Regularization of the Linearized Gravitational Self-Force for Branes

Abstract

We discuss the linearized, gravitational self-interaction of a brane of arbitrary codimension in a spacetime of arbitrary dimension. We find that in the codimension two case the gravitational self-force is exactly zero for a Nambu-Goto equation of state, generalizing a previous result for a string in four dimensions. For the case of a 3-brane, this picks out the case of a six-dimensional brane-world model as having special properties which we discuss. In particular, we see that bare tension on the brane has no effect locally, suppressing the cosmological constant problem.PACS numbers: 04.50,98.80The divergent self-force of a charged point particle, such as the electron, coupled to electromagnetism has been understood for many years. Its resolution via the inclusion of an ultra-violet (UV) cut-off, due to the finite radius of the particle, leads to a renormalization of the particle's mass and a suppression of the pole singularity at short distances (see, for example,This problem is not unique and, in fact, similar problems exist for any distributional source coupled to any kind of field in any spacetime dimension. An interesting case is that of a Nambu-Goto string coupled to linearized gravity. It has been shown [2,3,4] that the self-force, regularized in the UV by the core width of the string, ǫ, and in the infra-red (IR) by the inter-string separation, ∆, is exactly zero due to the fact that the induced linearized metric perturbation is orthogonal to the string worldsheet. This result can be shown to be true at all orders in perturbation theory, in the case of a static string [5].A similar result can be deduced when the string in four dimensions is coupled to an axion field, represented by a 2-form, and a dilaton, as well as linearized gravity [6,7,8]. For a special choice of couplings, which was predicted [9] in the context of N = 1, D = 10 Supergravity, one can show that the combined self-interaction is zero; the dilaton contribution is negative, which cancels the positive contribution from the axion field.It should be noted that the UV regularization of the self-field is not necessary in the codimension one case, the hypersurface, where the behaviour at the brane can be dealt with using junction conditions. The case of gravity can be dealt with exactly, at all orders, using the conditions often attributed to Israel [10] (although see ref.[11]). Similar lines of argument lead to the junction conditions at a surface in Maxwell's theory of electromagnetism [1].The extension of these ideas to higher dimensions has become more relevant recently with the interest that has arisen in brane-world models. In these models, the matter of the Standard Model of particle physics is confined to a four-dimensional subspace, or brane, of a higher-dimensional spacetime, often called the bulk. Two types of model have received particular attention: sixdimensional models with flat, compact extra dimensions, such as the Arkani-Hamed, Dimopoulos and Dvali (ADD) model [12], and five-dimensional models with warped extra dimensions, s...

Linearized self-forces for branes

“…In [16] it was shown that the regularized gradient operator applies more generally, and this was used in the case of purely gravitational fields in [17]. The cancellation of the gravitational self-force, already known for the case of a Nambu-Goto string in four spacetime dimensions, was seen to hold for any Nambu-Goto brane of co-dimension two, whatever the spacetime dimension.…”

“…We can deal with all these cases, leaving aside only the hypersurface case of codimension one, by introducing a regularization factor [16,17] of the form…”

Regularized braneworlds of arbitrary codimension

“…[32], the last term in (79) is a problem when we consider localizing gravity on the brane. To see why, we consider the Einstein equations (20), (21) and (22) for our metric (1). In the asymptotic limit r → ∞, we require a solution to the Einstein equations that is Anti-de-Sitter, that is we require R ∼ const.…”

Radiation Back-Reaction and Renormalization in Classical Field Theory with Singular Sources

“…But it follows from expressions (66) and (69) that the overall coefficient of this divergence is equal to zero. In the linearized gravity approximation, this fact was established in [12]. Therefore, the gravitational self-coupling of a codimension-two brane (and also of a hyperbrane) is not only renormalizable but also finite in all orders of the perturbation theory.…”

“…The effective equations of motion for a point charge in a curved background space-time [3], for a spinning particle [4], for a massive particle in higher dimensions [5], [6], for a massless charged particle in d = 4 [7], and for a massive particle coupled to the linearized gravity background [8] are currently known. Advances in string theory have motivated the study of the radiation back-reaction problem in models of extended relativistic objects (branes) [9]- [12]. Despite a variety of models, all these works share the common point that the fields considered satisfy linear equations of motion with a singular source (current) in the right-hand side.…”