We present a perturbative formulation of quantum gravity for asymptotically flat vacuum spacetimes based on the Null Surface Formulation (NSF), in which the expansion is ultraviolet-finite term by term up to the orders computed, without the need for renormalization. The outgoing Bondi shear operators are constructed explicitly up to fourth order, with interaction kernels determined recursively from on-shell gravitational data at null infinity. Ultraviolet finiteness at each order follows from the on-shell structure of the construction and the restriction of all integrations to the compact celestial sphere, eliminating off-shell propagators. The map between the in and out states admits a perturbative construction, and unitarity is verified explicitly up to fourth order. The outgoing operators satisfy the same commutation relations as the incoming ones, indicating that the transformation is canonical and consistent with the unitary implementation. Collinear configurations give rise to infrared singularities, as expected in massless quantum field theories, but do not affect the ultraviolet behavior established here. In coherent states, the expectation value of the shear reproduces the known finite classical graviton scattering at lowest nontrivial order. These results provide a perturbative framework for quantum gravity with improved ultraviolet behavior relative to the covariant approach.

This review examines the role of differential forms, Pfaffian systems, and hypersurfaces in general relativity. These mathematical constructions provide the essential tools for general relativity, in which the curvature of spacetime—described by the Einstein field equations—is most elegantly formulated using the Cartan calculus of differential forms. Another important subject in this discussion is the notion of conformal geometry, where the relevant invariants of a metric are characterized by Élie Cartan’s normal conformal connection. The previous analysis is then used to develop the null surface formulation (NSF) of general relativity, a radical framework that postulates the structure of light cones rather than the metric itself as the fundamental gravitational variable. Defined by a central Pfaffian system, this formulation allows the entire spacetime geometry to be reconstructed from a single scalar function, Z, whose level surfaces are null.

A quantum theory of asymptotically flat space-times is presented using the solutions of the Null Surface Formulation (NSF) field equations in a perturbative scheme. Free-field commutation relations are given for the null free data of NSF at future or past null infinity, and the main variables of NSF are then used to define the interior points of the space-time as labels and the metric of the space-time as a derived quantum operator. A non-trivial scattering of gravitons is given.

Using a set of field equations in the null surface formulation we obtain the linearized coupling between the gravitational and matter fields. We first derive a formula for the metric of the space time and then we use this formula to study the scattering of incoming gravitational waves when matter is present, obtaining explicit formulae relating the radiation modes at past and future null infinity for a general asymptotically flat spacetime. An example application is made at the end of this work when the matter field is a massless real scalar field. The relevance of this result for a perturbation procedure is discussed.

The nonlinear classical graviton solution of the Einstein equations is given to second order in the null surface formulation (NSF). We solve the NSF field equations in a perturbative scheme that gives geometrical and finite results at each order of the expansion obtaining the main variables of the formalism to second order. Then, using an algebraic relationship with the NSF variables, we obtain the metric components and the nonlinear scattering of incoming gravitons. We also analyze the relevance of this result at a classical and quantum level.

The evolution of global binary black holes variables such as energy or linear momentum are mainly obtained by applying numerical methods near coalescence, post-Newtonian (PN) expansions, or a combination of both. In this paper, we use a fully relativistic formalism presented several years ago that only uses global variables defined at null infinity together with the gravitational radiation emitted by the source to obtain the time evolution of such variables for binary black holes (BBH) systems. For that, we use the Rochester catalog composed of 776 BBHs simulations. We compute the final velocity, radiated energy, and intrinsic angular momentum predicted by the dynamical equations in this formalism for nonspinning, aligned and antialigned spins, and several different precessing configurations. We compare obtained values with reported values in numerical simulations. As BBHs parameter space is still not completely covered by numerical simulations, we fit phenomenological formulas for practical applications to the radiated energy and final velocities obtained. Also, we compare the fits with reported values. In conclusion, we see that our formulae and correlations for the variables described in this work are consistent with those found in the general literature.

The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contain the same geometrical meaning as the 4-D formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constructed on asymptotically flat space times and some of their kinematical aspects discussed. A particular example, which nevertheless contains most of the generic features is explicitly constructed and analyzed, revealing the behavior predicted in the full theory.

The asymptotic approach derived by Kozameh-Quiroga (K-Q) provides a modern framework to obtain the evolution of global variables of isolated sources of gravitational radiation. We test the K-Q formalism evolving the equations of motion for the center of mass, the intrinsic angular momentum, and several other global variables, for black hole binary coalescence. First we evolve the equations of motion using 777 simulations from the RIT catalogue numerical data of ψ4 [1]. We then analyze the trajectory of the center of mass and compute the final state of other physical variables after the coalescence has taken place. Finally, we show the results obtained from our equations of motion are consistent with those in the Rochester metadata.

Emanuel Gallo, Carlos Kozameh, Thomas Mädler, Osvaldo M. Moreschi and Alejandro Perez FaMAF, UNC; Instituto de Física Enrique Gaviola (IFEG), CONICET, Ciudad Universitaria, (5000) Córdoba, Argentina. Escuela de Obras Civiles and Núcleo de Astronomía, Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Avenida Ejército Libertador 441, Casilla 298-V, Santiago, Chile. and 3 Aix Marseille Univ, Université de Toulon, CNRS, CPT, 13000 Marseille, France (Dated: July 22, 2021)

The center of mass (c.m.) and spin for isolated sources of gravitational radiation that move at relativistic speeds are defined. As a first step we also present these definitions in flat space. This contradicts some general wisdom given in textbooks claiming that such definitions are not covariant and thus, have no physical meaning. We then generalize the definitions to asymptotically flat spacetimes giving their equations of motion when gravitational radiation is emitted by the isolated sources. The resulting construction has some similarities with the Mathisson-Papapetrou equations which describe the motion of the particle in an external field. We analyze the relationship between the c.m. velocity and the Bondi linear momentum and show they are not proportional to each other. A similar situation happens between the total and intrinsic angular momentum when the Bondi momentum vanishes. We claim that extra terms should be added in other approaches to adequately describe the time evolution of isolated sources of gravitational radiation.

We analyze the spin stability of a binary black hole coalescence when the binary system is described by the Post-Newtonian (PN) equations in the adiabatic regime. The main idea in this work is to make a massive exploration of the solution space in search of chaos. For that, we evolve the PN equations using a CUDA implementation of the RKF78 scheme and study the dynamical behavior of the system. Each initial spin configuration run in the GPU is composed by more than 80000 simulations. The chaos indicator used to characterize the degree of separation of two infinitesimally close trajectories is the Lyapunov exponent. We find zones in the solution space where the separation between nearby trajectories reaches several orders of magnitude bigger than the initial separation. Also, we note that the chaotic behavior can be observed in forward as well as in backward evolution.

In this work we analyze the similarities and differences between the equations of motion for the center of mass and intrinsic angular momentum for isolated sources of gravitational radiation obtained by two different formulations. One approach is based on the asymptotic formulation of the general relativity, whereas the other relies on post-Newtonian methods. Several conclusions are obtained which could be useful for further developments in both approaches.

The electro-optic modulators (EOM) are devices designed to modulate a laser beam. Depending on the configuration adopted by the EOM, they can be used to change the polarization state, to modulate the phase or the amplitude of the laser [1]. It is also possible to simultaneously modulate the amplitude and phase of the beam [2]. The EOM have multiple applications, for example frequency modulation spectroscopy [3, 4], modulation transfer spectroscopy [5, 6], two tone frequency modulation spectroscopy [7, 8], laser frequency stabilization and cavity length locking [9, 10]. Specifically, the EOM are used in the Laser Interferometer Gravitational-Wave Observatory (LIGO) where they play an important role. These observatories, which have recently achieved the first direct observation of gravitational waves emitted by black hole coalescence, are capable of detecting perturbations of the space time on the order of 10 m. For these outstanding observations LIGO has recently received the 2017 Nobel Prize in Physics. To achieve these sensitivity levels it is necessary to accurately control the length of the two Fabry-Perot cavities, (each cavity 4 Km in length) so that they are always in optical resonance. The length control system is done via a variation of the Pound-Drever-Hall technique to generate sidebands in the laser beams that go to the cavities. The sidebands are generated by EOM that produce a phase modulation in the laser beam [11]. To achieve phase modulation, the index of refraction of the crystal used in the EOM is modulated by periodic, slowly varying external electric field. This external field is perpendicular to the direction of the laser wave and both are aligned with the principal axis of the crystal. However, besides the required phase modulation, the experimental setup of the EOM also produced an unwanted modulation in the amplitude of the transmitted wave. This residual amplitude modulation (RAM) can be regarded as noise added to the laser beam. Depending on the spurious change in intensity, there is evidence that this could affect the calibration of the Fabry-Perot cavities [12]. This is important for future observations of Advanced LIGO when searching for much weaker sources of gravitational radiation like coalescence of neutrons stars. The RAM was attributed to deficiencies in the phase modulation process. According to several authors, there are many sources that could contribute to RAM:

The residual amplitude modulation (RAM) is a spurious effect of the phase modulation in the laser beam power. In this work we show that the optical modulators that are used to generate the sidebands on the laser beam generate a RAM that should be regarded as a fundamental level of noise in the optical setup.

The residual amplitude modulation ($\mathrm{RAM}$) is the undesired, non-zero amplitude modulation that usually occurs when a phase modulation based on the electro-optic effect is imprinted on a laser beam. In this work, we show that electro-optic modulators (EOMs) that are used to generate the sidebands on the laser beam also generate a $\mathrm{RAM}$ in the optical setup. This result contradicts standard textbooks, which assume the amplitude remains unchanged in the process and should be considered as a fundamental $\mathrm{RAM}$ ($\mathrm{RAM_{F}}$) for these devices. We present a classical model for the propagation of an infrared laser with frequency $\omega_{0}$ in a wedge-shaped crystal and an EOM with an RF modulating signal of frequency $\Omega$. Since ${\Omega}\ll \omega_{0}$, we solve Maxwell's equations in a time-varying media via a WKB approximation and we write the electromagnetic fields in terms of quasi-plane waves. From the emerging fields of the setup, we compute the associated $\mathrm{RAM_{F}}$ and show that it depends on the phase-modulation depth $m$ and the quotient $\left(\frac{\Omega}{\omega_{0}}\right)$. The $\mathrm{RAM_{F}}$ values obtained for the EOMs used in gravitational wave detectors are presented. Finally, the cancellation of $\mathrm{RAM_{F}}$ is analyzed.

Following the approach of Adamo–Newman–Kozameh (ANK) we derive the equations of motion for the center of mass and intrinsic angular moment for isolated sources of gravitational waves in axially symmetric spacetimes. The original ANK formulation is generalized so that the angular momentum coincides with the Komar integral for a rotational Killing symmetry. This is done using the Winicour–Tamburino Linkages which yields the mass dipole-angular momentum tensor for the isolated sources. The ANK formalism then provides a complex worldline in a fiducial flat space to define the notions of center of mass and spin. The equations of motion are derived and then used to analyse a very simple astrophysical process where only quadrupole and octupole contributions are included. The results are then compared with those coming from the post newtonian approximation.

The dynamical equations for the null surface formulation of general relativity for purely radiative spacetimes are derived. Those asymptotically flat spacetimes describe the nonlinear evolution of gravitational radiation and represent a classical graviton. The evolution equations constitute a set of three partial differential equations in a six-dimensional space and the source term is the free initial data of incoming gravitational radiation. The Huygens part of the wave propagation, backreaction terms, and source terms are identified in the resulting equations. An analysis of the range of validity of these equations based on the development of caustics is also given.

We define the center of mass and spin of an isolated system in General Relativity. The resulting relationships between these variables and the total linear and angular momentum of the gravitational system are remarkably similar to their Newtonian counterparts, though only variables at the null boundary of an asymptotically flat spacetime are used for their definition. We also derive equations of motion linking their time evolution to the emitted gravitational radiation. The results are then compared to other approaches. In particular one obtains unexpected similarities as well as some differences with results obtained in the Post Newtonian literature . These equations of motion should be useful when describing the radiation emitted by compact sources such as coalescing binaries capable of producing gravitational kicks, supernovas, or scattering of compact objects.