Newman-Penrose Formalism
Encyclopedia
The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman
and Roger Penrose
for General Relativity
. 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.
The NP formalism is itself a special case of the tetrad formalism
, 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 space-time, 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 asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime.
The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor
. In particular, it can be shown that one of these scalars--
in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.
(set of four vectors) is introduced. The first two vectors,
and 
are just a pair of standard (real) null vectors such that 
. For example, we can think in terms of spherical coordinates, and take 
to be the outgoing null vector, and 
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
The complex conjugate of this vector then forms the fourth element of the tetrad. The orthogonality relations between these vectors are then:
if we assume the usual
sign convention
for the metric.
Newman and Penrose then introduce some functions using this tetrad:
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.
was defined by Newman & Penrose as
(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
. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, 
. We can make the standard choice for the tetrad at infinity:
In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as
assuming propagation in the
direction. Combining these, and using the definition of 
above, we can write
Far from a source, in nearly flat space, the fields
and 
encode everything about gravitational radiation propagating in a given direction. Thus, we see that 
encodes in a single complex field everything about (outgoing) gravitational waves.
:
Here, prefixed superscripts indicate time derivatives. That is, we define
The components
and 
are the mass and current multipoles, respectively. 
is the spin-weight -2 spherical harmonic.
Ezra T. Newman
Ezra Ted Newman is an American physicist, known for his many contributions to general relativity theory. He is Professor Emeritus at the University of Pittsburgh...
and Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
for General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. Their notation is an effort to treat General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
in terms of spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
notation, which introduces complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
forms of the usual variables used in GR.
The NP formalism is itself a special case of the tetrad formalism
Tetrad formalism
The tetrad formalism is a non-coordinate approach to General Relativity.In the tetrad formalism all tensors are represented in terms of a chosen basis, a set of four vectors called the tetrad....
, 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 space-time, 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 asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime.
The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
. In particular, it can be shown that one of these scalars--
in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.Notation
The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric. At each point, a tetradCartan connection applications
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...
(set of four vectors) is introduced. The first two vectors,
and are just a pair of standard (real) null vectors such that . For example, we can think in terms of spherical coordinates, and take to be the outgoing null vector, and 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 isThe complex conjugate of this vector then forms the fourth element of the tetrad. The orthogonality relations between these vectors are then:
if we assume the usual
sign conventionSign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
for the metric.
Newman and Penrose then introduce some functions using this tetrad:
- twelve complex spin coefficients which describe the change in the tetrad from point to point:
. - five complex functions encoding various pieces of the Weyl tensor in the tetrad basis:
. - ten complex functions encoding pieces of the Ricci tensor in the tetrad basis:
.
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.
Radiation field
The Weyl scalar was defined by Newman & Penrose as(note, however, that the overall sign is arbitrary
Sign convention
In physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...
, and that Newman & Penrose worked with a "timelike" metric signature of
).In empty space, the Einstein Field Equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
reduce to
. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, . We can make the standard choice for the tetrad at infinity:In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as
assuming propagation in the
direction. Combining these, and using the definition of above, we can writeFar from a source, in nearly flat space, the fields
and encode everything about gravitational radiation propagating in a given direction. Thus, we see that encodes in a single complex field everything about (outgoing) gravitational waves.Radiation from a finite source
Using the wave-generation formalism summarised by Thorne, we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonicsSpin-weighted spherical harmonics
Spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U gauge fields rather than scalar fields: mathematically, they take...
:
Here, prefixed superscripts indicate time derivatives. That is, we define
The components
and are the mass and current multipoles, respectively. is the spin-weight -2 spherical harmonic.