Why does the second Weyl scalar describe electromagnetic radiation?

Question

I've been reading about the null tetrad, the Weyl tensor, and the Newman-Penrose identities, and so I found out about the Weyl scalars. While the zeroth, first, third, and fourth scalars describe ingoing and outgoing transverse or longitudinal waves, the second ($ \Psi_2$) describe electromagnetic radiation? Why not something completely different?

Answer 1

For the benefit of brevity I'll give a straight answer: the second Weyl scalar is not related to electromagnetic radiation in any way. As per my previous comment, a simple proof is the Schwarzschild (or Kerr) spacetime. It describes a stationary black hole, without electromagnetic (or even gravitational) radiation, yet there is a basis in which all Weyl scalars are identically zero but for $\Psi_2$.

A more extensive answer would require a lot of the the NP formalism, so I'll just point the way. First of all you should note that the Weyl scalars (or any other scalar in NP) are just components of tensors (or Christoffel symbols) in a given basis (i.e. a coordinate frame), and therefore they don't have any meaning in a general situation, since they are not invariants. Nevertheless sometimes the manifold has a special structure and one can find coordinates adapted to this fact, and interpret them accordingly. A simple example is the usual coordinates for Schwarzschild solution, which are adapted to the fact that the spacetime is both static and spherically symmetric (the coefficients of the metric are independent of $t$, and $\theta$ and $\phi$ except for the solid angle part).

Now for the Weyl scalars all spacetimes, at any point, possess one special structure, the so called principal null directions. Consider a general null tetrad $\{l^a,n^a,m^a,\bar{m}^a\}$, we would like to perform a linear transformation that preserves the vector $n^a$. The most general one is given by $l^a\rightarrow l^a+\bar{b}m^a+b\bar{m}^a+|b|^2n^a$ and $m^a\rightarrow m^a+bn^a$, where $b$ is a complex parameter that defines the transformation. Now the Weyl scalar $\Psi_0$ will, under this change of basis, transform as

$\Psi_0\rightarrow \Psi_0+4b\Psi_1+6b^2\Psi_2+4b^3\Psi_3+b^4\Psi_4$.

Since the transformation is quartic in $b$ there are exactly four roots for the equation $\Psi_0=0$. Therefore, in any given point in the spacetime, there are four null directions $l^a$ such that $\Psi_0=0$. This are the principal null directions. The physical relevance of this idea is that if you consider a null geodesic congruence with tangent vector $l^a$ the derivative of the shear along this direction is

$l^a\nabla_a\sigma=\sigma(\rho+\bar{\rho})+\Psi_0$

and therefore if the congruence has zero shear the principal null directions are the ones in which the derivative is also null. Now the interpretation of the Weyl scalars is connected with the Petrov classification. For Petrov Type I (algebraically general) we have four distinct roots $b$ for the transformation of $\Psi_0$. In Petrov Type II there are 3 distinct roots, one of multiplicity 2 (and the corresponding principal null direction is said to be of multiplicity 2), in which direction we have $\Psi_0=\Psi_1=0$. In Petrov Type III we have 2 distinct roots, one of them of multiplicity 3, in which direction we have $\Psi_0=\Psi_1=\Psi_2=0$. In Petrov Type N there is only one root, necessarily of multiplicity 4, in which direction we have $\Psi_0=\Psi_1=\Psi_2=\Psi_3=0$. And for Petrov Type D (the case of Schwarzschild and Kerr) there are two distinct roots, both of multiplicity 2, in which directions we have $\Psi_0=\Psi_1=\Psi_3=\Psi_4=0$. This exhaust all possibilities. The interpretation for the scalars relates what scalars are not null in a given direction, and the interpretation of Petrov classes described in the paper by Szekeres you mention (which is reproduced faithfully in wikipedia, and also in the book Introducing Einstein's Relativity by Ray D'Inverno).

The idea is that since, by Szekeres argument, in Type N we have transversal waves we connect $\Psi_4$ for a given principal null direction with transverse wave modes. For Type III Szekeres establishes that we have transversal and longitudinal waves. Since we identified $\Psi_4$ with the transverse modes we are induced to relate $\Psi_3$ with the longitudinal ones. Now when we change $l^a\leftrightarrow n^a$ we change $\Psi_0\leftrightarrow \Psi_4$ and $\Psi_1\leftrightarrow\Psi_3$ (and $\Psi_2$ is unnaltered). Remember that $l^a=t^a+r^a$ and $n^a=t^a-r^a$ so changing the real null elements of the tetrad amounts to changing the sign of the radial coordinate. Therefore if $\Psi_4$ describes transverse waves going in a certain direction, $\Psi_0$ describes waves going in the opposite direction, and the same for the longitudinal modes.

As for $\Psi_2$, in Petrov Type D Szekeres showed that a spherical arrangement of test particles is deformed in a elliptical one with increasing eccentricity, exactly the same behavior of test particles under an inverse-squared potential, as in Newtonian gravity. Therefore we relate this class, as well as $\Psi_2$, with the presence of Coulomb-like potential. This interpretation is furthered by the peeling off theorem for asymptotically flat spacetimes.

One can only speculate as to why McMahon wrote that $\Psi_2$ is connected to electromagnetic radiation. Maybe is the folklore that says that the Weyl tensor encodes the gravitational radiation (which is not true as showed, beyond grav. radiation it also encodes tidal forces from Newtonian-like gravity) and not finding a radiation description for the second scalar was induced to error. Nevertheless his analysis of Nariai spacetime is totally incorrect, this is a static blac khole in the presence of positive cosmological constant and there is no radiation of any kind whatsoever. Overall I've found his approach to the subject very poor and wouldn't recommend it as a source for Newman-Penrose formalism.

My answer was based in the standard literature in the subject. A classical account, with all the math one could wish, is in Chandrasekhar's Mathematical Theory of Black Holes. For the more physical interpretation I've used Penrose's Structure of space-time, in Battelle Rencontres, edited by Wheller and Dewitt-Morette. Although a bit more mathematical, I've found the approach in O'Neill's The geometry of Kerr black holes very elucidating.

Answer 2

To add to what cesaruliana wrote, I think the confusion lies in the fact that apart from the gravitational NP scalars $\Psi_0$ through $\Psi_4$, the NP formalism also defines analogous electromagnetic scalars, $\Phi_0$ through $\Phi_2$. These are contractions of the tetrad vectors with the electromagnetic field tensor $F_{\mu \nu}$. In this case, with the standard choice of null tetrad (the same one that encodes outgoing gravitational radiation in $\Psi_4$), the outgoing e-m radiation is contained in $\Phi_2$.

A useful reference might be Teukolsky's early-1970s papers on perturbations (scalar, electromagnetic, and gravitational) on a Kerr-black-hole background. The earliest of these is Physical Review Letters vol. 29, pages 1114-1118 (1972).