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Equations of Motion in Relativistic Gravity pp 327–398Cite as

Motion in Classical Field Theories and the Foundations of the Self-force Problem

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 179))

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 self-torque 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.

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Notes

  1. 1.

    Classical point particles are sometimes discussed as though they were the fundamental building blocks of all classical matter. This viewpoint is severely problematic on both mathematical and physical grounds, and is rejected here. That said, appropriately-regularized point particles do arise as mathematical structures obtained from certain well-defined limits involving families of extended bodies. All mention of point particles here is to be understood in this (effective) sense.

  2. 2.

    As is typical throughout physics, simple underlying principles do not imply simple applications to explicit problems. Applications typically do require significant computations.

  3. 3.

    It is common in the literature to use the words renormalization and regularization interchangeably, both implying the removal of unwanted infinities. This is not the usage here. Renormalization is intended in this review essentially as a synonym for “dynamical shift.” These shifts need not be infinite. Regularizations, by contrast, always refer to rules for handling singular behavior. Almost all discussion here focuses on finite renormalizations. Regularizations arise only in certain limiting cases.

  4. 4.

    Although the gradients of \(\phi \) and \(\hat{\phi }\) coincide at the center of a spherically-symmetric mass, they can be quite different in general. Consider, for example, a barbell constructed by joining two unequal spheres with a massless strut.

  5. 5.

    In simple cases, \(\varvec{v}\) represents a velocity field in the standard sense. More generally, it might be only an effective construction. This occurs, for example, if a body is composed of multiple interpenetrating fluids.

  6. 6.

    This follows from noting that any sufficiently small piece of matter with finite density responds to gravitational forces as though it were a test body.

  7. 7.

    For any single time function \(T: \mathcal {M} \rightarrow \mathbb {R}\) and any \(c, d \in \mathbb {R}\) such that \(c >0\), the map \(c T+ d\) is also an acceptable time function.

  8. 8.

    Large astrophysically-relevant objects like planets tend to be very nearly spherical due to the limited shear stresses which can be supported. The trace-free components of the moments, which are all that couple to the motion, are then much smaller than \(m \ell ^n\). These tend to be induced mainly by rotation and external tidal fields, and are typically modeled using Love numbers.

  9. 9.

    The notion of self-force used here is consistent with the usual Newtonian definition, but is unconventional in relativistic contexts. Its precise meaning is made clear below.

  10. 10.

    This is to be considered as a model problem. If interpreted as a theory of gravity, the type of scalar field theory described here is not compatible with observations. Of course, it is not necessary to interpret \(\phi \) as a gravitational potential (so \(\rho \) needn’t be a “mass density” in any sense).

  11. 11.

    In a Lagrangian formalism, the total stress-energy tensor considered here is derived from a functional derivative of the action with respect to the metric. It is conserved whenever the action is diffeomorphism-invariant [15].

  12. 12.

    Consider, e.g., the past-directed light cones associated with a timelike worldline.

  13. 13.

    It is also possible to introduce \((n+1)\)-point self-fields similar to (44). This is not considered any further here.

  14. 14.

    The term self-field is used in several different ways in the literature. The definition adopted here is uncommon, and is sometimes described as the “Coulomb-like” component of the self-field.

  15. 15.

    Recalling (76), \(T^{ab}_\mathrm {field}\) is quadratic in \(\phi \). The stress-energy tensor “associated with \( \phi _\mathrm {S} \)” is taken to mean that portion of \(T^{ab}_\mathrm {field}\) which is quadratic in \( \phi _\mathrm {S} \). Terms linear in \( \phi _\mathrm {S} \) are not included.

  16. 16.

    References [17, 31] derive the momentum-velocity relation using Dixon’s momenta without a scalar field, but in a spacetime which is not maximally-symmetric. Here, Dixon’s momenta are modified by \(E_s\), there is a scalar field, and the spacetime is maximally-symmetric. Despite these differences, the relevant tensor manipulations are identical.

  17. 17.

    The same geometric conditions can also be imposed in Lorentzian geometries. Physically, however, the vector fields discussed here are most useful in non-relativistic contexts. A more complicated structure described in Sect. 4.1.2 is better-suited to Lorentzian physics.

  18. 18.

    Dixon’s papers never considered matter coupled to scalar fields. The momenta associated with (131) are those which arise naturally for objects falling freely in curved spacetimes.

  19. 19.

    This type of renormalization fundamentally arises from the connection between \(\mathcal {L}_\xi G\) and \(\mathcal {L}_\xi g_{ab}\) which occurs for Green functions associated with the Klein-Gordon equation. In different theories, Lie derivatives of \(G\) can depend on fields other than the metric. Self-forces renormalize whichever moments are coupled to these fields.

  20. 20.

    More precisely, the coordinate components \(g_{ij}\) must not vary too rapidly when expressed in a Riemann normal coordinate system with origin \(z_s\). Physically, this is a significant restriction. It would be far better to perform multipole expansions only using effective metrics where gravitational self-fields have been appropriately removed. It is not known how to do to this for the full Einstein-Klein-Gordon system.

  21. 21.

    The two-point scalar \(V\) is to be understood here as equivalent to \(G_\pm \) when its arguments are timelike-separated. This defines it even in the presence of caustics and other potential complications.

  22. 22.

    The point particle field derived in [10] includes a derivative of the particle’s acceleration. A careful treatment of the perturbation theory shows that such terms refer only to accelerations at lower order [28]. The self-consistent discussion which is implicit here therefore requires that accelerations be simplified using the zeroth order equation of motion. This is taken into account in (163).

  23. 23.

    Some sign conventions in [2, 42] are different from those adopted here.

  24. 24.

    It is unclear that there is any sense in which an electron’s behavior can be modeled using equations derived for classical extended charges. Nevertheless, the example appears to be suggestive.

  25. 25.

    It would be more elegant to instead demand that \(K_G( \mathcal {Z}, \{ \mathfrak {B}_s \} ; \hat{g})\) and \(K_G( \mathcal {Z}, \{ \mathfrak {B}_s \} ; g)\) be identical or otherwise closely related. Such an assumption would restrict possible relations between \(g_{ab}\) and \(\hat{g}_{ab}\), and is an avenue which has not been explored.

  26. 26.

    It could also be interesting to consider reformulations where an effective connection is sought instead of an effective metric.

  27. 27.

    Analyzing the effect of a conformal factor on the laws of motion is similar to considering objects coupled to a particular type of nonlinear scalar field. Despite the nonlinearity, such systems can be understood exactly using only minimal adaptations of the formalism used to analyze the (linear) Klein-Gordon problem.

  28. 28.

    Quasi-local momenta have recently been proposed in general relativity which use isometric embeddings to lift flat Killing fields into arbitrary spacetimes [60, 61]. See also [62] for a proposal which allows conformal Killing vectors to be introduced in geometries without symmetries.

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Authors and Affiliations

  1. Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476, Golm, Germany

    Abraham I. Harte

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  1. Abraham I. Harte
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Correspondence to Abraham I. Harte .

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Editors and Affiliations

  1. ZARM University of Bremen, Bremen, Germany

    Dirk Puetzfeld

  2. ZARM University of Bremen, Bremen, Germany

    Claus Lämmerzahl

  3. Max-Planck-Institut für Gravitationsphysik, Golm, Brandenburg, Germany

    Bernard Schutz

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Harte, A.I. (2015). Motion in Classical Field Theories and the Foundations of the Self-force Problem. In: Puetzfeld, D., Lämmerzahl, C., Schutz, B. (eds) Equations of Motion in Relativistic Gravity. Fundamental Theories of Physics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-18335-0_12

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