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Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.
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Backgrounds are pervasive in almost every application of general relativity. Here we consider the Lagrangian formulation of general relativity for large perturbations with respect to a curved background spacetime. We show that Noethers theorem combined with Belinfantes symmetrization method applied to the group of displacements provide a conserved vector, a superpotential and a energy-momentum that are independent of any divergence added to the Hilbert Lagrangian of the perturbations. The energy-momentum is symmetrical and divergenceless only on backgrounds that are Einstein spaces in the sense of A.Z.Petrov.
In the present article we study the cosmological evolution of a two-scalar field gravitational theory defined in the Jordan frame. Specifically, we assume one of the scalar fields to be minimally coupled to gravity, while the second field which is the Brans-Dicke scalar field is nonminimally coupled to gravity and also coupled to the other scalar field. In the Einstein frame this theory reduces to a two-scalar field theory where the two fields can interact only in the potential term, which means that the quintom theory is recovered. The cosmological evolution is studied by analyzing the equilibrium points of the field equations in the Jordan frame. We find that the theory can describe the cosmological evolution in large scales, while inflationary solutions are also provided.
We find a method to rewrite the equations of motion of scalar fields, generalized DBI field and quintessence, in the autonomous form foremph{arbitrary} scalar potentials. With the aid of this method, we explore the cosmic evolution of generalized DBI field and quintessence with the potential of multiple vacua. Then we find that the scalars are always frozen in the false or true vacuum in the end. Compared to the evolution of quintessence, the generalized DBI field has more times of oscillations around the vacuum of the potential. The reason for this point is that, with the increasing of speed $dot{phi}$, the friction term of generalized DBI field is greatly decreased. Thus the generalized DBI field acquires more times of oscillations.
Gravitational waves are considered as metric perturbations about a curved background metric, rather than the flat Minkowski metric since several situations of physical interest can be discussed by this generalization. In this case, when the de Donder gauge is imposed, its preservation under infinitesimal spacetime diffeomorphisms is guaranteed if and only if the associated covector is ruled by a second-order hyperbolic operator which is the classical counterpart of the ghost operator in quantum gravity. In such a wave equation, the Ricci term has opposite sign with respect to the wave equation for Maxwell theory in the Lorenz gauge. We are, nevertheless, able to relate the solutions of the two problems, and the algorithm is applied to the case when the curved background geometry is the de Sitter spacetime. Such vector wave equations are studied in two different ways: i) an integral representation, ii) through a solution by factorization of the hyperbolic equation. The latter method is extended to the wave equation of metric perturbations in the de Sitter spacetime. This approach is a step towards a general discussion of gravitational waves in the de Sitter spacetime and might assume relevance in cosmology in order to study the stochastic background emerging from inflation.
It is well known that the Klein Gordon (KG) equation $Box Phi + m^2Phi=0$ has tachyonic unstable modes on large scales ($k^2<vert m vert^2$) for $m^2<m_{cr}^2=0$ in a flat Minkowski spacetime with maximum growth rate $Omega_{F}(m)= vert m vert$ achieved at $k=0$. We investigate these instabilities in a Reissner-Nordstrom-deSitter (RN-dS) background spacetime with mass $M$, charge $Q$, cosmological constant $Lambda>0$ and multiple horizons. By solving the KG equation in the range between the event and cosmological horizons, using tortoise coordinates $r_*$, we identify the bound states of the emerging Schrodinger-like Regge-Wheeler equation corresponding to instabilities. We find that the critical value $m_{cr}$ such that for $m^2<m_{cr}^2$ bound states and instabilities appear, remains equal to the flat space value $m_{cr}=0$ for all values of background metric parameters despite the locally negative nature of the Regge-Wheeler potential for $m=0$. However, the growth rate $Omega$ of tachyonic instabilities for $m^2<0$ gets significantly reduced compared to the flat case for all parameter values of the background metric ($Omega(Q/M,M^2 Lambda, mM)< vert m vert$). This increased lifetime of tachyonic instabilities is maximal in the case of a near extreme Schwarzschild-deSitter (SdS) black hole where $Q=0$ and the cosmological horizon is nearly equal to the event horizon ($xi equiv 9M^2 Lambda simeq 1$). The physical reason for this delay of instability growth appears to be the existence of a cosmological horizon that tends to narrow the negative range of the Regge-Wheeler potential in tortoise coordinates.
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