Black-hole scattering with numerical relativity: Self-force extraction and post-Minkowskian validation

The post-Minkowskian expansion of Einstein’s general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein–Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck’s constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck’s constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years.

The effective one-body formalism of the gravitational two-body problem in general relativity is reconsidered in the light of recent scattering amplitude calculations. Based on the kinematic relationship between momenta and the effective potential, we consider an energy-dependent effective metric describing the scattering in terms of an effective one-body problem for the reduced mass. The identification of the effective metric simplifies considerably in isotropic coordinates when combined with a redefined angular momentum map. While the effective energy-dependent metric as expected is not unique, solutions can be chosen perturbatively in the post-Minkowskian expansion without the need to introduce nonmetric corrections. By a canonical transformation, our condition maps to the one based on the standard angular momentum map. Expanding our metric around the Schwarzschild solution we recover the solution based on additional nonmetric contributions.

We show how approximate solutions of the two-body problem in General Relativity, and the approximate solutions of Einstein’s equations in vacuo can be constructed using small deformations of geodesics and of Einstein space-times embedded into a pseudo-Euclidean flat space of higher dimension. The method consists in using expansions of equations around a given simple solution (a circular orbit in the case of geodesics, and Minkowskian or Schwarzschild space in the case of Einstein’s equations) in a series of powers of small deformation parameter, and then solving by iteration the corresponding linear systems of differential equations.

The advantages of the analysis of the two-body problem from an initially static configuration are discussed. The stress which maintains the static equilibrium of two axially (but not spherically) symmetric bodies is removed, and the complementary phenomena of stress breaking and mass motion lead to a dynamic metric perturbation and radiation. The distinctions between the stress-breaking and free-fall phases of the motion are clearly delineated. A new contribution to the radiated energy which arises from the time variation of the tidal forces across the sources in the course of free fall is developed. A potentially very significant contribution from the nonlinearities is also discussed.

After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

We describe a method of implementing the axisymmetric evolution of general-relativistic hydrodynamics and magnetohydrodynamics through modification of a multipatch grid scheme. In order to ease the computational requirements required to evolve the post-merger phase of systems involving binary compact massive objects in numerical relativity, it is often beneficial to take advantage of these system’s tendency to rapidly settle into states that are nearly axisymmetric, allowing for 2D evolution of secular timescales. We implement this scheme in the spectral Einstein code and show the results of application of this method to four test systems including viscosity, magnetic fields, and neutrino radiation transport. Our results show that this method can be used to quickly allow already existing 3D infrastructure that makes use of local coordinate system transformations to be made to run in axisymmetric 2D with the flexible grid creation capabilities of multipatch methods. Our code tests include a simple model of a binary neutron star postmerger remnant, for which we confirm the formation of a massive torus which is a promising source of post-merger ejecta.

The notion of mass or energy has been one of the most challenging problems in general relativity since Einstein's time. As is well known from the equivalence principle, there is no well-defined concept of energy density for gravitation. On the other hand, when there is asymptotic symmetry, concepts of total energy and momentum can be defined. This is the ADM energy-momentum and the Bondi energy-momentum when the system is viewed from spatial infinity and null infinity, respectively. These concepts are fundamental in general relativity but there are limitations to such definitions if the physical system is not isolated and cannot quite be viewed from infinity where asymptotic symmetry exists.The positive energy conjecture states that the total energy of a nontrivial isolated physical system must be positive. This conjecture lies in the foundation of general relativity upon which stability of the system rests.This long standing conjecture had attracted many physicists and mathematician,but only very special cases were verified up until the seventies.

We present the first surrogate model for gravitational waveforms from the coalescence of precessing binary black holes. We call this surrogate model NRSur4d2s. Our methodology significantly extends recently introduced reduced-order and surrogate modeling techniques, and is capable of directly modeling numerical relativity waveforms without introducing phenomenological assumptions or approximations to general relativity. Motivated by GW150914, LIGO's first detection of gravitational waves from merging black holes, the model is built from a set of $276$ numerical relativity (NR) simulations with mass ratios $q \leq 2$, dimensionless spin magnitudes up to $0.8$, and the restriction that the initial spin of the smaller black hole lies along the axis of orbital angular momentum. It produces waveforms which begin $\sim 30$ gravitational wave cycles before merger and continue through ringdown, and which contain the effects of precession as well as all $\ell \in \{2, 3\}$ spin-weighted spherical-harmonic modes. We perform cross-validation studies to compare the model to NR waveforms \emph{not} used to build the model, and find a better agreement within the parameter range of the model than other, state-of-the-art precessing waveform models, with typical mismatches of $10^{-3}$. We also construct a frequency domain surrogate model (called NRSur4d2s_FDROM) which can be evaluated in $50\, \mathrm{ms}$ and is suitable for performing parameter estimation studies on gravitational wave detections similar to GW150914.

Coordinate approach to the energy conservation problem in general relativity

Tensor calculus serves as the mathematical framework underpinning the theory of General Relativity, bridging the gap between abstract mathematical concepts and their physical applications. This paper explores the fundamental principles of tensor calculus, its critical role in formulating Einstein's field equations, and its impact on our understanding of spacetime and gravity. We delve into key tensor operations, including differentiation and contraction, and examine their applications in solving problems in general relativity. By highlighting both historical developments and modern advancements, this study underscores the profound connection between tensor calculus and theoretical physics, demonstrating its significance in describing the universe's structure and dynamics.

We discuss some aspects about the computation of kinematic, spectroscopic, Fermi and astrometric relative velocities that are geometrically defined in general relativity. Mainly, we state that kinematic and spectroscopic relative velocities only depend on the 4-velocities of the observer and the test particle, unlike Fermi and astrometric relative velocities, that also depend on the acceleration of the observer and the corresponding relative position of the test particle, but only at the event of observation and not around it, as it would be deduced, in principle, from the definition of these velocities. Finally, we propose an open problem in general relativity that consists on finding intrinsic expressions for Fermi and astrometric relative velocities avoiding terms that involve the evolution of the relative position of the test particle. For this purpose, the proofs given in this paper can serve as inspiration.

This chapter contains a description of the Cauchy problem in the general theory of relativity. The intended audience consists of physicists with basic knowledge of general relativity. Beyond the formulation, there is a discussion of ways of measuring the distance between initial data sets. Finally the notions of linear and nonlinear stability are compared.

We improve the accuracy of the effective-one-body (EOB) waveforms that were employed during the first observing run of Advanced LIGO for binaries of spinning, nonprecessing black holes by calibrating them to a set of 141 numerical-relativity (NR) waveforms. The NR simulations expand the domain of calibration towards larger mass ratios and spins, as compared to the previous EOBNR model. Merger-ringdown waveforms computed in black-hole perturbation theory for Kerr spins close to extremal provide additional inputs to the calibration. For the inspiral-plunge phase, we use a Markov-chain Monte Carlo algorithm to efficiently explore the calibration space. For the merger-ringdown phase, we fit the NR signals with phenomenological formulae. After extrapolation of the calibrated model to arbitrary mass ratios and spins, the (dominant-mode) EOBNR waveforms have faithfulness --- at design Advanced-LIGO sensitivity --- above $99\%$ against all the NR waveforms, including 16 additional waveforms used for validation, when maximizing only on initial phase and time. This implies a negligible loss in event rate due to modeling for these binary configurations. We find that future NR simulations at mass ratios $\gtrsim 4$ and double spin $\gtrsim 0.8$ will be crucial to resolve discrepancies between different ways of extrapolating waveform models. We also find that some of the NR simulations that already exist in such region of parameter space are too short to constrain the low-frequency portion of the models. Finally, we build a reduced-order version of the EOBNR model to speed up waveform generation by orders of magnitude, thus enabling intensive data-analysis applications during the upcoming observation runs of Advanced LIGO.

. Penrose has described a method for computing a solution for the characteristic initial value problem for the spin-2 equation for the Weyl spinor. This method uses the spinorial properties in an essential way. From the symmetrized derivatives of the Weyl spinor which are known from the null datum on a cone one can compute all the derivatives by using the field equation and thus one is able to write down a power series expansion for a solution of the equation. A recursive algorithm for computing the higher terms in the power series is presented and the possibility of its implementation on a computer is discussed.

After reviewing the known stability results for vacuum, U(1) symmetric solutions to Einstein’s field equations, we shall describe some mathematical ideas and techniques that were unavailable during the earlier research in this area and which might conceivably be exploited to shed new light on the large data global Cauchy problem for this interesting class of spacetimes.