Outgoing Null Geodesic Families

Updated 24 August 2025

Outgoing Null Geodesic Families are collections of lightlike curves in Lorentzian geometries, vital for analyzing causal structures and horizon formation.
Their properties are defined through null foliations, shear-free congruences, and expansion scalars, utilizing variational principles and differential geometry.
Applications span from classic black hole horizon detection to quantum gravity and fluid dynamics, offering insights into both classical and quantum regimes.

Outgoing null geodesic families are collections of lightlike curves in Lorentzian or generalized geometries that propagate away from a given spacetime surface or region, playing a foundational role in causal structure, horizon theory, and the dynamical analysis of geometrical flows. Their paper encompasses the intrinsic and extrinsic geometric, analytic, and topological properties of null hypersurfaces and congruences, with the behavior and classification of these families governed by variational principles, symmetry considerations, and both classical and quantum propagation laws.

1. Geometric Foundations and Foliations

Outgoing null geodesic families are characterized in the context of null hypersurfaces and their foliations. In vacuum spacetimes, a non-expanding horizon (NEH) is defined as a null 3-submanifold $H$ with topology $\mathbb{R} \times S^2$ whose intrinsic generators satisfy the vanishing divergence and shear conditions ( $\rho = \sigma = 0$ ) (0908.0751). This ensures the "intrinsic" outgoing null geodesics have no distortion along the horizon. Null foliations can be constructed using a 2+2 decomposition $g_{ab} = -2\ell_{(a}n_{b)} + 2m_{(a} \bar{m}_{b)}$ , where $\ell^a$ and $n^a$ are null directions and $(m^a, \bar{m}^a)$ span transverse surfaces (Huber, 2019). The generators are often gradients of optical functions, e.g., $\ell_a = -d\sigma_a$ , automatically ensuring hypersurface-orthogonality and the eikonal condition.

The classification of outgoing and ingoing null geodesic families is central to several frameworks, such as dynamical system foliations of the Schwarzschild spacetime (Belbruno et al., 2011), where every null geodesic belongs to an invariant manifold distinguished by angular momentum and its causal origin—outgoing geodesics may flow from white hole singularities toward infinity or return to black hole horizons.

2. Null Geodesic Congruences and Shear-Free Formalism

A congruence is a smooth family of geodesics characterized by expansion, shear, and rotation (twist) tensors. For NEH and similar settings, the outgoing null geodesic congruences intersect the horizon with vanishing shear, a condition achieved by null rotations of the tetrad (0908.0751). The shear-free requirement leads to a "good cut" equation,

$ð^2 G(\tau, \zeta, \bar{\zeta}) = -\lambda^{(0)}(\zeta, \bar{\zeta}),$

where $G$ labels cuts of the horizon, and the solutions correspond to shear-free congruences determined by four complex parameters (one for each analytic world-line). These congruences induce CR (Cauchy-Riemann) structures on the horizon, and a privileged tetrad frame can be selected to remove specific harmonic parts of the Weyl tensor, uniquely associating a CR structure and a "center-of-mass" frame with the outgoing family.

In more general causal geometries, null geodesic families (including outgoing congruences) arise from Legendrian dynamics on regular conical subbundles (where null directions are not necessarily light cones of metrics but smooth degree- $k$ cones), and the resulting tangent spray defines unparametrized null geodesics (Holland et al., 2011).

3. Analytical Structures, Expansion Scalars, and Horizons

A critical analytic property is the expansion scalar ( $\Theta$ for congruences, or tr $\chi$ for null hypersurfaces), which determines local focusing and defocusing. The Raychaudhuri equation, specialized for null congruences,

$\frac{d\Theta}{d\lambda} = -\frac{1}{2} \Theta^2 - \sigma_{\mu\nu}\sigma^{\mu\nu} + \omega_{\mu\nu}\omega^{\mu\nu} - R_{\mu\nu}k^\mu k^\nu,$

demonstrates how expansion evolves under curvature and shear, often simplifying for algebraically special metrics (e.g., Kerr, where shear vanishes and expansion is $\Theta_+ = +2/r$ ) (Trujillo et al., 1 Mar 2024). In black hole mechanics, marginally trapped surfaces (apparent horizons) are characterized by the vanishing of the outgoing expansion ( $\theta_\ell = 0$ ), and the invariance of the product $\theta_\ell \theta_n$ under reciprocal rescaling of the null normals underlines the coordinate independence of horizon detection (Adler, 2021).

Canonical foliations on null hypersurfaces in low regularity settings have expansion scalars locally controlled by initial data and $L^2$ curvature fluxes through the hypersurface, with elliptic and Besov estimates guaranteeing uniform boundedness even under minimal assumptions (Czimek et al., 2019).

4. Symmetry, Contact/Engel Structures, and Topology

The symmetry algebra of outgoing null geodesic families is tightly linked to the underlying geometric structure. For manifolds admitting null G-structures, congruences are preserved up to conformal rescaling, and the corresponding diffeomorphism algebra interpolates between the Bondi-Metzner-Sachs (BMS) algebra at null infinity and Lorentz symmetry at the horizon (Papadopoulos, 2018). Outgoing congruences also generate orbit spaces with well-defined induced geometry (screen metrics, CR structures).

The topology and contact structure of the space of null geodesics (including outgoing families) in compact spacetimes can be understood via Engel geometry and the Cartan prolongation/deprolongation process. For spacetimes such as $(\mathbb{S}^2 \times \mathbb{S}^1, g_\circ - \frac{d^2}{c^2}dt^2)$ , the space of null geodesics is diffeomorphic to the lens space $L(2c,1)$ equipped with a pushforward contact structure; analytic formulas in local coordinates are available for both the Engel distribution and its kernel (Marín-Salvador et al., 2021).

5. Variational Equations, Integrability, and Quantum Extensions

Geodesic deviation and stability of outgoing families are analyzed via the Jacobi equation and its Hamiltonian formulation. The Morales-Ramis theorem guarantees that, if the geodesic flow is integrable, so is its deviation equation (in the sense of differential Galois theory), even in singular cases such as photon spheres in Schwarzschild spacetime (Morales-Ruiz et al., 2023). Specific cases (radial and circular outgoing null geodesics) are shown to yield integrable variational equations with solutions expressible in elementary functions with commutative Galois groups.

Quantum extensions involve associating propagators to congruences. In the Kerr metric, for principal equatorial outgoing null geodesic congruences, the classical divergence of expansion at singularities (ring singularity) does not propagate into the quantum regime: the associated Feynman propagator remains finite for these congruences, implying quantum regularization or softening of classical singularities (Trujillo et al., 1 Mar 2024). The methodology interprets transverse area evolution as a free-particle Lagrangian, with exact path integral expressions.

6. Generalizations, Applications, and Physical Interpretations

Generalizations of outgoing null geodesic families appear in Minkowski 3-space as "null similar curves" (curves that, under variable reparameterizations, preserve tangent directions), with geodesics and helices forming invariant families (Önder, 2012). In fluid dynamics, closed null-geodesics of tailored Lorentzian metrics delineate boundaries of coherent vortex structures—a method implemented numerically via reduced ODEs integrating over admissible initial conditions, with applications to oceanographic data (Serra et al., 2016).

Quantum spacetime models replace sharp null geodesics with probabilistic causal connections: the "tendency postulate" posits that free massless objects follow regions of large quantum causal fluctuations, offering a natural mechanism for information leakage through horizons without violating locality or permitting superluminal signaling (Jia, 2018).

Optical equations for null strings generalize congruence kinematics by encoding the full evolution in a complex scalar $Z = \theta_s + i\kappa_2$ , with universal features such as the freezing-out of string shapes at null infinity and memory effects from gravitational radiation embedded in subleading corrections (Fursaev, 2021).

Outgoing null geodesic families thus constitute a unifying concept across geometric analysis, general relativity, quantum gravity models, and applied mathematics, with their properties and dynamics encapsulating key aspects of causality, horizon structure, symmetry, and integrability. Their roles in horizon foliation, propagation of physical fields, and topological classification continue to drive developments in both theoretical and applied gravitational research.