Probing Strong Field Gravity Through Numerical Simulations

TL;DR

Probing Strong Field Gravity Through Numerical Simulations surveys how numerical relativity has illuminated the dynamical, strongly non-linear regime of general relativity, from historical roots to the present-day capabilities and challenges. The chapter emphasizes the mathematical formulations and numerical methods that enable stable evolutions, and recounts major physics results across critical phenomena, binary black hole and neutron star mergers, gravitational collapse, ultra-relativistic collisions, and gravity in higher dimensions, including AdS/CFT applications. It highlights the role of NR in predicting gravitational-wave signals for detectors and in exploring fundamental questions about singularities, cosmic censorship, and the behavior of gravity in diverse settings. Finally, it outlines unsolved problems and future directions for bringing simulations, theory, and observations into tighter synergy.

Abstract

This article is an overview of the contributions numerical relativity has made to our understanding of strong field gravity, to be published in the book "General Relativity and Gravitation: A Centennial Perspective", commemorating the 100th anniversary of general relativity.

Figures (10)

• Figure 1: Type II discretely self-similar critical solution computed from the collapse of a spherically symmetric distribution of massless scalar field. The figure shows the late time configuration of the scalar field from a marginally subcritical evolution where the family parameter has been tuned to approximately a part in $10^{15}$. The radial coordinate is logarithmic, making the discretely self-similar (echoing) nature of the solution evident: each successive echo represents a change in scale of $e^\Delta\approx31$. The data were generated using the axisymmetric code described in Choptuik:2003ac.
• Figure 2: Depictions of the gravitational waves emitted during the merger of two equal mass (approximately) non-spinning black holes Buonanno:2006ui. Left: The plus-polarized component $h_+$ of the wave measured along the axis perpendicular to the orbital plane. $t_{CAH}$ on the horizontal axis is the time a common apparent horizon is first detected. Right: A color-map of the real component of the Newman-Penrose scalar $\Psi_4$ (proportional to the second time derivative of $h_+$ far from the BH) multiplied by $r$ along a slice through the orbital plane (grey is $0$, toward white (black) positive (negative)). From top left to bottom right the time $(t-t_{CAH})/M$ of each panel is approximately $-150,-75,0,75$.
• Figure 3: Recoil velocities from equal mass, spinning binary black hole merger simulations (circles) together with analytical fitting functions. Each black hole has the same spin magnitude $\alpha$, equal but opposite components of the spin vector within the orbital plane, and $\theta$ is the initial angle between each spin vector and the orbital angular momentum. The dashed line corresponds to a fitting formula that depends linearly on the spins, while solid lines add non-linear spin contributions (from Lousto:2011kp).
• Figure 4: Left: Rest-mass density induced by a supermassive black hole binary interacting with a magnetized disk prior to when the binary "decouples" from the disk, namely when the gravitational wave backreaction timescale becomes smaller than the viscous timescale (from 2012PhRvL.109v1102F). Right: Poynting flux produced by the interaction of an orbiting binary black hole interacting with a surrounding magnetosphere. The "braided" jet structure is induced by the orbital motion of the black holes (from Palenzuela:2010nf).
• Figure 5: Examples of the "plus" polarization component of gravitational waves from binary neutron star mergers, measured $100$ Mpc from the source along the direction of the orbital angular momentum. The different curves correspond to different choices of the EOS of the neutron star matter, labeled APR4, ALF2, H4 and MS1. For a $1.4M_{\odot}$ neutron star, the APR4, ALF2, H4, MS1 EOS give radii of $11.1,12.4,13.6,14.4$km respectively. Left: Mergers of an equal mass binary neutron star system (with $m_1=m_2=1.4M_{\odot}$). A hypermassive neutron star (HMNS) is formed at merger, but how long it survives before collapse to a black hole strongly depends on the EOS. The H4 case collapses to a black hole $\approx 10$ms after merger; the APR and MS1 cases have not yet collapsed $\simeq 35$ms after merger when the simulations where stopped (the MS1 EOS allows a maximum total mass of $2.8 M_{\odot}$, so this remnant may be stable). The striking difference in gravitational wave signatures is self evident (from 2013PhRvD..88d4026H). Right: Emission from black hole-neutron star mergers, with $m_{{\rm BH}} = 4.05 M_{\odot}, m_{{\rm NS}} = 1.35 M_{\odot}$. Variation with EOS is primarily due to coalescence taking place earlier for larger radii neutron stars (from Kyutoku:2013wxa).