Spin-coefficient formalism

Post-publication activity

The spin-coefficient formalism (SC formalism) (also known in the literature as Newman-Penrose formalism (NP formalism) ) is a commonly used technique based on the use of null tetrads, with ideas taken from 2-component spinors, for the detailed treatment of 4-dimensional space-times satisfying the equations of Einstein's theory of general relativity.

Introduction

Since the inception, in 1915, of Einstein's equations for general relativity, there has been a variety of different (physically and mathematically equivalent) ways of writing them. They include the standard coordinate-basis version using the metric tensor components as the basic variable and the Christoffel symbols for the connection, the methods of Cartan using differential forms (Lovelock and Rund, 1975), the space-time (orthonormal) tetrad version of Ricci (Levy, 1925) and the spin-coefficient (SC) (Newman-Penrose) version (Newman and Penrose, 1962; Geroch et al., 1973; Penrose, 1968; Penrose and Rindler, 1984; Penrose and Rindler, 1986; Newman and Tod, 1980; Newman and Unti, 1962). (An unheralded preliminary move in this direction was given earlier by Jordan, Ehlers and Sachs (Jordan et al., 1961).) The last version is really a variation on Ricci's method where the orthonormal tetrad is replaced by null vectors or two-component spinors. Though all versions have significant domains of useful applicability, it appears that the SC version is particularly valuable in many situations that are important for general relativity, such as in the study of gravitational radiation and for the search for exact solutions. A stream-lined version of SC known as the GHP formalism will not be discussed here (Geroch et al., 1973; Penrose and Rindler, 1984). The integrability structure of the SC equations was studied by a variety of authors (S.B. Edgar, 1979).

The reason for this usefulness relies on several related facts: 1) all the equations are first order - and often they can be partially grouped together into sets of linear equations (Newman and Penrose, 1962; Newman and Unti, 1962), 2) all the equations are complex, and are thereby reducing the total number by half, 3) allows this reduced number of equations to be written out explicitly without the use of the index and summation conventions, 4) this in turn allows one to concentrate on individual 'scalar' equations with particular physical or geometric significance and then to see a natural hierarchical structure in the set of Einstein equations, 5) often it allows one to search for solutions with specific special features, such as the presence of one or two null directions that might be singled out by physical or geometric considerations.

The basic ingredients on the SC equations are the null tetrad (\(l^{a},n^{a},m^{a},\overline{m}^{a}\)) (with \(m\) and \(\overline{m}\) complex conjugates)

\[\tag{1} l^{a}l_{a} =n^{a}n_{a}=m^{a}m_{a}=\overline{m}^{a}\overline{m}_{a}=l^{a}m_{a}=n^{a}m_{a}=l^{a}\overline{m}_{a}=n^{a}\overline{m}_{a}=0,\ l^{a}n_{a} =-m^{a}\overline{m}_{a}=1. \]

Equivalently, we can think of the null tetrad as being determined by a spin-frame \(\ (o^{A},\iota ^{A})\) according to the following simple relations (Newman and Penrose, 1962; Penrose and Rindler, 1984)

\[\tag{2} l^{a}=o^{A}\overline{o}^{A^{\prime }},~\ m^{a}=o^{A}\overline{\iota } ^{A^{\prime }},~~~\overline{m}^{a}=\iota ^{A}\overline{o}^{A^{\prime }}, \ \ n^{a}=\iota ^{A}\overline{\iota }^{A^{\prime }}. \]

Here the 'abstract index notation' is being adopted, according to which an individual tensor (lower case) index may be re-interpreted as the corresponding pair of (capital) spinor indices, one unprimed and the other primed (e.g., \(a=AA^{\prime }\)), the primed spin-space being the complex conjugate of the unprimed (2-complex dimensional) spin-space. For the details of this procedure, see (Penrose 1967). Spinor indices are raised and lowered by means of skew-symmetric quantities \(\varepsilon ^{AB}\) and \( \varepsilon _{AB}\) (inverses of each other in the sense of \(\varepsilon ^{AC}\varepsilon _{BC}=\delta _{B}^{A}\)), according to

\[\tag{3} \xi _{B}=\xi ^{A}\varepsilon _{AB},\qquad \xi ^{A}=\varepsilon ^{AB}\xi _{B},\quad etc. \]

The quantities \(o^{A}\) and \(\iota ^{A}\) are normalized against each other, \( o_{A}\iota ^{A}=1\ ,\) (whence \(\iota _{A}o^{A}=-1)\) and we necessarily have \( o_{A}o^{A}=0=\iota _{A}\iota ^{A};\) accordingly the dual basis to \( (o^{A},\iota ^{A})\) is \((-\iota _{A},o_{A}).\) We have

\[\tag{4} \varepsilon _{AB}=o_{A}\iota _{B}-\iota _{A}o_{B}, \quad \mathrm{and} \quad \varepsilon ^{AB}=o^{A}\iota ^{B}-\iota ^{A}o^{B}. \]

In this notation the metric tensor takes the form

\[\tag{5} \begin{align} g_{ab} &=g_{AA'BB'} =\varepsilon _{AB}\varepsilon_{A'B'} =(o_{A}\iota _{B}-\iota _{A}o_{B}) (\overline{o}_{A'}\overline{\iota }_{B'}-\overline{\iota }_{A'}\overline{o}_{B'} )\\ &=o_{A}\overline{o}_{A'}\iota _{B}\overline{\iota }_{B'}+\iota _{A}\overline{\iota }_{A'} o_{B}\overline{o}_{B'} -o_{A}\overline{\iota }_{A'}\iota _{B}\overline{o}_{B'}-\iota _{A}\overline{o}_{A'}o_{B}\overline{\iota }_{B'}\\ &=l_{a}n_{b}+n_{a}l_{b}-m_{a}\overline{m}_{b}-\overline{m}_{a}m_b, \end{align} \]

the final line providing the expression for the metric in terms of the null tetrad.

The essential space-time structures that are used in the SC formalism are derived, in conjunction with the null tetrad or spin-frame, by use of the (unique Levi-Civita/Christoffel) torsion-free covariant derivative operator \( \nabla _{a}=\nabla _{AA'} \) which annihilates \(g_{ab}\) (and also \( \varepsilon _{AB} \) and \(\varepsilon _{A'B'}\)). These are, first, the twelve spin-coefficients which are the spin-frame components of the covariant derivatives of the spin-frame elements and can be re-interpreted as certain complex combination of tetrad components of the covariant derivatives of the tetrad vectors and second, the five spin-frame components of the Weyl 'gravitational spinor', \( \Psi _{ABCD} \) (see Eqs. (13)-(17)), which can be re-interpreted as complex tetrad components of the self-dual (anti-self-dual) Weyl tensor. The tetrad components of the stress tensor replace, via the Einstein equations, the Ricci tensor components of the rest of curvature tensor.

The present work will be confined to the vacuum Einstein equations, i.e., a vanishing stress tensor/Ricci tensor. However, the formalism has been extended to the Einstein-Maxwell (Newman and Penrose, 1962; Penrose and Rindler, 1984), Einstein-Yang-Mills and matter cases.

The standard version of the Einstein equations (the vanishing of the Einstein tensor) becomes, in this formalism, a very large number of complex first order differential equations which are naturally grouped into three different (interacting) sets: The metric equations, the spin-coefficient equations and the Bianchi identities. In essence, the metric equations relate the spin-coefficients to the derivatives of the tetrad components, the spin-coefficient equations describe the relationship of the curvature tensor to derivatives of the connection (the spin-coefficients), and the Bianchi identities being the tetrad version of the ordinary Bianchi identities. A novelty here is that these equations are integrated not one set at a time, but together, i.e., by going back and forth between the sets.

Notation

Some notation is needed before the spin-coefficient equations can be displayed.

Lower-case italic Latin letters from the beginning of the alphabet, \(a,b,c,\dots,h\ ,\) for tetrad indices, with the corresponding capital letters, \(A, A', B, B', C, C', \ldots, H, H'\) for spinor indices, where these can be taken as abstract indices (Penrose, 1968; Penrose and Rindler, 1984) and treated completely formally. When it becomes necessary, in later sections, to introduce specific coordinates and frames, the bold-face upright versions of these indices will be used \( \mathbf{a}, \mathbf{b}, \mathbf{c},\cdots, \mathbf{h}\ .\) We also need explicit spin-frame components, \(\mathbf{A}, \mathbf{A'}, \mathbf{B}, \mathbf{B'}, \mathbf{C}, \mathbf{C'},\cdots, \mathbf{H}, \mathbf{H}\ .\) These indices are necessarily numerical (where in the case of spin-frame indices, the numbers \(0\) and \(1\ ,\) or \(0'\) and \(1'\ ,\) will be used). Each index letter \( i, j, k,\cdots\) from the latter part of the alphabet, stands simply for one of the numbers \(1, 2, 3, 4,\) these integers respectively labelling one of the tetrad vectors \((l^{a},n^{a},m^{a},\overline{m}^{a}).\) These vectors are collectively denoted by \(\lambda _{i}^{a},\) with \(i= 1, 2, 3, 4\ :\)

\[\tag{6} \lambda _{i}^{a}=(l^{a},n^{a},m^{a},\overline{m}^{a}), \]

and the directional derivatives by

\[\tag{7} \nabla _{i}\equiv \lambda _{i}^{b}\nabla _{b}=(D,\Delta ,\delta ,\bar{\delta}). \]

The tetrad components of the covariant derivatives of tetrad vectors (Ricci rotation coefficients) are defined by:

\[\tag{8} \gamma^i{}_{jk} =\lambda _{j}^{b}\lambda _{k}^{a}\nabla_{a}\lambda _{b}^{i}, \]

\[\tag{9} \gamma _{ijk} =-\gamma _{jik}=\eta _{il}\gamma^l{}_{jk} \]

with

\[\tag{10} \eta _{ij}=\left[\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1& 0 \end{array}\right]. \]

Note that some \(\gamma^i{}_{jk}\) are real, some pure imaginary and some complex.

Finally the spin coefficients are defined by complex combinations of the \(\gamma_{ijk}\)'s as

\[\tag{11} \begin{array}{lll} \pi =-\gamma _{\,241}, & \epsilon =\frac{1}{2}(\gamma _{\,121}-\gamma _{\,341}), & \kappa =\gamma _{\,131} \\ \lambda =-\gamma _{\,244}, & \alpha =\frac{1}{2}(\gamma _{\,124}-\gamma _{\,344}), & \rho =\gamma _{\,134} \\ \mu =-\gamma _{\,243}, & \beta =\frac{1}{2}(\gamma _{\,123}-\gamma _{\,343}), & \sigma =\gamma _{\,133} \\ \nu =-\gamma _{\,242}, & \gamma =\frac{1}{2}(\gamma _{\,122}-\gamma _{\,342}), & \tau =\gamma _{\,132} \end{array} \]

The origin of these particular combinations comes from their expressions in terms of spinor components, with respect to the spin-frame (\(o^{A},\iota ^{A}\)), Writing \(\varepsilon_{\mathbf{B}}{}^{A}\) to mean

\[ o^{A}=\varepsilon_{\mathbf{0}}{}^{A}, \quad \iota^{A}=\varepsilon _{\mathbf{1}}{}^{A} \] and \( o_{A}=-\varepsilon_{A\mathbf{0}},\quad \iota_{A}=-\varepsilon _{A\mathbf{1}} \)

the four-indexed quantity \(\gamma _{\mathbf{AA'BC}}\) (with symmetry \( \gamma _{\mathbf{AA'BC}}=\gamma _{\mathbf{AA'CB}} \)), given by

\[ \gamma_{\mathbf{AA'BC}}=\varepsilon_{A\mathbf{B}}\nabla_{\mathbf{AA'}}\varepsilon _{\mathbf{C}}{}^{A} \]

are, in fact, the spin-coefficients defined directly in spin-frame terms. Explicitly, we have

\[\tag{12} \begin{array}{lll} \pi =\gamma _{\mathbf{00}^{\prime }\mathbf{11}}=\iota ^{A}\nabla _{\mathbf{00 }^{\prime }}\iota _{A}, & \epsilon =\gamma _{\mathbf{00}^{\prime }\mathbf{10} }=\iota ^{A}\nabla _{\mathbf{00}^{\prime }}o_{A}, & \kappa =\gamma _{\mathbf{ 00}^{\prime }\mathbf{00}}=o^{A}\nabla _{\mathbf{00}^{\prime }}o_{A}, \\ \lambda =\gamma _{\mathbf{10}^{\prime }\mathbf{11}}=\iota ^{A}\nabla _{ \mathbf{10}^{\prime }}\iota _{A}, & \alpha =\gamma _{\mathbf{10}^{\prime } \mathbf{10}}=\iota ^{A}\nabla _{\mathbf{10}^{\prime }}o_{A}, & \rho =\gamma _{\mathbf{10}^{\prime }\mathbf{00}}=o^{A}\nabla _{\mathbf{10}^{\prime }}o_{A},\\ \mu =\gamma _{\mathbf{01}^{\prime }\mathbf{11}}=\iota ^{A}\nabla _{\mathbf{01 }^{\prime }}\iota _{A}, & \beta =\gamma _{\mathbf{01}^{\prime }\mathbf{10} }=\iota ^{A}\nabla _{\mathbf{01}^{\prime }}o_{A}, & \sigma =\gamma _{\mathbf{ 01}^{\prime }\mathbf{00}}=o^{A}\nabla _{\mathbf{01}^{\prime }}o_{A}, \\ \nu =\gamma _{\mathbf{11}^{\prime }\mathbf{11}}=\iota ^{A}\nabla _{\mathbf{11 }^{\prime }}\iota _{A}, & \gamma =\gamma _{\mathbf{11}^{\prime }\mathbf{10} }=\iota ^{A}\nabla _{\mathbf{11}^{\prime }}o_{A}, & \tau =\gamma _{\mathbf{11 }^{\prime }\mathbf{00}}=o^{A}\nabla _{\mathbf{11}^{\prime }}o_{A}. \end{array} \]

The Ricci Identity, which establishes sign conventions, is given by

\[ \nabla _{\lbrack a}\nabla _{b]}\lambda _{c}^{i}=\frac{1}{2} R_{abc}{}^d\lambda _{d}^{i}. \]

(Although these conventions are opposite to those in (Penrose 1967), they are brought into line with them by the choice of minus sign in the following expressions, Eqs. (13)-(17).) These give the five complex self-dual components of the Weyl tensor

\[ C_{ab}{}^{cd}=R_{ab}{}^{cd}-2R_{[a}{}^{[c} g_{b]}{}^{d]}+\frac{1}{3} R g_{[a}{}^{c} g_{b]}{}^{d}. \]

The five complex self-dual components of the Weyl tensor are written as

\[\tag{13} \Psi _{0} =-C_{abcd}l^{a}m^{b}l^{c}m^{d}=-C_{1313}, \]

\[\tag{14} \Psi _{1} =-C_{abcd}l^{a}n^{b}l^{c}m^{d}=-C_{1213}, \]

\[\tag{15} \Psi _{2} =-C_{abcd}l^{a}m^{b}\overline{m}^{c}n^{d}=-C_{1342}, \]

\[\tag{16} \Psi _{3} =-C_{abcd}l^{a}n^{b}\overline{m}^{c}n^{d}=-C_{1242}, \]

\[\tag{17} \Psi _{4} =-C_{abcd}\overline{m}^{a}n^{b}\overline{m}^{c}n^{d}=-C_{4242}. \]

(Note that Eq. (15) is frequently written in the equivalent form

\[ \Psi _{2}=-\frac{1}{2}(C_{1212}-C_{1234}).) \]

In spinor terms, with

\[ -C_{abcd} = -C_{AA'BB'CC'DD'} = \Psi _{ABCD}\varepsilon _{A'B'} \varepsilon _{C'D'} + \overline{\Psi}_{A'B'C'D'} \varepsilon _{AB}\varepsilon _{CD}, \]

the gravitational spinor \(\Psi _{ABCD}\) being totally symmetric, we have the more systematic looking

\[ \Psi _{0} =\Psi _{0000}=\Psi _{ABCD}o^{A}o^{B}o^{C}o^{D}, \] \[ \Psi _{1} =\Psi _{0001}=\Psi _{ABCD}o^{A}o^{B}o^{C}\iota ^{D}, \] \[ \Psi _{2} =\Psi _{0011}=\Psi _{ABCD}o^{A}o^{B}\iota ^{C}\iota ^{D}, \] \[ \Psi _{3} =\Psi _{0111}=\Psi _{ABCD}o^{A}\iota ^{B}\iota ^{C}\iota ^{D}, \] \[ \Psi _{4} =\Psi _{1111}=\Psi _{ABCD}\iota ^{A}\iota ^{B}\iota ^{C}\iota^{D}. \]

Often it is of interest to consider functions defined on some 2-sphere \(\mathfrak{S}\ ,\) (or perhaps a family of spheres, or of space-like 2-surfaces, \(\mathfrak{S}\ ,\) which need not have the metric of a sphere) embedded in the space-time, where the null tetrad is to be chosen so that, at each point of \(\mathfrak{S}\) the real and imaginary parts of \(m^{a}\) are tangent to \(\mathfrak{S}\ ,\) whence \(l^{a}\) and \(n^{a}\) are normal to \(\mathfrak{S}\ .\) There is a freedom in the tetrad

\[ l^{a}\Rightarrow l'{}^a=Vl^{a},\quad n^{a}\Rightarrow n'{}^a = V^{-1} n^{a}, \quad m^{a}\Rightarrow m'{}^a=e^{i\chi } m^{a} \]

with \(V\) and \(\chi\) being real and \(V>0\ .\) One is concerned with functions \(\eta\ ,\) called spin-weighted and boost-weighted functions, which scale under such replacements according to

\[ \eta \Rightarrow \eta' = V^{w} e^{is\chi }\eta , \] where the real numbers \(s\) and \(w\) are, respectively, the spin-weight and the boost weight of \(\eta\ .\) Normally, \(s\) and \(w\) are such that the quantities

\[ p=w+s\] and \(q=w-s \)

are both integers. We note that the spin-frame elements \(o_{A}\) and \(\iota_{A}\) scale according to

\[ o_{A}\Rightarrow o'_{A} = \iota o_{A},\quad \iota _{A}\Rightarrow \iota' _{A} = \iota ^{-1}\iota _{A}, \]

where

\[ \iota ^{2}=Ve^{i\chi }, \]

so the spin- and boost-weighted quantity \(\eta\ ,\) above, scales as

\[ \eta \Rightarrow \eta' =\iota^{p}\overline{\iota }^{q}\eta . \]

The differential operators \(\eth\) and \(\overline{\eth }\) which act on spin-weighted functions are used frequently. (They do not depend on the boost weight.) For a general \(\mathfrak{S},\) the definition of \(\eth\) is given (Penrose and Rindler, 1984) by

\[\tag{18} \eth \eta =(\delta -p\beta -q\overline{\alpha }), \]

with \(p=w+s\) and \(q=w-s.\) In the case of a unit 2-sphere \(\mathfrak{S}\ ,\) it is convenient to use the following expressions (Penrose and Rindler, 1984; Newman and Penrose, 1966; Goldberg et al., 1967) for \(\eth\) and \(\overline{\eth }\ ,\) differing in certain minor respects (including a factor of \(\sqrt{2}\)) from Eq. (18), which are what is used in the remainder of this work, consistently with the other conventions employed here.

\[\tag{19} \eth \eta \equiv P^{1-s}\frac{\partial }{\partial \zeta }(P^{s}\eta ), \]

\[ \overline{\eth }\eta \equiv P^{1+s}\frac{\partial }{\partial \overline{ \zeta }}(P^{-s}\eta ), \] with

\[ P=1+\zeta \overline{\zeta }. \]

The variables (\(\zeta, \overline{\zeta}\)), which are the complex stereographic coordinates on the sphere, are related to the ordinary sphere coordinates (\(\theta,\phi\)) by

\[ \zeta =e^{i\phi }\cot (\theta/2), \]

while the quantity \(P\) is defined from the unit sphere metric in the \((\zeta ,\overline{\zeta})\) coordinates by

\[ ds^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}=\frac{4d\zeta d\overline{ \zeta }}{P^{2}}. \]

(If \((\theta,\phi)\) coordinates are used instead of \((\zeta,\overline{\zeta })\ ,\) there is a slight modification in the definition of \(\eth\) due to a rotation in the \(m^{a}\) vector.)

The spin-weighted functions on the sphere are often expanded in the orthonormal basis (Newman and Penrose, 1966; Goldberg et al., 1967; Newman and Silva-Ortigoza, 2006; Held et al., 1970; Durrer, 2008), the spin-\(s\) spherical harmonics \(_{s}Y_{lm}(\zeta ,\overline{\zeta }).\)

The spin-coefficient equations

The spin-coefficient equations (also known in the literature as Newman-Penrose equations or simply NP equations), broken into the three sets, are:

the metric equations:

\[\tag{20} \Delta l^{a}-Dn^{a} =(\gamma +\overline{\gamma })l^{a}+(\epsilon + \overline{\epsilon })n^{a}-(\tau +\overline{\pi })\overline{m}^{a}-( \overline{\tau }+\pi )m^{a}, \]

\[ \delta l^{a}-Dm^{a} =(\overline{\alpha }+\beta -\overline{\pi } )l^{a}+\kappa n^{a}-\sigma \overline{m}^{a}-(\overline{\rho }+\epsilon - \overline{\epsilon })m^{a}, \] \[ \delta n^{a}-\Delta m^{a} =-{\overline\nu} l^{a}+(\tau -\overline{\alpha }-\beta )n^{a}+\overline{\lambda }\overline{m}^{a}+(\mu -\gamma +\overline{\gamma } )m^{a}, \] \[ \overline{\delta }m^{a}-\delta \overline{m}^{a} =(\overline{\mu }-\mu )l^{a}+(\overline{\rho }-\rho )n^{a}-(\overline{\alpha }-\beta )\overline{m} ^{a}+(\alpha -\overline{\beta })m^{a}. \]

the spin-coefficient equations:

\[\tag{21} \Delta \lambda -\overline{\delta }\nu =-(\mu +\overline{\mu } + 3\gamma -\overline{\gamma }) \lambda + (3\alpha +\overline{\beta }+\pi -\overline{ \tau })\nu -\Psi _{4} \]

\[ \delta \rho -\overline{\delta }\sigma =\rho (\overline{\alpha }+\beta )-\sigma (3\alpha -\overline{\beta })+(\rho -\overline{\rho })\tau +(\mu - \overline{\mu })\kappa -\Psi _{1} \] \[ \delta \alpha -\overline{\delta }\beta =\mu \rho -\lambda \sigma +\alpha \overline{\alpha }+\beta \overline{\beta }-2\alpha \beta +\gamma (\rho - \overline{\rho })+\epsilon (\mu -\overline{\mu })-\Psi _{2} \] \[ \delta \lambda -\overline{\delta }\mu =(\rho -\overline{\rho })\nu +(\mu - \overline{\mu })\pi +\mu (\alpha +\overline{\beta })+\lambda (\overline{ \alpha }-3\beta )-\Psi _{3} \] \[ \delta \nu -\Delta \mu =\mu ^{2}+\lambda \overline{\lambda }+\mu (\gamma + \overline{\gamma })-\overline{\nu }\pi +\nu (\tau -3\beta -\overline{\alpha }) \] \[ \delta \gamma -\Delta \beta =\gamma (\tau -\overline{\alpha }-\beta )+\mu \tau -\sigma \nu -\epsilon \overline{\nu }-\beta (\gamma -\overline{\gamma } -\mu )+\alpha \overline{\lambda } \] \[ \delta \tau -\Delta \sigma =\mu \sigma +\rho \overline{\lambda }+\tau (\tau +\beta -\overline{\alpha })-\sigma (3\gamma -\overline{\gamma } )-\kappa \overline{\nu } \] \[ \Delta \rho -\overline{\delta }\tau =-(\rho \overline{\mu }+\sigma \lambda )+\tau (\overline{\beta }-\alpha -\overline{\tau })+(\gamma +\overline{ \gamma })\rho +\kappa \nu -\Psi _{2} \] \[ \Delta \alpha -\overline{\delta }\gamma =\nu (\rho +\epsilon )-\lambda (\tau +\beta )+\alpha (\overline{\gamma }-\overline{\mu })+\gamma (\overline{ \beta }-\overline{\tau })-\Psi _{3} \]

\[\tag{22} D\rho -\overline{\delta }\kappa =\rho ^{2}+\sigma \overline{\sigma } +(\epsilon +\overline{\epsilon })\rho -\overline{\kappa }\tau -\kappa (3\alpha +\overline{\beta }-\pi ) \]

\[ D\sigma -\delta \kappa =(\rho +\overline{\rho })\sigma +(3\epsilon - \overline{\epsilon })\sigma -(\tau -\overline{\pi }+\overline{\alpha } +3\beta )\kappa +\Psi _{0} \] \[ D\tau -\Delta \kappa =(\tau +\overline{\pi })\rho +(\overline{\tau }+\pi )\sigma +(\epsilon -\overline{\epsilon })\tau -(3\gamma +\overline{\gamma } )\kappa +\Psi _{1} \] \[ D\alpha -\overline{\delta }\epsilon =(\rho +\overline{\epsilon }-2\epsilon )\alpha +\beta \overline{\sigma }-\overline{\beta }\epsilon -\kappa \lambda - \overline{\kappa }\gamma +(\epsilon +\rho )\pi \] \[ D\beta -\delta \epsilon =(\alpha +\pi )\sigma +(\overline{\rho }-\overline{ \epsilon })\beta -(\mu +\gamma )\kappa -(\overline{\alpha }-\overline{\pi } )\epsilon +\Psi _{1} \] \[ D\gamma -\Delta \epsilon =(\tau +\overline{\pi })\alpha +(\overline{\tau } +\pi )\beta -(\epsilon +\overline{\epsilon })\gamma -(\gamma +\overline{ \gamma })\epsilon +\tau \pi -\nu \kappa +\Psi _{2} \] \[ D\lambda -\overline{\delta }\pi =\rho \lambda +\overline{\sigma }\mu +\pi ^{2}+(\alpha -\overline{\beta })\pi -\nu \overline{\kappa }-(3\epsilon -\overline{\epsilon })\lambda \] \[ D\mu -\delta \pi =\overline{\rho }\mu +\sigma \lambda +\pi \overline{\pi }-(\epsilon +\overline{\epsilon })\mu -\pi (\overline{\alpha }-\beta )-\nu \kappa +\Psi _{2} \] \[ D\nu -\Delta \pi =(\overline{\tau }+\pi )\mu +(\tau +\overline{\pi } )\lambda +(\gamma -\overline{\gamma })\pi -(3\epsilon +\overline{\epsilon } )\nu +\Psi _{3} \]

and finally the Bianchi identities:

\[\tag{23} \overline{\delta }\Psi _{0}-D\Psi _{1} =(4\alpha -\pi )\Psi _{0}-2(2\rho +\epsilon )\Psi _{1}+3\kappa \Psi _{2}, \]

\[ \overline{\delta }\Psi _{1}-D\Psi _{2} =\lambda \Psi _{0}+2(\alpha -\pi )\Psi _{1}-3\rho \Psi _{2}+2\kappa \Psi _{3}, \] \[ \overline{\delta }\Psi _{2}-D\Psi _{3} =2\lambda \Psi _{1}-3\pi \Psi _{2}+2(\epsilon -\rho )\Psi _{3}+\kappa \Psi _{4}, \] \[ \overline{\delta }\Psi _{3}-D\Psi _{4} =3\lambda \Psi _{2}-2(\alpha +2\pi )\Psi _{3}+(4\epsilon -\rho )\Psi _{4}, \]

\[\tag{24} \Delta \Psi _{0}-\delta \Psi _{1} =(4\gamma -\mu )\Psi _{0}-2(2\tau +\beta )\Psi _{1}+3\sigma \Psi _{2}, \]

\[ \Delta \Psi _{1}-\delta \Psi _{2} =\nu \Psi _{0}+2(\gamma -\mu )\Psi _{1}-3\tau \Psi _{2}+2\sigma \Psi _{3}, \] \[ \Delta \Psi _{2}-\delta \Psi _{3} =2\nu \Psi _{1}-3\mu \Psi _{2}+2(\beta -\tau )\Psi _{3}+\sigma \Psi _{4}, \] \[ \Delta \Psi _{3}-\delta \Psi _{4} =3\nu \Psi _{2}-2(\gamma +2\mu )\Psi _{3}+(4\beta -\tau )\Psi _{4}. \]

Though these equations appear to be quite formidable, it will be seen that by imposing both tetrad and coordinate conditions (frequently losing no generality) they can often be made amenable to both study and integration, with certain classes of metrics leading to particular simplification.

The following commutator relations that act on scalars have proven to be very useful: \[\tag{25} \begin{array}{lcl} \Delta D-D\Delta &=&(\gamma +\bar{\gamma})D+(\epsilon +\bar{\epsilon} )\Delta -(\bar{\tau}+\pi )\delta -(\tau +\bar{\pi})\bar{\delta}~, \\ \delta D-D\delta &=&(\bar{\alpha}+\beta -\bar{\pi})D+(\kappa )\Delta -(\bar{ \rho}+\epsilon -\bar{\epsilon})\delta -(\sigma )\bar{\delta}~, \\ \delta \Delta -\Delta \delta &=&(-\bar{\upsilon})D+(\tau -\bar{\alpha} -\beta )\Delta +(\mu -\gamma +\bar{\gamma})\delta +(\bar{\lambda})\bar{\delta }~, \\ \bar{\delta}\delta -\delta \bar{\delta} &=&(\bar{\mu}-\mu )D+(\bar{\rho} -\rho )\Delta +(\alpha -\bar{\beta})\delta -(\bar{\alpha}-\beta )\bar{\delta} ~. \end{array} \]

Tetrad conditions, coordinate conditions and some special cases

A small sample of the very numerous applications of SC is given in the following sections.

Null geodesic congruences

When the space-time contains a ray congruence, \(\mathfrak{C}\ ,\) (a foliation by null geodesics) that is singled out, many of the spin-coefficients take on simple geometric interpretation (Newman and Penrose, 1962; Geroch et al., 1973; Penrose and Rindler, 1984; Newman and Tod, 1980; Newman and Unti, 1962) when they are associated with the congruence. The null vector \(l^{a}\) may be chosen to point in the direction of the rays, whose geodicity can be stated as

\[\tag{26} \kappa =0. \]

Affine normalization (parallel propagation of \(l^{a}\)) is stated as \[\tag{27} \epsilon +\overline{\epsilon }=0. \]

The differential properties of \(\mathfrak{C}\) are described by the so-called (Sachs) optical parameters, namely the spin-coefficients, \(\rho \) and \(\sigma \ ,\) respectively referred to as the 'complex divergence' and the shear. They can be written in terms of the \(l^{a}\) derivatives as

\[ \rho =\frac{1}{2} \left(-\nabla _{a}l^{a}+i~|\mathrm{curl}\, l| \right), \] \[ |\mathrm{curl}\,l| \equiv \sqrt{\nabla _{\lbrack a}l_{b]}\nabla ^{\lbrack a}l^{b]}}, \] \[ \sigma \overline{\sigma } =\frac{1}{2}[\nabla _{(a}l_{b)}\nabla^{(a}l^{b)}-\frac{1}{2}(\nabla _{a}l^{a})^{2}]. \]

Using \(\mathfrak{C}\ ,\) the remaining vectors, \((n^{a},m^{a},\overline{m}^{a}),\) can be chosen to be parallelly propagated along the geodesics leading to

\[\tag{28} \epsilon -\overline{\epsilon }=\pi =0. \]

When \(\mathfrak{C}\) is a gradient field, i.e., \(l_{a}=\nabla _{a}\Phi ,\) then, equivalently,

\[\tag{29} \rho =\overline{\rho }, \]

\[\tag{30} \tau =\overline{\alpha }+\beta . \]

Algebraically special metrics and the Goldberg-Sachs theorem. Among the most studied vacuum space-times are those referred to as 'algebraically special' space-times, i.e., vacuum space-times that possess two or more coinciding principal null directions (PNDs). Principal null direction fields (Penrose and Rindler, 1984; Cartan, 1922; Majorana, 1932; Pirani, 1957; Penrose, 1960) (in general, four locally independent fields exist) are most easily described in the spinor formalism, since the gravitational spinor \(\Psi _{ABCD}\) can be written as the symmetrized product

\[\tag{31} \Psi _{ABCD}=\alpha _{(A}\beta _{B}\gamma _{C}\delta _{D)} \]

(the factors being unique up to proportionality), the null vectors, \(\alpha _{A}\overline{\alpha }_{A'},\) etc., being the PNDs. (Their directions may coincide in various arrangements, giving the so-called 'Cartan-Petrov-Pirani-Penrose' classification referred to below.) We may identify a PND from solutions of the equation

\[\tag{32} \Psi _{ABCD}\alpha ^{A}\alpha ^{B}\alpha ^{C}\alpha ^{D}=0 \]

which, with \(L^{a}=\alpha ^{A}\overline{\alpha }^{A'},\) is equivalent to finding solutions, \(L^{a},\) to the algebraic equation

\[ L^{b}L_{[e}C_{a]bc[d}L_{f]}L^{c}=0,\qquad L^{a}L_{a}=0. \]

Figure 1: The Cartan-Petrov-Pirani-Penrose classification of curvature types. Arrows show directions of specialization.

Comment. The Cartan-Petrov-Pirani-Penrose classification (Petrov 1954; Pirani, 1957; Penrose, 1960) describes the different degeneracies:

\[ \begin{array}{ll} \mathrm{Alg.\ General} & [1,1,1,1] \\ \mathrm{Type\ II} & [2,1,1] \\ \mathrm{Type\ D\ or}\,\,\mathrm{degenerate} & [2,2] \\ \mathrm{Type\ III} & [3,1] \\ \mathrm{Type\ IV\ or\ Null} & [4] \end{array}. \] The symbol \([1,1,1,1]\) means all four PND are independent, the \(2\ ,\) \(3\ ,\) or \(4\) tells the order of degeneracy. The \(0\) in the diagram means that the Weyl tensor has vanished.

In the SC language, if the tetrad vector \(l^{a}\) is a principal null direction, i.e., \(L_{a}=l_{a},\) then automatically:

\[ \Psi _{0}=0. \]

For the algebraically special metrics, the special cases are:

\[ \begin{array}{ll} \mathrm{Type\ II} & \Psi _{0}=\Psi _{1}=0 \\ \mathrm{Type\ III} & \Psi _{0}=\Psi _{1}=\Psi _{2}=0 \\ \mathrm{Type\ IV} & \Psi _{0}=\Psi _{1}=\Psi _{2}=\Psi _{3}=0 \\ \mathrm{Type\ D} & \Psi _{0}=\Psi _{1}=\Psi _{3}=\Psi _{4}=0 \\ &\mbox{with both } l^{a} \text{ and } n^{a} \text{ PNDs}. \end{array}. \] The algebraic classification of space-times is based on properties of the Weyl tensor, so that the classification extends naturally to the non-vacuum case.

An outstanding feature of the algebraically special metrics is contained in the beautiful Goldberg-Sachs theorem (Goldberg and Sachs, 1962; Newman and Penrose, 1962).

Theorem For a non-flat-vacuum space-time, if there is a null geodesic congruence that is shear-free, i.e., there is a null vector field with \((\kappa =0, \sigma =0)\ ,\) then the space-time is algebraically special and, conversely, if a vacuum space-time is algebraically special, there is a null geodesic congruence with \((\kappa =0,\sigma =0)\ .\) \(\blacksquare \)

Though the proofs for both parts of the theorem are relatively simple, only the proof for the second part is given.

Assuming that \(\Psi _{0}=\Psi _{1}=0,\) Eqs. (23) and (24) become

\[\tag{33} \begin{align} 0 &=3\kappa \Psi _{2},\\ -D\Psi _{2} &=-3\rho \Psi _{2}+2\kappa \Psi _{3},\\ \overline{\delta }\Psi _{2}-D\Psi _{3} &=-3\pi\Psi_2 +2(\epsilon -\rho )\Psi _{3}+\kappa\Psi _{4}, \end{align} \]

\[\tag{34} \begin{align} 0 &=3\sigma \Psi _{2},\\ -\delta \Psi _{2} &=-3\tau \Psi _{2}+2\sigma \Psi _{3},\\ \Delta \Psi _{2}-\delta \Psi _{3} &=-3\mu\Psi_2+2(\beta -\tau )\Psi _{3}+\sigma \Psi_{4}. \end{align} \]

It follows immediately that \((\kappa =0,\sigma =0)\) in any of the cases. The first part of the theorem is just slightly more complicated.

Null coordinates and associated tetrad conditions

One of the most important and productive areas of application of the SC equations lies in the study of asymptotically flat space-times. This area was born from the early brilliant work of H. Bondi (H. Bondi et al., 1962; R. Sachs, 1962; Newman and Penrose, 1962; Newman and Tod, 1980) where a one-parameter family of null (i.e., characteristic) surfaces, \(\mathfrak{C}_{u}\) labeled by a retarded time-coordinate \(u\) was introduced. Each of these surfaces is generated by a two-parameter family of null geodesics, \(\mathfrak{G,}\) each labeled by sphere coordinates \((\theta ,\phi )\) or equivalently by complex stereographic coordinates \((\zeta ,\overline{\zeta }),\) where \(\zeta =e^{i\phi }\cot (\theta/2).\) The 'length' along the geodesics was given by the affine parameter, \(r\ .\) This coordinate system, called Bondi coordinates (H. Bondi et al., 1962; R. Sachs, 1962; Newman and Penrose, 1962; Penrose and Rindler, 1984; Newman and Tod, 1980), (although Bondi himself did not use an affine parameter, but what is referred to as a "luminosity distance" parameter), is not unique; there is a large class of such coordinates. The transformations from one to another constitute what is referred to as the Bondi-Metzner-Sachs group (H. Bondi et al., 1962; R. Sachs, 1962; Held et al., 1970; Penrose, 1965; Penrose, 1963; Newman and Tod, 1980; Newman and Unti, 1962). The BMS group also arises geometrically as the symmetry group of the conformal boundary \(\mathfrak{I}^{+}\ ;\) see Remark, below. With the choice of (any) one set, there is a natural choice of null tetrad system. The vector \(l^{a}\) is taken as the tangent vector to the null geodesics, while the (\(m^{a},\overline{m}^{a}\)) are tangent to the null 3-surface. Their remaining freedom is greatly limited by having them parallel propagated along the null geodesics. When these restrictions are translated to the tetrad, metric and the spin-coefficients, the derivative operators become:

\[\tag{35} D =l^{\textstyle{\mathbf{a}}}\frac{\partial }{\partial x^{\textstyle{\mathbf{a}}}}=\frac{\partial }{\partial r} \]

\[\tag{36} \Delta =n^{\textstyle{\mathbf{a}}}\frac{\partial }{\partial x^{\textstyle{\mathbf{a}}}}=\frac{\partial }{\partial u} +U\frac{\partial }{\partial r}+X^{A}\frac{\partial }{\partial x^{A}} ,~~~~~x^{A}=(x^{3},x^{4})=(\zeta ,\overline{\zeta }), \]

\[\tag{37} \delta =m^{\textstyle{\mathbf{a}}}\frac{\partial }{\partial x^{\textstyle{\mathbf{a}}}}=\omega \frac{\partial }{\partial r}+\xi ^{A}\frac{\partial }{\partial x^{A}}, \]

\[\tag{38} \overline{\delta } =\overline{m}^{\textstyle{\mathbf{a}}}\frac{\partial }{\partial x^{\textstyle{\mathbf{a}}}}= \overline{\omega }\frac{\partial }{\partial r}+\overline{\xi }^{A}\frac{ \partial }{\partial x^{A}}, \]

so that the metric takes the form

\[\tag{39} g^{\textstyle{\mathbf{a}}\textstyle{\mathbf{b}}}=\left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & g^{22} & g^{2A} \\ 0 & g^{2A} & g^{AB} \end{array}\right] \]

with

\[\tag{40} g^{22} =2(U-\omega \overline{\omega }), \]

\[ g^{2A} =X^{A}-(\overline{\omega }\xi ^{A}+\omega \overline{\xi }^{A}), \] \[ g^{AB} =-(\xi ^{A}\overline{\xi }^{B}+\overline{\xi }^{A}\xi ^{B}), \]

and the spin-coefficients (see Eqs. (26)-(30) simplify to:

\[\tag{41} \kappa =\pi =\epsilon =0, \]

\[ \rho =\overline{\rho }, \] \[ \tau =\overline{\alpha }+\beta. \]

Remark. An important point should be made and emphasized here. The somewhat 'fuzzy' idea of a boundary for space-time found by passing to the limit of \(r\rightarrow \infty\) has been made more precise (Penrose, 1965; Penrose, 1963) by the conformal compactification of the space-time; \(g_{ab}\rightarrow \hat{g}_{ab}=\Omega ^{2}g_{ab}\) where \(\Omega =0\) on the conformal boundary. This boundary, known as \(\mathfrak{I}^{+},\) (spoken as 'scri-plus') is a null surface, that has the structure of \(S^{2}\times \mathbb{R}\) and a 'strong conformal geometry', which determines 'null angles' as well as ordinary space-like angles. The BMS group is the symmetry group preserving this geometry. It is usual to coordinatize it by the Bondi coordinates \((u,\zeta ,\overline{\zeta }\)).

The spin-coefficient equations simplified

Putting in the assumptions, Eq. (41), of the previous section the SC equations become (Geroch et al., 1973; Penrose and Rindler, 1984; Newman and Tod, 1980; Newman and Unti, 1962):

The metric equations:

\[\tag{42} DU =\tau \overline{\omega }+\overline{\tau }\omega -(\gamma +\overline{\gamma }), \]

\[\tag{43} DX^{A} =\tau \overline{\xi }^{A}+\overline{\tau }\xi ^{A} \]

\[\tag{44} D\omega =\sigma \overline{\omega }+\rho \omega -\tau , \]

\[\tag{45} D\xi ^{A} =\sigma \overline{\xi }^{A}+\rho \xi ^{A}, \]

\[\tag{46} \delta U-\Delta \omega =-\overline{\nu} +\overline{\lambda }\overline{\omega }+(\mu -\gamma +\overline{\gamma })\omega , \]

\[\tag{47} \delta X^{A}-\Delta \xi ^{A} =\overline{\lambda }\,\overline{\xi }^{A}+(\mu -\gamma +\overline{\gamma })\xi ^{A}, \]

\[\tag{48} \overline{\delta }\omega -\delta \overline{\omega } =(\overline{\mu }-\mu )-(\overline{\alpha }-\beta )\overline{\omega }+(\alpha -\overline{\beta })\omega, \]

\[\tag{49} \overline{\delta }\xi ^{A}-\delta \overline{\xi }^{A} =-(\overline{\alpha }-\beta )\overline{\xi }^{A}+(\alpha -\overline{\beta })\xi ^{A}. \]

The spin-coefficient equations:

\[\tag{50} D\rho =\rho ^{2}+\sigma \overline{\sigma } \]

\[ D\sigma =2\rho \sigma +\Psi _{0} \] \[ D\tau =\tau \rho +\overline{\tau }\sigma +\Psi _{1} \] \[ D\alpha =\rho \alpha +\beta \overline{\sigma } \] \[ D\beta =\alpha \sigma +\rho \beta +\Psi _{1} \] \[ D\gamma =\tau \alpha +\overline{\tau }\beta +\Psi _{2} \] \[ D\lambda =\rho \lambda +\overline{\sigma }\mu \] \[ D\mu =\rho \mu +\sigma \lambda +\Psi _{2} \] \[ D\nu =\overline{\tau }\mu +\tau \lambda +\Psi _{3} \]

\[\tag{51} \Delta \lambda -\overline{\delta }\nu =(\overline{\gamma }-\mu -\overline{\mu }-3\gamma) \lambda + 2\alpha \nu -\Psi _{4} \]

\[ \delta \rho -\overline{\delta }\sigma =\rho \tau-\sigma (3\alpha -\overline{\beta })-\Psi _{1} \] \[ \delta \alpha -\overline{\delta }\beta =\mu \rho -\lambda \sigma +\alpha\overline{\alpha }+\beta \overline{\beta }-2\alpha \beta -\Psi _{2} \] \[ \delta \lambda -\overline{\delta }\mu =\mu \overline{\tau}+\lambda (\overline{\alpha }-3\beta )-\Psi _{3} \] \[ \delta \nu -\Delta \mu =\mu^{2}+\lambda \overline{\lambda }+\mu (\gamma + \overline{\gamma })-2\beta \nu \] \[ \delta \gamma -\Delta \beta =\mu \tau -\sigma \nu -\epsilon\, \overline{\nu } -\beta (\gamma -\overline{\gamma }-\mu )+\alpha \overline{\lambda } \] \[ \delta \tau -\Delta \sigma =\mu \sigma +\rho \overline{\lambda }+2\tau\beta-\sigma (3\gamma -\overline{\gamma }) \] \[ \Delta \rho -\overline{\delta }\tau =-\sigma\lambda-2\tau\alpha+(\gamma +\overline{\gamma }-\overline{\mu})\rho -\Psi _{2} \] \[ \Delta \alpha -\overline{\delta }\gamma =\nu \rho -\lambda (\tau +\beta)+(\overline{\gamma }-\gamma-\overline{\mu })\alpha-\Psi _{3} \]

and finally the Bianchi identities:

\[\tag{52} \overline{\delta }\Psi _{0}-D\Psi _{1} =4\alpha \Psi _{0}-4\rho \Psi _{1}, \]

\[ \overline{\delta }\Psi _{1}-D\Psi _{2} =\lambda \Psi _{0}+2\alpha \Psi _{1}-3\rho \Psi _{2}, \] \[ \overline{\delta }\Psi _{2}-D\Psi _{3} =2\lambda \Psi _{1}-2\rho \Psi _{3}, \] \[ \overline{\delta }\Psi _{3}-D\Psi _{4} =3\lambda \Psi _{2}-2\alpha \Psi _{3}-\rho \Psi _{4}, \]

\[\tag{53} \Delta \Psi _{0}-\delta \Psi _{1} =(4\gamma -\mu )\Psi _{0}-2(2\tau +\beta )\Psi _{1}+3\sigma \Psi _{2}, \]

\[ \Delta \Psi _{1}-\delta \Psi _{2} =\nu \Psi _{0}+2(\gamma -\mu )\Psi _{1}-3\tau \Psi _{2}+2\sigma \Psi_{3}, \] \[ \Delta \Psi _{2}-\delta \Psi_{3} =2\nu \Psi _{1}-3\mu \Psi _{2}+2(\beta -\tau )\Psi _{3}+\sigma \Psi_{4}, \] \[ \Delta \Psi _{3}-\delta \Psi _{4} =3\nu \Psi _{2}-2(\gamma +2\mu )\Psi _{3}+(4\beta -\tau )\Psi _{4}. \]

By a simple transformation (see the discussion after Eq. (55)) the optical equations for \(\sigma\) and \(\rho\) are made linear and then using the known \(\rho\) and \(\sigma \) many other equations become linear.

Integrating the 'simplified' spin-coefficient equations

Though these simplified equations still appear formidable, at least for the asymptotic solutions there is a systematic procedure (Newman and Penrose, 1962; Geroch et al., 1973; Newman and Tod, 1980; Newman and Unti, 1962) for their integration. Several preliminary comments should first be made.

1. The basic program is to initially consider only the equations involving the \(D\) (or \(r\)) derivative. These equations, appropriately ordered, are integrated asymptotically (using theorems on the asymptotic behavior of linear equations) with a "function of integration", i.e., an arbitrary function of the remaining variables, appearing for each equation. The role of the remaining SC equations is to determine or establish relations between these integration functions and to provide for their evolution.

2. There are two of the \(D\) equations, the so-called optical equations (Sachs, 1961), that determine the optical parameters \(\rho\) and \(\sigma\ ,\) of the geodesic congruence \(\mathfrak{G}\ ,\) namely:

\[\tag{54} D\rho = \rho ^{2}+\sigma \overline{\sigma }, \]

\[ D\sigma =2\rho \sigma +\Psi _{0}. \]

They play a primary role in the sense that they 'drive' many of the other equations.

Comment. Note that Eq. (54) can be written as a matrix Riccati equation

\[\tag{55} D\mathbf{P} =\mathbf{P}^{2}+\mathbf{Q} \]

\[ \mathbf{P} =\left[ \begin{array}{ll} \rho & \sigma \\ \overline{\sigma } & \rho \end{array} \right] ,\qquad\mathbf{Q}=\left[ \begin{array}{ll} 0 & \Psi _{0} \\ \overline{\Psi}_{0} & 0\end{array}\right] \,\,\,\,. \]

This is very useful since it can be linearized by \(\mathbf{P}=-D\mathbf{Y}\ \cdot \mathbf{Y}^{-1}\ :\)

\[ D^{2}\mathbf{Y}=-\mathbf{QY}. \]

3. Most frequently, in the study of asymptotically flat space-times, the Weyl component \(\Psi _{0}\) is taken (from linear theory) to have the asymptotic behavior:

\[\tag{56} \Psi _{0}=\Psi _{0}^{0}r^{-5}+O(r^{-6}). \]

The results

By integrating, in the asymptotic region, all the equations containing \(D\) derivatives (Newman and Unti, 1962), the asymptotic behavior of all the variables is found.

For the Weyl tensor the following fall-off behavior, known as the peeling theorem (Sachs, 1961; Newman and Penrose, 1962), is found:

\[ \Psi _{0} =\Psi _{0}^{0}r^{-5}+O(r^{-6}), \] \[ \Psi _{1} =\Psi _{1}^{0}r^{-4}+O(r^{-5}), \] \[ \Psi _{2} =\Psi _{2}^{0}r^{-3}+O(r^{-4}), \] \[ \Psi _{3} =\Psi _{3}^{0}r^{-2}+O(r^{-3}), \] \[ \Psi _{4} =\Psi _{4}^{0}r^{-1}+O(r^{-2}). \]

The asymptotic behavior of the spin-coefficients and metric variables is:

\[ \kappa =\pi =\epsilon =0,\qquad \tau =\overline{\alpha}+\beta, \] \[ \rho =\overline{\rho }=-r^{-1}-\sigma ^{0}\bar{\sigma } ^{0}r^{-3}+O(r^{-5}), \] \[ \sigma =\sigma ^{0}r^{-2}+((\sigma^{0})^{2}\overline{\sigma } ^{0}-\tfrac12\Psi _{0}^{0})r^{-4}+O(r^{-5}), \] \[ \alpha =\alpha ^{0}r^{-1}+O(r^{-2}),\quad \beta =\beta^{0}r^{-1}+O(r^{-2}), \] \[ \gamma =\gamma ^{0}- \tfrac12\Psi _{2}^{0}\,r^{-2}+O(r^{-3}),\quad \lambda =\lambda ^{0}r^{-1}+O(r^{-2}), \] \[ \mu =\mu ^{0}r^{-1}+O(r^{-2}),\ \ \ \ \ \ \ \ \ \nu =\nu ^{0}+O(r^{-1}), \] \[ \xi ^{A} =\xi ^{0A}r^{-1}-\sigma ^{0}\overline{\xi }^{0A}r^{-2}+O(r^{-3}), \] \[ \omega =\omega ^{0}r^{-1}-(\sigma ^{0}\overline{\omega }^{0}+ \tfrac12\Psi _{1}^{0})r^{-2}+O(r^{-3}), \] \[ X^{A} =(\Psi _{1}^{0}\overline{\xi }^{0A}+\overline{\Psi }_{1}^{0}\xi ^{0A})(6r^{3})^{-1}+O(r^{-4}), \] \[ U =U^{0}-(\gamma ^{0}+\overline{\gamma }^{0})r-(\Psi _{2}^{0}+\overline{\Psi }_{2}^{0})(2r)^{-1}+O(r^{-2}). \]

By comparing powers of \(r\) in the non-radial equations (using, as well, some coordinate conditions and special choices of tetrad) the functions of integration are determined as:

\[ \xi ^{0\zeta } =-P,\qquad \overline{\xi }^{0\zeta }=0, \] \[ \xi ^{0\overline{\zeta }} =0,\qquad \overline{\xi }^{0\overline{\zeta }}=-P, \] \[ P =1+\zeta \overline{\zeta }, \] \[ \alpha ^{0} =-\overline{\beta }^{0}=-\frac{\zeta }{2}, \] \[ \gamma ^{0} =\nu ^{0}=0, \] \[ \omega ^{0} =-\bar\eth\sigma ^{0}, \] \[ \overline{\lambda }^{0} =\dot{\sigma}^{0}, \] \[ \mu ^{0} =U^{0}=-1, \] \[ \Psi _{3}^{0} =\eth\lambda ^{0}, \] \[ \Psi _{4}^{0} =-\dot{\lambda}^{0}. \]

One also has the physically very important reality condition on the mass aspect:

\[\tag{57} \Psi =\overline{\Psi }\equiv \Psi _{2}^{0\,}+\eth^{2}\overline{\sigma}+\sigma \dot{\bar\sigma}. \]

Finally from the non-radial Bianchi identities, Eqs. (53), the dynamical (or evolution) relations are obtained:

\[\tag{58} \dot{\Psi}_{2}^{0\,} =-\eth\Psi _{3}^{0\,}+\sigma ^{0}\Psi_{4}^{0\,}, \]

\[\tag{59} \dot{\Psi}_{1}^{0\,} =-\eth\Psi _{2}^{0\,}+2\sigma ^{0}\Psi_{3}^{0\,}, \]

\[\tag{60} \dot{\Psi}_{0}^{0\,} =-\eth\Psi _{1}^0+3\sigma ^{0}\Psi_{2}^{0\,}. \]

With these results the characteristic initial value problem can roughly be stated in the following manner: At \(u=u_{0},\) the initial 'values', i.e., functions only of (\(\zeta ,\overline{\zeta }\)), are chosen for (\(\Psi_{0}^{0},\Psi _{1}^{0},\Psi _{2}^{0});\) then an arbitrary 'free' function, \(\sigma ^{0}(u,\zeta ,\overline{\zeta }),\) known as the Bondi shear, is given. Since \(\Psi _{3}^{0}\) and \(\Psi _{4}^{0}\) are functions of \(\sigma^{0}\) and its derivatives, all the asymptotic variables can then be determined.

Exactly analogous to the situation in Maxwell theory where the total internal electric charge can be `measured' at infinity by a two-surface (sphere) integral over the asymptotic Maxwell field, Bondi showed that the total internal energy-momentum, (four-momentum) could be determined by a similar integral.

Specifically, by expanding the mass aspect \(\Psi\) in spherical harmonics, Bondi identified the mass and the three-momentum by

\[ \Psi =\Psi ^{0}+\Psi ^{i}Y_{1i}^{0}+\Psi ^{ij}Y_{2ij}^{0}+. \] \[ \Psi =-\frac{2\sqrt{2}G}{c^{2}}M-\frac{6G}{c^{3}}P^{i}Y_{1i}^{0}+\ldots \]

By rewriting Eq. (58), replacing the \(\Psi _{2}^{0\,}\) by \(\Psi\) via Eq. (57), as

\[ \dot{\Psi}=\overset{.}{\sigma }\overset{\overset{.}{\_}}{\sigma } \]

one immediately has the famous Bondi mass/energy loss theorem:

\[\tag{61} \dot{M}=-\frac{c}{2\sqrt{2}G}\int \overset{.}{\sigma }\overset{\overset{.}{\_}}{\sigma }d^{2}S, \]

the integral taken over the unit 2-sphere. This relationship is at the basis of all the contemporary work on the detection of gravitational radiation.

Often, with some modification, the imaginary part of the \(l=1\) spherical harmonic part of \(\Psi _{1}^{0\,}\) is taken as proportional to the total internal angular-momentum and Eq. (59) becomes the angular momentum conservation law (Kozameh et al., 2008).

The Robinson-Trautman metrics

A relatively simple application of the SC equations is to the twist-free algebraically special Type II metrics (Robinson and Trautman, 1962), the Robinson-Trautman metrics. This is a simple example of searching for metrics with special properties.

One begins with Eqs. (42)-(49), (50), (51), (52), (53) and the associated assumptions, the same as the Eq. (41),

\[ \kappa =\pi =\epsilon =0, \] \[ \rho =\overline{\rho }, \] \[ \tau =\overline{\alpha }+\beta , \]

and adds in the algebraically special type II condition,

\[ \Psi _{0} =\Psi _{1}=\sigma =0 \] \[ \Psi _{2}^{0} \neq 0. \]

The radial (or \(D\)) equations can now be easily integrated leading to

\[ l =\frac{\partial }{\partial r}, \] \[ n =\frac{\partial }{\partial u}+U\frac{\partial }{\partial r}, \] \[ m =-Pr^{-1}\frac{\partial }{\partial \zeta }, \] \[ U =U^{0}-2\gamma ^{0}r-\Psi _{2}^{0}r^{-1}. \]

\[ \Psi _{2} =\Psi _{2}^{0}r^{-3}, \] \[ \Psi _{3} =\Psi _{3}^{0}r^{-2}, \] \[ \Psi _{4} =\Psi _{4}^{0}\;r^{-1}+\overline{\eth }\Psi_{3}^{0}\;r^{-2}, \] \[ \rho =-r^{-1}, \] \[ \alpha =\alpha ^{0}r^{-1}, \] \[ \beta =-\overline{\alpha }^{0}r^{-1}, \] \[ \lambda =\tau =0, \] \[ \gamma =\gamma^{0}-\frac{1}{2}\Psi _{2}^{0}r^{-2}, \] \[ \mu =\mu ^{0}r^{-1}-\Psi _{2}^{0}r^{-2} \] \[ \nu =\nu ^{0}-\Psi _{3}^{0}r. \]

From the non-radial equations -- using different coordinate conditions -- with now \(P=P(u,\zeta ,\overline{\zeta })\) (for the time being) an arbitrary function, one obtains

\[ \Psi _{2}^{0} =\overline{\Psi }_{2}^{0}=\text{constant} \] \[ \Psi _{3}^{0} =\overline{\eth }K \] \[ \Psi _{4}^{0} =2\overline{\eth }^{2}\gamma ^{0} \] \[ \alpha ^{0} =-P,_{\overline{\zeta }} \] \[ \gamma ^{0} =-\frac{1}{2}\frac{\dot{P}}{P} \] \[ U^{0} =\mu ^{0}=-K=-\overline{\eth }\eth \log P \]

Finally, the dynamical equation (the Robinson-Trautman equation), an equation for the only independent variable, \(P,\) is

\[\tag{62} 3\Psi _{2}^{0}\frac{\dot{P}}{P}+\overline{\eth }\eth \overline{\eth }\eth\log P=0. \]

Often the variable \(P\) is replaced by \(V\) via

\[ P=P_{0}V=(1+\zeta \overline{\zeta })V. \]

The associated Robinson-Trautman metric is

\[\tag{63} ds^{2}=2(K-\frac{\dot{P}}{P}r+\frac{\Psi _{2}^{0}}{r})du^{2}+2dudr-\frac{r^{2}}{2P^{2}}d\zeta d\overline{\zeta }. \]

Further applications

The range of applications of the SC formalism is quite large (a simple search on Google for 'Applications of the Spin-Coefficient Formalism' yields about 1800 items) and there is no possibility of even trying to give a reasonable discussion of them.

A brief list of some more familiar or important examples of topics where SC has been applied:

1. Exact solutions,

2. Gravitational radiation,

3. Perturbation Calculations,

4. Numerical computations,

5. Algebraic Computing,

6. Black-Hole physics,

7. Maxwell theory,

8. Gravitational Lensing,

9. Equations of Motion,

10. Riemannian (Euclidean signature) geometry.

Some application references

I. A major example of the application of the SC formalism to black-hole theory is given in the book of S. Chandrasekhar: The Mathematical Theory of Black Holes. Oxford University Press, New York, USA, 1992.

II. The SC formalism has recently been playing a large role in numerical relativity: examples of references are:

a. C. Lousto and Y. Zlochower. Practical formula for the radiated angular momentum. Phys. Rev. D, 76:041502, 2007.

b. D. R. Fiske, J. G. Baker, J. R. van Meter, Dae-Il Choi and J. M. Centrella. Wave zone extraction of gravitational radiation in three-dimensional numerical relativity. Phys. Rev. D, 71:104036, 2005.

c. M, Campanelli and C. O. Lousto. Regularization of the Teukolsky Equation for Rotating Black Holes. Phys. Rev. D, 56:6363, 1997.

d. M. Campanelli and C. O. Lousto. Second order gauge invariant gravitational perturbations of a Kerr black hole. Phys. Rev. D, 59:124022, 1999.

III. Several miscellaneous examples, chosen at random, are:

a. M. Campanelli, C. O. Lousto. Non-null electromagnetic fields and compacted spin coefficient formalism in general relativity. Indian journal of Physics B, 75(6):565-570, 2001.

b. J. Frauendiener. The Sparling Form and its Relationship to the Spin-Coefficient Formalism. General Relativity and Gravitation, 22(12):1423, 1990.

c. R. Gambini and L. Herrera. Einstein-Cartan theory in the spin coefficient formalism. J. Math. Phys. 21, 1449 (1980).

d. R. Posadas and D. M.Yanaga. Liénard-Wiechert fields in the spin-coefficient formalism. The Philippine Journal of Science: 207, Jl-D, 1985.

e. S. Heidenreich, T. Chrobok,and H.-H. v. Borzeszkowski. Supersymmetry, exact Foldy-Wouthuysen transformation, and Gravity. Phys. Rev. D: 044026, 2006.

f. R. Lind. Shear-free, Twisting Einstein-Maxwell metrics in the Newman-Penrose formalism. General Relativity and Gravitation, 5(1):25, 1974.

g. S. B. Edgar and A. Hoglund. The Lanczos Potential for the Weyl Curvature Tensor: Existence, Wave Eq. and Algorithms. Proc. Math. Phys. and Engineering Sciences, 453 (1959): 1997.

h. C. Kozameh, E.T. Newman and G. Silva-Ortigoza. On Extracting Physical Content from Asymptotically Flat Spacetime Metrics. Class. Quantum Grav., 25, 2008.

i. M.P.M. Ramos and J.A.G. Vickers. A Spacetime Calculus invariant under Null Rotations. Proc. Roy. Soc. A 1940, 693 (1995).

M.P.M. Ramos and J.A.G. Vickers. A spacetime calculus based on a single null direction. Class. Quant. Grav. 13, 1579 (1996).

S.B. Edgar and M.P.M. Ramos. Obtaining a class of Type N pure radiation metrics using invariant operators. Class.Quant.Grav. 22 (2005) 791-802.

j W. Kinnersley. Type D vacuum metrics. J. Math. Phys. 10, 1195 (1969).

S.T.C. Siklos. Some Einstein spaces and their global properties. J. Phys. A 14, 395 (1981).

Appendix: the non-vacuum spin-coefficient equations

For completeness we present the full set of SC equations for arbitrary space-times. In their application to physical problems the Ricci tensor components must be replaced by the associated stress tensor components via the standard Einstein equations.

See: The non-vacuum spin-coefficient equations.

References

H. Bondi, M. van der Burg, and A. Metzner. Gravitational Waves in General Relativity, VII, Waves from Axi-Symmetric Isolated Systems. Proc. R. Soc. London Ser. A, 269:21, 1962.

É. J. Cartan. Sur les équations de la gravitation d'Einstein. Math. Pures et Appl., 1:141-203 (cf. p. 194), 1922.

Ruth Durrer. The Cosmic Microwave Background (Cambridge University Press, 2008).

S. B. Edgar. A New Approach to Einstein-Petrov Type I Spaces. International Journal of Theoretical Physics, Vol. 18, No. 4, 1979.

R. Geroch, A. Held, and R. Penrose. A Space-Time Calculus Based on Pairs of Null Directions, J. Math. Phys., 14:874, 1973.

J. Goldberg, A. J. MacFarlane, F. Rohrlich, E. T. Newman, and E. C. G. Sudarshan. Spin-s Spherical Harmonics and ð. J. Math. Phys., 8:2155, 1967.

J. Goldberg and R. Sachs. A Theorem on Petrov Types. Acta Phys. Pol (Suppl.), 13, 1962.

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Internal references

Teviet Creighton and Richard H. Price (2008) Black holes. Scholarpedia, 3(1):4277.

Spin-coefficient formalism

Contents

Introduction

Notation

The spin-coefficient equations

Tetrad conditions, coordinate conditions and some special cases

Null geodesic congruences

Null coordinates and associated tetrad conditions

The spin-coefficient equations simplified

Integrating the 'simplified' spin-coefficient equations

The results

The Robinson-Trautman metrics

Further applications

Some application references

Appendix: the non-vacuum spin-coefficient equations

References

Further reading

See also

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