Null Tetrads and Normalization

A null tetrad replaces four real orthonormal directions by two real null directions and one complex-conjugate transverse pair. The replacement is algebraic, but its normalization determines every later sign in the spin coefficients and curvature scalars. Here the ordered tetrad is $(\ell^a,n^a,m^a,\bar m^a)$, with a future-directed pair $\ell^a,n^a$ and with the bar denoting complex conjugation. All formulas below use the site’s $(-,+,+,+)$ metric and its fixed spacetime orientation.

Required background. Frames, coframes, and anholonomy supplies the frame/coframe construction; The Riemann curvature tensor fixes the curvature convention used later; The Einstein equation supplies the field equation whose projections become Ricci scalars.

Helpful background. Hypersurface normals and tangential projectors clarifies how null directions meet hypersurfaces; Timelike congruences and spatial projection supplies the congruence language used for the optical interpretation.

The normalized complex null frame

Let $(e_0^a,e_1^a,e_2^a,e_3^a)$ be a positively oriented orthonormal tetrad, with $e_0^a$ future directed. Define

$$\ell^a=\frac{e_0^a+e_1^a}{\sqrt2},\qquad n^a=\frac{e_0^a-e_1^a}{\sqrt2},\qquad m^a=\frac{e_2^a+i e_3^a}{\sqrt2},\qquad \bar m^a=\frac{e_2^a-i e_3^a}{\sqrt2}.$$

The only nonzero inner products are

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

Thus neither $m^a$ nor $\bar m^a$ is a real tangent vector, even though the metric reconstructed from them is real:

$$g_{ab}=-\ell_a n_b-n_a\ell_b +m_a\bar m_b+\bar m_a m_b.$$

Equivalently, the completeness relation is

$$\delta^a{}_b=-\ell^a n_b-n^a\ell_b +m^a\bar m_b+\bar m^a m_b.$$

These relations are the safest starting point for any projection: lower indices with the site metric first, and then contract. They also fix the volume-form convention. If $\epsilon_{abcd}e_0^ae_1^be_2^ce_3^d=+1$, direct substitution gives

$$\epsilon_{abcd}\ell^a n^b m^c\bar m^d=i.$$

Changing $m^a$ to its complex conjugate reverses this ordered complex orientation. It is therefore not an innocuous relabeling once pseudoscalars or dual curvature tensors occur.

Residual frame transformations

A boost of the real null pair and a spin of the transverse pair preserve the normalization:

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

A scalar of boost weight $b$ and spin weight $s$ transforms as

$$\eta\mapsto A^b e^{is\chi}\eta.$$

This two-parameter subgroup preserves each of the two null directions. It is the residual tetrad freedom handled covariantly by the Geroch–Held–Penrose calculus. The full connected Lorentz freedom is larger. For example, a null rotation about $\ell^a$, with complex parameter $a$, is

$$\begin{aligned} \ell'^a&=\ell^a,\\ m'^a&=m^a+\bar a\,\ell^a,\\ \bar m'^a&=\bar m^a+a\ell^a,\\ n'^a&=n^a+a m^a+\bar a\,\bar m^a+|a|^2\ell^a. \end{aligned}$$

There is an analogous rotation about $n^a$. A short contraction verifies that $n'^2=0$: the $-2|a|^2$ cross term from $n\mathbin{\cdot}\ell$ cancels the $+2|a|^2$ transverse term. All other normalization relations survive as well. These are changes of tetrad at the same spacetime point, not coordinate transformations and not diffeomorphisms.

The literature contains both metric signatures. In particular, the review by Gómez López and Quiroga normalizes the real pair with the opposite sign and uses the corresponding opposite transverse sign. Its definitions are therefore not imported by retaining the symbols and guessing an overall minus sign. One must translate each contraction using the displayed metric decomposition; see Gómez López and Quiroga 2017, §II.A, equations (3)–(24), pp. 2–3, PDF. The weighted spin-boost action used here can be compared with Andersson et al. 2022, Appendix A, Definition A.1, PDF, after making the same signature translation.

Independent normalization checks

Three checks catch most convention errors. First, reconstruct $g_{ab}$ and apply it to every pair of tetrad vectors; a sign error in either dyad appears immediately. Second, compute the determinant of the change-of-basis matrix from $(e_0,e_1,e_2,e_3)$ to $(\ell,n,m,\bar m)$; it is $i$, which gives the volume-form contraction above. Third, apply a nonconstant boost or spin and verify the inner products pointwise. Derivatives of $A$ and $\chi$ do not affect normalization, but they do enter connection coefficients; treating local and constant transformations as interchangeable is a later source of mistakes.

The construction assumes a local orthonormal tetrad and hence is local in general. A global null tetrad may fail to exist because of topology, and a tetrad adapted to a horizon or symmetry axis may become singular even when the geometry is regular. Such a failure calls for overlapping regular frames, not a change in the normalization equations.

Exercises

Starting from the orthonormal tetrad above, verify both the metric reconstruction and the orientation contraction. Then apply the boost-spin transformation and determine the weights of the tetrad vectors themselves.

Solution

The orthonormal products give $\ell^2=n^2=m^2=\bar m^2=0$, $\ell\cdot n=-1$, and $m\cdot\bar m=1$; all mixed real-transverse products vanish. Substitution into the dyadic expression therefore reproduces $g(e_0,e_0)=-1$ and $g(e_i,e_j)=\delta_{ij}$. The two $2\times2$ change-of-basis blocks have determinants $-1$ and $-i$, so their product is $i$ and hence $\epsilon(\ell,n,m,\bar m)=i$. Under boost-spin, $\ell$ has $(b,s)=(1,0)$, $n$ has $(-1,0)$, $m$ has $(0,1)$, and $\bar m$ has $(0,-1)$. Every nonzero inner product has total weight $(0,0)$, independently checking invariance.

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.