Newman-Penrose Field and Bianchi Equations

The Newman–Penrose equations are ordinary tensor identities projected onto a complex null tetrad. Frame commutators describe anholonomy, Ricci identities relate derivatives of spin coefficients to curvature, Einstein’s equation fixes the Ricci projections, and the differential Bianchi identity evolves the Weyl projections. Organizing them as first-order scalar equations is powerful, but copying an equation from a different signature or coefficient convention is unsafe; the restricted identities derived here use the definitions of this chapter.

Required background. Frames, coframes, and anholonomy supplies noncoordinate commutators; The Riemann curvature tensor supplies the Ricci and Bianchi identities; The Einstein equation supplies the matter equations for Ricci components.

Helpful background. Hypersurface normals and tangential projectors helps interpret characteristic directions; Timelike congruences and spatial projection provides the optical and focusing interpretation.

Directional derivatives and a frame commutator

For a normalized tetrad define

$$D=\ell^a\nabla_a,\qquad \Delta=n^a\nabla_a,\qquad \delta=m^a\nabla_a,\qquad \bar\delta=\bar m^a\nabla_a.$$

Although covariant derivatives commute on a scalar $f$, these directional operators need not, because the tetrad varies. The product rule gives the exact identity

$$[D,\delta]f =\left(\ell^a\nabla_a m^b-m^a\nabla_a\ell^b\right)\nabla_b f =[\ell,m]^b\nabla_b f.$$

Inserting the completeness relation expands the vector in parentheses along $\ell,n,m,\bar m$; its four coefficients are linear combinations of spin coefficients. This is the systematic derivation of every Newman–Penrose commutator, and it prevents a sign table from being used outside its conventions.

For the outgoing spherical tetrad in Minkowski spacetime,

$$D=\frac{\partial_t+\partial_r}{\sqrt2}, \qquad \delta=\frac1{\sqrt2r} \left(\partial_\theta+i\csc\theta\,\partial_\phi\right),$$

so differentiating the factor $r^{-1}$ gives

$$[D,\delta]f=-\frac1{\sqrt2r}\,\delta f=\rho\,\delta f.$$

The nonzero commutator is compatible with zero curvature: it measures anholonomy of the spherical tetrad, not failure of covariant derivatives to commute on a scalar.

A Ricci identity and Einstein projection

With the site’s curvature convention, a covector obeys

$$(\nabla_a\nabla_b-\nabla_b\nabla_a)\ell_c =-R^d{}_{cab}\ell_d.$$

Projecting this identity on the screen and taking its trace gives the null optical equation. Under the explicit hypotheses that $\ell^a$ is geodesic and affinely parametrized, so that $\kappa=0$ and the relevant nonaffinity vanishes, the result in this chapter’s signs is

$$D\rho=\rho^2+\sigma\bar\sigma+\Phi_{00}, \qquad \Phi_{00}=\frac12R_{ab}\ell^a\ell^b.$$

Its real part is the null Raychaudhuri equation after using the full screen-area expansion $\theta_{(\ell)}=-2\operatorname{Re}\rho$; $-\operatorname{Re}\rho$ is the averaged Newman–Penrose expansion. Its imaginary part propagates the twist. If $\ell$ is nonaffinely parametrized or not geodesic, additional terms involving $\epsilon$ or $\kappa$ must be retained. Thus the displayed equation is a controlled specialization, not a replacement for the full Ricci-identity set.

The same calculation shows directly how Weyl curvature drives shear. Choose an affine geodesic ray and a screen basis parallel transported along it, and let $B_{AB}=e_A{}^a e_B{}^b\nabla_b\ell_a$. Commuting the covariant derivatives gives the two-dimensional matrix equation

$$DB_{AB}=-B_{AC}B^C{}_B-R(\ell,e_A,\ell,e_B).$$

Its trace yields the equation for $\rho$. Its complex trace-free projection uses $\sigma=-m^am^b\nabla_b\ell_a$ and the chapter’s $\Psi_0=-C(\ell,m,\ell,m)$, giving

$$D\sigma=(\rho+\bar\rho)\sigma-\Psi_0.$$

The minus sign is consequential: sources that define the Weyl scalar with the opposite sign display a plus here. Ricci focusing changes the trace of the screen tensor; the transverse trace-free Weyl component sources its shear. Setting $\sigma=0$ at one point therefore does not make the whole congruence shear-free when $\Psi_0$ is nonzero.

Projecting

$$R_{ab}-\frac12Rg_{ab}+\Lambda_{\mathrm c}g_{ab}=8\pi G T_{ab}$$

twice along $\ell^a$ yields $R_{\ell\ell}=8\pi G T_{\ell\ell}$ because $g_{\ell\ell}=0$. Hence

$$\Phi_{00}=4\pi G T_{\ell\ell}.$$

This makes the focusing source algebraic. Other tetrad projections determine the remaining $\Phi_{ij}$ and $\Lambda_{\mathrm{NP}}$; the cosmological term appears only in projections for which the metric contraction is nonzero.

A Bianchi identity in a type-D vacuum

The differential Bianchi identity, projected in all independent tetrad directions, gives a first-order system for $\Psi_0,\ldots,\Psi_4$ with Ricci-source terms. An explicit vacuum system in background-adapted near-Kerr variables appears in Andersson et al. 2022, §3.7, Lemma 3.15, equations (3.36a)–(3.36h), PDF. That system already assumes vacuum and its specified frame gauge; it is not a source for arbitrary matter terms. Its opposite metric signature must also be translated as in the preceding pages.

A particularly transparent specialization is a Ricci-flat Petrov type-D spacetime in a principal tetrad. Then only $\Psi_2$ is nonzero, and the Bianchi projection along $\ell$ reduces to

$$D\Psi_2=3\rho\Psi_2.$$

This equation does not assert that every type-D tetrad has only $\Psi_2$: principal alignment is an assumption. It also does not apply unchanged with matter, where derivatives of Ricci scalars enter.

As an independent check, let $f(r)=1-2GM/r$ and use the static orthonormal frame in the Schwarzschild exterior $r\gt2GM$ to form a normalized principal null tetrad. Then, on stationary radial scalars,

$$D=\frac{\sqrt f}{\sqrt2}\partial_r, \qquad \rho=-\frac{\sqrt f}{\sqrt2r}, \qquad \Psi_2=+\frac{GM}{r^3}.$$

Both sides of the Bianchi equation equal $-3GM\sqrt f/(\sqrt2r^4)$. This check tests the derivative direction, the sign of $\rho$, and the normalization of $\Psi_2$ simultaneously. The positive Schwarzschild scalar is the value fixed by the site’s Riemann and Weyl-scalar definitions; its Kerr parent calculation gives the source-sign translation. By contrast, in Minkowski spacetime every curvature scalar vanishes, so all Bianchi equations reduce to identities while the frame commutators and spin coefficients can remain nonzero.

The review presentation in Gómez López and Quiroga 2017, §II.B, equations (25)–(28), pp. 3–4, PDF shows how rotation coefficients and curvature scalars enter the NP organization; its equations (40)–(43) are a further specialization to the asymptotic Bondi system, not the general four-dimensional Bianchi system.

Exercises

For the outgoing spherical Minkowski tetrad, verify the commutator $[D,\delta]=\rho\delta$ and the affine optical equation $D\rho=\rho^2$.

Solution

Only the factor $1/(\sqrt2r)$ in $\delta$ depends on $t$ or $r$. Acting on it with $D$ gives $-1/(2r^2)$, which is $\rho$ times $1/(\sqrt2r)$ because $\rho=-1/(\sqrt2r)$. Therefore $[D,\delta]=\rho\delta$. Next,

$$D\rho=\frac1{\sqrt2}\partial_r\left(-\frac1{\sqrt2r}\right) =\frac1{2r^2}=\rho^2.$$

Here $\sigma=0$ and $\Phi_{00}=0$, so this is precisely the restricted Ricci identity. The two calculations are independent: the first uses frame anholonomy, while the second uses the optical projection of curvature.

For a check with nonzero shear, consider the vacuum Kasner metric

$$ds^2=-dt^2+a_x^2dx^2+a_y^2dy^2+a_z^2dz^2, \qquad (a_x,a_y,a_z)= \left((t/t_0)^{2/3},(t/t_0)^{-1/3},(t/t_0)^{2/3}\right),$$

where $t,t_0\gt0$. Use the affinely parametrized null leg and parallel screen

$$\ell=a_x^{-1}\partial_t+a_x^{-2}\partial_x, \qquad m=\frac1{\sqrt2}\left(a_y^{-1}\partial_y+i a_z^{-1}\partial_z\right).$$

Here $n=(a_x/2)\partial_t-(1/2)\partial_x$ completes the normalization. The tensor contractions give $\rho=-1/(6a_xt)$, $\sigma=1/(2a_xt)$, and $\Psi_0=2/(3a_x^2t^2)$. Verify both affine optical equations and explain what a flat-space check would miss.

Solution

On functions of $t$, $D=a_x^{-1}\partial_t$, and $\partial_t(a_xt)^{-1}=-5/(3a_xt^2)$. Therefore

$$D\rho=\frac5{18a_x^2t^2} =\rho^2+|\sigma|^2, \qquad D\sigma=-\frac5{6a_x^2t^2}.$$

The shear equation gives the same result:

$$2\rho\sigma-\Psi_0 =-\frac1{6a_x^2t^2}-\frac2{3a_x^2t^2} =-\frac5{6a_x^2t^2}.$$

Replacing $-\Psi_0$ by $+\Psi_0$ fails this check. A shear-free spherical Minkowski tetrad has $\sigma=\Psi_0=0$ and cannot detect that error.

References

• Andersson, Lars, Thomas Bäckdahl, Pieter Blue, and Siyuan Ma. “Nonlinear Radiation Gauge for Near Kerr Spacetimes.” Communications in Mathematical Physics 396 (2022): 45–90. DOI.
• Gómez López, L. A., and G. D. Quiroga. “Asymptotic Structure of Spacetime and the Newman–Penrose Formalism: A Brief Review.” arXiv:1711.11381v1 (2017). arXiv.