction of any additional field on

Thus, g describes the geometry of space-time as a physical property among others and, at the same time, gravitation. Suitable field equations, relating g to matter, can be found by requiring Newton's theory as the limiting case of static, weak fields [81], from more formal considerations on tensor equations preferred by Einstein ([40], [104]), or from variational principles as first shown by Hilbert [104]. The result is, in any case, Einstein's field equation,

if units are chosen so that G = c = 1.2 The coupling of gravitation to matter involves only the energy-momentum tensor and, thus, different types of matter which have the same distribution of energy and momentum should generate the same gravitational field.

Soon after their discovery in 1915, these equations were found to be hyperbolic; see [81], [177], [136] and references therein. This implies that Cauchy's problem can be solved, at least locally, with suitable initial data given on a spacelike hypersurface,3 and thus the dynamical object of general relativity, as a theory of space and time, is three-dimensional space evolving in time [174]. Therefore, although space-time is the fundamental theoretical concept, space and time themselves should perhaps not be completely relegated to the world of shadows as eg in the view of Minkowski [105].

1.2 Einstein spaces

Space-time manifolds with Ricci's tensor proportional to the metric tensor are called Einstein spaces [136]. By (1.1), they correspond to the presence of a pure gravitational field. If, in particular, the cosmological constant ? is set to zero, (1.1) goes over into

called Einstein's equations for the pure gravitational field or, simply, the vacuum field equations.

To find exact solutions to these complicated equations one has to rely on some symmetry assumptions. One possibility is to assume that admits an isometry [81] or, in the terminology of Petrov [136], a motion, i e a mapping of onto itself which preserves the metric. The one-parameter group of transformations ft generated by a vector field K is a group of isometries if and only if K satisfies Killing's equation,

By means of this equation Petrov has given an extensive classification of Einstein spaces admitting various groups of motions [136]. Among these is, for example, the spherically symmetric one with three independent Killing vectors whose metric is given, in certain regions, by Schwarzschild's solution.

Another method of finding exact solutions, introduced by Newman and Penrose ([122], [12]), rests on the use of so called spin coefficients, a kind of complex Ricci rotation coefficients, together with tetrad components of the tensors, or an equivalent spinor formalism. This method has proved very powerful and has resulted in large classes of exact solutions, corresponding to various choices of spin coefficients [12]. Among the simplest ones of these are the Robinson-Trautman solutions, to be discussed later.

1.3 The large scale structure of space-time

Any solution gab of the field equations is defined within some domain which depends both on the space-time manifold and on the coordinate system used. Limitations due to the coordinate system only are considered unessential since additional coordinate patches allow an extension. In other words, the space-time manifold is isometric to a proper subset of some manifold

However, in curved space-time there also arises the possibility of a topology other than that of R4. This may give rise to space-times which are inextendible, but yet incomplete. More precisely, there may be curves which cannot be be extended to arbitrary values of an affine parameter, in which case the space-time manifold is said to be singular ([81], ch 8). Most of the known exact solutions are singular in this sense. The question therefore arises whether this is an artefact of the high symmetry of these solutions or whether generic space-times are singular.

Thanks to a very precise mathematical formulation of general relativity, including the application of differential topology to space-time manifolds ([133], [65]), the latter alternative has been established as the correct one. In a series of fundamental theorems Hawking and Penrose have shown that singularities must occur if certain conditions, having nothing to do with symmetry, are fulfilled ([81], ch 1, 8). These conditions include an inequality for the energy-momentum tensor, the existence of a sufficient amount of matter within a bounded region to create a trapped surface (i e a surface across which no signal can escape), and causality conditions. All of these are very reasonable and indicate that some breakdown of general relativity must occur.4

Among the results reached by means of the global techniques are also theorems on black holes, domains in space-time bounded by an event horizon, from which no signal can escape to future null infinity. These horizons are generally accepted as physically reasonable but some authors express another opinion ([119], [78]).

Finally, with a topology other than that of R4, it becomes possible to construct space-times with unusual causal properties, examples of which are also found among the solutions of Einstein's field equations [69]. A reasonable causal structure, more precisely so called stable causality ([81], ch 6) is essential for the existence of cosmic time functions, making precise the concept of a universal time, and useful in particular in cosmology. When such functions exist, they can be defined in many different ways, showing the extreme relativity of time, especially in the presence of singularities or horizons. For some space-times this is illustated in [110], also reproduced later in this book.

Singularities also may be of a very different nature [51], some of them corresponding to a point where the curvature tensor goes to infinity. The latter are of particular interest in connection with the motion and structure of particles as will be considered below. Some general theorems have been proved giving conditions for the singularities predicted by Hawking and Penrose to be of this kind.

1.4 The problem of motion

Given a matter distribution, the field equations determine the generated gravitational field from the energy-momentum tensor. Conversely, the behaviour of matter under the influence of a gravitational field is determined by a set of equations, the equations of motion, which can be found by means of the principle of equivalence. In the case of coherent matter, for example, this principle implies that the individual particles move so that their world-lines are geodesics in space-time. On the other hand, the left hand side of the field equations is divergence-free and thus Tab;b = 0, which also leads to geodesic motion. Further, the non-linearity means that the field equations can be satisfied only if the sources of the gravitational field interact. Thus, the field generated by several sources in mutual rest must have some kind of singularities on a line connecting the sources, as can be examplified by Weyl's static, axially symmetric solutions [159].5 All this indicates that the equations of motion may be implicit in the field equation, and...