Geroch-Held-Penrose Calculus

The Geroch–Held–Penrose calculus differentiates null-tetrad components without fixing the residual boost and spin of a preferred pair of null directions. Its basic fields are weighted scalars, best regarded locally as components that patch to sections of complex line bundles. Corrected directional derivatives absorb the inhomogeneous transformation of four connection coefficients, so every term in a GHP equation has a definite boost-spin weight.

Required background. Frames, coframes, and anholonomy supplies local frame transformations and connection coefficients; The Riemann curvature tensor supplies curvature projections; The Einstein equation supplies the Ricci sources in weighted field equations.

Helpful background. Hypersurface normals and tangential projectors helps identify preferred null directions; Timelike congruences and spatial projection supplies the optical quantities used in the example below.

Weighted scalars for a preferred null pair

Assume a time-oriented, oriented spacetime and a locally defined normalized tetrad $(\ell,n,m,\bar m)$ with

$$\ell\cdot n=-1, \qquad m\cdot\bar m=1.$$

Suppose the two real null directions are geometrically preferred, but their normalization and the phase of $m$ are not. Two aligned tetrads then differ by

$$(\ell,n,m,\bar m) \mapsto (A\ell,A^{-1}n,e^{i\chi}m,e^{-i\chi}\bar m), \qquad A>0.$$

A scalar of GHP type $\{p,q\}$ transforms as

$$\eta\mapsto A^{(p+q)/2}e^{i(p-q)\chi/2}\eta.$$

Thus its boost and spin weights are $b=(p+q)/2$ and $s=(p-q)/2$. Complex conjugation interchanges the type: if $\eta$ has type $\{p,q\}$, then $\bar\eta$ has type $\{q,p\}$. Examples are

$$\begin{array}{c|cccccc} \eta&\rho&\sigma&\kappa&\tau&\Psi_0&\Psi_2\\ \hline \{p,q\}&\{1,1\}&\{3,-1\}&\{3,1\}&\{1,-1\}&\{4,0\}&\{0,0\}. \end{array}$$

This construction is the central scope of GHP covariance: it controls the $\mathbb C^*$ spin-boost freedom preserving the selected null directions. The bundle interpretation and the relation between $(p,q)$ and $(b,s)$ are stated explicitly in Andersson et al. 2022, Appendix A, Definition A.1 and the discussion following it, PDF. It also explains why one can work with regular local transverse frames even when no global $m$ exists on a sphere.

Four weighted derivative operators

Let $D,\Delta,\delta,\bar\delta$ be the four tetrad directional derivatives. For a scalar $\eta$ of type $\{p,q\}$, define

$$\begin{aligned} \mathsf P\eta &=(D-p\epsilon-q\bar\epsilon)\eta,\\ \mathsf P'\eta &=(\Delta+p\epsilon'+q\bar\epsilon')\eta,\\ \eth\eta &=(\delta-p\beta+q\bar\beta')\eta,\\ \eth'\eta &=(\bar\delta+p\beta'-q\bar\beta)\eta. \end{aligned}$$

Here $\mathsf P$ is read as “thorn”; $\mathsf P'$ is thorn-prime. The coefficient definitions are those fixed earlier in this chapter, including their $(-,+,+,+)$ signature translation. The resulting type shifts are

$$\begin{array}{c|c} \text{operator}&\text{output type}\\ \hline \mathsf P&\{p+1,q+1\}\\ \mathsf P'&\{p-1,q-1\}\\ \eth&\{p+1,q-1\}\\ \eth'&\{p-1,q+1\}. \end{array}$$

For a quantity of type $\{0,0\}$, the correction terms vanish on the first derivative. The output still has the shifted type, so a second derivative usually does need connection terms. For example, $\mathsf P f=Df$ for an ordinary invariant scalar, but $\mathsf P(\mathsf P f)$ acts on a field of type $\{1,1\}$. Ignoring that change loses covariance.

The cancellation can be checked directly. Under a local boost-spin transformation,

$$\epsilon_{\mathrm{new}} =A\left(\epsilon+\frac12D\ln A+\frac{i}{2}D\chi\right).$$

When $D$ acts on $A^{(p+q)/2}e^{i(p-q)\chi/2}\eta$, it generates exactly the $D\ln A$ and $D\chi$ terms subtracted by $p\epsilon_{\mathrm{new}}+q\bar\epsilon_{\mathrm{new}}$. Therefore $\mathsf P\eta$ acquires the original factor and one additional power of $A$, as its type shift requires. The other three cancellations follow from the corresponding transformations of $\epsilon',\beta,\beta'$.

Angular derivatives on a round screen

The spherical Minkowski tetrad provides a concrete meaning for $\eth$. In the angular patch $0\lt\theta\lt\pi$, direct projection of the connection gives

$$\epsilon=\epsilon'=0, \qquad \beta=\beta'=\frac{\cot\theta}{2\sqrt2r}.$$

For spin weight $s=(p-q)/2$, the two angular operators reduce to

$$\begin{aligned} \eth\eta &=\frac1{\sqrt2r} \left(\partial_\theta+i\csc\theta\,\partial_\phi-s\cot\theta\right)\eta,\\ \eth'\eta &=\frac1{\sqrt2r} \left(\partial_\theta-i\csc\theta\,\partial_\phi+s\cot\theta\right)\eta. \end{aligned}$$

The extra $\cot\theta$ term compensates the rotation of the angular frame over the sphere. It is absent for a spin-zero scalar on its first derivative, but cannot be dropped after that derivative changes the spin weight. Different normalizations of the null tetrad or angular eth operator can move the factor $1/(\sqrt2r)$ or an overall sign; the expressions here follow from the displayed tetrad and coefficient definitions.

A covariant optical identity

For a geodesic null direction, $\kappa=0$, a projected Ricci identity can be written

$$\mathsf P\rho =\rho^2+\sigma\bar\sigma+\Phi_{00}.$$

Every term has type $\{2,2\}$. In a frame affinely and spin-parallel transported along $\ell$, one may set $\epsilon=0$, reducing $\mathsf P$ to $D$ and recovering the earlier Newman–Penrose optical equation. After a local boost or spin, $D\rho$ alone develops derivative terms, whereas $\mathsf P\rho$ continues to transform homogeneously. This illustrates what the weighted calculus achieves without changing the underlying tensor identity.

Geroch, Held, and Penrose introduced this calculus to exploit a pair of null directions while retaining covariance under their residual transformations; the primary account is Geroch, Held, and Penrose 1973, pp. 874–881. Modern presentations can differ in metric signature, priming conventions, and the signs used for spin coefficients, so the operator formulas must be read together with their definitions.

Scope and failure modes

GHP covariance is not covariance under null rotations. A null rotation changes one of the preferred directions and mixes the Weyl scalars polynomially; a new aligned pair must be chosen before the GHP types above apply. Nor is spin-boost gauge a substitute for diffeomorphism gauge: the former changes a tetrad over fixed spacetime points, while the latter acts on spacetime fields by pullback.

Weighted does not mean invariant. Only type $\{0,0\}$ quantities are unchanged under spin-boost, and even they can change under null rotations. Finally, a compact weighted equation is trustworthy only when all terms have the same type and the same signature convention. Weight matching is a strong error check, but it cannot detect a common overall sign error in a convention-translated curvature identity.

Exercises

Let $\eta$ have type $\{p,q\}$. Show explicitly that $\mathsf P\eta$ has type $\{p+1,q+1\}$, and verify the weights in the optical identity.

Solution

Write $b=(p+q)/2$ and $s=(p-q)/2$. Then

$$D_{\mathrm{new}}\eta_{\mathrm{new}} =A^{b+1}e^{is\chi} \left[D\eta+(bD\ln A+isD\chi)\eta\right].$$

Using the displayed transformation of $\epsilon$ and its conjugate, the correction $p\epsilon_{\mathrm{new}}+q\bar\epsilon_{\mathrm{new}}$ subtracts $bD\ln A+isD\chi$. Hence $(\mathsf P\eta)_{\mathrm{new}}=A^{b+1}e^{is\chi}\mathsf P\eta$, which is type $\{p+1,q+1\}$. For the optical equation, $\rho$ has type $\{1,1\}$, so $\mathsf P\rho$ and $\rho^2$ have type $\{2,2\}$. Since $\sigma$ has type $\{3,-1\}$, $\bar\sigma$ has type $\{-1,3\}$ and their product has type $\{2,2\}$. Finally $R_{ab}\ell^a\ell^b/2$ has boost weight two and spin weight zero, hence type $\{2,2\}$.

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.
• Geroch, Robert, A. Held, and Roger Penrose. “A Space-Time Calculus Based on Pairs of Null Directions.” Journal of Mathematical Physics 14, no. 7 (1973): 874–881. DOI.