Sequestration and Stabilization

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Dees ii Acknowledgements Over the course of compiling my thesis, I was helped by far too many people to name. Whether the help came in the form of listening patiently while I entertained my latest thought on black holes, or graciously allotting funds to my continued academic pursuits, I thank you all. Thank you James for your idea leading to my exploration of solar sails. Thank you Sara for your saintly patience with my astrophysical obsessions, which got in our way more than once. Thank you Robin for seeing mathematical avenues I had never considered, and for seeming just as excited with this project as me. To Jan & Ted, enough thanks can hardly be put into words. Finally, thank you George for believing in me.

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Dees SEQUESTRATION AND STABILIZATION: TAMING THE BLACK HOLE Patrick Dees New College of Florida, 2010 ABSTRACT In my undergraduate thesis I investigate the subjects of electromagnetic manipulation for the purpose of containing a macroscopic charged or magnetized object. With these results in mind, the character of black holes is uncovered to the best ability of current theory. Next, two containment designs are presented for a charged, rotating black hole. The first, called the Millikan chamber, is designed for the controlled dissipation of a black hole by levitation against gravity via electric and magnetic manipulation. The second, named the Djinn chamber, is designed for stabilization. In combination with controlling magnetic fields, the Djinn chamber also utilizes the relationship between a black hole and a charged, rotating accretion disk in order to achieve both containment and stabilization. Finally, the uses for black holes both naturally and placed within a containing chamber are discussed. Professor George Ruppeiner Department of Physics iv

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Dees 1 Chapter 0: Introduction Black holes are where God divided by zero. Dave Barry In 1783, geologist John Michell made a disturbing discovery using Newtons equations. Michells calculation showed for the first time that gravitys hold can become so strong it does not allow light to escape. If the object is a star, then light created at the surface is gravitationally bound, never to leave. Gravity has rendered the star invisible to any outside observer. These dark stars were largely ignored in Michells time due to the popular notion of light existing as a wave, unaffected by gravity. However, general relativity changed the picture entirely in 1915, by defining the ground rules of interactions in curved space-time and proving that gravity does bend the path of light rays. With the advent of relativity dark stars were a possibility again, galvanized by solutions to Einsteins newly formulated vacuum field equations. In 1916, Karl Schwarzschild proposed one such solution, building the gravitational field of point and spherical masses [1, 2] from Einsteins equations. Through Schwarzschilds work the existence of Michells dark stars was once again supported by theory. However, interest waned until 1930, when Subrahmanyan Chandrasekhar showed that 1.44 solar masses of electrons would collapse under gravitational pressure. Chandrasekhars calculation ended up describing white dwarfs, but discussion continued unabated. Almost a decade later in 1939 Robert Oppenheimer presented his calculation that 3 solar masses would become a gravitationally completely collapsed star, the first sound proof of Michells dark star. Although theory now predicted the existence of these stars, astrophysical data was

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Dees 13 Stable electric field levitation requires that a charged particle be at a point in the vector field which has lines of force all pointing toward the space it occupies. Gauss law states however that the divergence of an electric field must be zero in free space [11]. In other words, an electrical force F r deriving from potential U r is always divergenceless, F E U 2 U 2 U x 2 2 U y 2 2 U z 2 0 (1.17) Physically this makes sense since if electric field lines were to all direct toward one point they would inevitably cross over one another, which is not allowed. Due to this property of the electric field, there are no local minima or maxima required for stable equilibrium. The potential of a magnetic dipole was defined in equation 1.12 and must satisfy equation 1.16 in order to produce equilibrium. The Laplacian of this system [15] 2 U 2 m x B x m y B y m z B z x 2 2 m x B x m y B y m z B z y 2 2 m x B x m y B y m z B z z 2 (1.18) can have one or more terms negative, but must be overall positive for stability. If the dipole is assumed to have a constant field, equation 1.18 can be rearranged [15] to 2 U m x 2 B x x 2 2 B x y 2 2 B x z 2 m y 2 B y x 2 2 B y y 2 2 B y z 2 m z 2 B z x 2 2 B z y 2 2 B z z 2 (1.19) which can also be rearranged into the more convenient form