Static, cylindrical symmetry in general relativity and vacuum energy
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In the first section of my research, in analogy with the standard derivation of the spherically symmetric Schwarzschild solution of the Einstein field equations, I find all static, cylindrically symmetric solutions of the Einstein equations for vacuum. These include not only the well known cone solution, which is locally flat, but others in which the metric coefficients are powers of the radial coordinate and the space-time is curved. These solutions appear in the literature, but in different forms, corresponding to different definitions of the radial coordinate. I find expressions for transforming between these different metric forms and examine some special points of interest. I then examine some special cases of non-vacuum solutions of the equations as well. Because all the vacuum solutions are singular on the axis, I match them to interior solutions with nonvanishing energy density and pressure. In addition to the well known cosmic string solution joining on to the cone, we find some numerical solutions that join on to the other exterior solutions. I then consider only a static, flat, cylindrically symmetric space-time. I calculate the components of the stress-energy tensor in terms of the cylinder kernel and its derivatives. The cylinder kernel in cylindrical coordinates has been previously calculated and can be used to find the energy density and pressure on various cylindrical boundaries; future work will include finding these quantities for various cylindrically symmetric geometries.
Einstein field equations
Trendafilova, Cynthia (2011). Static, cylindrical symmetry in general relativity and vacuum energy. Texas A&M University. Available electronically from