#### by Lydia Bieri

It is a great pleasure to write this contribution for the volume in honor of Yvonne Choquet-Bruhat. I had the privilege to meet Yvonne at a conference in Cambridge (GB) in 2005, when I still was a Ph.D. student, and many times after that. Of course, I had known her name and had admired her work for years before meeting her. Thus, I have enjoyed very much our conversations over the years. It goes without saying that discussions with Yvonne always have been enlightening from mathematical and physical points of view, but also topics about society and life get explored by her inquiring mind. It is amazing how broad Yvonne’s expertise and experience are. I would like to thank Yvonne for many of these interesting conversations.

Clearly, Yvonne has inspired many, from colleagues in the field to students and people in the street. As a pioneer of mathematical general relativity, and as one of the very few women in science at the time, she has overcome professional as well as social barriers for women, thereby opening doors for female mathematicians and scientists to succeed in academia.

Not only has Yvonne pioneered many aspects of mathematical general relativity (GR), but she has also made ground-breaking contributions to mathematics and physics on a broader spectrum. Her work is characterized by deep intuition, highest quality and creative approaches and at the same time spans volumes, demonstrating her enormous productivity. Consequently, these brief lines cannot do any justice to the importance or the realm of her work. Rather, I will select two topics. The main part of this article will consider Choquet-Bruhat’s breakthrough results on the Cauchy problem in GR and put them into perspective vis-à-vis global problems and the latest questions including gravitational waves. For a comprehensive discussion of the Cauchy problem, see Choquet-Bruhat’s article [5]; for further aspects of her work and life, see her memoirs [6], or a review of the latter by the present author [e48]. Then we will discuss Choquet-Bruhat’s most original mathematical work on waves propagating in and interacting with a background, where the wavelengths of the waves are considerably shorter than the length scale at which the background varies. For a comprehensive discussion of this topic see Yvonne’s book [4].

The current choice of topics is somewhat personal and not meant to be comprehensive. We will outline some of the highlights in various directions, but also sketch how Choquet-Bruhat’s work has inspired or impacted the present author’s work.

An intriguing aspect of the Einstein equations is their geometric nature. In these equations mathematics and physics are deeply intertwined. Their hyperbolic character allows for the characteristic laws of General Relativity. Thus, it is not surprising that the deepest insights in this field require profound connections between mathematics and physics. Choquet-Bruhat pioneered and established such connections opening up the field to new areas of mathematical investigation.

#### The Cauchy problem for the Einstein equations

The Einstein equations read
__\begin{equation} \label{ET}
R_{\mu \nu} - \textstyle\frac{1}{2}
g_{\mu \nu} R = 8 \pi T_{\mu \nu}
\end{equation}__
for __\( \mu, \nu = 0,1,2,3 \)__
and setting the constants __\( G=c=1 \)__, where __\( G \)__ is Newton’s gravitational
constant, __\( c \)__ the speed of light. We solve the equations for the
unknown metric __\( g_{\mu \nu} \)__. In __\eqref{ET}__, __\( R_{\mu \nu} \)__ denotes the
Ricci curvature tensor, __\( R \)__ the scalar curvature tensor, and __\( T_{\mu
\nu} \)__ is the energy-momentum tensor. The Einstein equations __\eqref{ET}__
have to be complemented with the corresponding equations for the
fields incorporated into __\( T_{\mu \nu} \)__ on the
right-hand side of
__\eqref{ET}__.

We denote the solution spacetimes by __\( (M,g) \)__. These are
4-dimensional manifolds with a Lorentzian metric __\( g \)__ solving the
system of equations.

Even though one may find highly symmetric solutions fast, and various physical statements concerning solutions were derived in the early years, it was not obvious how to use these equations in an initial value problem. We want to understand how physical systems evolve, make predictions, and understand the dynamics of the gravitational field. To achieve these goals, new ideas and works were required.

Let us recall briefly the situation of 1915 and the following years. A. Einstein derived the Einstein equations and formulated the theory of general relativity in 1915 [e1], [e2]. Whereas exact solutions to the Einstein equations were found shortly after Einstein introduced the new theory (the first one by K. Schwarzschild in 1916), A. Eddington’s expedition of 1919 confirmed the bending of light predicted by GR, and G. Lemaître derived the expansion of the Universe in 1927 using his dynamical solutions to the cosmological Einstein equations and comparing these with the redshifts found in the observations of nebulae by V. Slipher. The exact history is, of course, much richer and cannot be part of this paper. We refer to [e36], [e43] for a more comprehensive discussion of the recognition of the expansion of the Universe. Despite the physical success and mathematical progress in the pioneering years of GR, the Cauchy problem (initial value problem) for the Einstein equations had not been understood for decades. It was not until 1952, after many steps by various contributors, that Choquet-Bruhat [1] set up the Cauchy problem for the Einstein equations in a general form and proved a local existence and uniqueness theorem. In her proof, Yvonne used wave coordinates, in which Einstein’s vacuum equations appear clearly as a hyperbolic system of nonlinear (quasilinear) partial differential equations. Earlier contributors include G. Darmois, A. Lichnerowicz, T. de Donder, C. Lanczos, J. Leray, D. Hilbert, H. Weyl, J. Schauder, S. Sobolev and more. For details see [5]. In 1969, Y. Choquet-Bruhat and R. Geroch [2] proved the global existence of a unique maximal future development for every given initial data set for the Einstein equations.

Not only did Choquet-Bruhat’s work [1] lay the foundations for further investigations of big questions in GR, but it was also the first proof establishing that for the nonlinear Einstein equations gravitational waves propagate at finite speed and that causality holds. Ideas about causality had been mentioned earlier by Hermann Weyl without being mathematically established. Einstein was very interested in the question about gravitational waves. In 1916, he looked at the linearized equations and found wave solutions. However, he knew that linearization may infer nonphysical artifacts. Thus, Einstein was very happy to see Yvonne’s rigorous proof establishing gravitational waves. It took roughly a century to detect these waves: the Laser Interferometer Gravitational-Wave Observatory (LIGO) team measured them for the first time in 2015. This marked the beginning of a new era, where gravitational waves will be the messengers from parts of the Universe that telescopes cannot see.

We recall that the dynamics of the gravitational field can only be understood via the Cauchy problem for the Einstein equations, considering physical initial data and exploring the solution spacetimes. Therefore, the importance of Yvonne’s results on the Cauchy problem cannot be overstated. All the big and global, dynamical questions that we would like to understand in this field build on this work. These include the highly active research areas of gravitational radiation and the questions regarding what systems (initial data) will develop singularities (black holes) in the future versus those evolving for all time as causally geodesically complete solutions (without any singularities). Concerning the latter, D. Christodoulou and S. Klainerman proved [e20] in 1993 for asymptotically flat, suitably small (and nontrivial) initial data, that there exists a unique, causally geodesically complete and globally hyperbolic solution to the Einstein equations that itself is globally asymptotically flat. Concerning the development of black holes, D. Christodoulou showed in 2009 [e38] that highly concentrated gravitational waves form a closed trapped surface and eventually a black hole.

Choquet-Bruhat’s original method was used by S. Klainerman and I. Rodnianski to prove a breakdown criterion for the Einstein equations in [e40]. The authors considered an Einstein vacuum spacetime foliated by a constant mean curvature, or maximal, foliation. They showed that the said spacetime can be extended as long as the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded for any size of the data.

Another important idea of Choquet-Bruhat’s — one which she realized early on and which we mention here only briefly — was that of using harmonic coordinates for numerical simulations in General Relativity. There were some issues to overcome. In 1985, H. Friedrich suggested [e18] to use generalized harmonic coordinates. This was then successfully put to work by F. Pretorius [e27], [e30], where his coordinates satisfy a wave equation with a source. He used a form of the equations suggested by C. Gundlach, G. Calabrese, I. Hinder, and J. M. Martin-Garcia in [e29]. Pretorius produced the first fully successful numerical simulation of binary black holes. Shortly after that, the problem was solved independently as well by two other groups using different methods. One group [e32] consisted of M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower and the other [e31] of J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter. A few years earlier, harmonic coordinates were used in numerical calculations as well by D. Garfinkle [e22], and by J. Winicour, B. Schmidt, M. C. Babiuc, and B. Szilágyi, [e23], [e26], [e33].

#### Gravitational radiation: Understood via the Cauchy Problem

Let us turn to gravitational waves now. We emphasize that
Choquet-Bruhat’s proof establishing the well-posedness of the Einstein
equations also proves the existence of gravitational waves in the
general nonlinear theory. In view of the LIGO–VIRGO and future
collaborations, we would like to decode the information in the
gravitational waves to reveal the physics of their sources. And
vice versa, we would like to predict and understand the wave signals
produced by specific sources. Thus, we consider classes of physical
initial data, and we solve the Einstein equations to produce classes
of solution spacetimes. Gravitational waves travel along null
hypersurfaces in these spacetimes. First, we need to determine the
properties of the curvature components at null infinity. This type of
issue was addressed already in the 1960s.
Trautman
[e4],
Bondi
[e5],
Bondi–van der Burg–Metzner
[e7],
Sachs
[e8],
and
Penrose
[e9]
pioneered the use of null
hypersurfaces to describe gravitational radiation. Other discussions
were given by
Pirani
[e3],
Newman
and Penrose
[e6],
Geroch
[e14],
Ashtekar
and
Hansen
[e15],
Ashtekar
and
Schmidt
[e16],
and Ashtekar
and
Streubel
[e17].
It turns out that one of the problems in studying future null infinity
__\( \mathcal{I}^+ \)__ arises when (as in some of the cited papers) one would
like to expand the metric in power series in __\( r^{-1} \)__ with
coefficients depending on retarded time __\( u \)__ and the angular
coordinates. That is, one may ask: How smooth should null infinity be?
In the latter papers, the assumptions about the power series expansion
got replaced by another assumption also requiring a minimal
regularity. If one conformally compactifies the boundary at null
infinity, this implies a minimal regularity of the data, which in the
aforementioned works would be at least __\( C^2 \)__. Yet Christodoulou
showed that for physical spacetimes __\( C^2 \)__ is impossible. Rather, in
the general case, the conformal factor extends to __\( \mathcal{I}^+ \)__ as a
function in __\( C^{1, \alpha} \)__. We point out that
Christodoulou–Klainerman’s work on stability
[e20]
is within this
regime. The smoothness implies a specific hierarchy in the fall-off
for the curvature components. This is called *peeling*. Under the
stronger assumptions of the Newman–Penrose picture the curvature
components peel in __\( r \)__ like __\( r^{-1} \)__, __\( r^{-2} \)__, __\( r^{-3} \)__, __\( r^{-4} \)__, __\( r^{-5} \)__
(ignoring the fall-off in retarded time __\( u \)__).
Christodoulou–Klainerman’s results in
[e20]
give peeling up to
and including __\( r^{-3} \)__ with the next best order being __\( o(r^{-
7/2}) \)__. Now, the question may be what happens if one assumes
more decay on the initial data. In
[e24]
Christodoulou
considered this situation where the tail of the initial metric falls
off towards infinity like __\( r^{-2} \)__. He then shows that for physical
initial data a __\( \log r \)__ term is picked up in the evolution, resulting
in a behavior of __\( r^{-4} \log r \)__ towards __\( \mathcal{I}^+ \)__ instead of
__\( r^{-4} \)__. More recently, the present author proved
[e51]
that
for more general and large data the peeling stops at the order
__\( r^{-3} \)__ respectively, __\( r^{-4}|u|^{+1} \)__; for small data we obtain
__\( o(r^{- 7/2}) \)__.

The latter results make use of an important fact that follows as a
corollary from the stability proofs. Not only did Christodoulou and
Klainerman in their work
[e20]
establish the global nonlinear
stability for Minkowski spacetime mentioned above, but also the method
developed in their proof allows for rigorous studies of gravitational
waves via the Cauchy problem for the Einstein equations.
N. Zipser
achieved the first generalization of
[e20]
in 2000 in
[e21],
[e37]
for the Einstein–Maxwell equations. The present author in
[e34],
[e39]
generalized
[e20]
to the borderline case
for the Einstein vacuum equations. In particular, the more general
theorem assumes control of only one derivative on the Ricci curvature
__\( \bar{R}_{ij} \)__ of the initial Cauchy ~~hypersurface ~~ that is control of
three derivatives of the metric __\( H_0 \)__,__\( \bar{g}_{ij} \)__ in the initial data, as
opposed to four in the original proof. Further, the initial data falls
off towards spatial infinity by one less power of __\( r \)__. That is, in
__\( H_0 \)__ for __\( r \to \infty \)__, the metric has the form __\( \bar{g}_{ij} =
\delta_{ij} + o_3 (r^{- 1/2}) \)__ and the second fundamental form
__\( k_{ij} = o(r^{- 3/2}) \)__. The investigations
[e34],
[e39]
indicate that this decay is borderline from the point of view
of fall-off at infinity. If the fall-off were relaxed further, then
crucial energy integrals would not be bounded any longer. In the
meantime, there have been many stability proofs in various directions.
We would like to point out
[e47]
by
P. Hintz
and
A. Vasy
for a study of polyhomogeneous data. See
[3],
[e28],
[e41],
[e25],
and
[e46]
for a comprehensive list of references. In this article, we
concentrate on the works relevant for the current discussion. In these
works
[e20],
[e21],
[e37],
[e34],
[e39],
the behavior of the
geometric quantities along null hypersurfaces is mainly independent
from the smallness assumptions. The latter was necessary to show the
existence of these solutions. Therefore, we can plug in large data and
still obtain a portion of null infinity where radiation is read off
and that we understand well through the above works and the works
based on these results.

In what follows, __\( S \)__ denotes a spacelike 2-surface in the Lorentzian
manifold __\( (\mathcal{M}, g) \)__ solving the Einstein vacuum equations. At
each point __\( p \)__ in the manifold __\( \mathcal{M} \)__, we identify in the
corresponding tangent space __\( T_p \mathcal{M} \)__ two orthogonal
future-directed null vectors. We label the outward pointing vector as
__\( L_p \)__, and the inward pointing vector as __\( {\underline{L}}_p \)__. We
denote the corresponding vectorfields defined in this way on __\( S \)__ by
__\( {L} \)__ respectively __\( {\underline{L}} \)__. Further, let __\( C \)__ and
__\( \underline{C} \)__ be the null hypersurfaces generated by the
corresponding sets of null geodesics orthogonal to __\( S \)__. Let __\( \{ e_A
\}_{A=1,2} \)__ be an orthonormal frame on __\( S \)__. Then together with __\( L \)__ and
__\( {\underline{L}} \)__ they form a null frame. Finally, we consider the
second fundamental form of __\( S \)__ as a hypersurface in __\( C \)__ respectively
in __\( \underline{C} \)__. Namely, denote the former by __\( \chi \)__ and the latter
by __\( \underline{\chi} \)__. Their traceless parts are called the shears for
which we write __\( \hat{\chi} \)__, __\( \hat{\underline{\chi}} \)__, respectively.
Thus, __\( \hat{\chi}, \hat{\underline{\chi}} \)__ are symmetric, traceless
2-tensors. The traces __\( \mbox{tr} \chi \)__ and __\( \mbox{tr}
\underline{\chi} \)__ are the expansion scalars. Denote by __\( t \)__ a time
function and by __\( u \)__ an optical function (retarded time). The former
yields a foliation of the spacetime __\( (\mathcal{M}, g) \)__ into spacelike
hypersurfaces __\( H_t \)__ and the latter a foliation into null hypersurfaces
__\( C_u \)__. We write the intersections as __\( S_{t,u} = H_t \cap C_u \)__. The
__\( S_{t,u} \)__ are diffeomorphic to __\( S^2 \)__. We refer to __\( \theta, \phi \)__ given
on __\( S_{t,u} \)__ as the spherical variables.

We now go back to the original question we raised above, namely using
the Cauchy problem to understand gravitational waves and their
sources: For the reasons outlined in the previous paragraphs, the most
natural way to study gravitational radiation is to set up classes of
physical initial data, solve the Einstein equations to produce
corresponding classes of spacetimes and read off from future null
infinity __\( \mathcal{I}^+ \)__ the desired information. The physically most
important quantities contained in this information are the limits of
the shears __\( \underline{\hat{\chi}} \)__, __\( \hat{\chi} \)__ and the leading
order curvature component __\( R(X, \underline L, Y , \underline L) =:
\underline \alpha(X,Y) \)__, where __\( X \)__ and __\( Y \)__ are tangent ~~to ~~
This curvature component is contracted twice with the incoming null
vectorfield __\( S_{t,u} \)__.__\( \underline{L} \)__ and falls off towards infinity like
__\( r^{-1} \)__. Note that __\( \underline \alpha \)__ is a __\( S \)__-tangent, symmetric,
traceless 2-tensor. The radiative amplitude per unit solid angle for
one of these key shear quantities is given by the limit
__\begin{equation}\label{Xi}
\Xi (u, \theta, \phi) = \lim_{C_u, t \to \infty} r
\underline{\hat{\chi}} ,
\end{equation}__
while that for the other shear quantity
is given by the limit
__\begin{equation}\label{Sigma}
\Sigma (u, \theta, \phi) = \lim_{C_u, t \to \infty} r^2 \hat{\chi} .
\end{equation}__
Similarly, we take the
limit for the curvature component __\( \underline{\alpha} \)__
__\begin{equation}\label{Alimit}
A (u, \theta, \phi) = \lim_{C_u, t \to \infty} r \underline{\alpha} .
\end{equation}__

Note that __\( \Xi (u, \theta, \phi) \)__ and __\( A (u, \theta, \phi) \)__ as defined
in __\eqref{Xi}__ and __\eqref{Alimit}__ are well-defined as shown in
[e20],
[e34],
[e39],
[e51],
whereas the quantity __\( \Sigma (u,
\theta, \phi) \)__ is well-defined only for systems resulting from initial
data with enough decay towards spatial infinity, that is,
if
__\( (\bar{g}_{ij} - \delta_{ij}) \)__ falls off like __\( r^{-1} \)__ or better (as
in
[e20],
[e51]).
However, for the spacetimes investigated in
[e34],
[e39]
where __\( (\bar{g}_{ij} - \delta_{ij}) = o (r^{-
1/2}) \)__, the limit in __\eqref{Sigma}__ does not exist because
__\( \hat{\chi} \)__ decays like __\( o(r^{- 3/2}) \)__. The following crucial
relations emerge from these studies:
__\begin{align}
\frac{\partial \Sigma}{\partial u} & = - \Xi
\label{shears2}, \\
\frac{\partial \Xi}{\partial u} & = - \textstyle\frac{1}{4} A
\label{curvatureshear1} .
\end{align}__
The relation __\eqref{shears2}__ holds for spacetimes as investigated
in
[e20],
[e51].
Even though the limit __\( \Sigma \)__ on the left-hand
side of __\eqref{shears2}__ is not defined in spacetimes as investigated
in
[e34],
[e39],
a corresponding notion of the left-hand side
can be introduced and a variant of __\eqref{shears2}__ holds for these
more general solutions as well; see
[e50],
[e49].
In
[e20],
[e34],
[e39],
[e51],
[e50],
[e49]
these limits are
derived and investigated in each of these settings.

We recall that for a gravitational wave experiment on Earth like
Advanced LIGO the relative acceleration of the test masses — that is, of
the nearby geodesics marked by the test masses — is expressed through
curvature in the Jacobi equation
__\[
\nabla^2_U V = R(U, V) V,
\]__
where __\( U \)__ denotes the tangent vector for an object in free fall
separated from a second object by a vector __\( V \)__. Integrating twice and
using the structures derived at __\( \mathcal{I}^+ \)__ we read off
information about gravitational radiation. In particular, using that
__\( \underline \alpha \)__ is the leading order curvature component on the
right-hand side of the Jacobi equation, as well as using relations
__\eqref{shears2}__ and __\eqref{curvatureshear1}__, we integrate the Jacobi
equation twice and derive for the permanent relative displacement
__\( \triangle x \)__ (memory) of nearby test masses, respectively nearby
geodesics,
__\begin{equation} \label{reschrmem****4}
\triangle x = - \frac{d_0}{r} ( \Sigma^+ - \Sigma^- ) ,
\end{equation}__
with __\( d_0 \)__ the initial separation, and __\( {\Sigma }^{\pm} \)__
denoting the limits of __\( \Sigma \)__ when __\( u \to + \infty \)__, respectively __\( u
\to - \infty \)__. In an experiment like LIGO the relative displacements
of the test masses are measured by laser interferometry. The __\( u \)__-rate
of change of this relative displacement is determined by __\( \Xi (u,
\theta, \phi) \)__. More precisely, we have
__\[
\Sigma (u) - \Sigma^- = - \frac{1}{2} \int_{- \infty}^{u} \Xi\, du^{\prime},
\quad
\Sigma^+ - \Sigma^- = - \frac{1}{2} \int_{- \infty}^{+ \infty} \Xi\, du.
\]__
Equation
__\eqref{reschrmem****4}__ describes the permanent displacement of the
test masses, that is the permanent change of the spacetime after the
wave packet has passed. This is in fact the “memory effect” of
gravitational waves
[e13],
[e19].
It is expected that this
effect will be measured in the near future. The instantaneous
displacements that have been measured already by LIGO and VIRGO are
correspondingly given by __\( \Sigma (u) - \Sigma^- \)__. Hidden behind
__\( (\Sigma (u) - \Sigma^-) \)__ and __\( (\Sigma^+ - \Sigma^-) \)__ lie interesting
structures carrying information about the distant sources of the
waves. Exploring these is a major goal in the studies of gravitational
waves.

The memory effect of gravitational waves was first derived in a
linearized setting by
Ya. B. Zel’dovich
and
A. G. Polnarev
[e13]
in 1974, and then in the fully nonlinear setting by D. Christodoulou
[e19]
in 1991. In 2014 the present author and
D. Garfinkle
proved
[e44]
that these are two different effects, the former
(called *ordinary memory*) being sourced by a change of a particular
electric component of the Weyl tensor, the latter (called *null
memory*) being sourced by radiation sent to infinity. For the latter
we write
__\begin{equation}
F = \mathcal{C} \int_{- \infty}^{+ \infty} | \Xi |^2 \,du,
\end{equation}__
with __\( F/(4\pi) \)__ being the total energy radiated away in a
given direction per unit solid angle.

Various new challenges lie ahead in this interesting journey to explore parts of our Universe via gravitational waves. More questions will be tackled via the Cauchy problem. In all these, Choquet-Bruhat’s groundbreaking work will be fundamental, as all these explorations will build on her result.

#### Short wavelength approximations

Next, we turn to Choquet-Bruhat’s work on short wavelength approximations. Over several years, Yvonne studied waves propagating in and interacting with a background, where the wavelengths of the waves are much shorter in comparison with the length scale of variation of the background. This includes gravitational waves propagating in a cosmological background. Yvonne produced various results on this topic starting in 1967. She developed a mathematical method to deal with these physical problems. For an extensive treatment of her work in this field we refer to her book [4].

A major challenge in cosmology and in particular when studying gravitational waves propagating from early periods in the history of our Universe lies in the fact that these cosmological spacetimes do not possess a “future null infinity”; rather the asymptotic region is spacelike. Hence there is no convenient way to read off radiation from the asymptotics of the spacetime.

In the cosmological case, we add to the original Einstein equations
the term containing __\( \Lambda \)__, the positive cosmological constant, to
obtain
__\begin{equation}\label{ETCO}
R_{\mu \nu} - \textstyle\frac{1}{2} g_{\mu \nu} R +
\Lambda g_{\mu \nu} = 8 \pi T_{\mu \nu} .
\end{equation}__
This new term
drives the expansion of the Universe. Observations in 1998 showed that
we live in a Universe that is expanding at an accelerated rate.
Today’s model is the __\( \Lambda \)__CDM (Cold Dark Matter with cosmological
constant __\( \Lambda \)__) cosmology. Whereas the very early Universe can be
described as a small perturbation of a
Friedman–Lemaître–Robertson–Walker spacetime, during later stages
this scenario evolved into a highly inhomogeneous picture that is best
described by the __\( \Lambda \)__CDM model. See
[e36],
[e43]
for
the full story of the discovery of the expanding Universe.

The main idea in Choquet-Bruhat’s method is to consider a background
solution of the Einstein equations. Let us call it __\( \bar{g}
(x^{\mu}) \)__. This background does not describe gravitational waves or
their sources. Instead, one introduces high-frequency deformations
__\( \hat{g} \)__ of this background as follows:
__\[
g_{\alpha \beta} =
\bar{g}_{\alpha \beta} (x^{\mu}) + \omega^{-2} \hat{g}_{\alpha \beta}
(x^{\mu}, \omega \phi (x^{\mu})),
\]__
where __\( \omega \)__ denotes the
frequency of the perturbations and __\( \phi (x^{\mu}) \)__ a scalar field.
These perturbations are uniformly bounded in __\( \omega \)__.
Correspondingly, one can introduce perturbations of the matter fields
under consideration. The only restriction is that the wavelengths of
the waves are much shorter in comparison with the length scale of
variation of the background. Then __\( g_{\alpha \beta} \)__ together with
possible other fields, depending on the problem to investigate, solve
the corresponding Einstein equations to the appropriate order.

More recently, the present author together with D. Garfinkle and
N. Yunes
applied this method to derive gravitational wave memory in
__\( \Lambda \)__CDM cosmology
[e45].
Thereby, the region was
divided into a
*wave zone* and a *cosmological zone*.
The former is
the region where the distance from the source is large compared to the
wavelength of the waves, but small compared to the Hubble distance.
The latter are regions where the distance is not small compared to the
Hubble distance. It is shown that gravitational radiation and memory
are affected differently in these zones. In particular, it was shown
that in the wave zone the memory is given via an expression involving
the radiated energy per unit solid angle, whereas in the cosmological
zone the memory is given by the memory computed for the wave zone
multiplied by __\( (1+z)M \)__, with __\( z \)__ being the redshift and __\( M \)__ a
magnification factor due to lensing and the Sachs–Wolfe effect
[e35].

Yvonne’s treatment of high-frequency gravitational waves writes the
Lorentzian metric as the sum of a nonoscillating part and a rapidly
varying piece depending on the large parameter __\( \omega \)__, the
frequency. (In the Einstein vacuum case, this is the only
perturbation. Correspondingly, this is generalized to Einstein-matter
fields and the cosmological equations as indicated above.) This method
has had huge impact throughout many branches of General Relativity.

#### Concluding remarks

Looking at the large field of General Relativity, Yvonne Choquet-Bruhat’s early work on the Cauchy problem stands out as the beginning of the mathematically rigorous approach to the Einstein equations. While this is significant for various reasons, it also sets the stage for future generations of mathematicians and physicists to explore big questions via the Cauchy problem. In some sense, she opened the door to mathematics. Through this connection many exciting mathematical tools have been applied in and developed for General Relativity.

Another important connection to mathematics, namely, geometry, was established later by R. Penrose [e11], introducing the concept of a closed trapped surface, and proving the incompleteness theorem (“singularity theorem”).

Yet another connection to mathematics that Yvonne Choquet-Bruhat initiated concerns gravitational waves and their propagation. Choquet-Bruhat not only laid the groundwork in this direction but also contributed considerably through the theory of hyperbolic partial differential equations.

These deep connections between the physics of General Relativity and mathematics have enabled many breakthroughs. In recent decades, geometric analysis has become very fruitful in this interplay. Choquet-Bruhat has contributed her breakthroughs, and many more build on her work.

Yvonne Choquet-Bruhat’s work includes many highly creative ideas, new mathematical methods, answers physical problems, and spans an enormous breadth of topics reaching far beyond General Relativity.