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Metric tensor
Bernhard Riemann
Bernhard Riemann, a German mathematician, it should be noted, was perhaps the first to consider geometries in higher dimensional space. Riemann felt that the flat 3-dimensional geometry of Euclid was based more on common sense and human experience than on actual mathematical logic. To Euclid, nothing can be 4-dimensional. There is no such thing. A point has 0 dimensions, a line has 1 dimension, a sheet or membrane would have 2 dimensions (x, y) and a solid construction would have 3 dimensions (x, y, z). That's it in the world of Euclid's geometry. Aristotle and Ptolemy also proposed that the 4th spatial dimension was not possible. Riemann was challenged by Carl Friedrich Gauss, whom he had met at the University of Gottingen, to re-propose the foundations of geometry. Sadly, this led to Riemann's nervous breakdown in 1854.
It was Riemann's lecture at the University of Gottingen and his essay: "On the Hypothesis Which Lie at the Foundation of Geometry." The lecture was eventually given and the work was published 12 years later in 1868. Riemann's idea was a metric. This metric was a collection of numbers at every point in space: a tensor. These numbers would describe how bent or curved that space was. Riemann proposed that 10 numbers at each point were necessary to describe the curvature of some 4-dimensional manifold. This worked, no matter how curved the manifold was. It seemed to work for any arbitrary kind of curvature. The greater the number on the tensor, the greater the curvature.
This will give birth to Riemannian geometry and to the mysterious 4th dimension used by Einstein in his theories of relativity. This is the birth of a new idea that will shock the world of physics: physical laws are simplified in higher dimensional space. Furthermore, Einstien's general theory of relativity will prove to be very successful at describing how gravitation is a consequence of the curvature of spacetime.
General relativity was so successful, in fact, that Einstein continued to work, in isolation, to incorporate electromagnetism into his theory of general relativity. This unified field theory would unite his theory of gravity with Maxwell's theory of electromagnetism. This was his quest for a classical unified field theory of fundamental interactions. This is a theory that would be able to explain all of the laws of nature, from atomic to cosmic scales. This theory of everything would have been the crowning achievement of Einstein's life. Einstein termed this sought after idea: the unified field theory. Einstein once summarized his quest for unification with the analogy of wood and marble. Marble was the beautiful world of geometry. Wood was the chaotic world of matter. Wood determined the structure of the marble. This was a problem for Einstein. The geometry of spacetime was determined by the presence of matter. Einstein wanted to create a universe of pure geometry. Sadly, after about 30 years of searching: Einstein was ultimately unsuccessful in his pursuit of unification, as he passed away in 1955.
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