Lectures on General Relativity E-Book

Bengt Månsson

• Herausgeber: Books on Demand
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 507

Veröffentlichungsjahr: 2019

Beschreibung

Do you know the basics of general relativity? Do you want to know something of what more there is? Do you wonder how the theory of relativity came into being? Then this book is for you! Partial contents: - Black holes and gravitational collapse - Cosmological solutions of Einstein's field equations - Gravitational waves - Space-time singularities - The problem of motion for massive particles - A collection of exact solutions of Einstein's field equations - A history of Einstein's creation of the theory of relativity in the years 1905-1915 - A short course for repetition of the basics of general relativity - Bibliography, references, and index The book, although not very advanced, covers a number of topics not often seen in text books. The selection, of course, refelects my own interests. The different chapters may to a large extent, though not completely, be read in any desired order. The author has a PhD in theoretical physics and is lecturer of mathematics. He has for many years taught physics and mathematics at senior high school as well as university level.

Leseprobe

Contents

Preface

The present book is based on lectures given at Lund University and Stockholm University, Sweden, excerpts from my PhD dissertation in theoretical physics [108], and a somewhat related but more basic swedish text [111]. The level is intermediate, it being assumed that the reader is somewhat acquainted with general relativity. The intention is to give those who have read som introcuctory text, like Lawden [100] or Schutz [150], a collection of texts showing a part of what there is more, before reading the comprehensive books of Misner-Thorne-Wheeler [107] and others. To a large extent the texts are independent, and can be read in any desired order. Also, I have added a short basic course suited to refresh knowledge. This is found in part VI, chapters 29–32. Section 30.6.4 as well as chapters 31 and 32 may be of interest also for those acquainted with the basics. Finally a number of problems are included at the end (ch 33). The intention with this is to present topics that may be of interest to think about so they are not ordinary exercises with known (to me) answers.

Part I consists of two fairly modern overviews based on Wheeler’s view of the theory and the modern form of differential geometry.

In part III the existence of black hole solutions to Einstein’s field equations is demonstrated based on the theory of spherically symmetric space-times following from the general theory of symmetric space-times. The inner size of black holes is also investigated. Another application of symmetric spaces lies in the proof of the existence of cosmological solutions, i e solutions that can be regarded as descriptions of the entire universe.

Part IV consists of a selection of more advanced topics covering exact solutions to Einstein’s field equations like Weyl’s axially symmetric solutions, and the Robinson-Trautman solutions. Further, the problem of motion is treated, i e the question of whether and how the motion of particles is implicit in Einstein’s gravitational field equations. After this follows a discussion on how to define and calculate the energy of a gravitational field, and a presentation of gravitational waves, including a comprehensive calculation of the expression for gravitational radiation power in the linearized theory. Finally, the concept of a space-time singularity is considered, ending with a short presentation of a typical theorem on the necessity of the existence of singularities.

In part V we turn to a historic and epistemological study of Einstein’s theories of relativity. To that end we also include the history of the creation of special relativity as well as Einstein’s so-called theory for the non-symmetric field, his last attempt to create a unified theory of gravitation and electrodynamics.

References, index and a list of literature conclude the book.

A number of appendices are included. These are placed at the end of the respective part, that is not the different chapters, except for chapters 18, 21, and 22 where the appendices are at the end of the respective chapter.

I want to thank professor Bengt E Y Svensson, Institute for Theoretical Physics, Lund University, Sweden for valuable comments and suggestions on the chapter on cosmological solutions. Any remaining shortcoming is, of course, my own responsibility.

Partille, Sweden, October 22, 2018

Bengt Månsson

E-mail: [email protected]

Notation and conventions

Indices

Latin indices take values (1,2,3,4) or (1,2,3,0).

Greek indices take values (1,2,3).

Units

Metric and signature

The curvature tensor

where denotes the usual symmetric, metric affinity given by the second kind Christoffel symbols.

Ricci’s tensor

Einstein’s tensor

This definition is used by all authors as far as I know. The construction is motivated by the desire for a divergent-free tensor. It may also be noted that it is unaffected by the choice of the metric signature since the affinity and thus and Rab are invariant under a change of sign of the metric while changes sign.

Einstein’s field equations

where the material energy density is T44≥ 0.

Symmetries

The symmetric and anti-symmetric parts of a two-index quantity are denoted

respectively

and similarly for quantities with more indices.

Coordinate terminology

Exceptions

In a few cases (mainly Robinson-Trautman and Weyl solutions) other conventions are used. These do not, however, interfere with other parts of the book and should cause no confusion.

Software used

Derive: A general purpose computer algebra system, owned by Texas Instrument but no longer maintained [225].

Mathematica: An advanced computer algebra system and knowledge basis, created and maintained by Wolfram Research [226].

Maxima: A free program with among other things tensor calculation features suitable for general relativity [230]. This is used, particularly in the chapter on exact solutions, to find the components of Einstein’s or Ricci’s tensors. The program uses the same sign conventions as above for the curvature tensor and Ricci’s tensor.

WolframAlpha: A free net and mobile app service with among other things decent integral evaluation capabilities [227].

Part I

TWO OVERVIEWS OF 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 C∞ 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,

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 spacelikehypersurface,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 Λ 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 φt 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

Idealizing the sources as mass points one does not need to specify any energy-momentum tensor, a point strongly emphasized by Einstein. Instead the sources must the be represented by space-time singularities, and the vacuum field equations can be assumed everywhere in the resulting singular space-time. With some specification of the type of singularity, eg spherical symmetry of the surrounding field, the motion can then be calculated by means of succesive approximations, from the above-mentioned integrals. In this way one obtains equations of motion in the Newtonian and post-Newtonian, or even higher approximations, as was first shown by Einstein, Infeld, and Hoffmann ([68], [46], [44]).

As a shortcut to theses results, Infeld introduced an energy-momentum tensor of δ−function type, corresponding to spherical symmetry ([85], [86], [87]). By means of the full field equaions together with the Bianchi identities, it then became possible to derive exact equations of motion resembling those of a geodesic, which express the acceleration in terms of regularized field quantities.

Certain problems remain in connection with these investigations. First, the concept of spherical symmetry is problematic ([94], [83]). A promising new approach, treating the equations of motion of singularities as coupled to those determining their structure, has been developed by Newman and collaborators [121].6 In this way, one avoids a priori assumptions, which may be incompatible with the field equations.

Secondly, regularization procedures for field quantities may not be independent of the choice of coordinates [66]. This is certainly so when the modern conception of a singularity is taken into account, a fact which makes it problemetic to formulate covariant equations of motion. A detailed investigation of this for certain classes of singularities forms the subject of [109], which is also reproduced in chapter 22.

It might seem possible to avoid the second difficulty through the use of extended sources instead of singularities, and Fock and Papapetrou have found equations of motion in this way ([54], [130], [131]). However, besides the problem of finding a suitable energy-momentum tensor, it should yet be of interest to describe particles as mass points as an idealization and this entails representation of the gravitational field by means of a singular space-time manifold. Further, and more serious, the theorems of Hawking and Penrose predict the occurence of singularities from very general, regular initial conditions. At least in ”particle like” cases these singularities should obey covariant equations of motion. Also, the approach of Newnman et al [121] does not give equations of motion in this sense, i e covariant, ordinary differential equations connecting the acceleration with field quanities, but describes the motion and structure of certain kinds of singularities in terms of the surrounding null cone structure. This does not affect the present problem.

1.5 The crisis in general relativity

Since the singularity theorems of Hawking and Penrose were established around 1970 [81], it is generally agreed that the theory of relativity must in some respect be of limited validity ([119], [78]). This is so because a singularity can be regarded as a place where the concept of a space-time manifold breaks down, and consequently all known laws of physics wíll break down.7 In particular, one does not know what may be emitted from a singularity and therefore the future of space-time cannot be predicted. For instance, it might be that a singularity always occurs inside a black hole so that it is surrounded by an event horizon (the ”cosmic censorship” hypothesis). While retaining causality outside the horizon this attempt does not, however, avoid the very singularity, and the problem remains for any observer moving inside the horizon. Another suggestion is that singularities may not appear in a quantized version of general relativity. However, as has been discovereed by Hawking [78], the application of quantum theory to the macroscopic domain of a black hole leads to a basic limitation of our ability to predict the future (Hawking’s ”randomicity principle”) which may be considered as equally disastreous [119].

It seems, therefore, that a new classical theory of space-time geometry and gravitation is desirable. One possibility has been pointed out by Møller who suggets that the metric tensor does not describe the gravitational field completely, and instead proposes a tetrad description. This opens up new possibilities to formulate field equations, the solutions of some of which may have neither singularities nor horizons.

We finally point out the relevance for these problems of the investigations on which the present thesis is based. As mentioned in the preceding section it seems difficult to find covariant equations of motion for space-time singularities ([109] and ch 22). On the other hand, material particles should obey such equations but, neglecting their spatial exension, they should also be represented by singularities. Threfore, the concept of a mass point does not fit general relativity, not even as an idealization.

The problems connected with the possible existence of event horizons are touched upon in [110]. The structure of null cones near a horizon makes it possible to choose surfaces of simultaneity in many different ways. In case of a spherically symmetric hole, for instance, they can be chosen so as to intersect the central singularity once,twice, or not at all. Indeed, about the same thing happens with a small but finite particle, lying inside an event horizon. Thus, different observers do not agree as to the number of particles or even whether there are any. If this strange situation is to be remedied the occurence of event horizons must, as well as that of space-time singularities, be regarded as signs of limitations in the classsical theory of general relativity.8

1 The present chapter is a reproduction from my PhD thesis ”Investigations on the motion and structure of space-time singularities”, Department of Theoretical Physics, University of Lund, Lund, Sweden, 1978 [108], based on [109] and [110], included in this book as chapters 18 and 21, and on a third part reproduced in chapter 22.

2 Otherwise the coefficient of Tab will be 8πG/c4 .

3 For details on hyperfaces, see section 30.6.4

4 Singularities are considered in chapter 24.

5 These solutions are treated in sections 17.2 and 20.2.

6 This is considered in section 20.4.

7 See chapter 24.

8 Also, see ”Inner size of a black hole”, sect 15.6.

Chapter 2

Second overview

2.1 Differential Geometry

2.1.1 Manifolds

By physical space-time we mean the collection of all point events. This is the stage on which physical events take place but is also in itself a physical object, described in General Relativity by Einsteins field equations for the gravitational field.

The seemingly smooth structure of space-time leeds to using a C∞differentiable manifold as mathematical model.9

Figure 2.1: For the definition of the concept of a manifold.

Definition

A C∞, n-dimensional manifold is a set together with an atlas {Uα,Φα} such that

Example 1:

This is a two-dimensional manifold (plane) with coordinates (x, y). The entire plane is a coordinate neighbourhood with these coordinates.

Example 2:

This is a two-dimensional manifold (unit sphere) which can be covered with the six coordinate neighbourhoods

A topology is defined as {∪Uα|Uα ∈ some atlas}. Thus the Φα are homeomorphisms.

The dimension is n. This is well-defined since open subsets of Rm and Rn cannot be homeomorphic if m -n.

The manifolds are assumed paracompact meaning that each atlas {Uα,Φα} has a subatlas {Vβ,Ψβ}, Vβ ⊆ Uα, such that each p ∈ M belongs to finitely many Vβ. This property makes integration possible.

Finally, the topology is assumed hausdorff meaning that different points belong to disjunct open neighbourhoods.

2.1.2 Tensors

Let M be a manifold and a curve in M, i e

A (contravariant) vectorV tangent to λ(t) is defined as the directional derivative, i e

So, a vector is an operator. However there is a relation to the usual definition dxa/dt as follows,

The set of vectors at p forms an n−dimensional linear space, the tangent space Tp to M at p. The mapping defines a vector field on M.

The commutator [V,W] is an operator on functions on M, defined by

In particular,

The commutator [V,W] is also called the Lie derivative, denoted LVW.

A one-form (covariant vector) ω at p is defined as a real-valued, linear function10 on Tp

The set of one-forms at p forms an n−dimensional linear space, the cotangent space T∗p to M at p. The mapping defines a one-form field on M.

A coordinate basis11 dxa of T∗p is uniquely defined by

Thus,

The differential of a function f is defined by

which is equivalent with

A tensorT of type (0, 2) at p is defined as a real-valued bilinear function on A coordinate basis is defined by

The components Tab of the tensor T in a coordinate basis are defined

so that

By a tensor field of type (0, 2) on M we mean a a mapping where is a tensor on

Similarly, a tensor of type (r, s) at p is a real-valued, multilinear function on the Cartesian product defined as

Vectors are tensors of type (1, 0) and one-forms are tensors of type (0, 1).

The components of the tensor T of type (r, s) (at p) with respect to the bases {E1, . . . ,En}, {E1, . . . ,En} are defined

In particular the components in coordinate bases are

Coordinate transformations

A coordinate transformation induces transformations of tensors. Coordinate basis vectors and one-forms transform according to

Also the components of a tensor of type (0, 2) transform according to

since and similarly for other tensors, eg

2.1.3 Metric structure

A metric tensor field is a type (0, 2) symmetric tensor field g, meaning that

for all vectors X,Y or in component form

In a manifold there exists a metric tensor such that the path length along a curve is given by

where

It is presumed that g is non-degenerate so that the determinant Then there exists a type (2, 0) tensor such that

The gab and are regarded as covariant and contravariant components respectively of one and the same tensor g. Applied to the components of any tensor they can be used to ”raise” and ”lower” indices.

Signature of the metric

For any point there exists a coordinate system such that at p, gab is the unit matrix. Now, there also exists a symmetric tensor g whose component matrix gab can be diagonalized as follows,

A bit of terminology: By metric we are usually referring to the quadratic form12

which, in our present terminology is the same as while the tensor itself can be written

The correponding path length defines the manifold topology. A positive definite metric exists if the manifold M is paracompact; for a proof see [81], p 38 f.

Lorentz metric

at p. This case will be of particular interest in general relativity where a manifold M with Lorentz metric will be used as a model for physical space-time. Also, such a manifold is usually called a space-time.

A Lorentz metric exists on any non-compact manifold but not always else, eg not on the unit sphere S2 but on the torus T2 it exists; see [81], p →.

A reasonable manifold representing physical space-time cannot be compact ([81], p →), and thus admits a Lorentz metric. Further, the existence of a Lorentz metric on a hausdorff manifold M implies that M is paracompact so that M also admits a positive metric defining the usual manifold topology.

Finally, a Lorentz metric divides the vectors into three disjunct classes named timelike, null and spacelike respectively:

The light cone stucture in general relativity rest on this classification.

2.1.4 Connection

In euclidean space using Cartesian coordinates the partial derivatives of a tensor field13 form another tensor field. In general this is, however, not so. Therefore a new operator is desired that maps tensor fields into tensor fields and which reduces to partial derivative in euclidean space and Cartesian coordinates. We will define this operator in two steps.

Firstly, let X be a vector field on a manifold M. Then by ∇X we will mean an operator acting on vector fields with the properties

where X,Y,Z are vector fields, f, g functions, and α, β real constants. The operator ∇X is called the covariant directional derivative in the direction of X. ∇XY is a vector field.

Secondly, the covariant derivative of a vecor field Y, ∇Y, is defined as that tensor field of type (1, 1) for which

From the last item in (2.1) follows

and then, using

where is defined by

Figure 2.2: Illustration of covariant derivatives.

Thus, the components with respect to the dual coordinate bases are

Under a coordinate transformation

Covariant derivatives are extended to arbitrary tensors accordning to

and in particular for functions, In a coordinate basis the are normally given by

See further ”Metric connection below”.

2.1.5 Parallel transport

A vector field Y is by definition parallel transported or is a parallel field along a curve if

Given a connection this gives unique corresponding vector (tensor) components at different points, along a curve In general path-dependent.14

Geodesics

A curve is said to be a geodsic curve or simply a geodesic if

With a suitable parameter transformation

This differential equation is called the geodesic equation and the parameter v an affine parameter. If then locally a geodesic exists and is unique such that

M is said to be geodesically complete if v takes all real values. This is essential in connection with the so called singularity theorems.

Metric connection

If in particular the connection fulfills the conditions

it is said to be a metric connection and is the one used in general relativity. The are given in terms of the metric tensor by

For every point there exists a coordinate system such that

Such coordinates are called geodesic coordinates.

2.1.6 Curvature

The Riemann curvature tensor is defined15

This tensor is uniquely defined by the commutator of covariant derivatives,

for any vector field Z.

Symmetries

The covariant components of the curvature tensor, defined by satisfy the symmetry relations

From this it is realized the there are 20 algebraically independent covariant components

Ricci’s tensor

is defined

and has 10 algebraically independent covariant components.

Weyl’s tensor

is defined

The Weyl tensor is conformally invariant meaning that a change of metric tensor of the form where Ω is a function, does not change

Bianchi’s identities

This is equivalent to the topological theorem ∂ ∂ M

(contracted).

A consequence of ii) is that Einstein’s tensor has vanishing divergence, something that is essential in general relativity.

Geodesic deviation

A geodesic congruence is by definition a set of geodesics, exactly one of which passes through each point of a [sub]manifold [of] M.

Let V be a unit vector field tangent to a geodesic congruence and W a connection vector, i e a unit vector field each vector of which connects points with the same affine parameter value v on two neighbouring geodesic of a congruence. This satisfies the equation of geodesic deviation

In general relativity this equation describes the tidal force of gravitation ([81], ch 4; [12], p →). For a proof derivation of the equation, see appendix III.B.

A more advanced study of geodesic deviation is found in [81], chapter 4.

Flat manifold

A manifold is said to be flat if there is a coordinate system such that everywhere. Equivalent statements are

Parallel transport is path-independent.

2.1.7 ”Surgery” and embeddings

Identification

Manifolds can be constructed through identification. In the examples below opposite sides are identified with respect to orientation.

Figure 2.3: Construction of cylinder through identification.

Figure 2.4: Construction of torus through identification.

Figure 2.5: Construction of Möbius strip through identification.

Figure 2.6: Construction of Klein bottle through identification.

Using a rectangular rubber sheet we can construct physical models of these. However, the model of the Klein bottle can not be glued together unless the rubber sheet intersects itself, like this:

Figure 2.7: Model of Klein bottle.

This means that the 2-dimensional manifold cannot be imbedded (see below) in R3 but requires R4.

Manifolds with Lorentz metric

Construction through identification

Figure 2.8: Space-time with timelike curves.

Cutting out compact sets

The resulting manifold is non-compact and, thus, admits a Lorentz metric.

Embeddings

An embedding is a homeomorphism onto in the induced topology.16 In the first three examples above the physical models can be regarded as models of the image of an embedding in The Klein bottle, however, requires more dimensions and can be embedded in R4.

Not all mappings are homeomorphisms. Below is shown an example. The curve together with a curve segment γ and the ray is the image of a one-one mapping of the straight line R with the usual topology but it is not an embedding. The obstruction is the neighbourhoods of points on These are open sets in the induced topology but since they consist of infinitely many curve segments they are not connected, and thus they are not homeomorphic to open sets in R2.

Figure 2.9: An immersion which is not an embedding.

Hypersurfaces

Let M be a manifold with Lorentz metric and dimension n and let S be a manifold with dimension n − 1.

Classification

A timelike hypersurface is a manifold with Lorentz metric while a spacelike one has positive definitive metric (induced from M), i e all curves in the hypersurface are spacelike.17

2.2 General relativity

2.2.1 Space-time manifolds

Physical space-time is represented by a four-dimensional C ∞ manifold with Lorentz metric, a so called space-time manifold.

In general relativity matter refers to everything except gravitation. Matter fields are represented by tensor fields, described by tensor equations, on M. Matter is considered to generate gravitation and the tensors describing matter are related to gravitation by Einstein’s field equation. The matter field equations are found by assuming local validity of special relativity according to the principle of equivalence. In the language of differential geometry a more precise condition is that special relativity holds at each point p in the tangent space Tp(M) of the space-time manifold M.

The experimentally known interaction of light and gravitation implies non-vanishing space-time curvature in the presence of gravitation. As the simplest possibility, in general relativity it is assumed that gravitation is completely described by the Lorentz metric g.

2.2.2 Local causality

If is a convex subset of the space-time manifold M and then a signal can be sent in from p to q if and only if and can be joined by p q a non-spacelike curve belonging entirely to This identifies gphysically.

2.2.3 Energy-momentum tensor

The matter fields are assumed such that there exists a symmetric tensor T such that

T can be constructed a from a lagrangian, according to

2.2.4 The field equations

The metric g represents a physical field, describing gravitation and at the same time the geometry of space-time. The field equations for g are

where G is Newton’s gravitational constant. The value of the constant on the right hand side is determined by the requirement that Newton’s theory should follow in a first approximation, i e for weak, slowly varying gravitational fields. In empty space (no matter, purely gravitational field) the equations reduce to

which are sufficient for calculating the gravitational field of the sun in studies of the motion of the planets and black holes.

The 10 field equations are not completely independent due to general covariance, and this is expressed in

following from Bianchi’s identities. Unique solutions requires coordinate conditions such as

Such conditions are, of course, non-covariant. Also, the condition implies local conservation of the energy-momentum tensor, as mentioned above.

2.3 The Cauchy problem

if we restrict ourselves to empty space. Firstly, Bianchi’s identities imply

Secondly, the components Rαβ of Ricci’s tensor can be written

Finally, we remark that this does not work if so S must not be a null hypersurface. The field equations are hyperbolic.

An advanced text on The Cauchy problem in general relativity can be found in [81] (ch 7); see also [169], p 163 f.

2.4 Symmetric spaces

2.4.1 Killing vectors

Let M be a manifold with metric g and let {φt} be a one-parameter group of one-one mappings of M onto itself generated by a vector field K, i e

or, in component form,

Here φ∗ is the transformation of vectors induced by φ. Expansion of (φt∗g)ab in powers of t gives

Thus, if {φt} is a group of isometries then K satisfies Killing’s equation

Conversely, if then the φt∗ form a group of isometries.

Thus, there exists a group of isometries if and only if there exists a Killing vector, i e a vector field satisfying Killing’s equation.18

2.4.2 Coordinate condition

There exists a Killing vector if and only if there is a coordinate system (x1, x2, . . . ,xn) such that

A common example is stationary space-time. This can be defined as a space-time that admits a timelike vector ∂ /∂ t. Equivalently, e the metric tensor is independent of a timelike coordinate. By a timelike coordinate is meant a coordinate t such that gtt < 0.19

2.4.3 Maximal symmetry

The Killing vectors Kn are said to be independent if

where the cn are constants. The maximal number of independent Killing vectors in an n−dimensional manifold M is n(n + 1)/2. M is then said to be maximally symmetric.

Example 1: Minkowski space

Example 2: Schwarzschild space-time MS which in the commonly used coordinates (r, θ, φ, t) has the metric

The manifold MS is then called spherically symmetric. In fact, the S(r, t) are spheres. Obviously MS is also stationary. Since the it is also called static. The covariant components Ka of the three vectors are found, and then Killing’s equation is verified20.

It should be remarked that every sperically symmetric space-time where Einsteins vacuum field equations are satisfied is static and by a suitable choice of coordinates the metric takes the Schwarzschild form (Birkhoff’s theorem).

For more on symmetric spaces, see chapter 13.

2.5 Spin coefficient formalism

This section is just intended to present the concepts. For those who want to study the subject closer, [12] is recommended.

This gives an alternative description of space-time geometry and gravitation, mathematically equivalent to the description in terms of a Lorentz metric. Identities involving so called spin coefficients, vector and curvature are found. Restritions of these identities give field equations.

First let V, W be vectors in a space-time manifold (M, g), i e a 4-dimensional manifold with Lorentz metric. We will use the ”scalar product” and also type (2, 0) tensors

2.5.1 Tetrad and spin coefficients

We assume that there exists a tetrad, i e a system of four vector fields ( and being complex-valued vectors, and conjugates of each other) such that

so, in particular, all four are null vectors. This has made them interesting in the study of gravitational radiation.

The tensor products form a basis for tensors of type (2, 0). In particular, the (contravariant) metric tensor can be expanded,

The coefficients can be found, and in component form it is found that

Further, setting

every tensor can be represented by tetrad components:

These are invariant with respect to coordinate transformations, but depend on the tetrad.

Weyl’s tensor

gives 5 complex-valued tetrad components ΨA,

Ricci’s tensor

is replaced by 9 real tensor fields and one scalar Λ, replacing the Ricci scalar,

Complex rotation coefficients:

Spin coefficients

are the following 12 complex-valued functions.

These are coordinate independent but depend on the choice of tetrad; cf

2.5.2 Identities

From the commutators for second order covariant derivatives and the Bianchi identities a number of identities for the spin coefficients can be derived. One example is

where we have introduced two differential operators,

2.5.3 Free field equations

are null geodesics, each lying in a hypersurface u =constant. and r is an affine parameter,

The contravariant components of the metric tensor in these coordinates are

Then, from

Next, by

Further, m and n are required to be parallel fields along l so that

Then, since l is a gradient

The field equations

In a free field (pure gravitational field) the Ricci tensor vanishes so that and thus

and similarly for

Here, in the first term,

where we make the replacement

and use the definitions of the spin coefficients. The final result is

What is the use of all this?

Secondly, the equations are differential equations of just first order.

Thirdly, the spin coefficients have a physical meaning and the field equations in some cases give a direct physical consequence. For example, the equation derived above shows that the appearance of Ψ00 causes a congruence of null geodesics to focus astigmatically.

Fourthly, exakt solutions can be found by starting out with conditions on some spin coefficients. These conditions can be chosen so as to, more or less intuitively, represent some physical application. Since the spin coefficients are related to null geodesics a near at hand thought is to find exact solutions that may represent gravitational waves. An example follows in the next section.22

The complete set of field equations can be found in [12], appendix D.

2.5.4 Robinson-Trautman solutions

The conditions for this case are

The field equations lead to

Since the matrix (gab) is the inverse of (gab) the covariant components of the metric tensor can be obtained.

For more on the spin coefficient formalism, see [12] or [157].

2.6 Incompleteness, extensions, and elementary singularities

2.6.1 Incompleteness, extension

Consider Schwarzschild’s metric,

two coordinate transformation are defined for r > 2m but the resulting metrics are defined for all r > 0. The resulting Finkelstein metrics are

We have found two different extensions of Schwarzschild’s space-time. These are subsets of a larger space-time, Kruskal’s space-time which consttutes the maximal extension. The metric of this can be written

in certain coordinates, (X, Y, θ,φ). Of these X is space-like and Y time-like.24 The coordinates (θ,φ) parametrize each sphere {X, Y }πr2. r is not a coordinate but a function of {X, Y } implicitly defined25 by the equation

Figure 2.10: Kruskal diagram.

Null geodesics and light cones are oriented as in Minkowski space (flat space-time)!

Remark 1. When matter is present in a spherically symmetric configuration the space-time outside matter is described by Kruskal’s metric but the manifold must be truncated, and the gravitational field inside the matter has another form.

Figure 2.11: Part of Kruskal’s space-time illustrating gravitational collapse.

Remark 2. Any locally given metric defines a maximal extension, uniquely except for topology.

Remark 3. Kruskal’s metric can be transformed into a metric where all coordinates are bounded. The correponding figure is called a Penrose diagram. See figure below where the notation means:

i+ – future timelike infinity

i− – past timelike infinity

i0 – spacelike infinity

I+ – future null infinity

I− – past null infinity

Figure 2.12: Penrose diagram

The light cones are oriented like in the Kruskal diagram. Similarly for other metrics. For more on this, see [81], p →. A good text on this is also [245]. Finkelstein’s and Kruskal’s metrics will appear at several places in this book.

2.6.2 Elementary singularities

If some component ΨA of the Weyl tensor goes to infinity when calculated along a geodesic when it is calculated in a parallely propagated tetrad then the space-time is said to have an elementary singularity.

This is not an exhaustive definition of space-time singularity which is rather involved; see eg [81], chapter 8. Our definition gives conditions that intuitively seem sufficient to speak of a singularity. Also, since space-time breaks down ”at” a singularity we cannot say that there is a singularity at some point. The proper concept is that of a singular space-time which is inextensible and, yet contains curves that are finite in some appropriate sense.

Example 2. If some curvature invariant goes to infinity at finite affine distance the space-time is singular. These invariants are defined

and

The high symmetry of most known solutions to the field equations is not essential for the occurence of singularities. For example a slightly non-spherically symmetric gravitational collapse still produces a singularity. But there is much more to this as will be considered in the next section and in chapter 24.

2.7 Singular space-times

2.7.1 The concept of singularity

The requirement that a space-time manifold be maximally extended means that no points should have been cut out. Incompletenes of some curve at finite distance then indicates a singularity.

First we will define what is to be meant by ”finite distance”. Also, not just any curve should considered. That a curve is ”incomplete” indicates intuitively that there is a (non-removable) ”hole” in space-time.

Generalized affine parameter

As a suitable interpretation of the concept of ”distance” along a curve we will define a so called generalized affine parameter u on the curve ([81], p →).

If, in particular, the curve is a geodesic the parameter u becomes an affine parameter.

Incomplete curves

A point p ∈ M is said to be an endpoint of the curve λ if λ enters and remains within any neighbourhood of p.

A curve in M which is finite as measured by an affine parameter but is lacking an end point in M is said to be incomplete.

If there is an incomplete curve the manifold is said to be singular.

As mentioned the lacking endpoint can be associated with a curvature singularity going to infinity. This is a common situation in applications. However, there are other possibilities. A cone with the top point excluded is singular. Another example is the ”strut” in Weyl metrics. The latter is treated in a section 17.2.

2.7.2 Background material for a singularity theorem

Terminology

We begin with some terminology.

Mtime-oriented: The non-spacelike vectors can be continuously divided into two classes, labelled future-directed and past directed respectively.

Cauchy surface in M: A spacelike hypersurface in M which every non-spacelike curve intersects at exactly one point.

Energy conditions: There are several. We will use for every timelike vector K.

Expansion of a geodesic congruence: is the unit future-directed tangent vector field to the congruence (t affine parameter).

The energy condition

Firstly, from the energy condition we will find a condition for the energy-momentum tensor. To this end the field equations are written,

From this,

so, from the energy condition

In coordinates, geodesic at p ∈ M and with we get

From this we see that the energy condition implies

and conversely.

Raychaudhuri’s equation

The behaviour of θ and C is governed by Raychaudhuri’s equation:

Proof:

2.7.3 The singularity theorem

Theorem: A space-time M is singular if

M

contains a Cauchy surface

H

,

M

is time-oriented,

for all timelike vectors

K

, and

the timelike geodesic congruence orthogonal to converges in one direction on .

Proof in outline:

9 The present chapter consists, firstly, of an overview of differential geometry used in the mathematical formulation of General Relativity in a fairly modern setting. More on this is presented in Hawking-Ellis’ classical text [81