Mechanics of Isolated Extended Bodies in Classical Field Theories

Open Access
Harte, Abraham
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
May 08, 2007
Committee Members:
  • Pablo Laguna, Committee Chair
  • Deirdre Shoemaker, Committee Member
  • Benjamin Owen, Committee Member
  • Victor Nistor, Committee Member
  • extended bodies
  • equations of motion
This thesis discusses a number of issues related to the description and motion of extended matter distributions in certain classical field theories. Particular emphasis is placed on general relativity and Maxwell theory, although many results also apply in related formalisms. They are obtained by extending and applying a series of ideas originally developed by W. G. Dixon to understand the mechanics of isolated bodies. Since this formalism is not well-known, we provide an extensive review in a unified form. This elucidates the structure of an object's stress-energy tensor and electromagnetic current vector. Multipole decompositions of these objects are also studied in considerable detail, and are designed to automatically "factor out" the relevant conservation laws. In the case of charge-current vectors, we show how to extend these results to also take into account the assumed smoothness and compactness of the physical matter. This allows essentially all reasonable current configurations with a given total charge to be parameterized without any reference to the spacetime structure. Such constructions provide natural methods for comparing the properties of current distributions in different systems. They also simplify the study of "rigid" currents. It is found that such objects cannot generally exist without allowing for the presence of singularities. An even stronger result applies to the nonexistence of rigid number density vectors in systems where the total number of particles is fixed. These formal developments are applied in various way to study the motions of various compact extended bodies. The first case considered here is that of an uncharged test mass embedded in a spatially-flat Friedmann-Robertson-Walker universe. It is shown that even with zero (global) momentum, such an object may adjust its mass and trajectory merely by changing shape. Mass shifts are in fact unavoidable in almost all cases, and could be significant for galactic superclusters. This effect is due to changes in an object's gravitational potential and internal energies; an explanation which is made precise without any appeal to linearizations or symmetry principles. Lastly, we study the influence of a charge's internal structure on its motion in flat spacetime. Interactions with a body's own electromagnetic field are taken into account, and allowances are made for a large class of shapes, density distributions, internal currents, elasticities, and spins. Even if the angular momentum remains small, many such objects are found to be affected by large self-interaction effects beyond the standard Abraham-Lorentz-Dirac force. Limits where these effects become negligible are found to be quite restrictive, and depend on the boundary conditions (e.g. retarded or radiative) used to derive the self-field.