Numerical Relativity

The following sections are included:

• Preface: What is numerical relativity?
• Contents
• Notation

Before we describe the details for the formulations and numerical methods of numerical relativity, in this chapter, we shall touch on general relativity, gravitational waves, general relativistic objects such as black holes, neutron stars, and binary neutron stars, and sources of gravitational waves, to which special attention is paid in numerical relativity and to which we often refer in the subsequent chapters.

The primary purpose of numerical relativity is to clarify the nature of nonlinear dynamics for a variety of spacetime by numerically solving Einstein's equation. For this, we need to solve Einstein's equation as an initial-value problem (we need to evolve spatial hypersurfaces by solving Einstein's equation forward in time). For this procedure, we first have to reformulate Einstein's equation to a form with which the evolution of geometric quantities forward in time is feasible. As already seen in section 1.1.2, Einstein's equation is basically a hyperbolic equation. However, to explicitly clarify its hyperbolic nature, a well-defined formulation is necessary.…

This chapter is devoted to describing several numerical methods for a solution of Einstein's evolution equations. In general, solving Einstein's equation is composed of two procedures to do. One is to solve Einstein's evolution equations, which are hyperbolic partial differential equations with an advection term associated with the shift vector. The other is to solve the constraint equations which are elliptic equations and are usually solved for providing an initial condition for numericalrelativity simulations. Therefore, numerical relativists need to solve both hyperbolic and elliptic partial differential equations. In this chapter, we focus only on describing typical methods for solving hyperbolic equations; in particular, we will describe several finite-differencing schemes. Methods for a solution of elliptic equations are described in chapter 5.8. Spectral and pseudo-spectral methods are getting popular in the field of numerical relativity since late 1990s [e.g., Bonazzola et al. (1999); Pfeiffer et al. (2003); Grandclément and Novak (2009); Boyle et al. (2007)], in particular for solving the constraint equations and vacuum Einstein's equation. We will also outline these methods in section 5.8.2.…

In the presence of a matter field, we have to solve equations of motion for it in addition to Einstein's equation. The basic equations for the matter field are derived primarily from equation (1.13), ∇^aT_ab = 0. Here, the stress-energy tensor, T_ab, is written as a function of the matter field supposed in the chosen problem. In addition to this equation, several equations inherent in each matter field, which are often derived from conservation laws, have to be solved. In this chapter, we describe a variety of basic equations and several numerical methods for a solution of these equations. In sections 4.1 and 4.2, we will outline basic equations and typical numerical methods for solving equations of motion for scalar fields and swarms of collisionless particles. In the rest of this chapter, we will describe basic equations and numerical methods for a solution of compressible hydrodynamics, magnetohydrodynamics, and radiation hydrodynamics in detail. As in chapter 2, α, β^a, γ_ab, K_ab, and D_a denote the lapse function, shift vector, spatial metric, extrinsic curvature, and covariant derivative with respect to γ_ab, respectively, throughout this chapter.

Initial data are the starting points for any numerical simulation. In numerical relativity, Einstein's equation constrains our choices of initial data because of the presence of the constraint equations as we described in section 2.1.5. The purpose of this chapter is to introduce several formulations used for determining initial data in the 3+1 and N+1 formulations of Einstein's equation.

One of the most important roles of numerical relativity is to accurately derive waveforms of gravitational waves emitted from strong gravitational-wave sources such as coalescing binary neutron stars and binary black holes. Here, gravitational waves are defined as time-dependent tensor parts (transverse and tracefree parts: see appendix C.2) in the spatial metric observed in a wave zone. The wave zone is defined as a domain satisfying r ≳ λ where r and λ are the distance from the gravitational-wave source and the wavelength of gravitational waves, respectively. In asymptotically flat spacetime, each component of the metric perturbation defined by g_ab − η_ab is much smaller than unity in the wave zone. Thus, for deriving gravitational waveforms in numerical-relativity simulations, we have to extract the tensor components of the metric perturbation in the wave zone.…

One of the most important purposes in numerical relativity is to clarify the formation and evolution processes of a variety of black holes. For this purpose, we have to determine the location of the black holes and then analyze their properties. This procedure has to be performed in general for non-stationary spacetime. In this chapter, we describe methods for finding and analyzing the black holes in numerical relativity.

The merger of coalescing compact binary objects, binary neutron stars or black hole-neutron star binaries or binary black holes, has not been directly observed yet (at the end of 2014). This indicates that observing the merger events is a great challenge in astronomy and astrophysics. For the direct observation of the merger events, we have to predict possible signals generated by these phenomena as accurately as possible. Therefore, not only the observational effort but also the theoretical study for the merger events has been the important subject in astronomy and astrophysics…

One of the most important roles of numerical relativity is to clarify the formation process of a variety of black holes. Stellar-mass black holes of mass in the range from ∼ 3M_⊙ to ≳ 1000M_⊙ will be formed by the core collapse of massive stars and massive neutron stars. Supermassive black holes of mass larger than ∼ 105M_⊙ may be formed as a result of the collapse of a supermassive star. In addition, in the very early epoch of the universe, primordial black holes may be formed from a highly nonlinear density fluctuation. For clarifying the formation processes of these black holes, numerical relativity is probably the unique approach. In the following, we will introduce some of the numerical-relativity studies for the formation process of these black holes performed so far.

Rapidly rotating objects in nature are often subject to non-axisymmetric deformation by some mechanism of instability [see, e.g., Chandrasekhar (1969); Shapiro and Teukolsky (1983)]. Rapidly rotating and non-axially deformed neutron stars emit gravitational waves that could be important targets for the detection by grandbased laser-interferometric detectors such as advanced LIGO, advanced VIRGO, and KAGRA (see section 1.2.6). This fact has stimulated a detailed study for the instability of rapidly rotating neutron stars.…

This section is devoted to introducing several representative simulations in higherdimensional numerical relativity performed in the early stage of this field. Higherdimensional numerical relativity has been an active field in numerical relativity since a possibility of production of mini black holes in large particle accelerators such as CERN Large Hardon Collider (LHC) was pointed out (see Preface). We here pick up three representative research topics; studies of the Gregory – Laflamme instability for a black string [Lehner and Pretorius (2010)], two black-hole collisions [Zilhao et al. (2010); Witek et al. (2010, 2011); Okawa et al. (2011)], and the instability for rapidly spinning Myers–Perry (MP) black holes with one spin parameter [Shibata and Yoshino (2010b,a)].

I have described a variety of topics in numerical relativity including basic formulations, methods for a solution of the basic equations, numerical schemes, and the latest scientific results of the applications. I wish the readers to find that numerical relativity is one of the exciting fields in theoretical physics and astrophysics. I will be happy if a graduate student, motivated by this volume, participates in this field and contributes to accelerating its progress.…

The following sections are included:

• Appendix A Killing vector and Frobenius' theorem
• Appendix B Numerical relativity in spherical symmetry
• Appendix C Decomposition by spherical harmonics
• Appendix D Lagrangian and Hamiltonian formulations of general relativity
• Appendix E Solutions of Riemann problems in special relativistic hydrodynamics
• Appendix F Landau – Lifshitz pseudo tensor
• Appendix G Laws of black hole and apparent horizon
• Appendix H Post-Newtonian results for coalescing compact binaries

The following sections are included:

• Bibliography
• Index