Lecture 37. Riemannian Geometry and theGeneral Relativity

In the 19th century, mathematicians, scientists and philosophers experienced an extraor-dinary shock wave. By the emergence of non-Euclidean geometry, the old belief that math-ematics offers external and immutable truths was collapse. The magnitude of the revolutionin thought on such change has been compared to the Darwinian theory of evolution.

Birth of differential geometry In Newtons time, the study of curves begins with In-finitesimal Analysis, and he began to study the curvature of plane curves. By the definition,for a curve, the curvature at a point measures the deviation from its tangent.

Figure 37.1 Curves and surfaces

The theory of surfaces in the euclidean space was developed mainly in the 18th and19th centuries, and the first half of the 20th century. In the early 19th century, Young and

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Laplace proved that, for a spherical surface, the inner pressure is always higher than theouter one, and that the difference increases when the radius decreases. The laws of Physicsdictate that liquids tend to minimize their surfaces. In the interior of a drop or a bubblein equilibrium, the inner pressure is bigger than the outer one. This difference of pressureis due to the curvature of the boundary surface. Intuitively, it can be concluded that thecurvature of a surface at a point measures its deviation from its tangent plane.

Although the contributions of Euler, Monge and Dupin were of great importance, theessential part to establish the concept of space is due to Gausss famous paper Disquisitionesgenerales circa superficies curvas, in which the concept of curvature of a surface at a pointwas introduced. Surprisingly the definition of curvature is intrinsic. To do this, Gauss studiesthe intrinsic properties of the geometry of a surface, by using the first fundamental form asa starting point. Gausss Theorema Egregium can be stated: At a point of the surface,the curvature is an isometric invariant.

As an application of Gausss Theorema Egregium, one of the most profound anddifficult formulas of Differential Geometry and Algebraic Topology: The Gauss-Bonnettheorem for surfaces is proved. Gauss proves it for geodesic polygons and Bonnet extendedit for polygons with edges of non vanishing geodesic curvature. This theorem was generalizedto an arbitrary dimension nearly a century later, by Allendoerfer, Weyl and Chern.

Figure 37.2 Local coordinates and global geometry

Birth of Riemannian geometry In 1854, Riemann generalizes Gausss studies tospaces of arbitrary dimension, which was in a not very rigorous way. As a result, a geometryon a manifold would be a positive-definite quadratic form (i.e., metric form, or the firstfundamental form) on each of its tangent spaces. This definition of Riemann allows togeneralize much of Gausss work.

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As we have mentioned in the previous lecture, Gauss himself had suggested this topicfor Riemanns habilitation thesis. On June 10, 1854, in his brilliant lecture entitled On theHypotheses That Lie at the Foundations of Geometry, Riemann started by saying that

geometry presupposes the concepts of space, as well as assuming the basicprinciples constructions in space. It gives only nominal definitions of these things,while their essential specifications appear in the form of axioms. ...... Therelationship between these presuppositions is left in the dark; we do not seewhether, or to what extent, any connection between them is necessary, or apriori whether any connection between them is even possible.

Riemanns spaces of variable curvature include, as particular cases, the space forms, whichhistorically gave rise to the non-euclidean geometries, that are as consistent as the euclideanone.

Figure 37.3 Immanuel Kant

Impact hitting philosophy The German philosopher Immanuel Kant, who is a centralfigure of modern philosophy, regarded Euclidean geometry as a status of absolute certaintyand unquestionable validity. According to Kant, if we perceive an object, then necessarythis object is spatial and Euclidean. Kant also asserted that information from our sense isorganized exclusively along Euclidean templates before it is recorded in our consciousness.1

In the new horizons that the 19th century opened for geometry. Kants ideas of spacedid not survive much longer. Geometers of the 19th century quickly developed intuition in

1Is God a mathematician ? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto-Sydney, 2010, p. 152.

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the non-Euclidean geometries and learned to experience the world along those lines. TheEuclidean perception of space turned out to be learned after all, rather than intuitive. Allof these dramatic developments let the great French mathematician Henri Poincare (1854-1912) to conclude that the exaioms of geometry are neither synthetic a priori intuitions norexperimental facts., but it remains free.

Newtons omission In Newton universal, mathematical law of motion and gravitation,there was one major question that Newton left completely unanswered: How does gravityreally work? How does the Earth, a quarter million miles away from the Moon, affect theMoons motion? Being aware of this deficiency in his theory, Newton admitted it in thePrincipia:

Hitherto we have explained the phenomena of the heavens and of our seaby the power of gravity, but have not yet assigned the cause of this power. Thisis certain, that it must proceed from a cause that penetrates to the very centresof the Sun and planets...... and propagates its virtue on all sides to immensedistances, decreasing always as the inverse square of the distances. But hithertoI have not been able to discover the cause of those properties of gravity fromphenomena, and I frame no hypotheses.

Figure 37.4 The Sun and the Earth

Paradox of disappearance of the Sun Even though with certain unanswered questions,Newtons laws were so successful that it took two hundred years for science to take the nextfateful step. The next fateful step was made by Albert Einstein (1879-1955) who decided tomeet the challenge posed by Newtons omission.

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Figure 37.5 Albert Einstein

Graduated in 1900 with a bachelors degree from the Polytechnic Institute in Zurich, hefound hard time to find a job. In a letter, he confessed that he even considered to endinghis life: The misfortune of my poor parents, who for so many years have not had a happymoment, weights most heavily on me ....... I am noting but a burden to my relatives .... Itwould surely be better if I did not live at all. 2 Late with a recommendation of a classmates,Einstein was able to find a job as a clerk at the Swiss Patent Office in Bern in 1902.

In 1905, Einstein proposed his special relativity, which determines the laws of physicsare the same for all non-accelerating observers, and that the speed of light in a vacuum wasindependent of the motion of all observers. In particular, any speed cannot be faster thanthe speed of light.

Let us go back the Newtons omission about how does gravity really work, from whichEinsteins new theory of special relativity appeared to be in direct conflict with Newtonslaw of gravitation.

According to Newton, it was assumed that gravitys action was instantaneous, i.e., it tookno time at all for planets to fell the Earths attraction. On the other hand, by Einsteinsspecial relativity, it should be that no object, energy, or information could travel faster thanthe speed of light. So how could gravity work instantaneously? Einstein had the followinglogical thought, which was so-called the paradox of disappearing the Sun.

Let us image that the Sun would suddenly disappear. Then without the force holdingit to its orbit, the Earth would (according to Newtons theory) immediately start movingalong a straight line. However, by the limitation of the light speed, the Sun would actuallydisappeared from view to an observer at the Earth only about eight minutes later, whichwas required for light to travel from the Sun to the Earth. Therefore, gravitys actionshould not be instantaneous, and the change in the Earths motion would precede the Sunsdisappearance.

2Michio Kaku, Parallel Worlds, Anchor books, New York, 2005, p.30.

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Einstein was seeking a new theory, which should answer Newtons unanswered question,should overcome the above contradiction, should preserve all the remarkable successes ofNewtons theory, and should be compatible with his newly discovered special relativity. It isa formidable task. In 1915, Einstein finally reached his goal to propose his theory of generalrelativity, which is regarded as one of the most beautiful theories ever formulated.

Figure 37.6 Gravity was more like a fabric.

Einsteins general relativity What is Einsteins general relativity? Let us quoteinteresting explanation by physicist Michio Kaku as follows. 3

Think of a bowling ball placed on a bed, gently sinking into the mattress.Now shoot a marble along the warped surface of the mattress. It will travel ina curved path, orbiting around the bowling ball. A Newtonian, witnessing themarble circling the bowling ball from a distance, might conclude that there wasa mysterious force that the bowling ball exerted on the marble. A Newtonianmight say that the bowling ball exerted an instantaneous pull which forced themarble toward the center.

To a relativist, who can watch the motion of the motion of the marble onthe bed from close up, it obvious that there is no force at all. There is just thebending of the bed, which forces the marble to move in a curved line. To therelativist, there is no pull, there is only a push, exerted by the curved bed onthe marble.

Replace the marble with Earth, the bowling ball with the Sun, and the bedwith empty spacetime, and we see that Earth moves around the Sun not becauseof the pull of gravity but because the Sun warps the space around Earth, creatinga push that forces Earth to move in a circle.

3Michio Kaku, Parallel Worlds, Anchor books, New York, 2005, p.35-36.

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Einstein was thus led to believe that gravity was more like a fabric thanan invisible force that acted instantaneously throughout the universe. If onerapidly shakes this fabric, waves are formed which travel along the surface ata definite speed. This resolves the paradox of the disappearing of the Sun. Ifgravity is a by-product of the bending of the fabric of space-time itself, then thedisappearance of the Sun can be compared to suddenly lifting the bowling ballfrom the bed. As the bed bounces back to its original shape, waves are sent downthe bed sheet travelling at a definite speed. Thus, by reducing gravity to thebedding of space and time, Einstein was able to reconcile gravity and relativity.

Geometry is the most ancient branch of physics For modern science, it became moreand more clear that a good physics theory must be expressed as crystal-clear, self-consistentmathematical relations. The best example was that Newton established his theory of lawsof motion and gravitation.

Now with his new theory of general relativity, Einstein badly needed a mathematicaltheory as a tool to express and to work for his theory of general relativity. In desperation,he turned to his old classmate the mathematician Marcel Grossmann (1878-1936) who helpedhim to find the first job: I have become imbued with great respect for mathematics, themore subtle parts of which I have previously regarded as sheer luxury. Grossmann pointedout that Riemmans non-Euclidean geometry was precisely the tool that Einstein needed ? ageometry of curved spaces in any number of dimensions. Einstein was quick to acknowledge:We may in fact regard [geometry] as the most ancient branch of physics, he declared.Without it I would have been unable to formulate the theory of relativity.

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