Celebratio Mathematica

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 Ein­stein equa­tions read $$\begin{array}{}\text{(1)}& R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=8\pi T_{\mu \nu }\end{array}$$ 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 $\text{(1)}$, $R_{\mu \nu }$ de­notes the Ricci curvature tensor, $R$ the scal­ar curvature tensor, and $T_{\mu \nu }$ is the en­ergy-mo­mentum tensor. The Ein­stein equa­tions $\text{(1)}$ have to be com­ple­men­ted with the cor­res­pond­ing equa­tions for the fields in­cor­por­ated in­to $T_{\mu \nu }$ on the right-hand side of $\text{(1)}$.

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].

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 ${\overline{R}}_{ij}$ of the ini­tial Cauchy hy­per­sur­face $H_0$, that is con­trol of three de­riv­at­ives of the met­ric ${\overline{g}}_{ij}$ 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 $r\to \mathrm{\infty }$, the met­ric has the form ${\overline{g}}_{ij}={\delta }_{ij}+o_3(r^{-1/2})$ 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 ${\underset{―}{L}}_p$. We de­note the cor­res­pond­ing vec­tor­fields defined in this way on $S$ by $L$ re­spect­ively $\underset{―}{L}$. Fur­ther, let $C$ and $\underset{―}{C}$ 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 $\underset{―}{L}$ they form a null frame. Fi­nally, we con­sider the second fun­da­ment­al form of $S$ as a hy­per­sur­face in $C$ re­spect­ively in $\underset{―}{C}$. Namely, de­note the former by $\chi$ and the lat­ter by $\underset{―}{\chi }$. Their trace­less parts are called the shears for which we write $\hat{\chi }$, $\widehat{\underset{―}{\chi }}$, re­spect­ively. Thus, $\hat{\chi },\widehat{\underset{―}{\chi }}$ are sym­met­ric, trace­less 2-tensors. The traces $\text{tr}\chi$ and $\text{tr}\underset{―}{\chi }$ 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 $\underset{―}{\hat{\chi }}$, $\hat{\chi }$ and the lead­ing or­der curvature com­pon­ent $R(X,\underset{―}{L},Y,\underset{―}{L})=:\underset{―}{\alpha }(X,Y)$, where $X$ and $Y$ are tan­gent to $S_{t,u}$. This curvature com­pon­ent is con­trac­ted twice with the in­com­ing null vec­tor­field $\underset{―}{L}$ and falls off to­wards in­fin­ity like $r^{-1}$. Note that $\underset{―}{\alpha }$ 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 $$\begin{array}{}\text{(2)}& \mathrm{\Xi }(u,\theta ,\phi )=\underset{C_u,t\to \mathrm{\infty }}{lim}r\underset{―}{\hat{\chi }},\end{array}$$ while that for the oth­er shear quant­ity is giv­en by the lim­it $$\begin{array}{}\text{(3)}& \mathrm{\Sigma }(u,\theta ,\phi )=\underset{C_u,t\to \mathrm{\infty }}{lim}{r}^2\hat{\chi }.\end{array}$$ Sim­il­arly, we take the lim­it for the curvature com­pon­ent $\underset{―}{\alpha }$ $$\begin{array}{}\text{(4)}& A(u,\theta ,\phi )=\underset{C_u,t\to \mathrm{\infty }}{lim}r\underset{―}{\alpha }.\end{array}$$

Note that $\mathrm{\Xi }(u,\theta ,\phi )$ and $A(u,\theta ,\phi )$ as defined in $\text{(2)}$ and $\text{(4)}$ are well-defined as shown in [e20], [e34], [e39], [e51], where­as the quant­ity $\mathrm{\Sigma }(u,\theta ,\phi )$ is well-defined only for sys­tems res­ult­ing from ini­tial data with enough de­cay to­wards spa­tial in­fin­ity, that is, if $({\overline{g}}_{ij}-{\delta }_{ij})$ falls off like $r^{-1}$ or bet­ter (as in [e20], [e51]). However, for the space­times in­vest­ig­ated in [e34], [e39] where $({\overline{g}}_{ij}-{\delta }_{ij})=o(r^{-1/2})$, the lim­it in $\text{(3)}$ does not ex­ist be­cause $\hat{\chi }$ de­cays like $o(r^{-3/2})$. The fol­low­ing cru­cial re­la­tions emerge from these stud­ies: $$\begin{array}{}\text{(5)}& \frac{\partial \mathrm{\Sigma }}{\partial u}& =-\mathrm{\Xi },\text{(6)}& \frac{\partial \mathrm{\Xi }}{\partial u}& =-\frac{1}{4}A.\end{array}$$ The re­la­tion $\text{(5)}$ holds for space­times as in­vest­ig­ated in [e20], [e51]. Even though the lim­it $\mathrm{\Sigma }$ on the left-hand side of $\text{(5)}$ 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 $\text{(5)}$ 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 $${\mathrm{\nabla }}_U^{2}V=R(U,V)V,$$ 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 $\underset{―}{\alpha }$ is the lead­ing or­der curvature com­pon­ent on the right-hand side of the Jac­obi equa­tion, as well as us­ing re­la­tions $\text{(5)}$ and $\text{(6)}$, we in­teg­rate the Jac­obi equa­tion twice and de­rive for the per­man­ent re­l­at­ive dis­place­ment $\mathrm{△}x$ (memory) of nearby test masses, re­spect­ively nearby geodesics, $$\begin{array}{}\text{(7)}& \mathrm{△}x=-\frac{d_0}{r}({\mathrm{\Sigma }}^{+}-{\mathrm{\Sigma }}^{-}),\end{array}$$ with $d_0$ the ini­tial sep­ar­a­tion, and ${\mathrm{\Sigma }}^{\pm }$ de­not­ing the lim­its of $\mathrm{\Sigma }$ when $u\to +\mathrm{\infty }$, re­spect­ively $u\to -\mathrm{\infty }$. 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 $\mathrm{\Xi }(u,\theta ,\phi )$. More pre­cisely, we have $$\mathrm{\Sigma }(u)-{\mathrm{\Sigma }}^{-}=-\frac{1}{2}{\int }_{-\mathrm{\infty }}^u\mathrm{\Xi }d{u}^{\prime },{\mathrm{\Sigma }}^{+}-{\mathrm{\Sigma }}^{-}=-\frac{1}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\mathrm{\Xi }du.$$ Equa­tion $\text{(7)}$ 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 $\mathrm{\Sigma }(u)-{\mathrm{\Sigma }}^{-}$. Hid­den be­hind $(\mathrm{\Sigma }(u)-{\mathrm{\Sigma }}^{-})$ and $({\mathrm{\Sigma }}^{+}-{\mathrm{\Sigma }}^{-})$ 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 $$\begin{array}{}\text{(8)}& F=\mathcal{C}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}|\mathrm{\Xi }{|}^{2}du,\end{array}$$ 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.

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 $\mathrm{\Lambda }$, the pos­it­ive cos­mo­lo­gic­al con­stant, to ob­tain $$\begin{array}{}\text{(9)}& R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+\mathrm{\Lambda }{g}_{\mu \nu }=8\pi T_{\mu \nu }.\end{array}$$ 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 $\mathrm{\Lambda }$CDM (Cold Dark Mat­ter with cos­mo­lo­gic­al con­stant $\mathrm{\Lambda }$) 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 $\mathrm{\Lambda }$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 $\overline{g}(x^{\mu })$. This back­ground does not de­scribe grav­it­a­tion­al waves or their sources. In­stead, one in­tro­duces high-fre­quency de­form­a­tions $\hat{g}$ of this back­ground as fol­lows: $$g_{\alpha \beta }={\overline{g}}_{\alpha \beta }(x^{\mu })+{\omega }^{-2}{\hat{g}}_{\alpha \beta }(x^{\mu },\omega \phi (x^{\mu })),$$ 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 $\mathrm{\Lambda }$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.

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.