Null Surface Formulation in General Relativity

• NSF is a geometric framework where the full conformal structure of spacetime is encoded by a scalar function whose level sets define families of null surfaces.
• It reformulates Einstein's equations into a system of scalar partial differential equations and Pfaffian structures, facilitating boundary and asymptotic quantization.
• The approach extends to higher dimensions and quantum regimes, linking to Cartan geometry, twistor theory, and providing insights for holography and matter coupling.

The Null Surface Formulation (NSF) of General Relativity is a geometric framework in which the spacetime conformal geometry is encoded, not by the metric, but by a family of null surfaces parametrized by a scalar function $Z(x^a; \zeta, \bar\zeta)$ defined on the bundle $M^4 \times S^2$ of spacetime points and null directions. The level sets $Z(x^a; \zeta, \bar\zeta)=\text{const}$ define a congruence of null hypersurfaces, and all local geometric and physical content of General Relativity, including the Einstein equations, are re-expressed in terms of this function and associated structures. NSF provides a natural setting for boundary/bulk duality, asymptotic quantization, and the holographic principle, and has been formulated and extended to higher dimensions, quantum theory, and coupled matter systems.

1. Geometric Foundations and Pfaffian System

At the heart of NSF is the assertion that the entire conformal geometry of spacetime can be encoded in a single real or complex scalar function $Z: M^4 \times S^2 \to \mathbb{R}$ (or $\mathbb{C}$), such that for every $(\zeta, \bar\zeta) \in S^2$, the hypersurfaces $Z(x^a; \zeta, \bar\zeta) = u$ are null in $M^4$ with respect to some Lorentzian metric $g_{ab}(x)$. The eikonal condition

$$g^{ab}(x)\, \partial_a Z(x; \zeta, \bar\zeta)\, \partial_b Z(x; \zeta, \bar\zeta) = 0$$

guarantees that $Z$ defines null surfaces for all directions. This structure is made manifest using a central Pfaffian system—a set of differential one-forms $(\beta^0, \beta^+, \beta^-, \beta^1)$ constructed from derivatives of $Z$ with respect to $(\zeta, \bar\zeta)$. The vanishing of these forms defines the NSF structure: $$\begin{aligned} \beta^0 &= dZ - W\,ds - W^*\,ds^* \ \beta^+ &= dW - S\,ds - R\,ds^* \ \beta^- &= dW^* - R\,ds - S^*\,ds^* \ \beta^1 &= dR - T\,ds - T^*\,ds^* \end{aligned}$$ with $W = \partial_s Z$, $R = \partial_{s s^*} Z$, $S = \partial_{ss} Z$, etc. This Pfaffian system and its integrability conditions provide the means to recover the spacetime conformal geometry and to specify the Einstein equations as compatibility requirements (Gallo et al., 16 Dec 2025).

2. Reconstruction of the Metric and Cartan Connection

The function $Z$ and its derivatives encode the conformal class $[g]$ via an explicit algebraic construction. By pulling back the parameter-space forms to $M^4$ and imposing the zero-torsion condition, one constructs a Lorentzian conformal metric

$$h = \theta^0 \otimes \theta^1 + \theta^1 \otimes \theta^0 - \theta^+ \otimes \theta^- - \theta^- \otimes \theta^+$$

with coframe components $\theta^i$ related to the $\beta^i$. The full physical metric is recovered as $g_{ab}(x) = \Omega^2(x)\, h_{ab}[Z]$, where the conformal factor $\Omega(x)$ is determined by further field equations (Gallo et al., 16 Dec 2025, Bordcoch et al., 2012, Kozameh et al., 27 Jun 2025).

This geometric structure is interpreted as a Cartan normal conformal connection: a $SO(4,2)$-valued one-form $\omega$ with curvature $\Omega = d\omega + \omega \wedge \omega$, whose torsion-free and curvature components (Weyl, Cotton-York) correspond to properties of the spacetime conformal geometry. NSF thus connects to Cartan geometry and twistor theory at a foundational level.

3. NSF Field Equations and Metricity Conditions

The fundamental field equations of NSF replace the ten Einstein equations with a system on $(Z,\Omega)$. Typically, three coupled PDEs are required (NSF 2.0 formulation):

• First metricity condition (nullness of level sets):

$$\eth \bar\eth (\Omega^2) = \Omega^2 \left( \partial_r \bar\eth^2 \Lambda - h^{ab} \partial_a \Lambda \partial_b \bar\Lambda \right)$$

• Second metricity (generalized Wünschmann) condition:

$$\frac{\partial}{\partial r} (\eth \Lambda) + 3 h^{wi} \partial_i \Lambda = 0$$

• Trace-free Einstein equation:

$2 \partial_r^2 \Omega = R_{rr}[h]\, \Omega$

where $\Lambda = \eth^2 Z$ encodes the shear data and $h^{ab}$ is an algebraic functional of $\Lambda$ (explicit forms depend on the coordinate choice and the derivative structure in $S^2$ directions) (Kozameh et al., 27 Jun 2025, Bordcoch et al., 2012). For NSF 2.0, the system is recast into three real, spin-weight zero PDEs, enhancing both geometric transparency and computational tractability.

Vacuum Einstein equations are thus encapsulated as compatibility (metricity) conditions on $Z$ and $\Omega$, together with the requirement that the Pfaffian system be integrable. In higher dimensions $n$, the formulation generalizes to a hierarchy of dual PDEs in an $(n-2)$-parameter family and a set of generalized Wünschmann-type constraints critical for defining the conformal class of the metric (Gallo, 2011).

4. Asymptotic Structure, Characteristic Data, and Boundary Approaches

NSF is naturally adapted to characteristic and boundary formulations. In asymptotically flat spacetimes, the free gravitational data (Bondi shear $\sigma_B(u,\zeta,\bar\zeta)$ at null infinity $\mathscr{I}^+$) enters as a source in the equations for $Z$ and $\Omega$. The large-$R$ limit of the NSF equations yields the "good cut" equation, a 4th-order PDE for $Z$ entirely driven by null data at infinity: $$\eth^2 Z = \eth^2 \sigma_B + 2 \eth \bar\eth \sigma_B + \int_u^\infty \eth(\dot\sigma_B \dot{\bar\sigma}_B) du'$$ where $\dot{}$ denotes retarded time derivative (Bordcoch et al., 2012, Kozameh et al., 27 Jun 2025).

At the quantum level, the fundamental free data are promoted to operators (Bondi news commutation relations), and the full metric and curvature are constructed as operator-valued distributions. The S-matrix is defined directly in terms of outgoing and incoming shear modes at future and past null infinity, with all gravitational scattering information encoded in the matching of these modes (Kozameh et al., 27 Jun 2025).

Boundary field theories derived from NSF—such as null boundary Chern-Simons theories for self-dual Ashtekar connections—capture the entire gravitational dynamics quasi-locally and realize a form of null holography. The bulk transition amplitudes can be assembled from gluings of boundary amplitudes by integrating over edge modes defined on two-dimensional corners, implementing a "general-boundary" quantum prescription (Wieland, 2019).

5. Quantum NSF and Scattering Theory

NSF provides a framework for boundary-to-bulk quantization of gravity:

• The Bondi shear fields $\sigma^\pm$ at $\mathscr{I}^\pm$ are quantized as operator-valued distributions with canonical commutation relations.
• The metric operator is built as a normal-ordered functional of fields $Z$, $\Omega$ and their derivatives, with higher-order corrections encoding nontrivial "graviton" interactions.
• Outgoing and incoming graviton creation/annihilation operators are constructed from $\sigma^+$ and $-\bar\sigma^-$ respectively; graviton scattering amplitudes are computed perturbatively via matching conditions of the full NSF equations.
• The resulting framework is manifestly gauge-invariant, preserves the full (null) radiative causal structure at each order, and naturally incorporates BMS symmetry and infrared properties (Kozameh et al., 27 Jun 2025, Bordcoch et al., 2012).

The NSF approach has been successfully extended to perturbative coupling with matter fields, such as a massless scalar. In this case, matter sources enter the NSF field equations for $\Omega$, and the integrated effect on the Bondi shear encodes tree-level graviton-matter scattering, with higher perturbative orders corresponding to classical "loop" corrections (Kozameha et al., 2023).

6. Higher-Dimensional Generalizations

NSF admits a systematic extension to $n$-dimensional Lorentzian geometry. One defines a function $Z(x^a,s,s^*,y^m)$ on $M^n \times \mathbb{P}$, where $\mathbb{P}$ is an $(n-2)$-parameter family of null directions. The dual system consists of $n(n-3)/2$ second-order PDEs in the parameter space, whose solutions reconstruct the conformal metric. The generalized Wünschmann conditions, essential for metricity and torsion-freedom, become a system of third-derivative PDEs on the dual functions $(S,S^*,P,P^*,I)$, ensuring the full coordinate-invariant construction of the $n$-dimensional conformal geometry (Gallo, 2011).

Einstein's equations reduce to a single scalar PDE for the conformal factor, closing the system and guaranteeing equivalence with traditional (vacuum or Einstein–$\Lambda$) general relativity.

7. Physical and Mathematical Implications

The Null Surface Formulation fundamentally shifts the perspective from metric fields to scalar generating functions of null surfaces. Key implications include:

• A fully characteristic and manifestly conformally invariant approach to general relativity.
• Direct accessibility to gravitational radiation and classical/quantum scattering processes via null data at infinity.
• The reduction of Einstein's equations to a system of equations on scalar functions over $M^4 \times S^2$, with substantial computational benefits, particularly in asymptotic or perturbative contexts.
• Foundations for quasi-local holography, edge modes, and general-boundary field theories of gravity (Wieland, 2019, Gallo et al., 16 Dec 2025).
• Gauge-invariant, nonlocal quantum observables and a natural S-matrix for quantum gravity (Kozameh et al., 27 Jun 2025, Bordcoch et al., 2012).
• Deep connections to Cartan geometry, twistor theory, and integrability structures, with applications extending to higher dimensions and coupled matter systems.

NSF is thus a central theoretical framework for re-expressing the dynamical content of gravity in terms of scalar fields associated with families of null surfaces, with wide-ranging repercussions in geometric analysis, quantum gravity, and holography.