Lecture 37. Riemannian Geometry and the General Relativity In the 19th century, mathematicians, scientists and philosophers experienced an extraordinary shock wave. By the emergence of non-euclidean geometry, the old belief that mathematics offers external and immutable truths was collapse. The magnitude of the revolution in thought on such change has been compared to the Darwinian theory of evolution. Birth of differential geometry In Newton s time, the study of curves begins with Infinitesimal 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 and 19th centuries, and the first half of the 20th century. In the early 19th century, Young and 257
Laplace proved that, for a spherical surface, the inner pressure is always higher than the outer one, and that the difference increases when the radius decreases. The laws of Physics dictate that liquids tend to minimize their surfaces. In the interior of a drop or a bubble in equilibrium, the inner pressure is bigger than the outer one. This difference of pressure is due to the curvature of the boundary surface. Intuitively, it can be concluded that the curvature 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, the essential part to establish the concept of space is due to Gauss s famous paper Disquisitiones generales circa superficies curvas, in which the concept of curvature of a surface at a point was introduced. Surprisingly the definition of curvature is intrinsic. To do this, Gauss studies the intrinsic properties of the geometry of a surface, by using the first fundamental form as a starting point. Gauss s Theorema Egregium can be stated: At a point of the surface, the curvature is an isometric invariant. As an application of Gauss s Theorema Egregium, one of the most profound and difficult formulas of Differential Geometry and Algebraic Topology: The Gauss-Bonnet theorem for surfaces is proved. Gauss proves it for geodesic polygons and Bonnet extended it for polygons with edges of non vanishing geodesic curvature. This theorem was generalized to 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 Gauss s studies to spaces of arbitrary dimension, which was in a not very rigorous way. As a result, a geometry on a manifold would be a positive-definite quadratic form (i.e., metric form, or the first fundamental form) on each of its tangent spaces. This definition of Riemann allows to generalize much of Gauss s work. 258
As we have mentioned in the previous lecture, Gauss himself had suggested this topic for Riemann s habilitation thesis. On June 10, 1854, in his brilliant lecture entitled On the Hypotheses That Lie at the Foundations of Geometry, Riemann started by saying that geometry presupposes the concepts of space, as well as assuming the basic principles constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms.... The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible. Riemann s spaces of variable curvature include, as particular cases, the space forms, which historically gave rise to the non-euclidean geometries, that are as consistent as the euclidean one. Figure 37.3 Immanuel Kant Impact hitting philosophy The German philosopher Immanuel Kant, who is a central figure of modern philosophy, regarded Euclidean geometry as a status of absolute certainty and unquestionable validity. According to Kant, if we perceive an object, then necessary this object is spatial and Euclidean. Kant also asserted that information from our sense is organized exclusively along Euclidean templates before it is recorded in our consciousness. 1 In the new horizons that the 19th century opened for geometry. Kant s ideas of space did not survive much longer. Geometers of the 19th century quickly developed intuition in 1 Is God a mathematician? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto- Sydney, 2010, p. 152. 259
the non-euclidean geometries and learned to experience the world along those lines. The Euclidean perception of space turned out to be learned after all, rather than intuitive. All of these dramatic developments let the great French mathematician Henri Poincaŕe (1854-1912) to conclude that the exaioms of geometry are neither synthetic a priori intuitions nor experimental facts., but it remains free. Newton s omission In Newton universal, mathematical law of motion and gravitation, there was one major question that Newton left completely unanswered: How does gravity really work? How does the Earth, a quarter million miles away from the Moon, affect the Moon s motion? Being aware of this deficiency in his theory, Newton admitted it in the Principia: Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the Sun and planets... and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, 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, Newton s laws were so successful that it took two hundred years for science to take the next fateful step. The next fateful step was made by Albert Einstein (1879-1955) who decided to meet the challenge posed by Newton s omission. 260
Figure 37.5 Albert Einstein Graduated in 1900 with a bachelor s degree from the Polytechnic Institute in Zurich, he found hard time to find a job. In a letter, he confessed that he even considered to ending his life: The misfortune of my poor parents, who for so many years have not had a happy moment, weights most heavily on me... I am noting but a burden to my relatives... It would 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 physics are the same for all non-accelerating observers, and that the speed of light in a vacuum was independent of the motion of all observers. In particular, any speed cannot be faster than the speed of light. Let us go back the Newton s omission about how does gravity really work, from which Einstein s new theory of special relativity appeared to be in direct conflict with Newton s law of gravitation. According to Newton, it was assumed that gravity s action was instantaneous, i.e., it took no time at all for planets to fell the Earth s attraction. On the other hand, by Einstein s special relativity, it should be that no object, energy, or information could travel faster than the speed of light. So how could gravity work instantaneously? Einstein had the following logical thought, which was so-called the paradox of disappearing the Sun. Let us image that the Sun would suddenly disappear. Then without the force holding it to its orbit, the Earth would (according to Newton s theory) immediately start moving along a straight line. However, by the limitation of the light speed, the Sun would actually disappeared from view to an observer at the Earth only about eight minutes later, which was required for light to travel from the Sun to the Earth. Therefore, gravity s action should not be instantaneous, and the change in the Earth s motion would precede the Sun s disappearance. 2 Michio Kaku, Parallel Worlds, Anchor books, New York, 2005, p.30. 261
Einstein was seeking a new theory, which should answer Newton s unanswered question, should overcome the above contradiction, should preserve all the remarkable successes of Newton s theory, and should be compatible with his newly discovered special relativity. It is a formidable task. In 1915, Einstein finally reached his goal to propose his theory of general relativity, which is regarded as one of the most beautiful theories ever formulated. Figure 37.6 Gravity was more like a fabric. Einstein s general relativity What is Einstein s general relativity? Let us quote interesting 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 in a curved path, orbiting around the bowling ball. A Newtonian, witnessing the marble circling the bowling ball from a distance, might conclude that there was a mysterious force that the bowling ball exerted on the marble. A Newtonian might say that the bowling ball exerted an instantaneous pull which forced the marble toward the center. To a relativist, who can watch the motion of the motion of the marble on the bed from close up, it obvious that there is no force at all. There is just the bending of the bed, which forces the marble to move in a curved line. To the relativist, there is no pull, there is only a push, exerted by the curved bed on the marble. Replace the marble with Earth, the bowling ball with the Sun, and the bed with empty spacetime, and we see that Earth moves around the Sun not because of the pull of gravity but because the Sun warps the space around Earth, creating a push that forces Earth to move in a circle. 3 Michio Kaku, Parallel Worlds, Anchor books, New York, 2005, p.35-36. 262
Einstein was thus led to believe that gravity was more like a fabric than an invisible force that acted instantaneously throughout the universe. If one rapidly shakes this fabric, waves are formed which travel along the surface at a definite speed. This resolves the paradox of the disappearing of the Sun. If gravity is a by-product of the bending of the fabric of space-time itself, then the disappearance of the Sun can be compared to suddenly lifting the bowling ball from the bed. As the bed bounces back to its original shape, waves are sent down the bed sheet travelling at a definite speed. Thus, by reducing gravity to the bedding 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 more and more clear that a good physics theory must be expressed as crystal-clear, self-consistent mathematical relations. The best example was that Newton established his theory of laws of motion and gravitation. Now with his new theory of general relativity, Einstein badly needed a mathematical theory 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 helped him to find the first job: I have become imbued with great respect for mathematics, the more subtle parts of which I have previously regarded as sheer luxury. Grossmann pointed out that Riemman s non-euclidean geometry was precisely the tool that Einstein needed? a geometry 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. 263