Motion in Classical Field Theories and the Foundations of the Self-force Problem
This article serves as a pedagogical introduction to the problem of motion in classical field theories. The primary focus is on self-interaction: How does an object’s own field affect its motion? General laws governing the self-force and self-torque are d
- PDF / 741,040 Bytes
- 72 Pages / 439.37 x 666.142 pts Page_size
- 80 Downloads / 151 Views
Abstract This article serves as a pedagogical introduction to the problem of motion in classical field theories. The primary focus is on self-interaction: How does an object’s own field affect its motion? General laws governing the self-force and selftorque are derived using simple, non-perturbative arguments. The relevant concepts are developed gradually by considering motion in a series of increasingly complicated theories. Newtonian gravity is discussed first, then Klein-Gordon theory, electromagnetism, and finally general relativity. Linear and angular momenta as well as centers of mass are defined in each of these cases. Multipole expansions for the force and torque are derived to all orders for arbitrarily self-interacting extended objects. These expansions are found to be structurally identical to the laws of motion satisfied by extended test bodies, except that all relevant fields are replaced by effective versions which exclude the self-fields in a particular sense. Regularization methods traditionally associated with self-interacting point particles arise as straightforward perturbative limits of these (more fundamental) results. Additionally, generic mechanisms are discussed which dynamically shift—i.e., renormalize—the apparent multipole moments associated with self-interacting extended bodies. Although this is primarily a synthesis of earlier work, several new results and interpretations are included as well.
1 Introduction How are charges accelerated by electromagnetic fields? How do masses fall in curved spacetimes? Such questions can be answered in many different ways. Consider, for example, the Newtonian n-body problem. This is typically solved using a certain system of ordinary differential equations which govern the locations of n points in R3 . Besides its location, each point is characterized only by its mass. This is a considerable abstraction from the stars or planets whose motion the n-body problem is intended A.I. Harte (B) Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476 Golm, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_12
327
328
A.I. Harte
to describe. Physically, each mass point is really an extended body described by the laws of continuum mechanics. The internal density distributions, velocity fields, and temperatures of these bodies might be governed by complicated sets of nonlinear partial differential equations. From one point of view, it is the solutions to these equations which represent “the motion” of each mass. This is not, however, the approach which is typically adopted in celestial mechanics. In that context, one instead focuses only on each body’s center of mass (and perhaps its spin angular momentum). These are observables which describe motion “in the large.” It is a central result of Newtonian gravity that much of the dynamics of these observa
Data Loading...
