The Newman–Penrose (NP) formalism[1][2] 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 forms of the usual variables used in GR.
Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables.
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.
in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.
[4] Newman and Penrose introduced the following functions as primary quantities using this tetrad:[1][2] 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 tensorial rather than spinorial version of NP formalism, because the former is easier to understand and more popular in relevant papers.
The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric.
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:
The former is the original one that was adopted when NP formalism was developed[1][2] and has been widely used[6][7] in black-hole physics, gravitational waves and various other areas in general relativity.
need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged.
In keeping with the formalism's practice of using distinct unindexed symbols for each component of an object, the covariant derivative operator
In NP formalism, instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient
in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex spin coefficients (in three groups),
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.
, which is to say if the vector is unchanged by parallel propagation or transportation in its own direction.
The 10 independent components of the Weyl tensor can be encoded into 5 complex Weyl-NP scalars,
The 10 independent components of the Ricci tensor are encoded into 4 real scalars
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 Cartan's first and second structure equations),[5][13] These equations in various notations can be found in several texts.
− β ) γ + μ τ − σ ν − ε
= ( ρ + ε ) ν − ( τ + β ) λ + (
can be calculated indirectly from the above NP field equations after obtaining the spin coefficients rather than directly using their definitions.
The six independent components of the Faraday-Maxwell 2-form (i.e. the electromagnetic field strength tensor)
(note, however, that the overall sign is arbitrary, and that Newman & Penrose worked with a "timelike" metric signature of
In empty space, the Einstein Field Equations reduce to
In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as
encodes in a single complex field everything about (outgoing) gravitational waves.
Using the wave-generation formalism summarised by Thorne,[17] we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonics: