Null Surface Formulation Of General Relativity Research Articles

Three-variable solution in the $$(2+1)$$ ( 2 + 1 ) -dimensional null-surface formulation

The null-surface formulation of general relativity (NSF) describes gravity by using families of null surfaces instead of a spacetime metric. Despite the fact that the NSF is (to within a conformal factor) equivalent to general relativity, the equations of the NSF are exceptionally difficult to solve, even in 2+1 dimensions. The present paper gives the first exact $$(2+1)$$ -dimensional solution that depends nontrivially upon all three of the NSF’s intrinsic spacetime variables. The metric derived from this solution is shown to represent a spacetime whose source is a massless scalar field that satisfies the general relativistic wave equation and the Einstein equations with minimal coupling. The spacetime is identified as one of a family of $$(2+1)$$ -dimensional general relativistic spacetimes discovered by Cavaglia.

Solution for the null-surface formulation of general relativity in 2+1 dimensions

The null surface formulation of general relativity (NSF) provides an alternative way of describing the gravitational field and focuses exclusively on families of null surfaces. The metric is not needed. The present paper presents an exact solution of the (2+1)-dimensional NSF by solving the third-order nonlinear field equations directly. The resulting solution is non-singular and is shown to describe a perfect fluid with vorticity.

Higher dimensional conformal metrics from PDEs and null-surface formulation of GR

We analyze the relationship between n-dimensional conformal metrics and a certain class of partial differential equations (PDEs) that are in duality with the eikonal equation. In particular, we extend the null-surface formulation of general relativity (GR) to higher dimensions and give explicit expressions for the components of the metric and the generalized Wünschmann-like metricity conditions. We also compute the equation that the conformal factor must satisfy in order the metric to be a solution of the Einstein equations.

Dynamics of Fermat potentials in nonperturbative gravitational lensing

We present a framework, based on the null-surface formulation of general relativity, for discussing the dynamics of Fermat potentials for gravitational lensing in a generic situation without approximations of any kind. Additionally, we derive two lens equations: one for the case of thick compact lenses and the other one for lensing by gravitational waves. These equations in principle generalize the astrophysical scheme for lensing by removing the thin-lens approximation while retaining the weak fields.

Quantization of the null-surface formulation of general relativity

We define and discuss various quantum operators that describe the geometry of spacetime in quantum general relativity. These are obtained by combining the Null-Surface Formulation of general relativity, recently developed, with asymptotic quantization. One of the operators defined describes a ``fuzzy'' quantum light cone structure. Others, denoted ``spacetime-point operators'', characterize geometrically-defined physical points. We discuss the interpretation of these operators. This seems to suggest a picture of quantum spacetime as made of ``fuzzy'' physical points. We derive the commutation algebra of the quantum spacetime point operators in the linearization around flat space.

Fuzzy spacetime from a null-surface version of general relativity

The null-surface formulation of general relativity - recently introduced - provides novel tools for describing the gravitational field, as well as a fresh physical way of viewing it. The new formulation provides `local' observables corresponding to the coordinates of points - which constitute the spacetime manifold - in a geometrically defined chart, as well as nonlocal observables corresponding to lightcone cuts and lightcones. In the quantum theory, the spacetime point observables become operators and the spacetime manifold itself becomes `quantized', or `fuzzy'. This novel view may shed light on some of the interpretational problems of a quantum theory of gravity. Indeed, as we discuss briefly, the null-surface formulation of general relativity provides (local) geometrical quantities - the spacetime point observables - which are candidates for the long-sought physical operators of the quantum theory.