My Personal Journey on the Geometric

Aspect of General Relativity

Shing-Tung YauHarvard University

The first annual meeting of ICCM 2017

Sun Yat-sen University, Guangzhou

December 28, 2017

This talk is based on joint workwith Mu-Tao Wang and Po-NingChen.

About 100 years ago, Einsteinaccomplished one of the mostspectacular work in physics andradically changed the view ofspace and time in the history ofmankind.

Einstein

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The foundation laid by Isaac New-ton on the theory of gravity wascompletely changed by the the-ory of general relativity. In thevery successful theory of Newton,space is static and time is inde-pendent of space. Newton

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By 1905, when Einstein estab-lished special relativity along withPoincare and others, it was re-alized that space and time arelinked and that the very founda-tion of special relativity, and thatinformation cannot travel fasterthan light, is in contradiction withNewtonian gravity where actionat a distance was used.

Poincare

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Einstein learnt from his teacher in1908 that special relativity is bestdescribed as the geometry of theMinkowski spacetime. He realizedgravitational potential cannot bedescribed by a scalar function. Itshould be described by a tensor. Minkowski

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After tremendous helps from his two friends inmathematics: Grossmann and Hilbert, Einsteinfinally wrote down the famous Einstein equation:

Rµν −1

2Rgµν = 8πTµν .

Grossmann Hilbert

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Note that Hilbert was the first one that write downthe action principle of gravity, which plays the mostimportant role in any attempts to quantize generalrelativity. The action is given by the total scalarcurvature of the metric tensor which is consideredto be the gravitational potential. If gravity iscoupled with other matter, we simply add thematter Lagrangian.

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The field equation was used by Einstein to calculatethe perihelion of mercury and the light bendingpredicted by the Schwarzschild solution of theEinstein equation, which was found shortly. Thiswas of course a great triumph of the Einstein theoryof general relativity.

Perihelion of mercury

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However, since the theory is highly nonlinear andthe geometry of spacetime is dynamical, the actualunderstanding of Einstein equation was verydifficult: even to Einstein himself.

Einstein thought that the equation determinedgravity completely. But that is actually not true aswe cannot tell what is the initial condition for thefield equation and we have difficulty to find theboundary condition.

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Most of the problems in general relativity can beformulated in terms of geometry. This is naturalbecause that was the original intention of Einstein.Gravity is described by the geometry of spacetime.

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Local information of spacetime is pretty muchgoverned by the full curvature tensor of thespacetime. Matter tensor is described by only partof the full curvature tensor: namely the Ricci tensorwhich is basically the trace of the full curvaturetensor.

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The dynamical equation forspacetime is described by theEinstein equation which saysexactly that the matter stresstensor is described by the Riccitensor. This is probably the mostfascinating equation in geometry.It has influenced the developmentof geometry in the past 100years!

Ricci

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I was very puzzled by the Einstein equation when Ilearnt about it 47 years ago. I was intrigued by thebeauty of the equation. But at the same time, Iwonder what happens when there is no matter inthe universe, it seems to me that the spacetime maystill be nontrivial. (Trivial spacetime is theMinkowski spacetime where curvature tensor isidentically zero.)

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There is certainly possibility that Ricci tensor of aspacetime is identically zero while its full curvaturetensor is not zero. I was wondering whether we canfind a global spacetime which exhibit the followingphenomenon: it is geodesically complete andnontrivial while the Ricci curvature is identicallyzero.

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The problem of finding and classi-fying geodesic complete Ricci flatmetric spaces is a problem that Ihave always been excited about.When I was a graduate student,there is not a single space of thissort that has no singularity. It wasa fascinating question that droveme to look into Kahler geometry. Kahler

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Even before the birth of general relativity in 1915,many mathematicians have been communicatingwith Einstein. The most notable one was his friendGrossmann, but there was also Levi Civita and alsoHilbert. In fact the equation describing vacuum wasalready correctly written by the joint paper ofEinstein and Grossmann in 1912.

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Herman Weyl, Kaluza and Elie Cartan followedEinstein on trying to unify gravity with other theoryin the framework of general relativity.

Weyl Cartan

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Two very major works were created. One is gaugetheory developed by Herman Weyl from 1918 to1928. The gauge group that proposed by Weyl wasnoncompact at the beginning and was criticized byEinstein.

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The birth of quantum mechanics gave the hint ofimportance of phase and Weyl changed the group tobe U(1). This is a spectacular achievement of Weylwhich has deep influence in fundamental physicsafter it was realized by Yang and Mills that it ismore fruitful to replace U(1) by SU(n).

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The other major work was due to anothermathematician Kaluza and later followed by aphysicist Klein, and is called Kaluza-Klein model.They want to take a model of gravity withspacetime to be five-dimensional. And they wantthe spacetime to have a circle symmetry.

Kaluza Klein

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They look at the Einstein equation with no matterin the five spacetime dimension. The reduction tofour-dimensional spacetime by the circle symmetrybecome an Einstein equation coupling gravity withelectromagnetism. That worked out like magic andEinstein was very much impressed by it.Unfortunately the theory also created an extrascalar particle which was not observed. Althoughthe theory was abandoned at the time, it never died.

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In fact, when it was found sixty years later that inorder for gravity and quantum mechanics to beconsistent on a spacetime with a symmetry calledsupersymmetry, the spacetime has to beten-dimensional.

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String theorists proposed that some six-dimensionalspace can curl up to be so tiny that theten-dimensional spacetime looks likefour-dimensional. The simplest way to curl up isthat the ten-dimensional spacetime is the product ofa four-dimensional spacetime with anothersix-dimensional space with some kind ofsupersymmetry.

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It was observed by Candelas-Horowitz-Strominger-Witten thatthis six-dimensional space is aKahler manifold with zero Riccicurvature. They called thissix-dimensional space Calabi-Yauspace in honor of what I did in1976 in proving the Calabi con-jecture.

Calabi-Yau space

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I was strongly motivated to prove the Calabiconjecture starting from the time when I was agraduate student in Berkeley. This was the story Imentioned earlier: I was fascinated by the questionof existence of a non singular metric which describesvacuum, but not the trivial spacetime wherecurvature is identically zero.

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I found later that Eugene Calabi proposed a way toconstruct such a metric many years ago using theframework of Kahler geometry. It depends on anansatz which requires to solve some highly nonlinearequation. It was in 1976 that I managed to solvethe equation with a great deal of hard works.

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The very important character of this Ricci flatmetric has extra quality that it enjoys a symmetrycalled supersymmetry and that was why the abovefour authors were so happy with them when theyknew such spacetime exists.

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Recently I found out that Kahler who was thefounder of Kahler geometry wrote about the ansatztwenty years before Calabi. He called themKahler-Einstein metric and I believe Kahler wasinfluenced by the strong desire to understandEinstein equation right after general relativity wascreated.

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The constructions of those Kahler-Einstein metricswere used by me to solve some old problems inalgebraic geometry eight years before they wererecognized by my friends in physics community. Butthose metrics are so beautiful that I have neverdoubt that should appear in nature.

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Extra dimension theory may still be weird for manyclassical physicists. But following Kalusa-Kleintheory, the topology of Calabi-Yau manifold can beused to compute particle contents of the universe ifthose Calabi-Yau spaces are actually the rightmodels of the physical universe. Nobody has reallysaid that the universe cannot be modeled by theCalabi-Yau spaces. The only problem is that we donot have a good selection principle to choose theright model among billions of such models.

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In any case ,while I was spending a lot of time onthis very rich subject of Calabi-Yau spaces, I wasalso very interested in the classical four-dimensionalspacetime where galaxies and black holes weremodeled. In order to model such isolated physicalsystem , we assume the spacetime metric isasymptotically trivial and in mathematically terms,the four-spacetime looks like Minkowski spacetimenear infinity.

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For such spacetime, there is asymptotic Poincaregroup acting at infinity of the spacetime. With sucha group, the concept of total mass and total linearmomentum can be defined based on the theory ofNoether. This concept was in fact noticed byEinstein himself. And was formulated more preciselyby Arnowitt-Deser-Misner many years later.

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But it left a very important ques-tion that they all wanted to solve:namely, in order for an isolatedphysical system to be stable, theenergy defined in such a way bet-ter be positive. This remained tobe unsolved for a long time untilRichard Schoen and myself solvedit in 1978-1979.

Schoen

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Our method has initiated a new direction in theinteraction between geometry and general relativity.The subsequent argument of Witten, using Diracequation to reprove the positive mass theorem,provided another important tool for the subject thatflourished in the last forty years .

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The development allows us to give rigoroustreatment for black hole theory. For example, in1983, Schoen and I were able to demonstrate afolklore statement that when matter density is largein a fixed region of the space, black hole will form.Our argument in fact gives an effective estimate ofthe density.

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In fact, some very important achievements weremade way before of this. Schwarzchild and Kerrfound the exact model of stationary black hole whenthere is circle symmetry and there is no matter. Thework of Kerr is probably one of the most remarkableachievement in the theory of nonlinear evolutionequation. Practically every scholar used this modelto build their theory of black holes.

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It was also remarkable that in the early 1970s,Israel, Carter and Hawking were able todemonstrate that under some smoothnessassumptions on the event horizon of the black hole,stationary black hole with no other matter must bethe one constructed by Kerr. This was coined to beno hair theorem by John Wheeler. This theorem hasbeen the foundation for most discussion of blackhole theory. On the other hand, whether thesmooth assumption is valid should be examined.

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Many more important achievements were made inthe last forty years . To mention a few, my formerstudent Robert Bartnik was able to construct anontrivial coupled static nonsingular solution ofEinstein-Yang-Mills equation.

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Felix Finster, Joel Smoller, and I demonstratedthere are infinite (discrete) number of static blackholes based on Einstein-Yang-Mills equations. Forthis natural classical system to exhibit quantumphenomenon is interesting. Their physicalsignificance has not been understood and need tobe explored.

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Then there was the important work ofChristodoulou-Kleinemann on the dynamicalstability of Minkowski spacetime, which gave amuch more precise understanding of the so calledPenrose compactification of spacetime. It also gaverise to the theory of Christodoulou on nonlinearmemory effect of gravitational waves.

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The theory of nonlinear memory effect wasgeneralized by Lydia Bieri, Po-Ning Chen, DavidGarfinkle and myself to include many other fields.Hopefully, such memory effect can be tested in therecent LIGO observations.

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Many basic nonlinear effects on general relativity arebeing established using rather deep works developedin geometric analysis. A very importantdevelopment is the establishment of the importantconcept of quasilocal physical quantities such asquasilocal mass in spacetime. Many authors workedon this subject. Most notable ones were due toPenrose, Hawking, Robert Bartnik, Brown-York andothers.

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Personally I am more satisfied with the definitiondue to Mu-Tao Wang and myself with recentcooperation with Po-Ning Chen. These areimportant concepts that enable us to discussNewtonian concepts even when the gravity field isvery strong.

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The necessity of the concept of quasilocal mass canbe seen from the following question: When twoblack holes are interacting, we know the concept oftheir total energy, according to Einstein. But whatabout their binding energy? We need to know theenergy of individual black hole before we can talkabout its binding energy.

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With the quasilocal energy defined, it would beinterested to see how binding energy of twointeracting black holes behave during collision. Itshould be related to the gravitational radiationwhich I believe is still a mysterious quantity, despitethe most recent discovery of it by LIGO.

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The concept of quasilocal energy can be used tounderstand the energy radiated by an isolatedphysical system in the following manner: If we usethe Kerr black hole as a model, the formation of theblack hole will release gravitational waves. Thepossible deformations were studied by Chandrasekarusing Regge-Wheeler equations.

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Po-Ning Chen, Mu-Tao Wang and myself proposedto use quasilocal mass to study gravitationalradiation by measuring quasilocal mass enclosed bya sphere of radius one far away from the source.The order of the quasilocal mass turns out to be(1/d)2, where d is the distance to the source.

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As a result, take a closed loop and vector field alongthe loop, we can define the flux of gravitationalenergy passing within the loop along that vectorfield by simply computing the energy carried by thehemispherical cell capped by this loop along thisvector field.

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The flux defined in this way is well defined and ithas order 1/d . In this way, we can measure thegravitational flux in any closed loop along a vectorfield. By changing the geometry of the loop, wehave a good way to confirm the measurement of thegravitational radiation.

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Thank you!

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