The N--P and 1+1+2 correspondence

Abstract: In this letter, we establish a complete correspondence between the Newman--Penrose and 1+1+2 semitetrad covariant formalisms by expressing all Newman--Penrose spin coefficients, Ricci scalars, and Weyl scalars in terms of the scalar, vector, and tensor variables of the 1+1+2 decomposition. This provides a direct dictionary between two widely used approaches to general relativity and gives a geometrical interpretation of Newman--Penrose quantities in terms of covariantly defined 1+1+2 variables. As an application, we derive necessary conditions for the existence of future outer trapping horizons in locally rotationally symmetric class II spacetimes, expressed in terms of the Ricci and Weyl Newman--Penrose scalars and the cosmological constant.

• The paper establishes a rigorous mapping between NP spin coefficients and 1+1+2 covariant variables, enabling translation of complex geometric quantities.
• It derives explicit expressions for Ricci and Weyl scalars in terms of covariant scalars, vectors, and tensors, facilitating clear horizon and perturbation analyses.
• The framework applies to LRS class II spacetimes, formulating precise horizon criteria where a positive cosmological constant limits future outer trapping horizons.

Correspondence between the Newman–Penrose and 1+1+2 Covariant Formulations

Introduction

The paper "The N--P and 1+1+2 correspondence" (2605.30255) establishes a rigorous, complete dictionary between the Newman–Penrose (N–P) formalism and the 1+1+2 semitetrad covariant approach to general relativity. This work characterizes all N–P spin coefficients, Ricci scalars, and Weyl scalars in terms of the scalar, vector, and tensor variables native to the 1+1+2 decomposition, yielding a unified framework amenable to both perturbative and horizon analyses. The implications are demonstrated in the context of locally rotationally symmetric (LRS) class II spacetimes, where necessary conditions for the existence of black hole horizons are formulated using exclusively N–P curvature scalars and the cosmological constant.

Technical Foundation

The N–P formalism employs a null tetrad $\{\ell^a,n^a,m^a,\bar{m}^a\}$ to recast geometric quantities into complex scalars with direct physical interpretation, facilitating analyses of radiation, perturbations, and horizon structure. In contrast, the 1+1+2 approach decomposes spacetime relative to a timelike direction $u^a$, a distinguished spacelike direction $e^a$, and a sheet orthogonal to both, yielding scalar, vector, and tensor variables manifestly covariant under local transformations and with clear geometric meaning. The full metric splits as $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$, where $q_{ab}$ is the projection onto the sheet.

The paper constructs explicit mappings for vector and tensor objects, directional derivatives (dot, prime, sheet), associated expansion, shear, vorticity, and acceleration variables. The energy-momentum tensor, Ricci, and Weyl tensors are similarly decomposed. The resulting structure enables a first-order system of Einstein equations with physically transparent constraints and evolution equations.

Explicit Mapping of N–P Quantities

The core technical contribution is the explicit algebraic correspondence for all N–P spin coefficients and curvature scalars:

• Spin coefficients: Each N–P spin coefficient ($\tilde{\kappa}$, $\tilde{\sigma}$, $\tilde{\nu}$, $\tilde{\lambda}$, $\tilde{\tau}$, $u^a$0, $u^a$1, $u^a$2, $u^a$3, $u^a$4, $u^a$5, $u^a$6) is derived as a precise function of 1+1+2 geometric variables including shear, vorticity, expansion, acceleration, and their covariant derivatives along $u^a$7, $u^a$8, and the sheet. Notably, null congruence expansions $u^a$9 are mapped to $e^a$0, $e^a$1 respectively, linking horizon structure directly to sheet expansions.
• Ricci scalars: All N–P Ricci scalars ($e^a$2, $e^a$3, $e^a$4, $e^a$5, $e^a$6, $e^a$7) are represented as functions of energy density ($e^a$8), pressure ($e^a$9), heat flux ($g_{ab} = -u_a u_b + e_a e_b + q_{ab}$0), anisotropic stress ($g_{ab} = -u_a u_b + e_a e_b + q_{ab}$1), cosmological constant ($g_{ab} = -u_a u_b + e_a e_b + q_{ab}$2), and their projections into the decomposition.
• Weyl scalars: The N–P Weyl invariants ($g_{ab} = -u_a u_b + e_a e_b + q_{ab}$3, $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$4, $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$5, $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$6, $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$7) are mapped to the electric and magnetic parts of the Weyl tensor ($g_{ab} = -u_a u_b + e_a e_b + q_{ab}$8, $g_{ab} = -u_a u_b + e_a e_b + q_{ab}$9) and their projections, with $q_{ab}$0 encoding Coulombic gravitational information and $q_{ab}$1, $q_{ab}$2 quantifying radiative content.

This mapping clarifies the geometric and physical interpretation of all N–P quantifiers within fully covariant 1+1+2 variables, yielding a robust toolkit for translating results across methods and for developing gauge-invariant gravitational perturbation theory.

Horizon Characterization in LRS Spacetimes

The correspondence is leveraged to refine the analysis of horizon formation in LRS class II geometries, where all sheet vectors and trace-free tensors vanish due to symmetry. The expansions of outgoing and ingoing null congruences, $q_{ab}$3 and $q_{ab}$4, serve as key invariants identifying marginally trapped surfaces (MOTS) and hence horizons.

The paper establishes that the Gaussian curvature $q_{ab}$5, matter fields, Weyl electric part, and null expansions are linked by:

$q_{ab}$6

Imposing $q_{ab}$7 (MOTS) and future outer trapping horizon conditions ($q_{ab}$8, $q_{ab}$9), the existence criteria for horizons reduce to explicit inequalities involving N–P scalars ($\tilde{\kappa}$0, $\tilde{\kappa}$1) and $\tilde{\kappa}$2:

$\tilde{\kappa}$3

If the strong energy condition holds, this inequality is both necessary and sufficient for horizon existence. Crucially, the results explicitly demonstrate that a positive cosmological constant obstructs the presence of future outer trapping horizons in LRS class II spacetimes—a geometric requirement independent of field content.

For isolated horizons in vacuum, the necessary condition simplifies to $\tilde{\kappa}$4. The analysis seamlessly incorporates Ricci and Weyl information, showing that horizon formation depends on balancing matter flux, gravitational field curvature, and cosmological constant.

Implications and Future Directions

The precise dictionary constructed between N–P and 1+1+2 formalisms enables the translation of radiative and horizon analyses, perturbative studies, and field equation dynamics between these frameworks. For gravitational perturbation theory, the mapping supports gauge-invariant approaches, suggesting utility in extending perturbative analyses to more general Petrov types and beyond LRS symmetry. The formalism may yield deeper insights in dynamical spacetimes, including the classification of perturbative geometries and causal dynamics of null horizons.

The explicit horizon criteria formulated in terms of N–P curvature scalars are immediately applicable in numerical relativity and in the study of cosmological black holes, facilitating stability, uniqueness, and existence checks in diverse gravitational scenarios. The approach lays groundwork for recasting established N–P perturbative results in the geometrically transparent language of covariant variables.

Conclusion

The paper establishes an exhaustive, algorithmic mapping between the Newman–Penrose and 1+1+2 semitetrad covariant formalisms, elucidating the geometric and physical meaning of N–P spin and curvature scalars within the covariant decomposition. This correspondence is shown to provide powerful analytic tools for the characterization of black hole horizons in LRS class II spacetimes, yielding directly computable existence and stability criteria expressed in terms of N–P scalars and the cosmological constant. The results suggest broad applicability to gravitational perturbation theory, horizon analysis in dynamical settings, and deeper classification of spacetime geometry, with anticipated future developments in geometric methods for general relativity and relativistic astrophysics.