Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein's equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy initial data in the 3 + 1 decomposition of Einstein's equations. We will then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coalescence.

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Initial Data for Numerical Relativity

This review aims mainly to disencumber the proper heuristic functions of the model from such misconceptions, avoidable in the light of the classic papers by Block (1933), Jensen (1937) and Samoilovich (1945). The boundary conditions and linearised differential equations are established without cutoff; they determine the normal modes and orthogonality relations. The model is quantised through its normal modes, and the equal-time commutation rules are discussed. The equations in Fourier space are found; it is argued that a cutoff, if required, should be imposed on the Hamiltonian in this representation and before diagonalisation, and the consequences are explored. With such a cutoff, surface though not bulk modes become dispersive even when beta =0. The formalism is applied briefly to image potentials, and in more detail to the attraction between two half-spaces; the role of bulk modes (when beta >0) is stressed; the asymptotics are discussed at long and short distances.

Some surface effects in the hydrodynamic model of metals

Shape dynamics is a completely background-independent universal framework of dynamical theories from which all absolute elements have been eliminated. For particles, only the variables that describe the shapes of the instantaneous particle configurations are dynamical. In the case of Riemannian three-geometries, the only dynamical variables are the parts of the metric that determine angles. The local scale factor plays no role. This leads to a shape-dynamic theory of gravity in which the four-dimensional diffeomorphism invariance of general relativity is replaced by three-dimensional diffeomorphism invariance and three-dimensional conformal invariance. Despite this difference of symmetry groups, it is remarkable that the predictions of the two theories – shape dynamics and general relativity – agree on spacetime foliations by hypersurfaces of constant mean extrinsic curvature. However, the two theories are distinct, with shape dynamics having a much more restrictive set of solutions. There are indications that the symmetry group of shape dynamics makes it more amenable to quantization and thus to the creation of quantum gravity. This introduction presents in simple terms the arguments for shape dynamics, its implementation techniques, and a survey of existing results.

Shape Dynamics. An Introduction

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t. scaling developed in the parallel particle dynamics paper by one of the authors. In spatially compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented. Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.

https://iopscience.iop.org/article/10.1088/0264-9381/20/8/311/pdf

Scale-invariant gravity: geometrodynamics

We compare the results of constructing binary black hole initial data with three different decompositions of the constraint equations of general relativity. For each decomposition we compute the initial data using a superposition of two Kerr-Schild black holes to fix the freely specifiable data. We find that these initial-data sets differ significantly, with the ADM energy varying by as much as 5% of the total mass. We find that all initial-data sets currently used for evolutions might contain unphysical gravitational radiation of the order of several percent of the total mass. This is comparable to the amount of gravitational-wave energy observed during the evolved collision. More astrophysically realistic initial data will require more careful choices of the freely specifiable data and boundary conditions for both the metric and extrinsic curvature. However, we find that the choice of extrinsic curvature affects the resulting data sets more strongly than the choice of conformal metric.

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Comparing initial-data sets for binary black holes

The initial-value problem of general relativity is treated in the case where the external sources are electromagnetic or neutrino fields. Taking into account the initial conditions that must be satisfied by these fields, we show that the resulting equations in each case form a quasilinear elliptic system of a type that has been treated extensively in previous work. We also treat the initial-value problem of the scalar-tensor theory of gravitation. Throughout this work we use first-order "canonical" gravitational variables. The principal mathematical tools are conformal transformations and a covariant decomposition of symmetric tensors.

Initial-value problem of general relativity. III. Coupled fields and the scalar-tensor theory

Recent investigations of the initial-value problem of general relativity have shown that the initial-value constraints can be formulated in all cases as a system of elliptic equations with well-defined physical and mathematical properties. The solutions of these equations can be regarded as generalized gravitational potentials. These potentials are interrelated and depend on their sources quasilinearly. They are particularly useful in analyzing asymptotically flat solutions of Einstein's equations. We have found from these results (1) a technique for constructing physically meaningful initial data in the integration of Einstein's equations, and (2) a method for characterization and analysis of the spacelike mass, momentum, angular momemtum, and multipole moments of gravitational fields.

Niall ó Murchadha

Papers

Initial-value problem of general relativity. III. Coupled fields and the scalar-tensor theory

Gravitational potentials: A constructive approach to general relativity