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The '''Newman–Penrose''' ('''NP''') '''formalism'''<ref name="NP1">{{cite journal | author= Ezra T. Newman and Roger Penrose | title=An Approach to Gravitational Radiation by a Method of Spin Coefficients | journal=Journal of Mathematical Physics | year=1962 | volume=3 | issue=3 | pages=566–768 | doi=10.1063/1.1724257 |bibcode = 1962JMP.....3..566N }}  The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.</ref><ref name="NP2">Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''.  Journal of Mathematical Physics, 1963, '''4'''(7): 998.</ref> is a set of notation developed by [[Ezra T. Newman]] and [[Roger Penrose]] for [[general relativity]] (GR). Their notation is an effort to treat general relativity in terms of [[spinor]] notation, which introduces [[complex number|complex]] forms of the usual variables used in GR. The NP formalism is itself a special case of the [[tetrad formalism]],<ref name="Chandra">{{cite book |last=Chandrasekhar |first=S. |author-link=Subrahmanyan Chandrasekhar |title=The Mathematical Theory of Black Holes |year=1998 |publisher=Oxford University Press |isbn=0-19850370-9 |edition=Oxford Classics Series |url=https://global.oup.com/academic/product/the-mathematical-theory-of-black-holes-9780198503705|access-date=31 May 2019 |page=40 |quote=The Newman–Penrose formalism is a tetrad formalism with a special choice of the basis vectors.}}</ref> where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The [[Weyl scalars]], derived from the [[Weyl tensor]], are often used. In particular, it can be shown that one of these scalars—<math>\Psi_4</math> in the appropriate frame—encodes the outgoing [[gravitational radiation]] of an asymptotically flat system.<ref>{{cite journal | author=Saul Teukolsky | title=Perturbations of a rotating black hole | journal=Astrophysical Journal | year=1973 | volume=185 | pages=635–647 | doi=10.1086/152444 | bibcode=1973ApJ...185..635T}}</ref>
Newman and Penrose introduced the following functions as primary quantities using this tetrad:<ref name="NP1" /><ref name="NP2" />
* Twelve complex spin coefficients (in three groups) which describe the change in the tetrad from point to point: <math>\kappa, \rho, \sigma, \tau\,;  \lambda, \mu, \nu, \pi\,; \epsilon, \gamma, \beta, \alpha. </math>.
* Five complex functions encoding Weyl tensors in the tetrad basis: <math>\Psi_0, \ldots, \Psi_4</math>.
* Ten functions encoding [[Ricci tensor]]s in the tetrad basis: <math> \Phi_{00}, \Phi_{11}, \Phi_{22}, \Lambda </math> (real); <math> \Phi_{01}, \Phi_{10}, \Phi_{02}, \Phi_{20}, \Phi_{12}, \Phi_{21} </math> (complex).
In many situations—especially algebraically special spacetimes or vacuum spacetimes—the Newman–Penrose formalism simplifies dramatically, as many of the functions go to zero.  This simplification allows for various theorems to be proven more easily than using the standard form of Einstein's equations.
In this article, we will only employ the [[tensor]]ial rather than [[spinor]]ial version of NP formalism, because the former is easier to understand and more popular in relevant papers. One can refer to ref.<ref name=ODonnell>Peter O'Donnell. ''Introduction to 2-Spinors in General Relativity''. Singapore: World Scientific, 2003.</ref> for a unified formulation of  these two versions.
==Null tetrad and sign convention==
The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric.  At each point, a [[Cartan connection applications|tetrad]] (set of four vectors) is introduced.  The first two vectors, <math>l^\mu</math> and <math>n^\mu</math> are just a pair of standard (real) [[null vectors]] such that <math>l^a n_a = -1</math>.  For example, we can think in terms of spherical coordinates, and take <math>l^a</math> to be the outgoing null vector, and <math>n^a</math> to be the ingoing null vector.  A complex null vector is then constructed by combining a pair of real, orthogonal unit space-like vectors.  In the case of spherical coordinates, the standard choice is
:<math>m^\mu = \frac{1}{\sqrt{2}}\left( \hat{\theta} + i \hat{\phi} \right)^\mu\ .</math>
The complex conjugate of this vector then forms the fourth element of the tetrad.
Two sets of signature and normalization conventions are in use for NP formalism: <math>\{(+,-,-,-); l^a n_a=1\,,m^a \bar{m}_a=-1\}</math> and <math>\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}</math>. The former is the original one that was adopted when NP formalism was developed<ref name="NP1"/><ref name="NP2"/> and has been widely used<ref name=NP3>Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Chicago: University of Chikago Press, 1983.</ref><ref name=GWaveNP>J B Griffiths. ''Colliding Plane Waves in General Relativity''. Oxford: Oxford University Press, 1991.</ref> in black-hole physics, gravitational waves and various other areas in general relativity. However, it is the latter convention that is usually employed in contemporary study of black holes from quasilocal perspectives<ref>Ivan Booth. ''Black hole boundaries''. Canadian Journal of Physics, 2005, '''83'''(11): 1073-1099. [arxiv.org/abs/gr-qc/0508107 arXiv:gr-qc/0508107v2]</ref> (such as isolated horizons<ref name=refIH>Abhay Ashtekar, Christopher Beetle, Jerzy Lewandowski. ''Geometry of generic isolated horizons''.  Classical and Quantum Gravity, 2002, '''19'''(6): 1195-1225. [https://arxiv.org/abs/gr-qc/0111067 arXiv:gr-qc/0111067v2]</ref> and dynamical horizons<ref>Abhay Ashtekar, Badri Krishnan. ''Dynamical horizons: energy, angular momentum, fluxes and balance laws''. Physical Review Letters, 2002, '''89'''(26): 261101. [arxiv.org/abs/gr-qc/0207080 arXiv:gr-qc/0207080v3]</ref><ref>Abhay Ashtekar, Badri Krishnan. ''Dynamical horizons and their properties''. Physical Review D, 2003, '''68'''(10): 104030. [arxiv.org/abs/gr-qc/0308033 arXiv:gr-qc/0308033v4]</ref>). In this article, we will utilize <math>\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}</math> for a systematic review of the NP formalism (see also refs.<ref name=refNP1>Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.</ref><ref name=refNP2>Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.</ref><ref name=refNP3>Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. ''Isolated horizons: Hamiltonian evolution and the first law''. Physical Review D, 2000, '''62'''(10): 104025. Appendix B. [https://arxiv.org/abs/gr-qc/0005083 gr-qc/0005083]</ref>).
It's important to note that, when switching from <math>\{(+,-,-,-)\,,l^a n_a=1\,,m^a \bar{m}_a=-1\}</math> to <math>\{(-,+,+,+)\,,l^a n_a=-1\,,m^a \bar{m}_a=1\}</math>, definitions of the spin coefficients,  Weyl-NP scalars <math>\Psi_{i}</math> and Ricci-NP scalars <math>\Phi_{ij}</math> need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged.
In NP formalism, the complex null tetrad contains two real null (co)vectors <math>\{\ell\,,n\}</math> and two complex null (co)vectors <math>\{m\,, \bar m\}</math>. Being ''null'' (co)vectors, ''self''-normalization of <math>\{\ell\,,n\}</math> naturally vanishes,
<br />
<math>l_a l^a=n_a n^a=m_a m^a=\bar{m}_a \bar{m}^a=0</math>,
so the following two pairs of ''cross''-normalization are adopted
<br />
<math>l_a n^a=-1=l^a n_a\,,\quad m_a \bar{m}^a=1=m^a \bar{m}_a\,,</math>
while  contractions between the two pairs are also vanishing,
<br />
<math>l_a m^a=l_a \bar{m}^a=n_a m^a=n_a \bar{m}^a=0</math>.
Here the indices can be raised and lowered by the global [[Metric tensor|metric]] <math>g_{ab}</math>  which in turn can be obtained via
<br />
<math>g_{ab}=-l_a  n_b - n_a  l_b +m_a  \bar{m}_b +\bar{m}_a  m_b\,,  \quad g^{ab}=-l^a  n^b - n^a  l^b +m^a  \bar{m}^b +\bar{m}^a  m^b\,.</math>
== NP quantities and tetrad equations ==
=== Four covariant derivative operators ===
In keeping with the formalism's practice of using distinct unindexed symbols for each component of an object, the [[covariant derivative]] operator <math>\nabla_a</math> is expressed using four separate symbols (<math>D, \Delta, \delta, \bar{\delta}</math>) which name a [[Directional derivative|directional]] [[covariant derivative]] operator for each tetrad direction. Given a linear combination of tetrad vectors,  <math>X^a=\mathrm{a}l^a+\mathrm{b}n^a+\mathrm{c}m^a+\mathrm{d}\bar{m}^a</math>, the covariant derivative operator in the <math>X^a</math> direction is <math>\nabla_X = X^a\nabla_a=(\mathrm{a}D+\mathrm{b}\Delta+\mathrm{c}\delta+\mathrm{d}\bar{\delta})</math>.
The operators are defined as
<br />
<math>D:= \nabla_\boldsymbol{l}=l^a\nabla_a\,,\; \Delta:= \nabla_\boldsymbol{n}=n^a\nabla_a\,, </math><br />
<math>\delta := \nabla_\boldsymbol{m}=m^a\nabla_a\,,\; \bar{\delta} := \nabla_\boldsymbol{\bar{m}}=\bar{m}^a\nabla_a\,,</math>
which reduce to <math>D=l^a\partial_a\,, \Delta=n^a\partial_a\,,\delta=m^a\partial_a\,,\bar{\delta}=\bar{m}^a\partial_a </math> when acting on ''scalar'' functions.
=== Twelve spin coefficients ===
In NP formalism, instead of using index notations as in [[orthogonal tetrad]]s,  each [[Christoffel symbols#Ricci rotation coefficients (asymmetric definition)|Ricci rotation coefficient]] <math>\gamma_{ijk}</math> in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex ''spin coefficients'' (in three groups),
<br />
<math> \kappa:= -m^aDl_a=-m^a l^b \nabla_b l_a\,,\quad \tau:= -m^a\Delta l_a=-m^a n^b \nabla_b l_a\,,</math><br />
<math>  \sigma:= -m^a\delta l_a=-m^a m^b\nabla_b l_a\,, \quad  \rho := -m^a\bar{\delta} l_a=-m^a \bar{m}^b \nabla_b l_a\,; </math>
<math>\pi:= \bar{m}^aDn_a=\bar{m}^al^b\nabla_b n_a\,, \quad \nu:= \bar{m}^a\Delta n_a=\bar{m}^a n^b\nabla_b n_a\,, </math><br />
<math>\mu:= \bar{m}^a\delta n_a=\bar{m}^a m^b\nabla_b n_a\,, \quad \lambda:= \bar{m}^a\bar{\delta} n_a=\bar{m}^a \bar{m}^b \nabla_b n_a\,;</math>
<math>\varepsilon:= -\frac{1}{2}\big(n^aDl_a-\bar{m}^aDm_a \big)=-\frac{1}{2}\big(n^al^b\nabla_b l_a-\bar{m}^al^b\nabla_b m_a \big)\,,</math><br />
<math>\gamma:= -\frac{1}{2}\big(n^a\Delta l_a-\bar{m}^a\Delta m_a \big)= -\frac{1}{2}\big(n^a n^b\nabla_b l_a-\bar{m}^a n^b\nabla_b m_a \big)\,,</math><br />
<math>\beta:= -\frac{1}{2}\big(n^a\delta l_a-\bar{m}^a\delta m_a \big)=-\frac{1}{2}\big(n^a m^b\nabla_b l_a-\bar{m}^am^b\nabla_b m_a \big)\,,</math><br />
<math>\alpha:= -\frac{1}{2}\big(n^a\bar{\delta} l_a-\bar{m}^a\bar{\delta}m_a \big)=-\frac{1}{2}\big(n^a\bar{m}^b\nabla_b l_a-\bar{m}^a\bar{m}^b\nabla_b m_a \big)\,.</math>
Spin coefficients are the primary quantities in NP formalism, with which all other NP quantities (as defined below) could be calculated indirectly using the NP field equations. Thus, NP formalism is sometimes referred to as ''spin-coefficient formalism'' as well.
=== Transportation equations: covariant derivatives of tetrad vectors ===
The sixteen directional covariant derivatives of tetrad vectors are sometimes called the ''transportation/propagation equations,''{{citation needed|reason=obvious references either don't list these at all, or else don't call them transportation equations|date=May 2019}} perhaps because the derivatives are zero when the tetrad vector is parallel propagated or transported in the direction of the derivative operator.
These results in this exact notation are given by O'Donnell:<ref name="ODonnell"/>{{rp|57–58(3.220)}}
<br />
<math>D l^a=(\varepsilon+\bar{\varepsilon})l^a-\bar{\kappa}m^a-\kappa\bar{m}^a\,,</math><br /> <math>\Delta l^a=(\gamma+\bar{\gamma})l^a-\bar{\tau}m^a-\tau\bar{m}^a\,,</math><br /> <math>\delta l^a =(\bar{\alpha}+\beta)l^a-\bar{\rho}m^a-\sigma\bar{m}^a\,,</math><br /> <math>\bar{\delta} l^a=(\alpha+\bar{\beta})l^a-\bar{\sigma}m^a-\rho\bar{m}^a\,;</math>
<math>D n^a=\pi m^a+\bar{\pi}\bar{m}^a-(\varepsilon+\bar{\varepsilon})n^a\,,</math><br /> <math>\Delta n^a=\nu m^a+\bar{\nu}\bar{m}^a-(\gamma+\bar{\gamma})n^a\,,</math><br /> <math>\delta n^a=\mu m^a+\bar{\lambda}\bar{m}^a-(\bar{\alpha}+\beta)n^a\,,</math><br /> <math>\bar{\delta} n^a=\lambda m^a+\bar{\mu}\bar{m}^a-(\alpha+\bar{\beta})n^a\,;</math>
<math>D m^a=(\varepsilon-\bar{\varepsilon})m^a+\bar{\pi}l^a-\kappa n^a\,,</math><br /> <math>\Delta m^a=(\gamma-\bar{\gamma})m^a+\bar{\nu}l^a-\tau n^a\,,</math><br /> <math>\delta m^a=(\beta-\bar{\alpha})m^a+\bar{\lambda}l^a-\sigma n^a\,,</math><br /> <math>\bar{\delta} m^a=(\alpha-\bar{\beta})m^a+\bar{\mu}l^a-\rho n^a\,;</math>
<math>D \bar{m}^a=(\bar{\varepsilon}-\varepsilon)\bar{m}^a+\pi l^a-\bar{\kappa} n^a\,,</math><br /> <math>\Delta \bar{m}^a=(\bar{\gamma}-\gamma)\bar{m}^a+\nu l^a-\bar{\tau} n^a\,,</math><br /> <math>\delta \bar{m}^a=(\bar{\alpha}-\beta)\bar{m}^a+\mu l^a-\bar{\rho} n^a\,,</math><br /> <math>\bar{\delta} \bar{m}^a=(\bar{\beta}-\alpha)\bar{m}^a+\lambda l^a-\bar{\sigma} n^a\,.</math>
==== Interpretation of <math>\kappa, \varepsilon, \nu, \gamma</math> from <math>D l^a</math> and <math>\Delta n^a</math> ====
The two equations for the covariant derivative of a real null tetrad vector in its own direction indicate whether or not the vector is tangent to a geodesic and if so, whether the geodesic has an affine parameter.
A null tangent vector <math>T^a</math> is tangent to an affinely parameterized null geodesic if <math>T^b\nabla_bT^a=0</math>, which is to say if the vector is unchanged by parallel propagation or transportation in its own direction.<ref>{{cite book|title=General Relativity|url=https://archive.org/details/generalrelativit0000wald|url-access=registration|author=Robert M. Wald|date=1984|isbn=9780226870335}}</ref>{{rp|41(3.3.1)}}
<math>D l^a=(\varepsilon+\bar{\varepsilon})l^a-\bar{\kappa}m^a-\kappa\bar{m}^a</math> shows that <math>l^a</math> is tangent to a geodesic if and only if <math>\kappa=0</math>, and is tangent to an affinely parameterized geodesic if in addition <math>(\varepsilon+\bar{\varepsilon})=0
</math>. Similarly, <math>\Delta n^a=\nu m^a+\bar{\nu}\bar{m}^a-(\gamma+\bar{\gamma})n^a</math> shows that <math>n^a</math> is geodesic if and only if <math>\nu=0</math>, and has affine parameterization when <math>(\gamma+\bar{\gamma})=0</math>.
(The complex null tetrad vectors <math>m^a=x^a+iy^a</math> and  <math>\bar{m}^a=x^a-iy^a</math> would have to be separated into the spacelike basis vectors <math>x^a</math> and <math>y^a</math> before asking if either or both of those are tangent to spacelike geodesics.)
=== Commutators ===
The [[Metric compatibility|metric-compatibility]] or [[Torsion tensor|torsion-freeness]] of the covariant derivative is recast into the ''[[commutator]]s of the directional derivatives'',
<br />
<math>\Delta D-D\Delta=(\gamma+\bar{\gamma})D+(\varepsilon+\bar{\varepsilon})\Delta-(\bar{\tau}+\pi)\delta-(\tau+\bar{\pi})\bar{\delta}\,,</math><br /> <math>\delta D-D\delta=(\bar{\alpha}+\beta-\bar{\pi})D+\kappa\Delta-(\bar{\rho}+\varepsilon-\bar{\varepsilon})\delta-\sigma\bar{\delta}\,,</math><br /> <math>\delta\Delta-\Delta\delta=-\bar{\nu}D+(\tau-\bar{\alpha}-\beta)\Delta+(\mu-\gamma+\bar{\gamma})\delta+\bar{\lambda}\bar{\delta}\,,</math><br /> <math>\bar{\delta}\delta-\delta\bar{\delta}=(\bar{\mu}-\mu)D+(\bar{\rho}-\rho)\Delta+(\alpha-\bar{\beta})\delta-(\bar{\alpha}-\beta)\bar{\delta}\,,</math>
which imply that
<br />
<math>\Delta l^a-D n^a=(\gamma+\bar{\gamma})l^a+(\varepsilon+\bar{\varepsilon})n^a-(\bar{\tau}+\pi)m^a-(\tau+\bar{\pi})\bar{m}^a\,,</math><br />
<math>\delta l^a-D m^a=(\bar{\alpha}+\beta-\bar{\pi})l^a+\kappa n^a-(\bar{\rho}+\varepsilon-\bar{\varepsilon}) m^a-\sigma\bar{m}^a\,,</math><br />
<math>\delta n^a-\Delta m^a=-\bar{\nu}l^a+(\tau-\bar{\alpha}-\beta)n^a+(\mu-\gamma+\bar{\gamma})m^a+\bar{\lambda}\bar{m}^a\,,</math><br />
<math>\bar{\delta}m^a-\delta\bar{m}^a=(\bar{\mu}-\mu)l^a+(\bar{\rho}-\rho)n^a+(\alpha-\bar{\beta})m^a-(\bar{\alpha}-\beta)\bar{m}^a\,.</math>
Note: (i) The above equations can be regarded either as implications of the commutators or combinations of the transportation equations; (ii) In these implied equations, the vectors <math>\{l^a,n^a,m^a,\bar{m}^a\}</math> can be replaced by the covectors and the equations still hold.
=== Weyl–NP and Ricci–NP scalars ===
The 10 independent components of the [[Weyl tensor]] can be encoded into 5 complex [[Weyl scalar|Weyl-NP scalars]],
<br />
<math>\Psi_0:= C_{abcd} l^a m^b l^c m^d\,,</math>
<math>\Psi_1:= C_{abcd} l^a n^b l^c m^d\,,</math>
<math>\Psi_2:= C_{abcd} l^a m^b\bar{m}^c n^d\,,</math>
<math>\Psi_3:= C_{abcd} l^a n^b\bar{m}^c n^d\,,</math>
<math>\Psi_4:= C_{abcd} n^a \bar{m}^b n^c \bar{m}^d\,.</math>
The 10 independent components of the [[Ricci tensor]] are encoded into 4 ''real'' scalars <math>\{\Phi_{00}</math>, <math>\Phi_{11}</math>, <math>\Phi_{22}</math>, <math>\Lambda\}</math> and 3 ''complex'' scalars <math>\{\Phi_{10},\Phi_{20},\Phi_{21} \}</math> (with their complex conjugates),
<br />
<math>\Phi_{00}:=\frac{1}{2}R_{ab}l^a l^b\,, \quad \Phi_{11}:=\frac{1}{4}R_{ab}(\,l^a n^b+m^a\bar{m}^b)\,, \quad\Phi_{22}:=\frac{1}{2}R_{ab}n^a n^b\,, \quad\Lambda:=\frac{R}{24}\,;</math>
<math>\Phi_{01}:=\frac{1}{2}R_{ab}l^a m^b\,, \quad\; \Phi_{10}:=\frac{1}{2}R_{ab}l^a \bar{m}^b=\overline{\Phi_{01}}\,,</math><br /> <math>\Phi_{02}:=\frac{1}{2}R_{ab}m^a m^b\,, \quad \Phi_{20}:=\frac{1}{2}R_{ab}\bar{m}^a \bar{m}^b=\overline{\Phi_{02}}\,,</math><br /> <math>\Phi_{12}:=\frac{1}{2}R_{ab} m^a n^b\,, \quad\; \Phi_{21}:=\frac{1}{2}R_{ab} \bar{m}^a n^b=\overline{\Phi_{12}}\,.</math>
In these definitions, <math>R_{ab}</math> could be replaced by its [[Trace-free Ricci tensor|trace-free]] part <math>\displaystyle Q_{ab}=R_{ab}-\frac{1}{4}g_{ab}R</math><ref name="refNP2"/> or by the [[Einstein tensor]] <math>\displaystyle G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R</math> because of the normalization relations. Also, <math>\Phi_{11}</math> is reduced to <math>\Phi_{11}=\frac{1}{2}R_{ab}l^a n^b=\frac{1}{2}R_{ab}m^a\bar{m}^b</math> for [[Electrovacuum solution|electrovacuum]] (<math>\Lambda=0</math>).
==Einstein–Maxwell–NP equations==
===NP field equations===
In a complex null tetrad, Ricci identities give rise to the following NP field equations connecting spin coefficients, Weyl-NP and Ricci-NP scalars (recall that in an orthogonal tetrad, Ricci rotation coefficients would respect [[Riemannian connection on a surface#Cartan structural equations|Cartan's first and second structure equations]]),<ref name="ODonnell"/><ref name="refNP2"/>
These equations in various notations can be found in several texts.<ref name="Chandra"/>{{rp|46–47(310(a)-(r))}}<ref name="refNP2"/>{{rp|671–672(E.12)}} The notation in Frolov and Novikov
<ref name="refNP2"/> is identical.
<br />
<math>D\rho -\bar{\delta}\kappa=(\rho^2+\sigma\bar{\sigma})+(\varepsilon+\bar{\varepsilon})\rho-\bar{\kappa}\tau-\kappa(3\alpha+\bar{\beta}-\pi)+\Phi_{00}\,,</math><br /> <math>D\sigma-\delta\kappa=(\rho+\bar{\rho})\sigma+(3\varepsilon-\bar{\varepsilon})\sigma-(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa+\Psi_0\,,</math><br /> <math>D\tau-\Delta\kappa=(\tau+\bar{\pi})\rho+(\bar{\tau}+\pi)\sigma+(\varepsilon-\bar{\varepsilon})\tau-(3\gamma+\bar{\gamma})\kappa+\Psi_1+\Phi_{01}\,,</math><br />
<math>D\alpha-\bar{\delta}\varepsilon=(\rho+\bar{\varepsilon}-2\varepsilon)\alpha+\beta\bar{\sigma}-\bar{\beta}\varepsilon-\kappa\lambda-\bar{\kappa}\gamma+(\varepsilon+\rho)\pi+\Phi_{10}\,,</math><br /> <math>D\beta-\delta\varepsilon=(\alpha+\pi)\sigma+(\bar{\rho}-\bar{\varepsilon})\beta-(\mu+\gamma)\kappa-(\bar{\alpha}-\bar{\pi})\varepsilon+\Psi_1\,,</math><br /> <math>D\gamma-\Delta\varepsilon=(\tau+\bar{\pi})\alpha+(\bar{\tau}+\pi)\beta-(\varepsilon+\bar{\varepsilon})\gamma-(\gamma+\bar{\gamma})\varepsilon+\tau\pi-\nu\kappa+\Psi_2+\Phi_{11}-\Lambda\,,</math><br /> <math>D\lambda-\bar{\delta}\pi=(\rho\lambda+\bar{\sigma}\mu)+\pi^2+(\alpha-\bar{\beta})\pi-\nu\bar{\kappa}-(3\varepsilon-\bar{\varepsilon})\lambda+\Phi_{20}\,,</math><br /> <math>D\mu-\delta\pi=(\bar{\rho}\mu+\sigma\lambda)+\pi\bar{\pi}-(\varepsilon+\bar{\varepsilon})\mu-(\bar{\alpha}-\beta)\pi-\nu\kappa+\Psi_2+2\Lambda\,,</math><br /> <math>D\nu-\Delta\pi=(\pi+\bar{\tau})\mu+(\bar{\pi}+\tau)\lambda+(\gamma-\bar{\gamma})\pi-(3\varepsilon+\bar{\varepsilon})\nu+\Psi_3+\Phi_{21}\,,</math><br /> <math>\Delta\lambda-\bar{\delta}\nu=-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})\lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu-\Psi_4\,,</math><br /> <math>\delta\rho-\bar{\delta}\sigma=\rho(\bar{\alpha}+\beta)-\sigma(3\alpha-\bar{\beta})+(\rho-\bar{\rho})\tau+(\mu-\bar{\mu})\kappa-\Psi_1+\Phi_{01}\,,</math><br /> <math>\delta\alpha-\bar{\delta}\beta=(\mu\rho-\lambda\sigma)+\alpha\bar{\alpha}+\beta\bar{\beta}-2\alpha\beta+\gamma(\rho-\bar{\rho})+\varepsilon(\mu-\bar{\mu})-\Psi_2+\Phi_{11}+\Lambda\,,</math><br /> <math>\delta\lambda-\bar{\delta}\mu=(\rho-\bar{\rho})\nu+(\mu-\bar{\mu})\pi+(\alpha+\bar{\beta})\mu+(\bar\alpha-3\beta)\lambda-\Psi_3+\Phi_{21}\,,</math><br />
<math>\delta\nu-\Delta\mu=(\mu^2+\lambda\bar{\lambda})+(\gamma+\bar{\gamma})\mu-\bar{\nu}\pi+(\tau-3\beta-\bar{\alpha})\nu+\Phi_{22}\,,</math><br /> <math>\delta\gamma-\Delta\beta=(\tau-\bar{\alpha}-\beta)\gamma+\mu\tau-\sigma\nu-\varepsilon\bar{\nu}-(\gamma-\bar{\gamma}-\mu)\beta+\alpha\bar{\lambda}+\Phi_{12}\,,</math><br /> <math>\delta\tau-\Delta\sigma=(\mu\sigma+\bar{\lambda}\rho)+(\tau+\beta-\bar{\alpha})\tau-(3\gamma-\bar{\gamma})\sigma-\kappa\bar{\nu}+\Phi_{02}\,,</math><br /> <math>\Delta\rho-\bar{\delta}\tau=-(\rho\bar{\mu}+\sigma\lambda)+(\bar{\beta}-\alpha-\bar{\tau})\tau+(\gamma+\bar{\gamma})\rho+\nu\kappa-\Psi_2-2\Lambda\,,</math><br /> <math>\Delta\alpha-\bar{\delta}\gamma=(\rho+\varepsilon)\nu-(\tau+\beta)\lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma-\Psi_3\,.</math>
Also, the Weyl-NP scalars <math>\Psi_{i}</math> and the Ricci-NP scalars <math>\Phi_{ij}</math> can be calculated indirectly from the above NP field equations after obtaining the spin coefficients rather than directly using their definitions.
===Maxwell–NP scalars, Maxwell equations in NP formalism===
The six independent components of the Faraday-Maxwell 2-form (i.e. the [[Electromagnetic tensor|electromagnetic field strength tensor]]) <math>F_{ab}</math> can be encoded into three complex Maxwell-NP scalars<ref name="refNP1"/>
<br />
<math>\phi_0:= F_{ab}l^a m^b \,,\quad \phi_1:= \frac{1}{2} F_{ab}\big(l^an^b + \bar{m}^a m^b \big)\,, \quad \phi_2 := F_{ab} \bar{m}^a n^b\,,</math>
and therefore the eight real [[Maxwell equations]] <math>d\mathbf{F}=0</math> and <math>d^{\star}\mathbf{F}=0</math>  (as <math>\mathbf{F}=dA</math>) can be transformed into four complex equations,
<br />
<math>D\phi_1 -\bar{\delta}\phi_0=(\pi-2\alpha)\phi_0+2\rho\phi_1-\kappa\phi_2\,,  </math><br /> <math> D\phi_2 -\bar{\delta}\phi_1=-\lambda\phi_0+2\pi\phi_1+(\rho-2\varepsilon)\phi_2\,, </math><br /> <math>  \Delta\phi_0-\delta\phi_1=(2\gamma-\mu)\phi_0-2\tau\phi_1+\sigma\phi_2\,, </math><br /> <math>\Delta\phi_1-\delta\phi_2=\nu\phi_0-2\mu\phi_1+(2\beta-\tau)\phi_2\,, </math>
<br />
with the Ricci-NP scalars <math>\Phi_{ij}</math> related to Maxwell scalars by<ref name="refNP1"/>
<br />
<math>\Phi_{ij}=\,2\,\phi_i\,\overline{\phi_j}\,,\quad (i,j\in\{0,1,2\})\,.</math>
It is worthwhile to point out that, the supplementary equation <math>\Phi_{ij}=2\,\phi_i\, \overline{\phi_j}</math> is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be <math>\Phi_{ij}=\,\text{Tr}\,(\digamma_i \,\bar{\digamma}_j)</math>  where <math>\digamma_i (i\in\{0,1,2 \})</math> are Yang-Mills-NP scalars.<ref>E T Newman, K P Tod. ''Asymptotically Flat Spacetimes'', Appendix A.2. In A Held (Editor): ''General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein''. Vol(2), page 27.  New York and London:  Plenum Press, 1980.</ref>
To sum up, the aforementioned transportation equations, NP field equations and Maxwell-NP equations together constitute the Einstein-Maxwell equations in Newman–Penrose formalism.
==Applications of NP formalism to gravitational radiation field==
The Weyl scalar <math>\Psi_4</math> was defined by Newman & Penrose as
:<math>\Psi_4 = -C_{\alpha\beta\gamma\delta} n^\alpha \bar{m}^\beta n^\gamma \bar{m}^\delta</math>
(note, however, that the overall sign is [[sign convention|arbitrary]], and that Newman & Penrose worked with a "timelike" metric signature of <math>(+,-,-,-)</math>).
In empty space, the [[Einstein Field Equations]] reduce to <math>R_{\alpha\beta}=0</math>.  From the definition of the Weyl tensor, we see that this means that it equals the [[Riemann tensor]], <math>C_{\alpha\beta\gamma\delta} = R_{\alpha\beta\gamma\delta}</math>.  We can make the standard choice for the tetrad at infinity:
:<math>l^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{t} + \hat{r} \right)\ ,</math>
:<math>n^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{t} - \hat{r} \right)\ ,</math>
:<math>m^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{\theta} + i\hat{\phi} \right)\ .</math>
In transverse-traceless gauge, a simple calculation shows that linearized [[gravitational waves]] are related to components of the Riemann tensor as
:<math> \frac{1}{4}\left( \ddot{h}_{\hat{\theta}\hat{\theta}} - \ddot{h}_{\hat{\phi}\hat{\phi}} \right) = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}} = -R_{\hat{t}\hat{\phi}\hat{r}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{t}\hat{\phi}\hat{t}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{r}\hat{\phi}\hat{r}\hat{\phi}}\ ,</math>
:<math> \frac{1}{2} \ddot{h}_{\hat{\theta}\hat{\phi}} = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{r}\hat{\theta}\hat{t}\hat{\phi}}\ ,</math>
assuming propagation in the <math>\hat{r}</math> direction.  Combining these, and using the definition of <math>\Psi_4</math> above, we can write
:<math> \Psi_4 = \frac{1}{2}\left( \ddot{h}_{\hat{\theta} \hat{\theta}} - \ddot{h}_{\hat{\phi} \hat{\phi}} \right) + i \ddot{h}_{\hat{\theta}\hat{\phi}} = -\ddot{h}_+ + i \ddot{h}_\times\ . </math>
Far from a source, in nearly flat space, the fields <math>h_+</math> and <math>h_\times</math> encode everything about gravitational radiation propagating in a given direction.  Thus, we see that <math>\Psi_4</math> encodes in a single complex field everything about (outgoing) gravitational waves.
===Radiation from a finite source===
Using the wave-generation formalism summarised by Thorne,<ref>{{cite journal|title=Multipole expansions of gravitational radiation | author=Thorne, Kip S. | journal=Rev. Mod. Phys. | volume=52 |  issue=2 | pages=299–339 |date=April 1980|doi=10.1103/RevModPhys.52.299 | bibcode=1980RvMP...52..299T|url=http://authors.library.caltech.edu/11159/1/THOrmp80a.pdf }} A broad summary of the mathematical formalism used in the literature on gravitational radiation.</ref> we can write the radiation field quite compactly in terms of the [[mass multipole]], [[current multipole]], and [[spin-weighted spherical harmonics]]:
:<math>\Psi_4(t,r,\theta,\phi) = - \frac{1}{r\sqrt{2}} \sum_{l=2}^{\infty} \sum_{m=-l}^l \left[ {}^{(l+2)}I^{lm}(t-r) -i\ {}^{(l+2)}S^{lm}(t-r) \right] {}_{-2}Y_{lm}(\theta,\phi)\ . </math>
Here, prefixed superscripts indicate time derivatives.  That is, we define
:<math>{}^{(l)}G(t) = \left( \frac{d}{dt} \right)^l G(t)\ .</math>
The components <math>I^{lm}</math> and <math>S^{lm}</math> are the mass and current multipoles, respectively.  <math>{}_{-2}Y_{lm}</math> is the spin-weight -2 spherical harmonic.
== See also ==
* [[Light-cone coordinates]]
* [[GHP formalism]]
* [[Tetrad formalism]]
* [[Goldberg–Sachs theorem]]
==References==
<references/>
* {{cite book
| last = Wald
| first = Robert
| author-link = Robert Wald
| title = General Relativity
| url = https://archive.org/details/generalrelativit0000wald
| url-access = registration
| publisher = [[University of Chicago Press]]
| year = 1984
| isbn = 0-226-87033-2 }} Wald treats the more succinct version of the Newman–Penrose formalism in terms of more modern spinor notation.
* {{cite book
| author = S. W. Hawking and G. F. R. Ellis
| title = The large scale structure of space-time
| publisher = [[Cambridge University Press]]
| year = 1973
| isbn = 0-226-87033-2
| url-access = registration
| url = https://archive.org/details/generalrelativit0000wald
}} Hawking and Ellis use the formalism in their discussion of the final state of a collapsing star.
==External links==
*[http://www.scholarpedia.org/article/Spin-coefficient_formalism Newman–Penrose formalism on Scholarpedia]
{{Roger Penrose}}
{{DEFAULTSORT:Newman-Penrose Formalism}}
[[Category:Theory of relativity]]
[[Category:Mathematical notation]]