The Newman-Penrose Pseudoscalars Ψ

Table Of Contents

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In Newtonian mechanics gravitation is described by a single quantity, the “strength” of the gravitational field as caused by the distribution of masses in space and time. Each massive particle in the universe contributes to the gravitational field at a certain point in space proportional to its mass and inverse proportional to its distance.

Newtonian gravity $\Phi(p)$ at a point $p$ is described as

$\Phi(p) = \sum_i G \frac{M_i}{\left| r_i - p \right|}$

where $r_i$ , $M_i$ denotes the location and mass of the i^th particle. For a smooth distribution of mass, the sum is replaced by an integral. The acceleration of a test mass is given by the gradient of the gravitational field $\nabla \Phi(p)$ .

In General Relativity gravitation is described via the “change” (aka curvature) of space-time. Spacetime itself is described by the distances between points in space and time. These distances are modeled via the metric tensor field, which in four dimensions encompasses 10 independent quantities. The changes of the metric tensor field are described via the Riemann tensor, which in four dimensions encompasses 20 independent quantities, built from the metric tensor and its first and second derivatives. These 20 components can be separated the Ricci and Weyl tensors, each carrying 10 independent quantities. In vacuum – such as gravitational waves – all components of the Ricci tensor are zero. The remaining 10 components of the Weyl tensor describe the gravitational field in vacuum and can be written as five complex quantities, the Newman-Penrose pseudo-scalar fields Ψ_i.

The mathematics of the Newman-Penrose formalism have been reviewed using the simplifying context of Geometric Algebra in the article Using Geometric Algebra for Navigation in Riemannian and Hard Disc Space.

Quadrupole Radiation and Ψ

Spherically symmetric and stationary rotating mass distributions do not radiate, thus there is no monopole momentum in gravitational waves. Neither is there a dipole momentum like in electro-magnetism since there is no “negative mass”. The lowest order is a quadrupole momentum.

+ – Polarization of the gravitational wave as indicated by Re Ψ4

× – Polarization of the gravitational wave as indicated by Im Ψ4

The figure illustrates gravitational waves, in their two possible linear polarizations, + and ×, in a locally inertial system of points a ﬁxed proper distance apart. The ellipses represent a system of freely-falling test masses whose positions, initially on a circle, are being perturbed by a gravitational wave moving in a direction perpendicular to the screen. The proper distance between freely-falling test masses oscillates.

Tetrad Frame

The NP formalism starts with the rest frame of an observer, and applies two tricks to it. The axes, or tetrad, of the observer’s locally inertial frame form an orthonormal basis of tangential vectors in the geometric algebra {γ_t , γ_x, γ_y, γ_z} with the Minkowski metric (-,+,+,+) as inner product such that (the signature (+,-,-,-) is an alternative convention):

(1) $\begin{equation*} \gamma_m \cdot \gamma_n = \left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \end{equation*}$

with indices $m$ , $n$ running over $t, x, y, z$ . The NP formalism chooses one axis, typically the $z$ -axis, to be the direction of propagation of the wave.

The first NP trick is to replace the transverse axes $\gamma_x$ and $\gamma_y$ by spinor axes $\gamma_{+}$ and $\gamma_{-}$ defined by

(2) $\begin{equation*} \gamma_{+} := \frac{1}{\sqrt{2}} \left( \gamma_x + \im \gamma_y \right) \ , \quad \gamma_{-} := \frac{1}{\sqrt{2}} \left( \gamma_x - \im \gamma_y \right) \ . \end{equation*}$

The second NP trick is to replace the time $t$ and propagation $z$ axes with outgoing and ingoing null axes $\gamma_v$ and $\gamma_u$ , defined by

(3) $\begin{equation*} \gamma_v := \frac{1}{\sqrt{2}} \left( \gamma_t + \gamma_z \right) \ , \quad \gamma_u := \frac{1}{\sqrt{2}} \left( \gamma_t - \gamma_z \right) \ . \end{equation*}$

The resulting outgoing, ingoing, and spinor axes form a NP null tetrad {γ_u , γ_v, γ₊, γ_–} with the metric

(4) $\begin{equation*} \gamma_m \cdot \gamma_n = \left( \begin{array}{cccc} 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right) \end{equation*}$

where indices $m$ , $n$ run over $v, u, +, -$ . The NP metric (4) has zeros down the diagonal. This means that each of the four NP axes $\gamma_m$ is null: the scalar product of each axis with itself is zero. In a profound sense, the null, or lightlike, character of each the four NP axes explains why the NP formalism is well adapted to treating fields that propagate at the speed of light.

Three kinds of transformation take a particularly simple form in the NP tetrad: reflections through the transverse axis y, rotations about the propagation axis z and boosts along the propagation axis z. See the article referenced at the end of this page for details.

Gravitational Radiation

The Riemann curvature tensor has symmetries which can be designated shorthandly as $R_{([kl][mn])}$ where [] denotes antisymmetry, and () symmetry. This designation thus signifies that the Riemann curvature tensor $R_{klmn}$ is antisymmetric in its first two indices $kl$ , antisymmetric in its last two indices $mn$ , and symmetric under exchange of the first and last pairs of indices, $kl \leftrightarrow mn$ . In addition to these symmetries the Riemann curvature tensor also provides the totally antisymmetric property

(5) $\begin{equation*} R_{klmn} + R_{kmnl} + R_{knlm} = 0 \ . \end{equation*}$

The symmetries on the index pairs imply that the Riemann curvature tensor is a symmetric matrix of antisymmetric tensors, which is to say, a $6 \times 6$ symmetric matrix operating on bi-vectors, remembering that a bi-vector has six components in 4D spacetime (three time-like “electrical” bi-vectors plus three space-like “magnetic” bi-vectors), i.e. R(u,v) is a real number for any bi-vectors u,v.

A $6 \times 6$ symmetric matrix has 21 independent components. The additional condition (5) eliminates one more degree of freedom, leaving the Riemann curvature tensor with 20 independent components.

Electric and Magnetic Parts of the Riemann Tensor

The geometric algebra shows that with respect to a locally inertial frame the 6 bivectors can be organized into 3 electric (E) bivectors ${ \gamma_{tx} , \gamma_{ty} , \gamma_{tz} }$ , and 3 magnetic (B) bivectors ${ \gamma_{yz} , \gamma_{zx} , \gamma_{xy} }$ . The Riemann tensor can thus be organized into a $2 \times 2$ matrix of $3 \times 3$ blocks, with the structure

(6) $\begin{equation*} \left( \begin{array}{cc} EE & EB \\ BE & BB \end{array} \right) \ . \end{equation*}$

The condition of being symmetric implies that $EE$ and $BB$ are symmetric, while $BE = (EB)^\top$ . The condition (5) states that the $3 \times 3$ block $EB$ (and likewise $BE$ ) is traceless.

The natural complex structure of bivectors in the geometric algebra suggests recasting the $6 \times 6$ Riemann curvature matrix (6) into a $3 \times 3$ complex matrix, which would have the structure $( E + \operatorname{Im} B ) ( E + \operatorname{Im} B )$ , or equivalently

(7) $\begin{equation*} EE - BB + \text{Im}( EB + BE ) \end{equation*}$

, which is a complex linear combination of the four $3 \times 3$ blocks of the Riemann matrix (6). However, it turns out that the complex symmetric $3 \times 3$ matrix (7) encodes only part of the Riemann curvature tensor, namely the Weyl tensor. More specifically, the Riemann curvature tensor decomposes into a trace part, the Ricci tensor $R_{km}$ , and a totally traceless part, the Weyl tensor $C_{klmn}$ . The Ricci tensor, which is symmetric, has 10 independent components. The Weyl tensor, which inherits the symmetries (??) and (5) of the Rieman tensor, and in addition vanishes on contraction of any pair of indices, also has 10 independent components. Together, the Ricci and Weyl tensors account for the 20 components of the Riemann tensor. The components of the Ricci and Weyl tensors, though algebraically independent, are related by the differential Bianchi identities.

The end result is that the Weyl tensor, the traceless part of the Riemann curvature tensor, can be written as a $3 \times 3$ complex traceless symmetric matrix 7). Such a matrix has 5 distinct complex components.

The Weyl Tensor as Complex Quantities

In empty space (vanishing energy-momentum tensor), the Ricci tensor vanishes identically. Thus the properties of the gravitational field in empty space are specified entirely by the Weyl tensor. In particular, gravitational waves are specified entirely by the Weyl tensor.

When the 5 complex components of the Weyl tensor are expressed in a NP null tetrad (??), the results are 5 complex components, of spins respectively $-2$ , $-1$ , $0$ , $+1$ , and $+2$ :

(8) $\begin{eqnarray*} -2 &:& C_{u\minus u\minus} \nonumber \\ -1 &:& C_{uv u\minus} \nonumber \\ 0 &:& \frac{1}{2} \left( C_{vuvu} + C_{vu\minus\plus} \right) \nonumber \\ +1 &:& C_{vu v\plus} \\ +2 &:& C_{v\plus v\plus} \ . \end{eqnarray*}$

These 5 complex components exhaust the degrees of freedom of the Weyl tensor.

For outgoing gravitational waves, only the spin $- 2$ component propagates, carrying gravitational waves to far distances. This propagating, outgoing $-2$ component has spin $-2$ , but its complex conjugate has spin $+2$ , so effectively both spin components, or helicities, or polarizations, of an outgoing wave gravitational wave are embodied in the single complex component. The remaining 4 complex NP components (spins $-1$ to $2$ ) of an outgoing gravitational waves are short range, describing the gravitational field near the source.

Conventionally (Chandrasekhar 1983), the 5 complex spin components of the Weyl tensor in the NP formalism are impenetrably denoted

(9) $\begin{eqnarray*} -2 &:& \psi_4 \ , \nonumber \\ -1 &:& \psi_3 \ , \nonumber \\ 0 &:& \psi_2 \ , \nonumber \\ +1 &:& \psi_1 \nonumber \\ +2 &:& \psi_0 \ . \end{eqnarray*}$

Thus the component $\psi_4$ represents propagating, outgoing gravitational waves. The real part of $\psi_4$ represents the $\cos(2\chi)$ , or $+$ , polarization of the propagating gravitational wave, while (minus) its imaginary part represents the $\sin(2\chi)$ ,

Literature

Benger, Werner & Hamilton, Andrew & Folk, Mike & Koziol, Quincey & Su, Simon & Schnetter, Erik & Ritter, Marcel & Ritter, Georg. (2009). Using geometric algebra for navigation in Riemannian and hard disc space. International Workshop on Computer Graphics, Computer Vision and Mathematics, GraVisMa 2009 – Workshop Proceedings.
E. T. Newman and R. Penrose, An Approach to Gravitational Radiation by a Method of Spin Coefficients, J. Math. Phys. 3, 566–79 (1962).
S. Chandrasekhar, The Mathematical Theory of Black Holes, Clarendon Press, Oxford, 1983.
A. Held, A formalism for the investigation of algebraically special metrics. I‘, Commun. Math. Phys. 37, 311–26 (1974).