As documented in the following, very substantial results have already been obtained from helio- and asteroseismology, based on huge observational projects, such as the GONG project, the helioseismic instruments on the Solar and Heliospheric Observatory (SOHO) and Solar Dynamics Observatory (SDO) satellites, and asteroseismology with the CoRoT and Kepler missions. But this is just the beginning. New asteroseismic projects are on the way, and there are also plans for new developments in observational helioseismology, including multi-wavelength observations (Hill et al., 2013). For local helioseismology a potential breakthrough will come from ESA's Solar Orbiter mission (Gandorfer et al., 2011), which will for the first time allow detailed investigations of the regions near the solar poles.

However, there is much to be done with the existing data, and much to be done further to develop the analysis techniques. The helioseismic data have been far from fully exploited, with the inverse analyses being typically based just on a small fraction of the available observations. In particular, the high-resolution observations from the Helioseismic and Magnetic Imager (HMI) instrument on SDO may provide reliable frequency data on high- degree modes, which would allow more detailed investigations of the thermodynamics of the region of helium ionization (e.g., Rabello-Soares et al., 2000). Also, the base of the solar convective envelope certainly deserves further investigation, including an updated search for possible latitude variations (Monteiro and Thompson, 1998) and a follow-up to the hint of solar-cycle variations found by Baldner and Basu (2008). The helioseismic data on frequency splittings have been extensively used to study variations of solar internal rotation during the solar cycle (for a recent example, see Howe et al., 2013). However, further work is certainly required on constraining the rotation of the solar core, where improvements in the inversion techniques may allow a better localization and balance between resolution and error.

Except for specialists, planetary seismology refers first to Earth seismology, and stimulates thoughts about earthquakes, tsunamis, and all such rare and brutal events that everyone knows by experience and/or fears. According to etymology, seismology shakes and shocks our bodies and our minds. According to semantics, all languages make the difference between “superficial” and “deep” things, so that everyone turns out to be a seismologist, encouraged to prefer a deep to a superficial analysis, to get to the bottom of things, to look at an issue in more depth, to delve into something, to get to the root of the matter, to seek out the underlying issues…

We have basically a 2D topological experience of the Earth since we live on it and are attached to its surface. Of course, in tunnels and by plane, we are invited to explore the third dimension, a thin envelope in fact, so shallow with respect to the Earth radius. Even this Earth radius is an ethereal concept (sorry for the hiatus), since we lack normal circumstances where we can simply feel that the Earth is not as flat as a pancake. When the Earth was a pancake, there was a great debate about the nature of its edge. In a 3D spherical view, defining and measuring the Earth radius implies necessarily prospecting the medium between the surface and the center.

Hidden signatures

The favorite activity of a planetary seismologist consists in seeking to look well below the planetary surface, as done by Athanasius Kircher, a German Jesuit (1601–1680), in his model of the Earth's interior. Volcanoes were seen as outlets of the Earth's internal fires. Similarly, any trace on a planetary surface can be related to properties of the interior structure. On the Moon, the absence of surface faulting indicates tectonic inactivity.

Introduction

For a long time, (geo)physicists have considered the Earth as a spherically symmetric body, compositionally divided into an iron core, a silicate mantle, and a crust mostly built from melts rising to the surface. However, the surface of the Earth, divided into oceans and continents, betrays the existence of lateral differences that cannot be ignored. None of our oceans is older than 200 million years, whereas there are indications that the first continents started to form 4.3 billion years ago. The processes that impose such differences must operate at depth. The fact that heat cannot escape sufficiently fast by conduction, yet the mantle is not molten, leaves no doubt that some kind of convective engine operates to cool the mantle and to drive the observable motions of the much more rigid plates covering the surface (Jaupart and Mareschal, 2010).

To study the dynamic processes in the interior, we can inspect various physical fields for signs of lateral heterogeneity: the gravity field, the magnetic field, the response of the electromagnetic field to forcings in the atmosphere and beyond, and the seismic wavefield after a strong earthquake. We are often interested in anomalies, i.e., in deviations from spherical symmetry. Here we shall concentrate on the seismic field, for the simple reason that it has the most powerful information about variations in elastic properties in the interior that are themselves influenced by temperature variations and – probably to a lesser degree – by chemical differences. The analysis of seismic waves has given us detailed images of wave speed variations in the interior of the Earth and Sun. On Mars, Mercury, or one of the many moons in the Solar System, we shall at best be able to observe the seismic field with a handful of seismometers, and we wish to extract as much information from such seismic recordings as possible. Fortunately, recent developments in terrestrial seismic tomography have led to significant progress in analyzing the observed field.

Short review of impact seismic records in the Solar System

Impact structures affect all planetary surfaces in the Solar System and especially those not resurfaced by recent tectonic processes or fast erosion. Planetary seismology and remote sensing are the tools used to monitor the dynamics of large impact processes and associated shock waves, in a variety of conditions, from the airless Moon to the atmosphere-protected Earth.

Meteorites and bolides on Earth

The largest impact ever instrumentally recorded occurred in the early days of seismology. This was the famous great Siberian meteor (Ben-Menahem, 1975) with an energy estimated to be about 12.5 megatons (1 ton of TNT = 4.185 × 109 J). It was recorded by two Russian seismic stations at about 1000 km and 5000 km, in addition to pressure records.

The development of worldwide infrasound and seismic networks today allows the detection of much smaller impacts, down to kilogram size in mass. Most of these impacts are detected by their generated airwaves (see Edwards et al., 2008; Edwards, 2008 for a review), either with infrasound sensors or by seismometers detecting the surface displacement generated by the airwave. This acoustic detection can be done not only locally but also at larger distances, thanks to the tropospheric waveguide (Edwards, 2008).

Airwaves are also generating seismic waves through conversion processes at the Earth's surface. This is illustrated by Figure 18.1, which shows the recent seismic observations made by Tauzin et al. (2013) following the Chelyabinsk meteor blast of February 15, 2013. This event generated large Rayleigh surface waves propagating at between 2.7 and 3.5 km/s, with amplitudes corresponding to an event with surface wave magnitude Ms ∼ 3.7. Comparable precursor Rayleigh waves, arriving well before the atmospheric air wave, are also found in many of the impacts reported by Edwards et al. (2008), but of course with smaller amplitudes.

Introduction

Wavefields in the Sun and Earth, measured using various instruments, display small- amplitude fluctuations, essentially zero-mean random processes. More careful investigation reveals that these are seismic fluctuations generated by stochastic processes such as convection in the Sun and by anthropogenic activity and non-linear ocean wave interactions on Earth. The distinguishing feature of imaging with noise is that we do not know the dynamical details of the source (unlike in classical earthquake tomography) and we consequently seek a statistical description of the wavefield.

Here, we discuss the interpretation of the cross-correlation measurement in solar and terrestrial scenarios. The most common measurement in the context of noise is the ensemble- averaged cross-correlation. The physics of such measurements can be markedly different from raw wavefield measurements (as in classical tomography). The helioseismic theory of stochastic oscillations in conjunction with computational adjoint optimization, widely used in terrestrial seismic applications, provides a unified framework for addressing outstanding inverse problems in these areas. The seismic methodologies and challenges in each of these fields, despite the vastly different physics, are intimately related and we highlight these similarities in this article.

Seismology of the Sun

The Sun, our nearest star, serves as an astrophysical benchmark, contributing to our understanding of stellar evolution, stellar interiors and atmospheres, etc. Life on Earth is directly affected by the magnetic variability of the Sun. Our climate is sensitive to variations in the irradiance of the Sun caused by cyclic magnetic activity. Space instrumentation and telecommunications are susceptible to solar high-energy eruptive events (e.g., for reviews, see, Schrijver and Zwaan, 2000). Predictive models of solar magnetic activity (e.g., Pesnell, 2008) depend on our knowledge of internal global-scale fluid motions of the Sun, constraints that can only be obtained seismically.

Subsequent to the discovery of oscillations on the surface of the Sun some 50 years ago (Leighton et al., 1962), there has been significant progress in uncovering its properties.

The nature of quantum field theory in curved spacetime

Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M, g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holes – provided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.

It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done – except in formal and/or heuristic ways – simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken in general situations. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the “basic principles” of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poincaré invariance, do not even make sense in the context of curved spacetime, and therefore cannot be considered as “fundamental” from the viewpoint of QFTCS.

Introduction

Gravitational waves are a consequence of Einstein's General Theory of Relativity, first presented in 1915 and published in 1916 [1]. Einstein himself linearized his theory and derived wave equations and calculated the gravitational radiation produced by sources in the weak-field, slow-motion limit [2]. As described in the following Chapter, this initial insight has been greatly expanded so that, in general, it is possible to calculate either numerically or analytically the details of the gravitational radiation for a broad range of potential astronomical sources. Much later, in the 1970s, the discovery of the binary neutron star system PSR1913+16 by Hulse and Taylor [3] demonstrated through this natural experiment that gravitational waves carry away energy and angular momentum, causing the neutron star orbit to decay at precisely the predicted rate. Early cosmological gravitational waves imprint a polarization signature in the electromagnetic microwave background that several sensitive instruments may detect. See [4] but also [5] and references therein for further discussion.

These brief remarks gloss over a more complex history where it was unclear whether gravitational waves were real or just gauge artifacts. The theory was finally settled on the side of reality [6]. The standard next step in physics – to build a receiver to directly detect gravitational waves – proved to be extremely challenging. The analog of the Hertz experiment where artificially generated waves are detected within the wave zone will fail because of the undetectably small amplitude (see for example [7]). Astrophysical sources are much stronger but are, of course, more distant. Yet their detection may be possible because gravitational wave receivers respond to amplitude and not to intensity. Nonetheless, the numbers are daunting.

In the early 1960s, J. Weber followed through on a bold vision – that gravitational waves were detectable – by measuring the resonant excitation of acoustic modes in heavy metallic bars, as would be caused by a passing gravitational wave from relatively nearby astrophysical sources [8].

Introduction

From a purely analytical perspective, Einstein's equations constitute a formidable PDE system. They mix constraint equations with evolution equations, their manifest character (hyperbolic or not) depends on the choice of coordinates, they are defined and studied on a spacetime manifold which is field-dependent and therefore not fixed, and the system is nonlinear in a serious way. These features make it challenging to study Einstein's equations using the analytical techniques and ideas which have been successfully applied to other nonlinear PDE systems. This is especially true of those analyses concerned with global, evolutionary aspects of solutions of Einstein's equations, which are the focus of interest in this chapter.

During the past thirty years, it has become apparent that the most successful way to meet these challenges and understand the behavior of solutions of Einstein's equations is to recognize the fundamental role played by spacetime geometry in general relativity and exploit some of its structures. Indeed, the Christodoulou–Klainerman proof of the stability of Minkowski spacetime [1] provides a good example of this: It relies strongly on the use of spacetime geometric structures such as null foliations, maximal hypersurfaces, “almost Killing fields”, and the Bel–Robinson tensor, combined with sophisticated use of standard analytical tools such as the control of energy functionals and hyperbolic radiation estimates. The more recent work of Christodoulou and others, which has discovered sufficient conditions for the formation of trapped surfaces and black holes, also relies on a strong alliance between geometric insight and the mastery of analytical technique.

As we saw in Chapters 3 and 4, gravitational effects play an important role both in astrophysics and cosmology. However, the two areas feature two fairly distinct branches of the mathematical analysis of solutions of Einstein's equations: One works with asymptotically flat solutions to Einstein's equations and often focuses on issues related to black holes, while the other works predominantly with spatially closed solutions and is more focused on the nature of cosmological singularities and possible mechanisms for isotropization and long-distance correlation.

This chapter describes what has been learned about the dynamical, strong-field regime of general relativity via numerical methods. There is no rigorous way to identify this regime, in particular since notions of energies, velocities, length and timescales are observer- dependent at best, and at worst are not well-defined locally or even globally. Loosely speaking, however, dynamical strong-field phenomena exhibit the following properties: there is at least one region of spacetime of characteristic size R containing energy E where the compactness 2GE/(c4R) approaches unity, local velocities approach the speed of light c, and luminosities (of gravitational or matter fields) can approach the Planck luminosity c5 /G. A less physical characterization, though one better suited to classifying solutions, involves spacetimes where even in “well-adapted” coordinates the non-linearities of the field equations are strongly manifest. In many of the cases where these conditions are met, numerical methods are the only option available to solve the Einstein field equations, and such scenarios are the subject of this chapter.

Mirroring trends in the growth and efficacy of computation, numerical solutions have had greatest impact on the field in the decades following the 1987 volume [1] celebrating the 300th anniversary of Newton's Principia. However, several pioneering studies laying the foundation for subsequent advances were undertaken before this, and they are briefly reviewed in Section 7.1 below. Though this review focuses on the physics that has been gleaned from computational solutions, there are some unique challenges in numerical evolution of the Einstein equations; these as well as the basic computational strategies that are currently dominant in numerical relativity are discussed in Section 7.2. As important as computational science has become in uncovering details of solutions too complex to model analytically, it is a rare moment when qualitatively new physics is uncovered. The standout example in general relativity is the discovery of critical phenomena in gravitation collapse (Section 7.3.1); another noteworthy example is the formation of so-called spikes in the approach to cosmological singularities (Section 7.3.7).