Geometry of Spacetime and Mass in

General Relativity

Shing-Tung YauHarvard University

Roads to AcademicTsinghua University

May 8, 2018

Einstein’s theory of generalrelativity is based on the desireto merge the newly developedtheory of special relativity andNewton’s theory of gravity. Heaccomplished this daunting taskin 1915. Most physicists considerthis to be the most creativework in science in the history ofmankind. Let me now explainsome part of this theory to you.

Einstein

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A very important ingredient is the concept ofequivalence principle, the development of which hada long history:

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Galileo used experiment toshow that the acceleration of atest mass due to gravitation isindependent of the amount ofmass being accelerated.

Galileo

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Kepler once said “If two stoneswere placed in any part ofthe world near each other, andbeyond the sphere of a thirdcognate body, those stones, liketwo magnetic needles, wouldcome together in the intermediatepoint, each approaching the otherby a space proportional to thecomparative mass of the other”.

Kepler

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Then in 1907, Einstein said:

“We assume the complete physical equivalent of agravitational field and a corresponding accelerationof the reference system. The gravitational motion ofa small test body depends only on its initial positionin spacetime and velocity, and not on itsconstitution. The outcome of any local experiment(gravitational or not) in a freely falling laboratory isindependent of the velocity of the laboratory and itslocation in spacetime”.

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Hence Einstein realized that in the new theory ofgravity that he would like to develop, the laws ofgravity should be independent of the observers. Buthe needed a framework to build such a theory ofgravity that can connect philosophy withobservations.

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Since gravity is equivalent to the acceleration of thereference system and acceleration of a particle canbe described by the curvature of its trajectory,Einstein speculated that the new theory of gravityshould have something to do with a new concept ofspace. He knew that the static space (with onefixed coordinate system) is not adequate.

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Einstein’s great work benefited from the help ofmany geometers. He, together with Grossmann, wasstudent of the great geometer and physicistMinkowski. He also interacted with Levi-Civita, andeventually Hilbert and Noether.

Grossmann Hilbert

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But most importantly, Einstein owed hisepoch-making contribution to the concept of spaceby the great 19th century mathematician Riemann.

Riemann

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Before Riemann, there were only three types ofspace: the Euclidean space, the sphere space, andthe hyperbolic space which were all described by asingle coordinate system.

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This was very similar to the timeof Newton where the universewas supposed to be static.Riemann, however, radicallychanged the notion of spacein his famous essay “On thehypotheses which lie at thefoundations of geometry” in1854.

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His space was totally different from the three spacesabove, and it could exist without referring to a fixedcoordinate system. He also knew that when twopoints are very close, we do not feel muchacceleration and so to first order effect, we do notfeel presence of curvature and thereforeinfinitesimally, the space should look like the flatEuclidean space. On the other hand, the secondorder effect of gravity should come fromacceleration of the particles. Therefore our spaceshould show curvature if it is used to describedynamics of gravity.

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We do not really know what the space should looklike globally. On the other hand, our space shouldbe general enough to allow many different observerswithout changing the essence of the physics ofgravity. Observers can propagate information fromone to another one.

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Hence Riemann demanded that we can use a varietyof different coordinate systems to observe the basicproperties of the space. However, the onlymeaningful properties of space should beindependent of choice of the coordinate systems.This point of view of space is very importantbecause it is the crucial principle of equivalence ingeneral relativity.

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Riemann defined the concept ofcurvature in his introduction ofabstract space. In fact, laterdevelopment of gravitationalfield in general relativity ismeasured by the curvature,while the material distributionis represented by a part of thecurvature. The distribution ofmatter changes over time and sodoes the curvature.

Riemann

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The dynamics of curvature shows the effect ofvibration of spacetime. And because of that,Einstein came to the conclusion that thegravitational wave, though small, should exist. InEinstein’s equation, the gravitational field and thegeometry of spacetime are inseparable, as a unifiedentity.

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It is remarkable that already in 1854 in his speech,Riemann developed the new concept of spacebecause of the need to understand physicalphenomena. He even suggested that the smallestand greatest parts of space should be described indifferent ways. From a modern physical point ofview, Riemann is looking for the possible structureof quantum space! Riemann once considered usingdiscrete space to explain this problem.

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Riemann started his scientific publication at the ageof 25 and died of lung disease at the age of 39.Three years before his death, he went to Italy everyyear to escape the cold, thus affecting a number ofItalian and Swiss geometers, including Christoffel,Ricci and Levi-Civita.

Christoffel Ricci Levi-Civita

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They generalized Riemann’s ideas, defined tensorsand connections rigorously, both of which wereindispensable for general relativity and gauge fieldtheory. Ricci introduced the Ricci curvature tensor,and proved that this tensor can produce a tensorthat satisfies the conservation law. All of theseworks, accomplished by geometers in themid-to-late 19th century, provided the most crucialtools for general relativity.

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Einstein wrote a paper in 1934entitled “Notes on the origin ofthe general theory of relativity”(see Mein Weltbild, Amsterdam:Querido Verlag), in which hereviews the development ofgeneral relativity.

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The first stage, of course, is the special theory ofrelativity. In addition to Einstein himself, the mainfounders of this theory include Lorentz and Poincare.

Lorentz Poincare

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One of the most important resultsis that the distance is affectedby time. But Einstein learnedthat the action at a distancebetween the special theory ofrelativity and Newton’s theory ofgravity is incompatible and mustbe rectified! Newton

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At first, physicists did not realize that the conceptof space had undergone fundamental changes afterthe breakthrough of Riemann. They attempted tocorrect Newton’s gravitational theory in theframework of three-dimensional space in line withthe special relativity just discovered. This idea ledEinstein to go astray three years!

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Einstein said in the essay “Notes of the origin of thegeneral theory of relativity” (pp. 286–287):

I was of course acquainted with Mach’s view,according to which it appeared conceivable thatinertial resistance counteracts is not acceleration assuch but acceleration with respect to the masses ofthe other bodies existing in the world. There wassomething fascinating about this idea to me, but itprovided no workable basis for a new theory.

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The simplest thing was, of course, to retain theLaplacian scalar potential of gravity, and tocomplete the equation of Poisson in an obvious wayby a term differentiated with respect to time in sucha way that the special theory of relativity wassatisfied. The law of motion of the mass point in agravitational field had also to be adapted to thespecial theory of relativity. The path was not sounmistakably marked out here, since the inert massof a body might depend on the gravitationalpotential. In fact, this was to be expected onaccount of the principle of the inertia of energy.

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These investigations, however, led to a result whichraised my strong suspicions.

The principle of equality of inertial and gravitationalmass could now be formulated quite clearly asfollows: In a homogeneous gravitational field allmotions take place in the same way as in theabsence of a gravitational field in relation to auniformly accelerated coordinate system.

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If this principle held good for any events whatever(The “principle of equivalence”), this was anindication that the principle of relativity needed tobe extended to coordinate systems in non-uniformmotion with respect to each other, if we were toreach a natural theory of the gravitational fields.Such reflections kept me busy from 1908 to 1911. . .

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When Einstein was a student in Zurich, hismathematical professor was Minkowski, who was agreat mathematician on a par with Hilbert andPoincare. Minkowski once said “There was a lazystudent in my class who had recently done animportant work which I had come up with ageometric interpretation”.

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Minkowski learned physics fromHelmholtz, J.J. Thomson andHeinrich Hertz. He held thatbecause of a ”preestablishedharmony between mathematicsand nature”, geometry could beused a key to physical insight.He ascribes physical reality to thegeometry of spacetime.

Minkowski

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This lecture entitled “Space and Time” wasdelivered by Minkowski in the eightieth meeting ofthe Assembly of Natural Scientists and Physicians inCologne in Sep. 21, 1908.

The ideas of space and time developed here wereapplied in a major work on the laws ofelectrodynamics by Minkowski “The fundamentalequations for electromagnetic phenomena in movingbodies” published in 1908. (Minkowski died in1909.)

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This last paper gave the first relativistically correctpresentation of Maxwell equation in a ponderablemedium and its mathematical formalism in terms oftensor calculus. Einstein actually wrote severalpapers with Planck on this work of Minkowski.

Planck and Einstein

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Minkowski wrote:

“The views of space and time which I wish to laybefore you have sprung from the soil ofexperimental physics, and therein lies their strength.They are radical. Henceforth space by itself, andtime by itself, are doomed to fade away into mereshadows, and only a kind of union of the two willpreserve an independent reality.”

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It may be interested to note that Minkowskiacknowledged his concept of spacetime owes a greatdeal to Poincare’s work in 1906, where Poincarenoticed that by changing time to imaginary time,Lorentz transformations become rotations.

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However, Poincare did not think thefour-dimensional representation has much physicalsignificance. Even at 1908, Poincare said that “Thelanguage of three dimensions seems the better fittedto our description of the world although thisdescription can be rigorously made in anotheridiom.”

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This is very different from Minkowski’s point of viewwhere he said directly “The world in space and timein a certain sense is a four-dimensional,non-Euclidean manifold. In truth, we are dealingwith more than merely a new conception of spaceand time. The claim is that it is rather a quitespecific natural law, which, because of itsimportance – since it alone deals with the primitiveconcepts of all natural knowledge, namely space andtime – can claim to be called the first of all laws ofnature.”

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In his essay of 1908, Minkowski constructed afour-dimensional space, introducing a metric tensor,following Riemann, to give geometric meaning ofspecial relativity. The Lorenzian group, thefundamental symmetric group of special relativity,became the group of isometries of this spacetime ofMinkowski.

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For the first time in history, we learned fromMinkowski that we live in a four-dimensionalspacetime. Hence in 1908, Einstein got the mostimportant inspiration for general relativity fromMinkowski: that he has to construct his new theoryof gravity based of the fact that the space should befour-dimensional.

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It is generally believed that the most importantthing Einstein did in the year is his thoughtexperiment. The thought experiment taughtEinstein the importance of equivalence principle andthe need of new geometry to exhibit gravity. Heknew that he need a new concept of space toachieve this. The static space of Newtonian gravityis not adequate any more.

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Why is Minkowski’s article so important? Not onlythat there is a conceptual breakthrough fromthree-dimensional space to four-dimensional space,but also that only within a four-dimensionalspacetime, gravity can have enough room to showits dynamical nature! Newton’s theory of gravity isstatic, in that a function is sufficient to describe thephenomenon of gravity.

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Minkowski spacetime gave the most importantconcept that we need a tensor to describe gravity.Tensor is a newly invented concept consisting ofmany functions which together can transformconsistently so that the principle of equivalence isobeyed. Minkowski’s tensor perfectly describes thespecial theory of relativity, but Einstein wanted tofurther combine Newtonian mechanics withMinkowski space, so his new theory of spacetimeshould be equal to Minkowski spacetimeinfinitesimally.

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Hence when two points are very close to each other,up to first order, the gravity rule governing themshould be the one of Minkowski spacetime.However, this is no more true when we countsecond order effect of gravity, curvature becomesimportant. At the time, physicists knew nothingabout the notion of tensors (in fact, only a fewgeometers knew about tensor analysis.)

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Einstein knew from the principle of equivalence thatthe new potential of gravity should depend on apoint and the tangent vector of space at that point(velocity vector), but he has no idea what kind ofmathematical tool can be used. So he asked hisclassmate Marcell Grossmann for help and finallyfigured out that the gravitational field should bedescribed by a metric tensor. The tensor varies inspacetime, but at every point it can beapproximated by a first-order Minkowski metric.

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Grossmann is a geometer, who helped Einstein todo homeworks in geometry in Zurich. He went tothe library and found the ideas of tensor. However,the idea of introducing metric tensor alone is notenough to describe the gravitational field. We needto know how to differentiate tensor in a curvedspace. We would like the result of differentiation isalso independent of choice of coordinate system(the requirement of equivalence principle). This isthe connection theory of Christoffel and Levi-Civita.

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Einstein said in his memoirs of general relativitymentioned above that this was his first question,and was found to have been solved by Levi-Civitaand Ricci. Einstein’s second question was how togeneralize Newton’s law of gravitation in this newframework. Newton’s equation is simple, that is, thesecond derivative of the gravitational potential isequal to matter density.

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At that time, neither Einstein nor Grossmann knewhow to differentiate metric tensors so that the resultis still a tensor which is independent of the choice ofcoordinate. Grossmann, at Einstein’s repeatedrequests, managed to find Ricci’s work in the library.

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It turned out that Ricci had already contractedRiemann’s curvature tensor to a symmetricsecond-order tensor. It has the same degree offreedom as the metric tensor and can be regardedas the second derivative of metric tensor. Einsteinimmediately realized that it must be the left-handside of the field equation, while the right-hand sideis the tensor of the general matter distribution (inflat space, this tensor has been well studied.)Einstein and Grossmann proposed this equation intwo articles published in 1912 and 1913.

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However when Einstein tried to solve this equationby an asymptotic approach, he did not recover theastronomical phenomena (e.g., light deflection,Mercury’s anomalous perihelion shift) that he wastrying to explain. This made him very frustrated.

Perihelion of mercury

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In the following years, in order to explainastronomical phenomena, he tried to choose specialcoordinates, essentially giving up the preciouslysimple principle of equivalence. The manycommunications between him and Levi-Civita couldnot help either.

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I would like to support what I said in the above bythe essay of Einstein’s own word in: Notes of theorigin of the general theory of relativity (1934, pp.288–289).

I soon saw that the inclusion of non-lineartransformations, as the principle of equivalencedemanded, was inevitably fatal to the simplephysical interpretation of the coordinates - i.e. thatit could no longer be required that coordinatedifferences should signify direct results ofmeasurement with ideal scales or clocks.

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I was much bothered by this piece of knowledge, forit took me a long time to see what coordinates atall meant in physics. I did not find the way out ofthis dilemma until 1912, and then it came to me asa result of the following consideration:

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A new formulation of the law of inertia had to befound which in case of the absence of a “realgravitational field” passed over into Galileo’sformulation for the principle of inertia if an inertialsystem was used as coordinate system. Galileo’sformulation amounts to this: A material point,which is acted on by no force, will be represented infour-dimensional space by a straight line, that is tosay, by a shortest line, or more correctly, anextremal line.

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This concept presupposes that of the length of aline element, that is to say, a metric. In the specialtheory of relativity, as Minkowski had shown, thismetric was a quasi-Euclidean one, i.e., the square ofthe “length” ds of a line element was a certainquadratic function of the differentials of thecoordinates.

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If other coordinates are introduced by means of anon-linear transformation, ds2 remains ahomogeneous function of the differentials of thecoordinates, but the coefficients of this function(gµν) cease to be constant and become certainfunctions of the coordinates. In mathematical termsthis means that physical (four-dimensional) spacehas a Riemannian metric.

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The timelike extremal lines of this metric furnishthat law of motion of a material point which is actedon by no force apart from the forces of gravity. Thecoefficients (gµν) of this metric at the same timedescribe the gravitational field with reference to thecoordinate system selected. A natural formulationof the principle of equivalence had thus been found,the extension of which to any gravitational fieldwhatever formed a perfectly natural hypothesis.

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The solution of the above-mentioned dilemma wastherefore as follows: A physical significance attachesnot to the differentials of the coordinates but onlyto the Riemannian metric corresponding to them. Aworkable basis had now been found for the generaltheory of relativity. Two further problems remainedto be solved, however.

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1. If a field-law is expressed in terms of the specialtheory of relativity, how can it be transferred to thecase of a Riemannian metric?

2. What are the differential laws which determinethe Riemannian metric (i.e., gµν) itself?

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As for problem 2, its solution obviously required theconstruction (from the gµν) of the differentialinvariants of the second order. We soon saw thatthese had already been established by Riemann (thetensor of curvature). We had already considered theright field-equation for gravitation two years beforethe publication of the general theory of relativity,but we were unable to see how they could be usedin physics.

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Einstein struggled from 1913to 1915. It is amusing thatthe equation that Einstein andGrossmann wrote down wasactually correct if there is nomatter. Indeed, Schwarzschildwas able to solve Einsteinequation for a spherical star in1916, right after Einstein andHilbert wrote down the right fieldequation.

Schwarzschild

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Schwarzschild solution assumed that there is nomatter. And it was enough to calculate lightbending due to the gravity of the sun. ThereforeEinstein and Grossmann could have made theobservation in 1913, if they found the exactspherical symmetric solution. Apparently Einsteingot discouraged when his approximate solution didnot give him the right answer compatible with thephysical observation. He was very depressed andwas attempting to use special coordinate and hencegave up the principle of equivalence. The followingwriting of him shows his frustration:

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On the contrary, I felt sure that they could not dojustice to experience. Moreover I believed that Icould show on general considerations that a law ofgravitation invariant with respect to arbitrarytransformations of coordinates was inconsistent withthe principle of causality. These were errors ofthought which cost me two years of excessively hardwork, until I finally recognized them as such at theend of 1915, and after having ruefully returned tothe Riemannian curvature, succeeded in linking thetheory with the facts of astronomical experience.

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In the light of knowledge attained, the happyachievement seems almost a matter of course, andany intelligent student can grasp it without toomuch trouble. But the years of anxious searching inthe dark, with their intense longing, theiralternations of confidence and exhaustion and thefinal emergence into light - only those who haveexperienced it can understand that.

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Let us now go back to what happened in the finalstage of Einstein’s work on general relativity. In thespring of 1915, he visited the great mathematicianDavid Hilbert in Gottingen. Hilbert certainly knowsgeometry well, but above all, he is the founder ofmodern geometric invariant theory. He alsogathered a large group of outstandingmathematicians in Gottingen. Some of them can bedescribed in the following:

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Felix Klein was a pioneer in classifying geometriesby using symmetry groups, Hilbert’s studentHerman Weyl was the founder of gauge field theory,along with Emmy Noether, the greatest femalemathematician in history.

Klein Weyl Noether

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During the period of 1915 to 1918, Noether wasdeveloping her theory of current where one can usegroup of continuous symmetries to deduceequations of motions. (In general relativity, thecontinuous group of symmetry is the group ofcoordinate transformations.)

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Einstein’s visit was just at the right time! Hilbertdiscovered the Hilbert action in November of thesame year, and deriving the correct gravitationalequation quickly from this action. Upon hearing thenews and receiving Hilbert’s postcard on theequation, Einstein quickly got his equation, andbased on this equation, deduced astronomicalphenomena he had been trying to solve.

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In the beginning, Einstein was unhappy withHilbert’s priority. But Hilbert quickly declared thatthe work should belong entirely to Einstein, andthat turned Einstein happy. This is anepoch-making work. Later generations of physicistsand mathematicians should all pay their highesttribute to Einstein. But I shall hope history willremember the group of Geometers who helpedEinstein achieved his great theory of gravity. Muchof what I discussed here is written by Einsteinhimself. It is unfortunate that in that article, he didnot mention the contribution of Hilbert.

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Looking backward, the correct equation of motionderived by Hilbert and Einstein could have beenfound by Grossmann and Einstein in 1913. Theleft-hand side of the equation in 1913 consists ofRicci tensor while the right-hand side is the mattertensor. The right-hand side is familiar and itsatisfies conservation law. But the left-hand side ofthe 1913 equation is only the Ricci tensor whichdoes not satisfy the conservation law. Hence theycannot be equal.

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The left-hand side should therefore be replaced bysome form of curvature tensor that satisfiesconservation law. This was actually found by Ricciusing Bianchi identity. One simply subtracts theRicci tensor by some multiple of the metric tensorby the trace of the Ricci tensor. If Einstein andGrossmann trust the beauty of geometry and triedto complete the equations based on its internalconsistency, Einstein would not have to wait until1915 to write down the right equations.

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After completing the general theory of relativity,Einstein believed that the most basic part of physicsshould be guided by thought experiment and theelegance of mathematics. At the end of the article,he said that after finding the equation of generalrelativity, everything came so natural and so simplethat it was a breeze for a capable scholar. However,before finding the truth, he tried his best, afteryears of hard work, suffered pains day and night,which was hard to tell. Einstein’s work can be saidto be the greatest scientific work ever undertaken bymankind.

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The success of general relativity left us anotherdaunting task to explain natural phenomena ofgravity. The task is difficult because the system ofequation is truly nonlinear and the backgroundspacetime is changing dynamically. Physics ofgravity does not give a precise description of theinitial data or the boundary conditions of thecomplicated nonlinear system.

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There is no global symmetry of the dynamicalchanging spacetime. Nonexistence of global time, ornonexistence of timelike translation symmetry, gavegreat difficulty to define many important physicalquantities that we learned in Newtonian mechanics.Noether’s theory of current allows us to define massand linear momentum four-vector if we havetimelike translation that preserves the system. Butfor a generic system in general relativity, continuousgroup of symmetry does not exist!

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Nonexistence of continuous symmetries causeddifficulties to define classical concepts such as mass,linear momentum and angular momentum that arefundamental in understanding physics of gravity.When we watch two neutron stars interacting witheach other, we need to know the mass of each starand the binding energy of the whole systemcounting contributions from matter and gravitytogether. This problem arises in general relativitybecause the concept of energy density is notpossible in this theory of gravity.

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The reason is that if the density exists, it willdepend only on the first order information of thepotential of the gravity which is the metric tensor.Yet we can always find a coordinate system so thatthe first order differentiation of the metric tensor iszero at one point. This will mean that the energydensity is zero.

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Einstein already realized such questions one hundredyears ago. He proposed a definition of energy basedon a concept of pseudo-tensor drawing analog withthe definition of Newtonian mechanics. Thisdefinition was clarified more precisely by the work ofArnowitt, Deser, and Misner in 1962. Nowadays it iscalled ADM mass.

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This definition works well for isolated physicalsystem of gravity where the total mass of the wholesystem is defined. From the point of view ofNoether, this is natural because for an isolatedphysical system, we expect existence of asymptoticsymmetry at infinity and the time translation atinfinity captures the total energy of the system.This is a good definition of total energy. However, itcaptures the total energy only and there are detailedinformation of partial energy we need to explore.

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The very important question went back to Einsteinagain. He proposed the concept of gravitationradiation: the vibration of spacetime will radiatewave which gives energy. The energy comes fromthe binding energy of the gravity of the system.This concept was clarified by Bondi-van derBurg-Metzner and Trautman where they defined amass along some null hypersurface. Such a mass iscalled Bondi mass and it has a pleasant propertythat it decreases when the null hypersurface movesto the future.

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The decreases of the Bondi mass is interpreted asthe energy carried away by the gravitationalradiation. The definition of Bondi mass is importantas it describes the dynamics of spacetime. However,the definition presumes some structure of spacetimethat depends on the dynamics of the Einsteinequation.

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Both ADM and Bondi mass are total mass innature. It cannot capture mass of bodies that areinteracting with some other bodies. An importantcase is how to define binding energy of two neutronstars interacting with each other. Hence we need aconcept of quasilocal mass: Given a closedtwo-dimensional (spacelike) surface S in spacetime,what is the total energy it encloses?

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If S is the boundary of a three-dimensional spacelikethree-manifold M in spacetime, we like to measurethe total mass enclosed by S within M . Since welike to make sure the energy to be conserved, thequantity that we want should depend only on theinformation of S in spacetime and independent ofthe choice of M . This is the conservation law forquasilocal mass.

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The existence of such quantity has been a seriousproblem for a long time. The very first thing thatEinstein and later workers in general relativity wasinterested in whether the total ADM mass for anisolated physical system is positive?

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In fact, in 1957, Bondi and other well knownphysicists had a meeting and discussed thepossibility of negative mass in general relativity.Einstein’s theory could not tell whether this ispossible or not. But if the total mass is negative,the system may collapse and it will mean Einstein’stheory of gravity may create a rather undesirableeffect of unstable system.

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The positivity of ADM mass was proved by Schoenand myself in 1979, the full proof published in 1981.Our proof is more geometric in nature.

With Schoen

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Subsequently, Witten gavea proof depending on Diracoperator which is moretransparent to physicists. Shortlyafterwards, Bondi mass was alsoproved to be positive and thestate of affair for total mass ofan isolated physical system ingravity is pretty satisfactory. Witten

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Schoen and I also used our method to prove in aneffective way that when matter density is largeenough, black hole will form. It is probably the firstrigorous statement that black hole forms whenmatter density is large.

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The demonstration of existence of a good definitionof quasilocal mass took a long time, after works ofmany people including Penrose, Hawking,Brown-York, Geroch, Bartnik, Horowitz andShi-Tam.

Penrose Hawking

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Since it is supposed to be trivial for any closedsurface in the flat Minkowski spacetime and yetnonnegative for general spacetime, it is a miraclethat such a definition can exist which is compatiblewith the previous works of ADM and Bondi. Twoimportant definitions were proposed: one is due toRobert Bartnik and the other due to Mu-Tao Wangand myself.

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The quasilocal mass allows us to define bindingenergy related to binary black holes and is related tothe energy of the gravitational radiation. Theapproach of Wang-Yau allows them to definequasilocal angular momentum with Po-Ning Chen.It helps to clarify the former definitions of totalangular momentum.

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By the works of Richard Schoen and his coauthors,we know the Bartnik mass is different fromWang-Yau mass. It would be interesting to knowwhich one is more useful to describe physicaldynamics of gravity.

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Many other important classical concepts in gravityhas their counterparts in general relativity. But thenonlinear nature of general relativity makes itdifficult to define. But I think the recent progressmade by many geometric analysts has been veryimportant.

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The concept of quasilocal mass and angularmomentum has opened a window on studying thephysics and geometry of spacetime. A great dealmore efforts need to be spent in their study.

The definitions are most successful for objectswithin an isolated physical system. It would still beuseful to understand a more general situationincluding higher dimensional analogue. Ratherintricate geometry are involved in the study of suchconcepts.

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Einstein’s theory of gravitation has initiated a deepunderstanding of geometry through physical insightand vice versa, in the last century. We expect thisto continue in this century. The very interestingactivities in this conference in Stony Brookdemonstrate the exciting future of this interplaybetween geometry, analysis and physics.

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Thank you!

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