1 Introduction↩︎

The broad utility of the Newman–Penrose (N–P) formalism [1], [2] has been demonstrated through its extensive application in gravitational perturbation theory (see, for example, [3]) and in the analysis of black hole horizons and their perturbations [4]–[6]. On the other hand, the introduction of the (1+1+2) semitetrad covariant formalism has provided a powerful framework for studying gravitational systems with preferred spatial directions and inhomogeneities [7], [8]. Each of these approaches possesses distinct advantages.

The strength of the N–P formalism lies in its relative simplicity, elegance, and broad range of applications. The (1+1+2) formalism, however, offers several important advantages. First, the Einstein field equations may be written as a first-order system governing the curvature and dynamical variables along preferred spacetime directions, supplemented by a set of constraints, in contrast to the second-order systems that commonly arise in coordinate-based approaches. Second, the variables arising from the decomposition admit clear geometric and physical interpretations, since they are naturally associated with the temporal and spatial congruences. Third, in perturbation theory, gauge invariance follows directly from the Stewart–Walker lemma [9]. The second point is particularly significant, as it allows geometrical interpretations to be assigned to structures arising in other formulations of general relativity. This feature has motivated, for example, the application of the (1+1+2) formalism to the study of horizons in exact solutions [10]–[12]. In [13], the effect of perturbations on a null horizon was also investigated within this framework. It is therefore natural to expect that establishing a direct connection between the N–P and (1+1+2) formalisms could provide useful insights into gravitational systems, particularly in clarifying the geometrical interpretation of horizon-related structures formulated within the language of the N–P approach.

The purpose of this letter is to establish a complete correspondence between the N–P and (1+1+2) formalisms by expressing all N–P spin coefficients, Ricci scalars, and Weyl scalars in terms of the covariant scalar, vector, and tensor variables of the (1+1+2) decomposition. In this way, we obtain a direct dictionary between two widely used approaches to general relativity.

Some aspects of this correspondence have previously been obtained in more restricted settings. For example, Pratten [14] exploited relations between the two formalisms in the study of gravitational perturbations, where perturbation variables were related to the electric and magnetic parts of the Weyl tensor and connected to the N–P Weyl scalars (\(\Psi_0\)), (\(\Psi_2\)), and (\(\Psi_4\)). This provided an elegant framework for analyzing perturbations of the Schwarzschild spacetime. Similarly, in [15], the Ricci N–P scalars were computed for a restricted subclass of locally rotationally symmetric spacetimes in the context of energy transfer between matter and the free gravitational field.

In the present work, however, we derive the complete set of relations for all N–P scalars. As an illustration of the utility of this correspondence, we apply the resulting dictionary to the study of black hole horizons in locally rotationally symmetric spacetimes. In particular, we obtain necessary conditions for the existence of future outer trapping horizons expressed entirely in terms of Newman–Penrose curvature scalars and the cosmological constant.

The paper is organized as follows. In Section 2, we briefly review the (1+1+2) semitetrad covariant formalism and establish the notation and conventions used throughout. In Section 3, we derive the complete correspondence between the N–P and (1+1+2) variables. In Section 4, we apply these relations to derive existence criteria for black hole horizons in locally rotationally symmetric spacetimes. Finally, in Section 5, we summarize our results and discuss possible future applications of the correspondence established here.

Throughout this work we use the metric signature \((-+++)\). The spacetime volume form is denoted by \(\eta_{abcd}\), while \(\bar{\eta}_{abc}=\eta_{dabc}u^d\) is the spatial volume form orthogonal to \(u^a\), and \(\tilde{\eta}_{ab}=\bar{\eta}_{abc}e^c\) is the area form on the two-sheet. Overbars on projected \(1+1+2\) quantities denote sheet projection, while an overbar on a complex quantity denotes complex conjugation. Tildes on Newman–Penrose quantities are used to distinguish spin coefficients from similarly named \(1+1+2\) variables.

2 A quick guide to the 1+1+2 formulation↩︎

We introduce the 1+1+2 semitetrad formulation of general relativity here, briefly. We curtail many of the intricate details as there is a wealth of literature which comprehensively introduces the formulation [7], [8], which we shall follow. To begin with, one considers a spacetime with metric tensor \(g_{ab}\) and compatible covariant derivative \(\nabla_a\), foliated into spacelike slices, and suppose these slices are themselves foliated by “surfaces” (these spaces/pseudo-surfaces are referred to as sheet in the literature since in general they are rather a collection of tangent planes, and only under certain conditions they are what is referred to as a genuine surface). The surfaces have unit normal \(e^a\), orthogonal to the timelike direction, which allows us to decompose the spacetime metric as \[\begin{align} q_{ab}=h_{ab}-e_ae_b=g_{ab}+u_au_b-e_ae_b, \end{align}\] where \(q_{ab}\) and \(h_{ab}\) are respectively the projection tensors onto the sheet and the 3-space orthogonal to \(u^a\).

A 3-vector \(\psi_a\) splits as \[\begin{align} \psi_a=\bar\psi e^a+\bar\psi^a,\quad\bar\psi=\psi_ae^a,\quad\bar\psi^a=q^b_{\;a}\psi_b.\label{3v} \end{align}\tag{1}\]

Similarly, a projected, symmetric, trace-free 3-tensor \(\Psi_{ab}\) may be decomposed as \[\begin{align} \label{split3} \Psi_{ab} = \bar{\Psi} \left(e_a e_b-\frac{1}{sec:2}q_{ab}\right) +2\bar{\Psi}_{(a}e_{b)} +\bar{\Psi}_{ab}, \end{align}\tag{2}\] where \[\begin{align} \bar{\Psi}&=\Psi_{ab}e^ae^b,\quad\bar{\Psi}_a=q_a{}^b\Psi_{bc}e^c,\\ \bar{\Psi}_{ab}&=\left(q_{(a}{}^c q_{b)}{}^d-\frac{1}{2} q_{ab}q^{cd}\right)\Psi_{cd}. \end{align}\]

Three directional derivatives result from the decomposition:

• Along \(u^a\) (“dot” derivative): \(\dot{\psi}^{a\cdots b}_{\;\;\;c\cdots d}=u^f\nabla_f\psi^{a\cdots b}_{\;\;\;c\cdots d}\),

• Along \(e^a\) (“prime” derivative): \((\psi^{a\cdots b}_{\;\;\;c\cdots d})'=e^f\nabla_f\psi^{a\cdots b}_{\;\;\;c\cdots d}\),

• Along the sheet/surface (“sheet/surface” derivative): \(\mathcal{D}_f\psi^{a\cdots b}_{\;\;\;c\cdots d}=q^{\bar a}_{\;a}\cdots q^{\bar b}_{\;b}q^{\bar c}_{\;c}\cdots q^{\bar d}_{\;d}q^e_{\;f}\nabla_e\psi^{\bar a\cdots \bar b}_{\;\;\;\bar c\cdots \bar d}\),

for any tensor \(\psi^{a\cdots b}_{\;\;\;c\cdots d}\), with \(\mathcal{D}_a\) denoting the derivative compatible with the 2-metric \(q_{ab}\). Finally, a scalar \(\psi\) will have its gradient decomposed as \[\begin{align} \nabla_a\psi=-\dot{\psi}u_a+\psi'e_a+\mathcal{D}_a\psi. \end{align}\]

Moving on to the geometry, the energy momentum tensor, under the splitting, decomposes as \[\begin{align} T_{ab}=\varrho u_au_b+ph_{ab}+2q_{(a}u_{b)}+\pi_{ab}, \end{align}\] with the 3-vector \(q_a\) and 3-tensor \(\pi_{ab}\) decomposing according to 1 and 2 , so that from the Einstein Field equations \[\begin{align} R_{ab}-\frac{1}{sec:2}(R-2\Lambda)g_{ab}=T_{ab}, \end{align}\] one obtains the Ricci tensor as \[\begin{align} R_{ab}&=\frac{1}{sec:2}(\varrho+3p-2\Lambda)u_au_b+\frac{1}{sec:2}(\varrho-p+2\Lambda+2\Pi)e_ae_b\nonumber\\ &+\frac{1}{sec:2}(\varrho-p+2\Lambda-\Pi)q_{ab}+2(Qu_{(a}+\Pi_{(a})e_{b)}\nonumber\\ &+2Q_{(a}u_{b)}+\Pi_{ab}. \end{align}\] The various scalars, vectors and tensor introduced above have the following definitions: \(\varrho=T_{ab}u^au^b\) is the (local) energy density, \(3p=-T_{ab}h^{ab}\) is the (isotropic) pressure, \(q_a=h^b_{\;a}T_{bc}u^c\) is the heat 3-flux vector, and \(\pi_{ab}\) captures deviation from isotropy.

The Weyl tensor takes the decomposed form \[\begin{align} \label{wt1} C_{abcd}&=4u_{[c}u^{[a}E^{b]}_{\;d]}+4E_{[c}^{\;[a}h^{b]}_{\;d]}\nonumber\\ &-2\bar\eta^{abf}H_{f[c}u_{d]}-2\bar\eta_{cd}^{\;\;f}H_{f}^{\;[a}u^{b]}, \end{align}\tag{3}\] where the square brackets indicate antisymmetrization, and the 3-tensors \[\begin{align} \label{ghsyecpw} E_{ab}=C_{acbd}u^cu^d,\quad H_{ab}=\frac{1}{sec:2}\bar\eta_a^{\;ef}C_{efbd}u^d, \end{align}\tag{4}\] respectively represent the electric and magnetic sectors of the Weyl tensor, and \(\bar\eta_{abc}=\eta_{dabc}u^a\), with \(\bar\eta_{abc}\) and \(\eta_{abcd}\) being the volume form on the 3-space and the full spacetime, respectively. These are 3-tensors that can be appropriately decomposed according to 2 .

The evolution and propagation equations of \(u^a\) and \(e^a\) are \[\begin{align} \dot{u}^a&=\mathcal{A}e^a+\mathcal{A}^a,\nonumber\\ \hat{u}^a&=\left(\frac{1}{sec:3}\theta+\Sigma\right)e^a+\left(\Sigma^a+\tilde{\eta}^{ab}\Omega_b\right),\\ \dot{e}^a&=\mathcal{A}u^a+\alpha^a,\quad\hat{e}^a=a^a, \end{align}\] and their full covariant derivatives are given by \[\begin{align} \nabla_au_b&=-u_a\left(\mathcal{A}e_b+\mathcal{A}_b\right)+\left(\frac{1}{sec:3}\theta+\Sigma\right)e_ae_b\nonumber\\ &+\frac{1}{sec:2}\left(\frac{sec:2}{sec:3}\theta-\Sigma\right)q_{ab}+\Omega\tilde{\eta}_{ab}+\Sigma_{ab}\tag{5}\\ &+2\left(e_{(a}\Sigma_{b)}+e_{[a}\tilde{\eta}_{b]c}\Omega^c\right),\nonumber\\ \nabla_ae_b&=-u_a\left(\mathcal{A}u_b+\alpha_b\right)+\left(\frac{1}{sec:3}\theta+\Sigma\right)e_au_b\nonumber\\ &+\frac{1}{sec:2}\phi q_{ab}+\xi\tilde{\eta}_{ab}+\zeta_{ab}+e_aa_b\tag{6}\\ &+\left(\Sigma_a-\tilde{\eta}_{ac}\Omega^c\right)u_b.\nonumber \end{align}\] \(\mathcal{A}\) is the acceleration scalar, \(\mathcal{A}_a\) the acceleration 2-vector, \(\theta=h^{ab}\nabla_au_b\) is the expansion, \(\phi=q^{ab}\nabla_ae_b\) is referred to as the sheet/surface expansion, \(2\Omega=\tilde{\eta}^{ab}\nabla_au_b\), \(2\xi=\tilde{\eta}^{ab}\nabla_ae_b\) are the respective vorticities of \(u^a\) and \(e^a\), (there is a 2-vector, the vorticity 2-vector \(\Omega_a\), which results from decomposing the usual vorticity 3-vector associated to \(u^a\) according to the first relation of 1 ), \(\Sigma,\Sigma_a,\Sigma_{ab}\) are scalar, 2-vector, and 2-tensor obtained by decomposing the 3-shear \(\sigma_{ab}\) of \(u^a\) according to 2 , and \(\zeta_{ab}=\mathcal{D}_{\{a}e_{b\}}\) is the shear of \(e^a\), with curly brackets indicating fully projected and trace-free with respect to \(q_{ab}\). Finally, \[\begin{align} \tilde{\eta}_{ab}=\bar\eta_{abc}e^c=\eta_{dabd}u^de^c\label{area2form} \end{align}\tag{7}\] is the area form associated with \(q_{ab}\).

The Einstein field equations can then be written as a set of first order equations in the convective derivatives and constraints for the covariant quantities appearing here, using the Ricci identities for \(u^a\) and \(e^a\) \[\begin{align} 2\nabla_{[a}\nabla_{b]}u_c=R_{abc}^{\;\;\;d}u_d,\quad 2\nabla_{[a}\nabla_{b]}e_c=R_{abc}^{\;\;\;d}e_d. \end{align}\] We will not write down this huge set of equations as they are not needed for the current purpose of this work, but there is no shortage of papers in the literature containing and systematically deriving them [7], [8].

3 Mapping the variables↩︎

In this section we obtain the full set of relationships between scalars of the N–P formalism in terms of the scalars vectors and tensors of the 1+1+2 covariant formalism.

We begin by constructing the real components of the null pair \(\{\ell^a,n^a\}\), defining tangent vectors to null geodesics, from the unit directions \(u^a\) and \(e^a\) as \[\begin{align} \ell^a=\frac{1}{\sqrt{sec:2}}(u^a+e^a),\quad n^a=\frac{1}{\sqrt{sec:2}}(u^a-e^a), \end{align}\] cross normalized to \(\ell_an^a=-1\). In the N-P formalism, a pair of complex conjugate vectors \(\{m^a,\bar m^a\}\) satisfying \[\begin{align} m_am^a=\bar m_a\bar m^a=0,\quad m_a\bar m^a=1 \end{align}\] completes the full tetrad. Then, the 2-sheet metric \(q_{ab}\) and the complex pair \(m^a\) and \(\bar m^a\) are related as \[\begin{align} q_{ab}=2m_{(a}\bar m_{b)}, \end{align}\] so that the spacetime metric decomposes as \[\begin{align} g_{ab}=-2\ell_{(a}n_{b)}+2m_{(a}\bar m_{b)}, \end{align}\] The area 2-form \(\tilde{\eta}_{ab}\) associated to \(q_{ab}\) will then take the form \[\begin{align} \tilde{\eta}_{ab}=2i \bar m_{[a}m_{b]}, \end{align}\] with \(i^2=-1\).

The N–P scalars are defined by appropriate projections of the covariant derivatives of the null vectors, and contractions of the Ricci and Weyl tensors. The spin coefficients are given by

\[\begin{align} \tilde{\kappa}&=-m^a\ell^b\nabla_b\ell_a,\quad\tilde{\sigma}=-m^am^b\nabla_b\ell_a,\\ \tilde{\nu}&=\bar m^an^b\nabla_bn_a,\quad\tilde{\lambda}=-\bar m^a\bar m^b\nabla_bn_a,\\ \tilde{\tau}&=-m^an^b\nabla_b\ell_a,\quad\tilde{\rho}=-m^a\bar m^b\nabla_b\ell_a,\\ \tilde{\pi}&=\bar m^a\ell^b\nabla_bn_a,\quad\tilde{\mu}=-\bar m^am^b\nabla_bn_a,\\ \tilde{\alpha}&=-\frac{1}{sec:2}(n^a\bar m^b\nabla_b\ell_a-\bar m^a\bar m^b\nabla_bm_a),\\ \tilde{\beta}&=-\frac{1}{sec:2}(n^am^b\nabla_b\ell_a-\bar m^am^b\nabla_bm_a),\\ \tilde{\gamma}&=-\frac{1}{sec:2}(n^an^b\nabla_b\ell_a-\bar m^an^b\nabla_bm_a),\\ \tilde{\epsilon}&=-\frac{1}{sec:2}(n^a\ell^b\nabla_b\ell_a-\bar m^a\ell^b\nabla_bm_a). \end{align}\] The Ricci N–P scalars are given by \[\begin{align} \Phi_{00}&=\frac{1}{sec:2}R_{ab}\ell^a\ell^b,\quad\Phi_{11}=\frac{1}{sec:4}R_{ab}(\ell^an^b+m^a\bar m^b),\\ \Phi_{22}&=\frac{1}{sec:2}R_{ab}n^an^b,\quad\Phi_{01}=\bar\Phi_{10}=-\frac{1}{sec:2}R_{ab}\ell^am^b,\\ \Phi_{02}&=\bar\Phi_{20}=\frac{1}{sec:2}R_{ab}m^am^b,\\ \Phi_{12}&=\bar\Phi_{21}=\frac{1}{sec:2}R_{ab}\bar m^an^b \end{align}\] Finally, the Weyl N–P scalars are given by \[\begin{align} \Psi_0&=C_{abcd}\ell^am^b\ell^cm^d,\quad\Psi_1=C_{abcd}\ell^an^b\ell^cm^d,\\ \Psi_2&=C_{abcd}\ell^am^b\bar m^cn^d,\quad\Psi_3=C_{abcd}\ell^an^b\bar m^cn^d,\\ \Psi_4&=C_{abcd}n^a\bar m^bn^c\bar m^d. \end{align}\]

So far these are the needed ingredients for the computations. We now compute all of the N-P quantities in terms of the 1+1+2 quantities.

3.1 Spin coefficients↩︎

The N–P spin coefficients are calculated as \[\begin{align} \tilde{\kappa}&=-\frac{1}{sec:2}m_aY^a,\tag{8}\\ \tilde{\sigma}&=-\frac{1}{\sqrt{sec:2}}(\Sigma_{ab}+\zeta_{ab})m^am^b,\tag{9}\\ \tilde{\nu}&=-\frac{1}{sec:2}(2(\mathcal{A}_a+a_a)-Y_a)\bar m^a,\tag{10}\\ \tilde{\lambda}&=-\frac{1}{\sqrt{sec:2}}(\Sigma_{ab}-\zeta_{ab})\bar m^a\bar m^b,\tag{11}\\ \tilde{\tau}&=-\tilde{\kappa}+(\Sigma_a+\tilde{\eta}_{ab}\Omega^b)m^a,\tag{12}\\ \tilde{\rho}&=-\frac{1}{sec:2}\theta_{(\ell)}\nonumber\\ &-\frac{1}{\sqrt{sec:2}}((\Omega+\xi)\tilde{\eta}_{ab}-(\Sigma_{ab}-\zeta_{ab}))\bar m^a m^b,\tag{13}\\ \tilde{\pi}&=\tilde{\nu}+(\mathcal{A}_a-\alpha_a)\bar m^a,\tag{14}\\ \tilde{\mu}&=-\frac{1}{sec:2}\theta_{(n)}\nonumber\\ &-\frac{1}{\sqrt{sec:2}}((\Omega-\xi)\tilde{\eta}_{ab}-(\Sigma_{ab}+\zeta_{ab}))m^a\bar m^b,\tag{15},\\ \tilde{\alpha}&=-\frac{1}{sec:2}\left(\mathring{m}_a-\frac{1}{sec:2}(\Sigma_a-\tilde{\eta}_{ab}\Omega^b)\right)\bar m^a,\\ \tilde{\beta}&=-\frac{1}{sec:2}\left(\check{m}_a-\frac{1}{sec:2}(\Sigma_a-\tilde{\eta}_{ab}\Omega^b)\right)m^a,\\ \tilde{\gamma}&=\frac{1}{2\sqrt{sec:2}}\left(\mathcal{A}-\left(\frac{1}{sec:3}\theta+\Sigma\right)+(\dot{m}_a-\hat{m}_a)\bar m^a\right),\\ \tilde{\epsilon}&=\frac{1}{2\sqrt{sec:2}}\left(\mathcal{A}+\left(\frac{1}{sec:3}\theta+\Sigma\right)+(\dot{m}_a+\hat{m}_a)\bar m^a\right), \end{align}\] where we have defined \[\begin{align} Y_a&=\mathcal{A}_a+\alpha_a+\Sigma_a+\tilde{\eta}_{ab}\Omega^b+a_a,\\ \theta_{(\ell)}&=q^{ab}\nabla_a\ell_b=\frac{1}{\sqrt{sec:2}}(\frac{sec:2}{sec:3}\theta-\Sigma+\phi)=-2\Re(\tilde{\varrho}),\tag{16}\\ \theta_{(n)}&=q^{ab}\nabla_an_b=\frac{1}{\sqrt{sec:2}}(\frac{sec:2}{sec:3}\theta-\Sigma-\phi)=-2\Re(\tilde{\mu})\tag{17}, \end{align}\] with the \(\mathring{*}\) and \(\check{*}\) derivatives being the directional derivatives along \(\bar m^a\) and \(m^a\), respectively, and where \(\Re(\;)\) denotes “the real part of”.

3.2 N–P Ricci scalars↩︎

The N–P Ricci scalars are calculated as \[\begin{align} \Phi_{00}&=\frac{1}{sec:4}(\varrho+p+\Pi-2Q),\tag{18}\\ \Phi_{11}&=\frac{1}{sec:4}(\varrho-\frac{1}{sec:2}\Pi-\Lambda+\Pi_{ab}m^a\bar m^b),\tag{19}\\ \Phi_{22}&=\frac{1}{sec:4}(\varrho+p+\Pi+2Q),\tag{20}\\ \Phi_{01}&=-\frac{1}{2\sqrt{sec:2}}(\Pi_a-Q_a)m^a,\tag{21}\\ \Phi_{02}&=\bar\Phi_{20}=\frac{1}{sec:2}\Pi_{ab}m^am^b,\tag{22}\\ \Phi_{12}&=\bar\Phi_{21}=-\frac{1}{2\sqrt{sec:2}}(\Pi_a+Q_a)\bar m^a.\tag{23} \end{align}\]

There appear to be minor errors in the computation of the Ricci NP scalars in [14] except for \(\Phi_{02}\) and \(\Phi_{12}\). \(\Phi_{01}\) of [14] differs from the one here by a sign. The differences in \(\Phi_{00}\), \(\Phi_{11}\) and \(\Phi_{22}\), however, involve missing and misplaced variables, as well as scaling differences.

3.3 Weyl N–P scalars↩︎

The Weyl N–P scalars are calculated as \[\begin{align} \Psi_0&=(\mathcal{E}_{ab}-\tilde{\eta}_{ac}\mathcal{H}^c_{\;b})m^am^b=(\mathcal{E}_{ab}+i\mathcal{H}_{ab})m^am^b,\tag{24}\\ \Psi_1&=-\frac{1}{\sqrt{sec:2}}(\mathcal{E}_a-\tilde{\eta}_{ab}\mathcal{H}^b)m^a\nonumber\\ &=-\frac{1}{\sqrt{sec:2}}(\mathcal{E}_a+i\mathcal{H}_a)m^a,\tag{25}\\ \Psi_2&=\frac{1}{sec:2}(\mathcal{E}-i\mathcal{H}),\tag{26}\\ \Psi_3&=\frac{1}{\sqrt{sec:2}}(\mathcal{E}_a+i\mathcal{H}_a)\bar m^a,\tag{27}\\ \Psi_4&=(\mathcal{E}_{ab}+\tilde{\eta}_{ac}\mathcal{H}^c_{\;b})\bar m^a\bar m^b=(\mathcal{E}_{ab}+i\mathcal{H}_{ab})\bar m^a\bar m^b.\tag{28} \end{align}\]

The above scalars have also been computed in [14], although now there are sign modifications to 24 and 28 .

In Table 1, we highlight some key N–P scalars and their corresponding geometric interpretations in the 1+1+2 formulation.

4 Brief comments on some implications for black hole horizons↩︎

In this section, we briefly illustrate how the relations established above may be used to characterize black hole horizons in terms of N–P quantities. We restrict attention to the class of locally rotationally symmetric (LRS) spacetimes, specifically the LRS class II subclass, which naturally admits the \(1+1+2\) decomposition [7], [8]. Owing to the underlying symmetry, all sheet vectors and projected trace-free tensors vanish identically, and the spacetime is completely described by covariantly defined scalar quantities. For LRS class II spacetimes, the vorticities \(\Omega\) and \(\xi\) also vanish.

One of the principal advantages of the \(1+1+2\) formalism is that it allows the Einstein field equations to be analyzed directly at the level of the preferred two-surfaces. A particularly important invariant associated with these surfaces is the Gaussian curvature scalar \(\mathcal{K}\). In LRS class II spacetimes, \(\mathcal{K}\) is related to the matter variables, the electric Weyl scalar, and the null expansions through \[\begin{align} \mathcal{K}=\frac{1}{sec:3}(\varrho+\Lambda)-\left(\mathcal{E}+\frac{1}{sec:2}\Pi\right)-\frac{1}{sec:2}\theta_{(\ell)}\theta_{(n)},\label{gauss1} \end{align}\tag{29}\] where \(\theta_{(\ell)}=-2\Re(\tilde{\rho})\) and \(\theta_{(n)}=2\Re(\tilde{\mu})\), as defined in 16 and 17 , denote the expansions associated with the outgoing and ingoing null congruences, respectively. Rearranging 29 gives \[\begin{align} \mathcal{E}-\frac{1}{sec:3}\varrho+\frac{1}{sec:2}\Pi=\frac{1}{sec:3}\Lambda-\mathcal{K}-\frac{1}{sec:2}\theta_{(\ell)}\theta_{(n)}.\label{gaussn1} \end{align}\tag{30}\] It will be assumed throughout that the local energy density \(\varrho\) is nonnegative. Now, the directional derivatives of the left hand side of 30 along \(u^a\) and \(e^a\) are given respectively by [8] \[\begin{align} \dot{\mathcal{E}}-\frac{1}{sec:3}\dot{\varrho}+\frac{1}{sec:2}\dot{\Pi}&=\left(\frac{sec:2}{sec:3}\theta-\Sigma\right)\left(-\frac{sec:3}{sec:2}\mathcal{E}-\frac{1}{sec:4}\Pi+\frac{1}{sec:2}(\varrho+p)\right)\nonumber\\ &+\frac{1}{sec:2}\phi Q,\tag{31}\\ \mathcal{E}'-\frac{1}{sec:3}\varrho'+\frac{1}{sec:2}\Pi'&=-\frac{1}{sec:2}\left(\frac{sec:2}{sec:3}\theta-\Sigma\right)Q\nonumber\\ &-\frac{sec:3}{sec:2}\phi\left(\mathcal{E}+\frac{1}{sec:2}\Pi\right).\tag{32} \end{align}\] Taking the corresponding derivatives of 30 and then combining the resulting expressions yields \[\begin{align} &\sqrt{sec:2}\theta_{(\ell)}\mathcal{Z}_1+\phi\mathcal{Z}_2\nonumber\\ &=-2\sqrt{sec:2}\left(\mathcal{L}_{\ell}\mathcal{K}+\frac{1}{sec:4}(\theta_{(n)}\mathcal{L}_{\ell}\theta_{(\ell)}+\theta_{(\ell)}\mathcal{L}_{\ell}\theta_{(n)}\right),\label{sol1} \end{align}\tag{33}\] where \(\mathcal{L}_{\ell}\) denotes the Lie derivative along \(\ell^a\), and \[\begin{align} \mathcal{Z}_1&=-Q-2\mathcal{E}+\mathcal{K}-\frac{sec:2}{sec:3}\varrho-p-\frac{1}{sec:3}\Lambda+\frac{1}{sec:4}\theta_{(\ell)}\theta_{(n)},\\ \mathcal{Z}_2&=2(Q+\mathcal{E}+\mathcal{K})+\left(\frac{sec:5}{sec:3}\varrho+p\right)+\frac{sec:4}{sec:3}\Lambda+\frac{1}{sec:2}\theta_{(\ell)}\theta_{(n)}. \end{align}\] For LRS class II spacetimes, the Gaussian curvature evolves according to \[\begin{align} \dot{\mathcal{K}}=-\left(\frac{sec:2}{sec:3}\theta-\Sigma\right)\mathcal{K},\quad \mathcal{K}'=-\phi\mathcal{K}. \end{align}\] from which it follows that \[\begin{align} \sqrt{sec:2}\mathcal{L}_{\ell}\mathcal{K}=-\theta_{(\ell)}\mathcal{K}. \end{align}\]

As we are interested in horizons, let us introduce the notion of a marginally outer trapped surface (MOTS) which generalizes cross sections of horizons. Consider a closed surface \(\mathcal{S}\) in the decomposition under consideration. Then, the surface \(\mathcal{S}\) is said to be marginally outer trapped if \[\begin{align} \theta_{(\ell)}=0 \end{align}\] everywhere on \(S\). Assuming that such surfaces foliate a three-surface \(\mathcal{T}\), which is referred to as a marginally outer trapped tube (MOTT), one may define an everywhere tangent vector to \(\mathcal{T}\) \[\begin{align} x^a=\ell^a-Cn^a, \end{align}\] where the scalar function \(C\) determines the causal character of the horizon. In spherical symmetry, \[\begin{align} C=\frac{\mathcal{L}_{\ell}\theta_{(\ell)}}{\mathcal{L}_n\theta_{(\ell)}},\label{ccc} \end{align}\tag{34}\] since \(\mathcal{L}_{x}\theta_{(\ell)}=0\) along the horizon [16], [17]. In this case the sign of \(C\) is constant over \(\mathcal{T}\) and \(\mathcal{T}\) is a horizon (see the reference [18]). A spacelike horizon corresponds to \(C>0\) (a dynamical horizon), whereas \(C=0\) characterizes an isolated horizon. In context of LRS spacetimes the reader is referred to [11], [12] for explicit forms of the expression for \(C\).

We consider a future outer trapping horizon (FOTH) which contains trapped surfaces just to the inside, since we are interested in black hole horizons. In this case the “outer” is usually dropped and a MOTS is simply a MTS. For a future outer trapping horizon (FOTH), one requires \[\begin{align} \theta_{(n)}<0, \quad \mathcal{L}_{n}\theta_{(\ell)}<0. \end{align}\]

Imposing the MOTS condition \(\theta_{(\ell)}=0\), the constraint equation 33 reduces to \[\begin{align} 2\phi\mathcal{Z}_2=-\sqrt{sec:2}\theta_{(n)}\mathcal{L}_{\ell}\theta_{(\ell)}.\label{sol2} \end{align}\tag{35}\] We emphasize that all expressions are evaluated at the horizon. Using the relations derived in Section 3, this expression may be rewritten entirely in terms of N–P quantities: \[\begin{align} (\Phi_{00}+\mathcal{Z}_2)\tilde{\mu}=0.\label{sol3} \end{align}\tag{36}\]

Of course, for the case of a minimal cross section, i.e. \(\tilde{\mu}=0\), the above is identically verified. Otherwise, for a FOTH we have \[\begin{align} \Phi_{00}+\mathcal{Z}_2=0.\label{sol4} \end{align}\tag{37}\] Assuming the null energy condition, \(\Phi_{00}\geq0\), equation 37 implies that \[\begin{align} \mathcal{Z}_2\leq0. \end{align}\] Using 18 and 20 , we have \(\Phi_{22}-\Phi_{00}=Q\), and we may write \(\mathcal{Z}_2\) in terms of the N–P scalars \[\begin{align} \mathcal{Z}_2&=2\mathcal{F}+\left(\frac{sec:5}{sec:3}\varrho+p\right),\label{z2np} \end{align}\tag{38}\] where \[\begin{align} \mathcal{F}=(\Phi_{22}+2\Psi_2+\mathcal{K})+\frac{sec:2}{sec:3}\Lambda. \end{align}\] If one further imposes the strong energy condition (SEC), which implies the NEC \(\varrho+3p\geq0\), then the parenthesized term of 38 is positive, and since \(\mathcal{Z}_2\leq0\), necessarily \[\begin{align} \mathcal{F}\leq0. \end{align}\] Substituting 38 into 37 yields \[\begin{align} \frac{1}{sec:2}\Phi_{00}&=\mathcal{F}+\frac{1}{sec:2}\left(\frac{sec:5}{sec:3}\varrho+p\right).\label{constraint1} \end{align}\tag{39}\] Therefore, we have that \[\begin{align} \frac{1}{sec:2}\Phi_{00}\geq\mathcal{F}.\label{sol5} \end{align}\tag{40}\]

Since we are considering black hole horizons (FOTH), spherical topology is necessary for the MOTS (this follows from a notion of stability for MOTS for which the interested reader is referred to the references [19], [20] for details). Therefore, the condition \[\begin{align} \Phi_{22}+2\Psi_2+\frac{sec:2}{sec:3}\Lambda\leq0,\label{sol6} \end{align}\tag{41}\] is both necessary and sufficient to guarantee the nonpositivity of \(\mathcal{F}\) under the SEC assumption. If 41 fails, \(\mathcal{F}>0\) and this fails the SEC as \(\mathcal{Z}_2\) is required to be nonpositive. Equation 41 therefore provides a purely geometric criterion, expressed in terms of Newman–Penrose scalars, that constrains the existence of black hole horizons in LRS class II spacetimes. In particular, if this inequality fails in some region of spacetime satisfying the SEC, then that region cannot admit a future outer trapping horizon. Since \(\Psi_2\) encodes the Coulombic part of the free gravitational field together with contributions from the heat flux, the condition illustrates that horizon formation depends on a delicate balance between matter flux and Weyl curvature.

Consider, for example, a LRS II spacetime that is anisotropic but with vanishing heat flux (\(Q=0,\Pi\neq0\)). The the scalars \(\Phi_{00}\) and \(\Phi_{22}\) coincide. the SEC then imposes \[\begin{align} \Psi_2+\frac{1}{sec:3}\Lambda\leq0. \end{align}\] Thus, in the absence of a cosmological constant, these spacetimes will not admit a FOTH for a positive \(\Psi_2\).

The case of a generic isolated horizon embedded in a vacuum configuration admits an especially simple characterization entirely reliant on the sign of the cosmological constant. For such a case, an important invariant characterizing the horizon geometry is the complex scalar [4], [21] \[\begin{align} \bar\psi=-\frac{1}{sec:2}\mathcal{K}-i\Im(\Psi_2),\label{psi2951} \end{align}\tag{42}\] where \(\Im(\;)\) denotes “the imaginary part of”, which in our formulation takes the form \[\begin{align} \bar\psi=-\frac{1}{sec:2}(\mathcal{K}-i\mathcal{H}).\label{psi2952} \end{align}\tag{43}\] This complex scalar relates to \(\Psi_2\) via [4], [21] \[\begin{align} \Psi_2=\bar\psi+\frac{\Lambda}{6}=-\frac{1}{sec:2}(\mathcal{K}-i\mathcal{H})+\frac{\Lambda}{6}.\label{psi2953} \end{align}\tag{44}\] Comparing this with the relation obtained for \(\Psi_2\) yields \[\begin{align} \mathcal{H}=0,\quad \frac{1}{sec:3}\Lambda=\mathcal{E}+\mathcal{K}=2\Psi_2+\mathcal{K}.\label{iso1} \end{align}\tag{45}\] For LRS II solutions the vanishing condition on the magnetic Weyl scalar \(\mathcal{H}\) is trivially met, and from the second condition of 45 and noting that \(\Phi_{22}=0\) for vacuum, we have \[\begin{align} \mathcal{F}=\Lambda. \end{align}\] Therefore, the condition 41 now simply becomes \[\begin{align} \Lambda\leq0.\label{forih} \end{align}\tag{46}\] Thus, in vacuum LRS II geometry satisfying the SEC, a positive \(\Lambda\) obstructs the existence of an isolated horizon, except when the MOTS are minimal (in this case the equation 36 holds trivially). Of course, necessarily, \(\Psi_2<0\) on the horizon by 45 .

In fact, it turns out that this horizon existence constraint from the cosmological constant may be generic for \(\tilde{\mu}\neq0\), for LRS spacetimes: using the Gauss curvature, after some basic algebra, we may rewrite the function \(\mathcal{F}\) in terms of a Ricci N–P scalar, cosmological constant and curvature variables as \[\begin{align} 2\mathcal{F}=2\Lambda-\Phi_{00}+\left(\frac{sec:5}{sec:3}\varrho+p\right).\label{constraint2} \end{align}\tag{47}\] which we compare to 39 to obtain \[\begin{align} \Lambda=2\mathcal{F}.\label{constraint3} \end{align}\tag{48}\] Therefore, a positive \(\Lambda\) implies \(\mathcal{F}>0\). Thus, for a LRS II geometry obeying the SEC, a positive \(\Lambda\) obstructs the existence of a FOTH. This result is true whether or not we are in a vacuum. Of course, generally, if \(\Lambda\leq0\), we still have to go back and verify 41 .

Notice that on an isolated horizon, if the SEC strictly holds, \(\mathcal{F}>\Lambda\). For these solutions then, a strictly negative \(\Lambda\) is necessary for the existence of an FOTH. It follows that, for a LRS II geometry with a vanishing cosmological constant and obeying the SEC strictly, any FOTH in such geometry is spacelike.

These results demonstrate how the mapping between the N–P and \(1+1+2\) formalisms provides a geometrically transparent framework for analyzing black hole horizons. More broadly, they suggest that several well-known horizon conditions expressed in the N–P language may admit natural geometric reinterpretations within the covariant \(1+1+2\) approach.

5 Summary and outlook↩︎

In this work, we have, to our knowledge, for the first time, derived the complete set of N–P scalars in terms of the scalar, vector, and tensor quantities of the \(1+1+2\) semitetrad covariant formulation of general relativity. In doing so, we have established a direct mapping between two widely used approaches to the study of gravitational systems in general relativity. Given the complementary strengths of these formalisms, we expect that this correspondence may prove useful in a variety of contexts within general relativity and relativistic astrophysics.

As a simple application of this mapping, we have considered conditions governing the existence of black hole horizons in the class of LRS II spacetimes. In particular, we obtained necessary conditions for the existence of dynamical and isolated horizons expressed purely in terms of the Ricci and Weyl N–P scalars together with the cosmological constant, specifically \(\Phi_{22}\) and \(\Psi_2\). It is also found that the existence of a future outer trapping horizon in these spacetimes favors a nonpositive cosmological constant. These conditions provide a simple criterion for ruling out the existence of black hole horizons in regions where the inequalities fail to hold.

There are several promising directions in which the mapping presented in this letter may be further exploited. For example, the horizon analysis outlined here could potentially be extended to more general classes of black hole horizons beyond the LRS setting. Another natural application concerns gravitational perturbation theory, particularly in light of the work of Pratten [14] in the context of \(f(R)\) gravity. Recasting known perturbative results formulated in the N–P approach in terms of the geometric variables of the \(1+1+2\) formalism may provide deeper insight into the interplay between spacetime geometry and gravitational perturbations.

Another interesting direction arises from the work of [13], where the dynamics of a null horizon subjected to linear perturbations was investigated. It is conceivable that the equations governing the perturbation dynamics of the horizon may admit a more transparent geometric interpretation, or perhaps even simplification, when reformulated within the present framework. Finally, it would be worthwhile to investigate the classification of general \(1+1+2\) spacetimes according to Petrov type, which would naturally encompass perturbations of locally rotationally symmetric geometries and could provide further insight into the geometric structure of perturbative spacetimes.

References↩︎