4 - Curved space
Published online by Cambridge University Press: 05 June 2012
Summary
We have, so far, studied classical mechanics from a variety of different points of view. In particular, the notion of length extremization, and its mathematical formulation as the extremum of a particular action, were extensible to define the dynamics of special relativity. As we went along, we developed notation sufficient to discuss the generalized motion of particles in an arbitrary metric setting. Most of our work on dynamics has been done in an explicitly “flat” space, or in the case of special relativity, space-time. But, we have pushed the geometry as far as we can without any fundamentally new ideas. Now it is time to turn our attention to the characterization of geometry that will allow us to define “flat” versus “curved” space-times. In order to do this, we need less familiar machinery, and will develop that machinery as we push on toward the ultimate goal: to find equations governing the generation of curvature from a mass distribution that mimic, in some limit, the Newtonian force of gravity.
Remember that it is gravity that is the target here. We have seen that the classical gravitational force violates special relativity, and that a modified theory that shares some common traits with electricity and magnetism would fix this flaw. For reasons that should become apparent as we forge ahead, such a modified theory cannot capture all of the observed gravitational phenomena, and, as it turns out, a much larger shift in thinking is necessary.
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- Advanced Mechanics and General Relativity , pp. 127 - 166Publisher: Cambridge University PressPrint publication year: 2010