# Self-force on an arbitrarily coupled scalar charge in cylindrical thin-shell spacetimes

## Abstract

We consider the arbitrarily coupled field and self-force of a static massless scalar charge in cylindrical spacetimes with one or two asymptotic regions, with the only matter content concentrated in a thin-shell characterized by the trace of the extrinsic curvature jump \(\kappa \). The self-force is studied numerically and analytically in terms of the curvature coupling \(\xi \). We found the critical values \(\xi _c^{(n)} = n/\left( \rho (r_s)\,\kappa \right) \), with \(n \in {\mathbb {N}}\) and \(\rho (r_s)\) the metric’s profile function at the position of the shell, for which the scalar field is divergent in the background configuration. The pathological behavior is removed by restricting the coupling to a domain of stability. The coupling has a significant influence over the self-force at the vicinities of the shell, and we identified \(\xi =1/4\) as the value for which the scalar force changes sign at a neighborhood of \(r_s\); if \(\kappa (1-4\xi )>0\) the shell acts repulsively as an effective potential barrier, while if \(\kappa (1-4\xi )<0\) it attracts the charge as a potential well. The sign of the asymptotic self-force only depends on whether there is an angle deficit or not on the external region where the charge is placed; conical asymptotics produce a leading attractive force, while Minkowski regions produce a repulsive asymptotic self-force.

## 1 Introduction

A field theory including a complex scalar field coupled to a gauge field predicts that spontaneous symmetry breaking can lead to cylindrical topological defects known as local or gauge cosmic strings [1]. The gravitational effects of such objects have been considered of particular physical interest within the study of structure formation in the early Universe, since they could have acted as possible “seeds” for density fluctuations [2, 3, 4]; besides, strings could be, in principle, observed by gravitational lensing effects. Though in the present day theoretical framework cosmic strings are not considered as the main source of the primordial cosmological matter fluctuations, they are still taken as a possible secondary source of fluctuations [5]; this motivates a recently renewed interest in their study.

Local or gauge strings are characterized by having an energy-momentum tensor whose only non null components are \(T_0^{\, 0}=T_z^{\, z}\). The metric around a gauge string was first calculated by Vilenkin [6] in the linear approximation of general relativity, using a Dirac delta to approximate the radial distribution of the energy-momentum tensor for a cosmic string along the *z* axis. Thus, it was proposed \( {\tilde{T}}_\mu ^{\, \nu } = \delta (x)\delta (y) \,\hbox {diag}(\mu ,0,0,\mu ),\) where \(\mu \) is the linear mass density, which is determined by the energy scale at which the symmetry breaking process took place. Under this assumption, and working up to first order in \(G\mu \), Vilenkin obtained a spacetime around the string which is flat but presents a deficit angle \(\Delta \varphi =8\pi G\mu \). Some years later, Hiscock [7], motivated by the possibility of theories which may lead to large values of \(G\mu \) (i. e. \(G\mu \sim 1\)), considered a thick cylinder of radius *a* with uniform tension and linear mass density, whose energy-momentum tensor is \( T_\mu ^{\, \nu }(x,y)=\hbox {diag}(\mu ,0,0,\mu )\theta (r-a)/ a^2,\) and solved the full Einstein equations in the interior and matched the resulting static metric with the vacuum solution for the exterior; no matter layer was assumed to exist at the radius *a*, so that the adopted matching conditions imposed the continuity of both the metric and the extrinsic curvature. His work showed that Vilenkin’s results for the metric outside the string core were actually valid to all orders in \(G\mu \), that is, the conical geometry was not an artifact introduced by the linearized approximation, but an essential feature of gauge strings. Since the corresponding metric has \(g_{00}=1\), i.e. the Newtonian potential is null, non charged rest particles would not be affected by the gauge string gravity. However, strings moving at relativistic speeds would induce waves which could eventually lead to observable matter density variations.

Things change considerably, however, when charged particles are taken into account. The local flatness of a conical background manifold does not imply zero force on a rest charge. A vanishing force requires a field symmetric around the charge, and this is not possible when a deficit angle exists. While the field equations of a given point source are locally those of Minkowski spacetime, the globally correct solution in a conical geometry is not symmetric around the source, and as a result a self-force appears on the charged particle [8]. Moreover, as shown in [9, 10, 11] for electrically charged point particles, the behavior of such force in relation with the position of the charge allows to distinguish between two locally identical geometries which differ in their global topological aspects. This provides a sort of tool for detecting wormholes of the thin-shell kind [12], whose geometries are locally the same of those of the type which they would connect. These wormholes are characterized by connecting two exterior geometries of a given class by a throat, which is a minimal area or circumference surface (see below), where a matter layer of negligible thickness is placed [13, 34, 35]. Hence, for example, by studying the force on a static charged particle an observer could determine if the background is that of a gauge cosmic string or that of a thin-shell wormhole connecting two conical spacetimes. Also, in the case of topologically trivial spacetimes, an analogous analysis would allow to distinguish different interior geometries by studying the self-force on a charge placed at the exterior region beyond a matter shell [11, 14, 15]. The problem of the self-force on a scalar point charge appears as a natural but not at all trivial extension of previous work.

Scalar fields are widely used in cosmological models to explain the early Universe and the present accelerated expansion; non-minimally coupled to gravity scalar fields are included in inflationary models [16], while the unknown nature of dark matter is sometimes described by classical light or massless scalar fields [17, 18]. Besides the physical interest of scalar charges and the self-force problem [19, 20, 21, 22, 23, 24], a central aspect motivating a detailed analysis is the different form in which the scalar field couples to gravity: while the field of an electric charge is continuous across a non charged matter shell separating two regions of a spacetime, this is not the case for a scalar charge, and matter shells induce a sort of new sources for the corresponding scalar field; this necessarily translates to an, in principle, different behaviour of the self-force. Some peculiarities due to non minimal coupling of the scalar and the gravitational fields were pointed out in [19, 25, 26, 27, 28, 29]. In particular, some authors who studied the scalar self-force and propagation of a scalar field in a fixed background found that the stability regions of solutions for the non minimally coupled wave equation are restricted by the value of the coupling constant. An anomalous divergence was first realized by Bezerra and Khusnutdinov [30] who reported an infinite scalar self-force for some critical values of the coupling constant in a class of spherically symmetric wormhole spacetimes. For example, in a spherical wormhole with an infinitely short throat of the thin-shell kind they found critical values at \(\xi =n/4\), for \(n \in {\mathbb {N}}\). Continuing to previous work, Taylor [31] notice that solutions to the scalar wave equation sourced by a point charge are unstable in the Ellis wormhole for the discrete set of couplings \(\xi =n^2/2\). The pathological behavior was removed by restricting the coupling constant to the domain of stability where no poles are encountered [32, 33]. In the present work we find that a similar feature occurs in cylindrical wormholes and also in trivial spacetimes.

In this article we study the self-force acting on a static point like scalar charge in a background constructed by cutting and pasting two cylindrical manifolds with different deficit angle, and we analyze the dependance on the non minimal coupling to gravity. The considered geometries have all the matter content isolated in a thin-shell (see for example [34, 35]) and in this way we could analyze the influence of the coupling on the scalar field solutions and the consequences over the self-force. Additionally, the obtained results clarify some aspects of the self-force in terms of the global properties of a given background geometry. On the other hand, resonances of the field appear at critical values \(\xi _c\) of the coupling which depend on the shape of the profile function of the metric or, more specifically, on the value of the deficit angles. We established that the stable domain of the coupling is related to the kind of matter to which it couples or, equivalently, to the sign of the trace of the extrinsic curvature tensor jump over the shell.

We shall consider two models which are presented in Sect. 2; in the first one, an interior and an exterior submanifolds are joined at an hypersurface where a thin-shell is placed and, in the second one, two exterior submanifolds are joined. The thin-shell is characterized by the value of \(\kappa \), the trace of the jump on the extrinsic curvature tensor. Section 3 is divided in three sections; in Sects. 3.1 and 3.2 the field equation sourced by the static scalar charge is solved in both types of manifolds by the method of separation of variables in cylindrical coordinates. The field \(\Phi \) is split in two terms, one homogeneous and the other inhomogeneous at the position of the charge. Resonant configurations for which the field diverges at critical values \(\xi _c\) of the coupling are found in Sect. 3.3. These are associated to instabilities of the coupled scalar field equation in the corresponding background geometries. The scalar field is regularized at the position of the particle in Sect. 4 by subtracting the Detweiler–Whiting singular Green function to the actual field. Finally the self-force over the scalar charge is calculated evaluating the gradient of the regular field at the position of the particle. The results are analyzed analytically and numerically in terms of the coupling constant for different configurations of the background spacetime in Sects. 4.1 and 4.2. Throughout the article the geometrized unit system is used where \(c=G=1\).

## 2 Approach

*g*is its determinant and \(\xi \) the arbitrary coupling of the field to the curvature scalar

*R*. The particle action \( S_{m_0} \) is

*q*is written as

*q*at rest; this yields the inhomogeneous equation

*q*, and \(\dot{z}^{0} = dt/d\tau \) . On the other hand, demanding the action to be stationary under a variation \(\delta z^{\alpha }(\tau )\) of the world line yields the equations of motion

## 3 Scalar field in conical spacetimes

^{1}

*z*and \(\theta \). Then, the radial functions \(\chi _{n}(k,r)\) are obtained from the radial equation

*n*and

*k*. Requiring finiteness if \(r\rightarrow 0\) and \(r\rightarrow \infty \), the continuity at \(r=r'\) and its derivative discontinuity (17), we obtain the scalar field in the infinitely thin straight cosmic string spacetime given as a series expansion with the radial solutions:

### 3.1 Scalar field in Type I spacetimes

*n*and

*k*. From (15), the respective equations become

### 3.2 Scalar field in Type II thin-shell wormholes

*n*and

*k*. With the particle placed in \({\mathcal {M}}_{+}\), the inhomogeneity is at \(r_2' > r_+\), the radial equations are

### 3.3 Resonant configurations

*q*in Type I or Type II spacetimes is

*n*presents a pole at \(k_p > 0\), the divergency can be circumvented splitting the integral at \(k_p \mp \epsilon \), with \(\epsilon \rightarrow 0\), by canceling out the contributions of the lateral limits. As an alternative to the last method, contour integrals for the complex-valued integrand function can be used to obtain the positive real half-line integral. But if the denominator in the integrand coefficient is null for \(k=0\) we have, inevitably, a divergent mode. We can examine directly the integrand of each mode in the limit \(k \rightarrow 0^+\) to identify these divergencies. For example, in Type I geometries with the particle placed at \(r_1=r_1'\), we obtained the internal radial solutions \(\chi ^{i}_n(k,r_1) = \, \chi ^{\omega _i}_n(k,r_1) + \frac{1}{\omega _i} I_{\lambda }(k r_1) A_n(k)\), and the external ones \(\chi ^{e}_n(k,r_2) = \frac{1}{\omega _i} K_{\nu }(k r_2) C_n(k)\), given in (26) and (27), from which we can compute:

*n*th mode of the scalar field in Type I spacetimes. For \(n=0\) there is no value of \(\xi \) associated to this kind of resonances (the denominator of radial coefficients do not vanish for \(k=0\) in the \(n=0\) mode). In terms of the extrinsic curvature on the thin-shell in Type I spacetimes; if \(\kappa < 0\) (ordinary matter thin-shell) the coupling can take values \(\xi > \xi _c^{(n=1)} = 1/(\kappa \omega _i r_i)\) to avoid encountering a resonant mode which makes the field divergent and, on the other hand, if \(\kappa >0\) (exotic matter thin-shell) couplings \(\xi < \xi _c^{(n=1)} = 1/(\kappa \omega _i r_i)\) ensure that the configuration does not produce some resonant mode. Repeating the analysis in Type II geometries, with the radial solutions obtained with the particle placed at \(r_2=r_2'\): \(\chi ^{-}_n(k,r_1) = \frac{1}{\omega _+} K_{\lambda }(k r_1) E_n(k)\) for region \({\mathcal {M}}_-\), and \(\chi ^{+}_n(k,r_2) = \, \chi ^{\omega _+}_n(k,r_2) + \frac{1}{\omega _+} K_{\nu }(k r_2) W_n(k)\) for region \({\mathcal {M}}_+\), given in (44) and (45), we can see the same dependance

## 4 Regular field and scalar self-force

*x*to the charge’s position \(x'\) along the shortest geodesic connecting them [39]. The field \(\Phi ^S = q \, 4 \pi \, G_{DW}(x ; x')\) for a static particle in a static spacetime is constructed with the Detweiler–Whiting singular Green function in three dimensions over the normal convex neighborhood of \(\mathbf x'\) [40]. This Green function has the same singularity structure as the particle’s actual field, exerts no force on the particle and, in the considered static problem, can be calculated as [36]

### 4.1 Self-force in Type I geometries

In Fig. 1 we show the results in a Type I geometry with an ordinary matter thin-shell and coupling values in the stable domain \(\xi > \xi _c^{(n=1)} = -2\). The self-force vanishes at the central axis of the geometry because the first term in (70) is null after regularization in a Minkowski interior. When approaching the shell from either sides we observe a divergent force, being attractive or repulsive depending on the value of the coupling, Fig. 1a. Nevertheless, in the conical exterior region the force is asymptotically attractive to the center for every value of the coupling constant due to the leading term \(\sim - L_{\omega _e} r^{-2}\) obtained after renormalization, Fig. 1b.

The solid line in Fig. 1 corresponds to minimal coupling; in the interior region with \(r<r_i\) the force increases from cero at the center to infinity as \(r \rightarrow r_i^-\), while in the exterior we see a negative force diverging at the vicinities of the shell and vanishing at the asymptotic conical infinity. This shows that for \(\xi =0\) the curvature jump at the ordinary matter shell acts attracting the particle. We observed that negative values of \(\xi \) present qualitatively the same results but with an intensified force near the shell. A richer spectrum appears for positive couplings. If \(0< \xi \leqslant 1/4\) the force changes sign in either region but becomes attractive to the shell in its vicinities. For \(\xi > 1/4\), the force diverges if \(r \rightarrow r_i\) but with the opposite sign *i*.*e*., the particle is repeled in the vicinities of the ordinary shell. Despite this local difference, the leading asymptotic term proportional to \(- L_{\omega _e=1/2}\) in (70) produces an attraction sufficiently far from the shell in the deficit angle exterior region.

In Fig. 2 we show the results in a Type I geometry with a shell of exotic matter and couplings in the stable range \(\xi < \xi _c^{(n=1)} = 2\). The self-force diverges at the center with the particle been attracted to the conical singularity at the interior, and approaching the shell from either sides it diverges showing different behaviors depending on the value of the coupling, Fig. 2a. In an exterior region without angle deficit \(L_{\omega _e=1} = 0\), and the force becomes repulsive from the center sufficiently far from the shell for any value of \(\xi \), Fig. 2b, differing from the previous case shown in Fig. 1b of a deficit angle exterior.

The solid plotted line corresponds to minimal coupling; the force is negative in the interior, *i*.*e*. attractive to the conical axis and repulsive from the shell, and in the exterior region the force is positive. For \(\xi =0\) the exotic shell repels the particle and the force diverges if \(r \rightarrow r_i\). Negative values of \(\xi \) present similar results but with a greater intensity of the force. The richer spectrum appears, again, for \(\xi >0\). With couplings in the range \(0< \xi \leqslant 1/4\), the force may vanish and change sign in a same region, but sufficiently near to the shell it becomes repulsive. Oppositely, for \(\xi > 1/4\) we see that the shell attracts the particle at its vicinities and sufficiently far from it, in the exterior region, a leading asymptotic repulsion from the center is observed due to the influence of the angle deficit of the core geometry.

### 4.2 Self-force in Type II geometries

The solid line in Fig. 3a corresponds to minimal coupling; the force is repulsive from the throat and diverges approaching the shell in the limit \(r \rightarrow 0\). We see that negative values of \(\xi \) produce the same behavior with an increasing repulsion with decreasing value of the coupling. Couplings in the range \(0< \xi < 1/4\) show an attractive force in some finite region but produce a repulsion in the vicinities of the infinitely short throat as well. Oppositely, the particle is attracted in the vicinities of the shell for couplings \(\xi > 1/4\), as shown in Fig. 3b, diverging in the limit \(r \rightarrow 0\). Despite the different local behavior near the throat for \(\xi > 1/4\), we find a leading asymptotic repulsion associated to the positive jump \(\kappa >0\) and the non trivial topology of this spacetime.

The solid line in Fig. 4a corresponds to minimal coupling; the force is repulsive in the vicinities of the throat, diverges in the limit \(r \rightarrow 0\) at the thin-shell, and sufficiently far it becomes attractive. Negative values of \(\xi \) present the same qualitative behavior with an increasing force with decreasing value of the coupling. Figure 4b shows results for couplings in the range \(0< \xi < 1/4\); the force is also repulsive near the infinitely thin throat but it may vanish in one or three positions in an intermediate region becoming asymptotically attractive as \(r^{-2}\). In Fig. 5 we show the self-force for couplings \(\xi > 1/4\); the force is attractive in the vicinities of the shell for all these cases, we find an intermediate region with a repulsive force and the leading asymptotic attraction associated to the deficit angle term. We plotted cases with couplings beyond the stable domain; this is \(3/4< \xi < 3/2\), between the critical values for the \(n=1\) and \(n=2\) modes, which show qualitatively the same self-force of those with couplings in the interval \(1/4< \xi < 3/4\). This last feature is repeated in the successive intervals \(3n/4< \xi < 3(n+1)/4\).

### 4.3 The \(\xi =1/4\) coupling from the radial equation

*n*and

*k*, to take it to the form

*point-like*infinite term disappears from the potential; for this specific value the contribution from the curvature coupling at the thin-shell exactly cancels out the effect of the extrinsic curvature discontinuity in the potential for \(\psi \). Without track of the delta function, the potential

^{2}

*q*, in Type I geometries with \(\kappa < 0\); a coupling \(\xi > 1/4\) (barrier) repels the particle from the shell in a neighborhood of \(r=r_i\), and a coupling \(\xi < 1/4\) (well) attracts the particle to the shell. This conclusion is in accordance with the results found in the plots of Fig. 1a of a shell with ordinary matter. Similarly, the opposite was found in terms of the coupling in the examples plotted on Fig. 2a; the shell of exotic matter and \(\kappa > 0\) produces an infinite potential with negative sign if \(\xi > 1/4\) (well) which attracts the particle, or with positive sign if \(\xi <1/4\) (barrier) which repels it, in a neighborhood of \(r=r_i\). In wormhole spacetimes, Type II, the analogous analysis applies; the potential well or barrier with \(\xi \ne 1/4\) is manifested in the sign of the self-force at the vicinities of the throat of \(\kappa >0\), producing an attraction (\(\xi > 1/4\)) or repulsion (\(\xi < 1/4\)) from the shell.

## 5 Summary and conclusions

The arbitrarily coupled massless scalar field produced by a static point charge in cylindrically symmetric backgrounds with a thin-shell of matter was found in spacetimes with one (Type I) or two (Type II) asymptotic regions. The self-force over the charged particle was calculated from the scalar field and studied numerically and analytically in terms of the curvature coupling \(\xi \). The fixed background geometries used are everywhere flat except over the thin-shell where the trace of the extrinsic curvature jump is \(\kappa \) and, for Type I spacetimes only, over the central axes of the geometry in case of conical interiors. Type I geometries have deficit angle interior and exterior regions, characterized by parameters \(\omega _i\) and \(\omega _e\) respectively, with \(\kappa = (\omega _e - \omega _i)/\rho (r_s)\), where \(\rho (r_s)\) is the metric’s profile function at the position of the shell. Type II wormhole spacetimes, with exterior regions of parameters \(\omega _-\) and \(\omega _+\) respectively, have \(\kappa = (\omega _- + \omega _+)/\rho (r_s)\). We found the critical values \(\xi _c^{(n)} = n/\left( \rho (r_s)\,\kappa \right) \), with \(n \in {\mathbb {N}}\), for which the coupled to curvature scalar field is unstable in the background configuration. For a Type I geometry with an ordinary matter thin-shell (\(\kappa <0\)) the safety domain of the coupling is \(\xi > \xi _c^{(n=1)}\), while for exotic matter shells (\(\kappa >0\)), either in geometries of Type I or II, the field is stable if the coupling takes values in the range \(\xi < \xi _c^{(n=1)}\). These results add to those in [30, 31, 32, 33] (mentioned in the introduction) relative to the stable domain of solutions for a scalar field coupled to spherically symmetric wormhole backgrounds but, in our case, studying spacetimes with cylindrical symmetry and in trivial topologies as well.

The only force over the charged particle in these geometries is the scalar self-force, which we obtained from the regularization of its own scalar field. The sign of the asymptotic force does not depend on the coupling nor on the topological difference between Type I and Type II, it only depends on whether there is an angle deficit or not on the external region where the charge is placed; conical asymptotics produce a leading attractive force, while Minkowski regions produce a repulsive asymptotic self-force. This clarify some aspects of the self-force in terms of global properties of the given background geometry. In thin-shell spherical wormholes the scalar self-force of a massless particle changes sign at the value \(\xi =1/8\) (corresponding to the conformal flatness of the 3D section of the constant time manifold, see [30, 31, 32, 33]), in our cylindrical geometries there is no value of \(\xi \) for which this occurs globally and the asymptotic sign of the force only depends on the conicity or not of the external region. On the other hand, there is a relevant local influence of the coupling over the self-force at the vicinities of the shell of matter. The specific coupling \(\xi =1/4\) was identified as the value for which the scalar force changes sign at a neighborhood of the shell; if \(\kappa (1-4\xi )>0\) the shell acts repulsively as an effective potential barrier, while if \(\kappa (1-4\xi )<0\) it attracts the charge as a potential well. Finally we note that beyond the stable domain of the curvature coupling, in the intervals \(\xi \in (\xi _c^{(n)}; \xi _c^{(n+1)})\) between two successive critical values, the self-force shows qualitatively the same behavior as the one produced with couplings in some interval of the safety range; this refers, precisely, to the interval \(\xi _c^{(n=1)}< \xi < 0\) in cases with \(\kappa < 0\), or to the interval \(1/4< \xi < \xi _c^{(n=1)}\) in cases with \(\kappa >0\).

## Footnotes

- 1.The axis at \(r=0\) of the conical geometry, where the infinitely thin cosmic string would be placed, is intentionally excluded. The field in the manifold which includes the axis couples to the Ricci scalar \(R(r) = -\, 2 (\omega - 1) \delta (r)/(\omega r)\) at \(r=0\) and it can be shown to be \(\Phi = \Phi _{\omega } + \Phi _{\xi }\), with where the term in brackets becomes identically null,
*i.e.*the complete solution which accounts for the coupling at the conical peak does not affect the relevant part of the field [38]. - 2.
The value \(\xi =1/4\) is known as the coupling that eliminates the Robin boundary energy for a scalar field [41].

## Notes

### Acknowledgements

This work was supported by the National Scientific and Technical Research Council of Argentina.

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