Reading for Nov 15

Reading for Nov 15 - PWOct11appell-3 16:27 Page 36 Feature...

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Einstein’s equations of general relativity are like the Himalayan mountains – beautiful and majestic when viewed from a distance, but slippery and full of crevasses when explored up close. Of those who venture into them, not everyone comes back alive. A set of 10 inde- pendent, nonlinear partial differential equations, Einstein’s equations relate the energy and matter in a region of space to its geometry. Astonishingly simple when expressed in the geometric, coordinate-indepen- dent language of tensors that Einstein ultimately hit upon, the equations – when applied to real situations – unfortunately become coupled beasts unlike anything physicists had tamed since the days of Newton. Einstein’s equations of general relativity can be solved exactly only in a handful of cases – with one of the first such solutions, and perhaps its most famous, being that derived by the German astronomer Karl Schwarzschild in 1916 for the simple case of a static, spherical, uncharged mass in a vacuum. Schwarzs- child’s assumptions, and his mathematical wizardry, reduced the Einstein equations to a single, ordinary dif- ferential equation that he was able to readily solve, though even the master himself was surprised at the possibility of an exact solution. The “Schwarzschild solution” leads naturally to the concept of a black hole (see box on p40), although Schwarzschild himself never grasped the significance of the singularity in his solu- tion, dying four months later on the Russian Front dur- ing the First World War. Even Einstein thought the Schwarzschild singularity – the radius where the solu- tion is invalid because of division by zero – was physi- cally meaningless, and it was only decades later that the depths of the Schwarzschild solution were plumbed in general relativity’s first golden age, which ran from about 1960 to 1975, by Roger Penrose, Kip Thorne, Stephen Hawking and many others besides. As theories go, general relativity has been a great suc- cess. Most famously, its early approximate solutions accounted for a well-known discrepancy in the orbit of the planet Mercury that could not be completely accounted for using classical Newtonian physics, yield- ing a value for the difference that agreed spot-on with astronomical measurements. Einstein’s equations have also predicted that light bends in a gravitational field and even that radar signals are delayed when bounced off one of our solar system’s inner planets. However, these successes are all based on the “post-Newtonian” approx- imation of the full Einstein equations, where speeds are small compared with that of light and gravitational fields are weak. Einstein’s general relativity has never been tested in the vastly different “strong field” regime. Thanks, however, to fast and powerful supercom-
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This note was uploaded on 01/11/2012 for the course AST 101 taught by Professor Rosenzweig during the Fall '08 term at Syracuse.

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Reading for Nov 15 - PWOct11appell-3 16:27 Page 36 Feature...

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