2

ELEMENTS OF GENERAL RELATIVITY

It is not the purpose of this chapter to introduce or explain Einstein’sgeneral theory of relativity, since the reader who is not already familiarwith it is unlikely to gain much from this book. The main purpose here issimply to clarify the notation that will be used. It is also appropriate inthis chapter to briefly introduce the Newman–Penrose formalism whichfacilitates the geometrical analysis of the colliding plane wave problemand which will be used in Chapter 6 to derive the field equations.

2.1 Basic notation

Basically, we will be following a very traditional approach, and the nota-tion adopted will be that of the well known paper of Newman and Penrose(1962).

Accordingly, a space-time will be represented by a connected C∞

Hausdorff manifold M together with a locally Lorentz metric gµν withsignature (+,−,−,−) and a symmetric linear connection Γλ

µν . Greekindices are used to indicate the values 0,1,2,3, and the covariant derivativeof a vector is given by

Aλ;ν = Aλ

,ν + ΓλµνAµ (2.1)

where a comma denotes a partial derivative.The curvature tensor is given in terms of the connection by

Rλκµν = Γλ

κν,µ − Γλκµ,ν + Γλ

αµΓακν − Γλ

ανΓακµ. (2.2)

The Ricci tensor, which is the first contraction of the curvature tensor, isgiven by

Rµν = −Rαµαν . (2.3)

The curvature tensor has twenty independent components. These can beconsidered as the ten independent components of the Ricci tensor, andthe ten independent components of the Weyl tensor, which is the tracefree part of the curvature tensor, and is given by

Cκλµν = Rκλµν − 12 (Rλµgκν − Rλνgκµ − Rκµgλν + Rκνgλµ)

+ 16R(gλµgκν − gλνgκµ)

(2.4)

4

2.2 Components of the curvature tensor 5

where R = R αα is the curvature scalar.

These two groups of components have different physical interpreta-tions. The components of the Ricci tensor are related to the energy-momentum tensor Tµν of the matter field present, through Einstein’sequation

Rµν − 12gµνR = −8πTµν . (2.5)

These components can be considered to define the amount of curvaturethat is directly generated by the matter fields that are present at anylocation. For a vacuum field they will be zero, but they will be non-zerowhen electromagnetic waves or other fields are present.

The components of the Weyl tensor, on the other hand, define the‘free gravitational field’. They may be considered as describing the com-ponents of curvature that are not generated locally. In this sense theydescribe the pure gravitational field components. They may be inter-preted as the components of gravitational waves, or of gravitational fieldsgenerated by non-local sources.

2.2 Components of the curvature tensor

It is convenient to represent the curvature tensor in terms of distinctsets of components. Not only may it be divided into the Weyl and Riccitensors, but each of these tensors may be described in terms of distinctcomponents. The appropriate notation here is that of Newman and Pen-rose (1962).

It is found to be convenient to introduce a tetrad system of nullvectors. These include two real null vectors lµ and nµ, a complex nullvector mµ, and its conjugate. They are defined such that their only non-zero inner products are

lµnµ = 1, mµmµ = −1, (2.6)

and they must satisfy the completeness relation

gµν = lµnν + nµlν − mµmν − mµmν . (2.7)

Having defined a tetrad basis, the Ricci and Weyl tensors may now beexpressed in terms of their tetrad components. The ten independent com-ponents of the Ricci tensor can conveniently be divided into a componentΛ representing the curvature scalar and the nine independent componentsof a Hermitian 3 × 3 matrix ΦAB which represents the trace free part ofthe Ricci tensor and satisfies

ΦAB = ΦBA (2.8)

6 Elements of general relativity

where A, B = 0, 1, 2. These components are defined by

Φ00 = − 12Rµν lµlν

Φ01 = − 12Rµν lµmν

Φ02 = − 12Rµνmµmν

Φ11 = − 14Rµν(lµnν + mµmν)

Φ12 = − 12Rµνnµmν

Φ22 = − 12Rµνnµnν

Λ = 124R.

(2.9)

The ten independent components of the Weyl tensor, representingthe free gravitational field, can more conveniently be expressed as the fivecomplex scalars

Ψ0 = −Cκλµν lκmλlµmν

Ψ1 = −Cκλµν lκnλlµmν

Ψ2 = −Cκλµν lκmλmµnν

Ψ3 = −Cκλµνnκlλnµmν

Ψ4 = −Cκλµνnκmλnµmν .

(2.10)

These components have distinct physical interpretations that will be men-tioned below. They also have particular convenience when considering thealgebraic classification of the space-time.

Gravitational fields are usually classified according to the Petrov–Penrose classification of the Weyl tensor. This is based on the numberof its distinct principal null directions and the number of times these arerepeated. This classification is most conveniently described using a spinorapproach. However, there is no need to introduce spinors here, as tetradsare sufficient.

According to the tetrad approach, a null vector kµ is said to describea principal null direction of the gravitational field with multiplicity 1, 2,3 or 4 if it satisfies respectively

k[ρCκ]λµ[νkσ]kλkµ = 0

Cκλµ[νkσ]kλkµ = 0

Cκλµ[νkσ]kµ = 0

Cκλµνkµ = 0

(2.11)

where square brackets are used to denote the antisymmetric part. Thereare at most four principal null directions.

2.3 Spin coefficients 7

If all four principal null directions are distinct, the space-time is saidto be algebraically general, or of type I. If there is a repeated principalnull direction, then the space-time is said to be algebraically special. If ithas multiplicity two, three or four, the space-time is said to be of types II,III or N respectively. If a space-time has two distinct repeated principalnull directions, it is said to be of type D. If the Weyl tensor is zero, thespace-time is conformally flat or of type O.

If either of the basis vectors lµ or nµ are aligned with principal nulldirections then either Ψ0 = 0 or Ψ4 = 0 respectively. If the vector lµ

is aligned with the repeated principal null direction of an algebraicallyspecial space-time, then Ψ0 = Ψ1 = 0. If this principal null directionis repeated two, three or four times, then the only non-zero componentsof the Weyl tensor are the sets (Ψ2,Ψ3,Ψ4), (Ψ3, Ψ4) or Ψ4 respectively.Finally, if lµ and nµ are both aligned with the distinct principal nulldirections of a type D space-time, then the only non-zero component ofthe Weyl tensor is Ψ2.

The physical meaning of the different components of the Weyl tensorhas been investigated by Szekeres (1965), and may be summarized asfollows:

Ψ0 denotes a transverse wave component in the nµ direction.Ψ1 denotes a longitudinal wave component in the nµ direction.Ψ2 denotes a coulomb component.Ψ3 denotes a longitudinal wave component in the lµ direction.Ψ4 denotes a transverse wave component in the lµ direction.

This interpretation will be very useful when we come to analyse the in-teraction between two gravitational waves. It is possible to align the twobasis vectors lµ and nµ with the two waves. The interaction to be con-sidered here is between transverse waves, so the problem is to find theinteraction between the Ψ4 and Ψ0 components.

2.3 Spin coefficients

In some situations it has been found convenient to modify the notationfor tetrads, spinors and spin coefficients in recent years.1 However, it ismost convenient here to continue to use the original notation of Newmanand Penrose (1962).2 Those not familiar with this notation need not be

1 See Geroch, Held and Penrose (1973), Penrose and Rindler (1985).2 For a detailed introduction to the Newman–Penrose formalism see Pi-

rani (1965), Carmeli (1977), Alekseev and Klebnikov (1978), Frolov (1979) and

Kramer et al. (1980).

8 Elements of general relativity

too alarmed, as we have no need here to introduce spinors or their moreintricate properties. Effectively, we are only introducing the notationbecause of its convenience for describing certain geometrical properties ofthe solutions.

The main feature of the Newman–Penrose formalism is the intro-duction of spin coefficients. These are complex linear combinations ofthe Ricci rotation coefficients associated with the null tetrad. They aredefined by

κ = lµ;νmµlν ν = −nµ;νmµnν

ρ = lµ;νmµmν µ = −nµ;νmµmν

σ = lµ;νmµmν λ = −nµ;νmµmν

τ = lµ;νmµnν π = −nµ;νmµlν

ε = 12 (lµ;νnµlν − mµ;νmµlν)

α = 12 (lµ;νnµmν − mµ;νmµmν)

β = 12 (lµ;νnµmν − mµ;νmµmν)

γ = 12 (lµ;νnµnν − mµ;νmµnν).

(2.12)

The spin coefficients have the following geometrical interpretations.If κ = 0, then lµ is tangent to a geodesic null congruence. If, in addi-tion, Re ε = 0, then lµ is the tangent vector corresponding to an affineparametrization, and −Re ρ, Im ρ and |σ| define the expansion, twistand shear of the congruence respectively. Also, arg σ determines the shearaxes. With a change of sign, a positive value for Re ρ is more appropri-ately referred to as the contraction of the congruence.

It may also be noted that lµ is propertional to the gradient of a scalarfield if, and only if, it is tangent to a twist-free null geodesic congruence(κ = 0, ρ− ρ = 0). Also, lµ is equal to the gradient of a scalar field whenκ = 0, ρ − ρ = 0, ε + ε = 0 and α + β = τ .

In the congruence defined by nµ the coefficients −ν, −γ, −µ, −λcorrespond to κ, ε, ρ, σ respectively. The geometrical properties of thesolutions given later can be very conveniently analysed in terms of thesespin coefficients.

We also need to define the intrinsic derivatives. These are directionalderivatives in the directions of the four tetrad vectors, and are defined by

D = lµ∇µ, ∆ = nµ∇µ, δ = mµ∇µ, δ = mµ∇µ (2.13)

where ∇µ is the covariant derivative operator, previously denoted by asemicolon.

2.3 Spin coefficients 9

Basic to their formalism is the set of Newman–Penrose identities.The first group of these are linear combinations of the Ricci identitiesapplied to the tetrad vectors. These will only be referred to occasionallyin the following sections, but it is still worth quoting them here as follows:

Dρ − δκ = ρ2 + σσ + ρ(ε + ε) − κτ − κ(3α + β − π) + Φ00 (2.14a)Dσ − δκ = σ(ρ + ρ) + σ(3ε − ε) − κ(π − π + α + 3β) + Ψ0 (2.14b)Dτ − ∆κ = ρ(τ + π) + σ(τ + π) + τ(ε − ε) − κ(3γ + γ)

+ Ψ1 + Φ01 (2.14c)Dα − δε = α(ρ + ε − 2ε) + βσ − βε − κλ − κγ + π(ρ + ε) + Φ10 (2.14d)Dβ − δε = σ(α + π) + β(ρ − ε) − κ(µ + γ) − ε(α − π) + Ψ1 (2.14e)Dγ − ∆ε = α(τ + π) + β(τ + π) − γ(ε + ε) − ε(γ + γ) + τπ − νκ

+ Ψ2 − Λ + Φ11 (2.14f)Dλ − δπ = ρλ + σµ + π2 + π(α − β) − νκ − λ(3ε − ε) + Φ20 (2.14g)Dµ − δπ = ρµ + σλ + ππ − µ(ε + ε) − π(α − β) − νκ + Ψ2 + 2Λ (2.14h)Dν − ∆π = µ(π + τ) + λ(π + τ) + π(γ − γ) − ν(3ε + ε)

+ Ψ3 + Φ21 (2.14i)∆λ − δν = −λ(µ + µ) − λ(3γ − γ) + ν(3α + β + π − τ) − Ψ4 (2.14j)δρ − δσ = ρ(α + β) − σ(3α − β) + τ(ρ − ρ) + κ(µ − µ)

− Ψ1 + Φ01 (2.14k)δα − δβ = ρµ − σλ + αα + ββ − 2αβ + γ(ρ − ρ) + ε(µ − µ)

− Ψ2 + Λ + Φ11 (2.14l)δλ − δµ = ν(ρ − ρ) + π(µ − µ) + µ(α + β) + λ(α − 3β)

− Ψ3 + Φ21 (2.14m)δν − ∆µ = µ2 + λλ + µ(γ + γ) − νπ + ν(τ − 3β − α) + Φ22 (2.14n)δγ − ∆β = τ(µ + γ) − γα − σν − εν − β(2γ − γ − µ) + αλ + Φ12 (2.14o)δτ − ∆σ = µσ + ρλ + τ(τ + β − α) − σ(3γ − γ) − κν + Φ02 (2.14p)∆ρ − δτ = −ρµ − σλ + τ(β − α − τ) + ρ(γ + γ) + κν − Ψ2 − 2Λ (2.14q)∆α − δγ = ν(ρ + ε) − λ(τ + β) + α(γ − µ) + γ(β − τ) − Ψ3. (2.14r)

We will also require the commutation relations between the intrinsicderivatives. When applied to scalar functions, these are given by

∆D − D∆ = (γ + γ)D + (ε + ε)∆ − (τ + π)δ − (τ + π)δδD − Dδ = (α + β − π)D + κ∆ − σδ − (ρ + ε − ε)δδ∆ − ∆δ = −νD + (τ − α − β)∆ + λδ + (µ − γ + γ)δδδ − δδ = (µ − µ)D + (ρ − ρ)∆ + (β − α)δ + (α − β)δ.

(2.15)

10 Elements of general relativity

These can be applied to the coordinates to give the so-called metricequations, and also to the spin coefficients and the curvature tensorcomponents.

An essential part of the general Newman–Penrose formalism is theset of Bianchi identies. These, however, will not be used explicitly in thistext and therefore do not need to be repeated here.

2.4 Einstein–Maxwell fields

In this formalism, Einstein’s field equations are applied simply by replac-ing the expressions for the Ricci tensor components in the above identitiesby the appropriate components of the energy-momentum tensor accordingto equation (2.5). For an electromagnetic field in a vacuum, for example,it is convenient to represent the electromagnetic field tensor Fµν by threecomplex scalars defined by

Φ0 = Fµν lµmν

Φ1 = 12Fµν(lµnν + mµmν)

Φ2 = Fµνmµnν .

(2.16)

It is also possible to scale the electromagnetic field tensor such that Ein-stein’s gravitational field equations are then given by

ΦAB = ΦAΦB , Λ = 0, (2.17)

and, in this case, Maxwell’s equations take the form

DΦ1 − δΦ0 = (π − 2α)Φ0 + 2ρΦ1 − κΦ2

DΦ2 − δΦ1 = −λΦ0 + 2πΦ1 + (ρ − 2ε)Φ2

δΦ1 − ∆Φ0 = (µ − 2γ)Φ0 + 2τΦ1 − σΦ2

δΦ2 − ∆Φ1 = −νΦ0 + 2µΦ1 + (τ − 2β)Φ2

(2.18)

It is also possible to classify the electromagnetic field in a similar wayto the classification of the Weyl tensor. An electromagnetic field is saidto be non-null or null if it has two distinct or one repeated principal nulldirection kµ satisfying respectively

Fµ[νkλ]kµ = 0

Fµνkµ = 0, F[µνkλ] = 0.(2.19)

Aligning lµ with a principal null direction of the electromagneticfield makes Φ0 = 0. If lµ is a repeated principal null direction, thenΦ0 = Φ1 = 0, and the only non-zero component of the null field is Φ2.