# Why does the second Weyl scalar describe electromagnetic radiation?

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?

• I've never seen such statement, could you provide a reference? Additionally it appears that the statement is wrong. Consider the case of Schwarzschild spacetime. It is well known that there exists a null tetrad such that all Weyl scalars are null except $\Psi_2$. So we have no electromagnetic radiation and non zero second Weyl scalar. In fact, since $\Psi_2$ decays as $\lambda^{-3}$ (where $\lambda$ is the affine parameter of a null geodesic) it cannot contain information on radiation of any kind, gravitational or electromagnetic. Aug 23, 2014 at 21:06
• It was mentioned in the book Relativity Demystified (by David McMahon). Aug 23, 2014 at 21:13
• It is worth noting that the Wikipedia page on the Weyl scalars mentions an interpretation by Szekeres of $\Psi_2$ as a "Coulomb term", which could mean something completely different - or something very similar. Unfortunately, I haven't been able to et to the original paper. Aug 23, 2014 at 21:17
• I could interpret that to mean something like telling you about the total mass causing the curvature (see the Schwarzschild case, for instance, which is characterized only by the mass), the way the Coulomb field is characterized by only the total charge of the point charge. Aug 23, 2014 at 21:44
• I've taken a look and McMahon gives zero reference to the connection between $\Psi_2$ and electromagnetic radiation. As for the latter, the Nariai (is a typo in the book) is the Schwarzschild-de Sitter metric, a black hole in a universe with positive cosmological constant. Since the solution is static there can be no electromagnetic radiation of any sort. In fact by being a solution with zero stress-energy tensor it cannot have any kind of electromagnetic field. This statement in the book is wrong. I'll try to post a detailed answer on the Weyl scalars latter Aug 25, 2014 at 18:19

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.

• This is a great answer. Just one addendum. There is one possible type for the Weyl tensor not mentioned: it can be 0, called Petrov type O. Type O is not very relevant to vacuum metrics though, since all type O vacuum metrics are known: they are (anti-)de Sitter or Minkowski. Aug 27, 2014 at 0:16
• @RobinEkman, thanks for the comment, I should have mentioned the Type O, in which case all Weyl scalars (i.e. the Weyl tensor) are identically null. The reason for my omission is that some references do not include it in the classification, since, as you mentioned, the Petrov classification is more interesting in the vacuum case, and for Type O the possibilities are trivial. Nevertheless I must add that all those types assume strict vacuum, that is $R_{ab}=0$, or no cosmological constant. With a non-zero cosmological constant one cannot separate Types I from D and Types II from N Aug 27, 2014 at 14:39
• @cesaruliana ... I love you man...No homo
– raul
Apr 7, 2016 at 20:45

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).