Covariant Perturbations

Branes are highly relevant in the description of different physical scenarios where the relevant physical degrees of freedom are confined to an appropriate subregion of interest. Branes are described by a local action that is a functional of the geometry of the worldvolume spanned by the brane in its evolution. The guiding principles in the construction of the action are reparametrization invariance and background diffeomorphism invariance. This turns the action into a geometric model. The term `brane mechanics' was introduced by Carter [1, 2], where it is emphasized the range of applications of the subject. Since then, the range has been widely extended in important ways. At the classical level, even the simplest geometric models produce equations of motion that are highly non- linear. An important special case are strings propagating in a at background, where the equations of motion turn out to be linear, in an appropriate gauge [3]. Besides this special case, in general one confronts a serious challenge in the analysis of the possible solutions of the equations of motion. In addition, the non-linearity can be made even worse if one considers a non trivial background spacetime, with non vanishing curvature, see e.g. [4] for the case of a string, where the background curvature plays the role of an external force. In face of these difficulties, one has to resort to perturbation theory. The first step is to encounter a relevant exact solution to perturb about. To attain this, symmetries are invoked and exploited, to make the problem tractable. The next obvious step is a direct linearization of the equations of motion that produce a set of equations for  field deviations of fields that (on-shell) reduce to coupled quadratic oscillators, and so on. This `direct approach' is the one more commonly used in the literature [5]. There is also a `covariant direct approach' that exploits a perturbation taken as a directional covariant derivative along a deviation vector, or deformation, and that is especially advantageous in a curved background, see e.g [6].

To offer a complete covariant variational approach that does not use any gauge-fixing at any stage, where by gauge-fixing it is meant a split of brane perturbations in their normal and tangential modes with respect to the worldvolume. This approach sets brane mechanics as a covariant classical field theory, trying to avoid any idiosyncratic notation, and using standard familiar language, that hopefully will be useful to the reader. The covariance is with respect to both background diffeomorphisms and worldvolume reparametrization invariance. The main tool that we exploit is a covariant variational derivative, inspired by the pioneering work of Bazanski [7], and its higher order extensions. From this point of view, conservation laws become easily accessible via Noether's theorem, as well as how they behave under perturbations [8].

References

[1] B. Carter, Perturbation dynamics for membranes and strings governed by the Dirac-Nambu-Goto action in curved space, Phys. Rev. D 48 (1993).

[2] B. Carter, Essentials of classical brane dynamics, Int. J. Theor. Phys. 40 (2001).

[3] B. Zwiebach, A First Course in String Theory, 2nd ed. (Cambridge University Press, 2009).

[4] A. L. Larsen and V. P. Frolov, Propagation of perturbations along strings, Nucl. Phys. B 414, 129 (1994).

[5] A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects (Cambridge University Press, 1994).

[6] R. Capovilla and J. Guven, Geometry of deformations of relativistic membranes, Phys. Rev. D 51 (1995).

[7] S. L. Bazanski, Relative dynamics of the classical theory of fields, Acta Phys. Polon. B 7 (1976).

[8] G. Arreaga, R. Capovilla, and J. Guven, Noether currents for bosonic branes, Annals of Physics 279 (2000).

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