The grand challenges of contemporary fundamental physics-dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem-all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on 'Black holes, Gravitational waves and Fundamental Physics'. © 2019 IOP Publishing Ltd.

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Black holes, gravitational waves and fundamental physics: a roadmap

This is the first major textbook on the methods of numerical relativity. The selection of material is based on what is known to work reliably in astrophysical applications and would therefore be considered by many as the 'mainstream' of the field. This means spacelike slices, the BSSNOK or harmonic formulation of the Einstein equations, finite differencing for the spacetime variables, and high-resolution shock capturing methods for perfect fluid matter. (Arguably, pseudo-spectral methods also belong in this category, at least for elliptic equations, but are not covered in this book.) The account is self-contained, and comprehensive within its chosen scope. It could serve as a primer for the growing number of review papers on aspects of numerical relativity published in Living Reviews in Relativity (LRR). I will now discuss the contents by chapter. Chapter 1, an introduction to general relativity, is clearly written, but may be a little too concise to be used as a first text on this subject at postgraduate level, compared to the textbook by Schutz or the first half of Wald's book. Chapter 2 contains a good introduction to the 3+1 split of the field equations in the form mainly given by York. York's pedagogical presentation (in a 1979 conference volume) is still up to date, but Alcubierre makes a clearer distinction between the geometric split and its form in adapted coordinates, as well as filling in some derivations. Chapter 3 on initial data is close to Cook's 2001 LRR, but is beautifully unified by an emphasis on how different choices of conformal weights suit different purposes. Chapter 4 on gauge conditions covers a topic on which no review paper exists, and which is spread thinly over many papers. The presentation is both detailed and unified, making this an excellent resource also for experts. The chapter reflects the author's research interests while remaining canonical. Chapter 5 covers hyperbolic reductions of the field equations. Alcubierre's excellent presentation is less technical than Reula's 1998 LRR or the 1995 book by Gustafsson, Kreiss and Oliger, but covers the key ideas in application to the Einstein equations. The reviewer (admittedly riding a hobby-horse) would argue that the hyperbolicity of the ADM and BSSNOK equations should have been investigated without introducing a specific first-order reduction. Chapter 6 covers gauge problems in numerical black hole spacetimes, black hole excision, and apparent horizons. Like chapter 4 it is both exhaustive and pedagogical. Perhaps more space than necessary is given here to work the author was involved in, while the section on slice stretching could have been more detailed, given that there is no good overview in the literature. Chapter 7 on relativistic hydrodynamics is, quite simply, excellent. Among many other useful things it contains some elementary material on equations of state that is not written up at this level elsewhere, a good mini-introduction to weak solutions of conservation laws, and a brief review of imperfect fluids in GR (Israel--Stewart theory). This chapter complements Font's 2008 LRR. Chapter 8 on gravitational wave extraction provides a welcome pedagogical introduction to a topic in which the original research papers are less than inviting and where notation is not uniform. The mathematical techniques described here are in constant use in numerical relativity codes, but are never fully described in research papers. Chapter 9 on numerical methods covers finite difference and high-resolution shock capturing methods. It is similar in presentation to Leveque's 1992 book and Kreiss and Busenhart's 2001 book, but gives a good selection of that material, concisely presented. It certainly impresses the importance of convergence testing on the reader. Chapter 10 covers methods for spherically symmetric and axisymmetric spacetimes. The former is excellent, reflecting the author's recent research work. The axisymmetry section would have been better if it had been based on a formal Geroch reduction, the method that has been the key to recent progress. This book is bound to become a standard text for beginning graduate students. In an overview for this audience, I would have liked to see a little more detail on null slicings and on the conformal field equations, and brief introductions to the theory of elliptic equations and to pseudo-spectral and finite element methods. One may also regret the many typographical errors. Nevertheless, this excellent book fills a real gap, and will be hard to follow.

Introduction to 3+1 Numerical Relativity

https://physicstoday.scitation.org/doi/pdf/10.1063/1.1881902

Spacetime and Geometry: An Introduction to General Relativity

The Hubble diagram is one of the cornerstones of observational cosmology. It is usually analysed assuming that, on average, the underlying relation between magnitude and redshift matches the prediction of a Friedmann-Lemaitre-Robertson-Walker model. However, the inhomogeneity of the Universe generically biases these observables, mainly due to peculiar velocities and gravitational lensing, in a way that depends on the notion of average used in theoretical calculations. In this article, we carefully derive the notion of average which corresponds to the observation of the Hubble diagram. We then calculate its bias at second-order in cosmological perturbations, and estimate the consequences on the inference of cosmological parameters, for various current and future surveys. We find that this bias deeply affects direct estimations of the evolution of the dark-energy equation of state. However, errors in the standard inference of cosmological parameters remain smaller than observational uncertainties, even though they reach percent level on some parameters; they reduce to sub-percent level if an optimal distance indicator is used.

/pdf/how-does-the-cosmic-large-scale-structure-bias-the-hubble-ygolr77ctm.pdf

How does the cosmic large-scale structure bias the Hubble diagram?

In the standard approach to studying cosmological structure formation, the overall expansion of the Universe is assumed to be homogeneous, with the gravitational effect of inhomogeneities encoded entirely in a Newtonian potential. A topic of ongoing debate is to what degree this fully captures the dynamics dictated by general relativity, especially in the era of precision cosmology. To quantitatively assess this, we directly compare standard N-body Newtonian calculations to full numerical solutions of the Einstein equations, for cold matter with various magnitude initial inhomogeneities on scales comparable to the Hubble horizon. We analyze the differences in the evolution of density, luminosity distance, and other quantities defined with respect to fiducial observers. This is carried out by reconstructing the effective spacetime and matter fields dictated by the Newtonian quantities, and by taking care to distinguish effects of numerical resolution. We find that the fully general relativistic and Newtonian calculations show excellent agreement, even well into the nonlinear regime. They only notably differ in regions where the weak gravity assumption breaks down, which arise when considering extreme cases with perturbations exceeding standard values.

Comparing Fully General Relativistic and Newtonian Calculations of Structure Formation

The apparent properties of distant objects encode information about the way the light they emit propagates to an observer, and therefore about the curvature of the underlying spacetime. Measuring the relationship between the redshift z and the luminosity distance DL of a standard candle, for example, yields information on the Universe's matter content. In practice, however, in order to decode this information the observer needs to make an assumption about the functional form of the DL(z) relation; in other words, a cosmological model needs to be assumed. In this work, we use numerical-relativity simulations, equipped with a new ray-tracing module, to numerically obtain this relation for a few black-hole-lattice cosmologies and compare it to the well-known Friedmann-Lema{ȋtre-Robertson-Walker case, as well as to other relevant cosmologies and to the Empty-Beam Approximation. We find that the latter provides the best estimate of the luminosity distance and formulate a simple argument to account for this agreement. We also find that a Friedmann-Lema{ȋtre-Robertson-Walker model can reproduce this observable exactly, as long as a time-dependent cosmological constant is included in the fit. Finally, the dependence of these results on the lattice mass-to-spacing ratio μ is discussed: we discover that, unlike the expansion rate, the DL(z) relation in a black-hole lattice does not tend to that measured in the corresponding continuum spacetime as 0μ → .

https://iopscience.iop.org/article/10.1088/1475-7516/2017/03/014/pdf

Light propagation through black-hole lattices

The apparent properties of distant objects encode information about the way the light they emit propagates to an observer, and therefore about the curvature of the underlying spacetime. Measuring the relationship between the redshift $z$ and the luminosity distance $D_{\rm L}$ of a standard candle, for example, yields information on the Universe's matter content. In practice, however, in order to decode this information the observer needs to make an assumption about the functional form of the $D_{\rm L}(z)$ relation; in other words, a cosmological model needs to be assumed. In this work, we use numerical-relativity simulations, equipped with a new ray-tracing module, to numerically obtain this relation for a few black-hole--lattice cosmologies and compare it to the well-known Friedmann-Lemaitre-Robertson-Walker case, as well as to other relevant cosmologies and to the Empty-Beam Approximation. We find that the latter provides the best estimate of the luminosity distance and formulate a simple argument to account for this agreement. We also find that a Friedmann-Lemaitre-Robertson-Walker model can reproduce this observable exactly, as long as a time-dependent cosmological constant is included in the fit. Finally, the dependence of these results on the lattice mass-to-spacing ratio $\mu$ is discussed: we discover that, unlike the expansion rate, the $D_{\rm L}(z)$ relation in a black-hole lattice does not tend to that measured in the corresponding continuum spacetime as $\mu \to 0$.

Daniel Gerlicher

Papers

Light propagation through black-hole lattices

Light propagation through black-hole lattices