# Arbitrarily coupled massive scalar field in conical thin-shell spacetimes

## Abstract

We study the massive scalar field of a charged particle held at rest in conical thin-shell spacetimes with one or two asymptotic regions. Resonant and stable scalar Green’s functions are characterized in terms of the coupling of the field to the trace of the extrinsic curvature jump at the shell. Stable coupling values, within the safety domain of the configuration parameters, are used to analyze the self-force of static point charges.

## 1 Introduction

Despite their mathematical simplicity, conical spacetimes are of physical interest because they are associated to astrophysical objects which may have appeared in phase transitions in the early Universe and could have played a role in processes of cosmological relevance. Spontaneous symmetry breaking in a system including a complex scalar field coupled to a gauge field would lead to the formation of practically one-dimensional topological defects called gauge cosmic strings or local cosmic strings [1]. In their most simple configuration, these strings are straight, and they can be described by the approximate energy-momentum tensor \({\tilde{T}}_\mu ^{\, \nu }= \delta (x)\delta (y) \text{ diag }(\mathcal {E},0,0,\mathcal {E})\), where \(\mathcal {E}\) is the linear mass density [2]. This density is determined by the energy scale at the moment of the symmetry breaking process. From the point of view of the gravitational effects, the equality of the only two non vanishing components of the energy-momentum tensor has the consequence of a metric with \(g_{00}=1\), i.e. a null Newtonian force; thus gauge strings could not be detected by their gravitational attraction on rest or non relativistic particles^{1}. However, though locally flat, the geometry induced by a straight gauge cosmic string is conical, that is, it presents a deficit angle around its symmetry axis: the line element reads \(ds^2=-dt^2+dr^2+(1-8G\mathcal {E})r^2d\theta ^2+dz^2\) (see [2] and also [4]). As a result of this deficit angle, a gauge string deflects light and relativistic massive particles. Then such string would act as a gravitational lens which generates double images, and if moving at relativistic speed through a dust cloud or a gas it would generate a matter wake [5, 6, 7]. The interest on cosmic strings has been mainly driven by the prediction of these observable effects. In particular, they have been considered as candidates for the generation of density fluctuations leading to structure formation; while in the last years they have been discarded as the main source for such cosmological process, in the present framework they are considered as possible secondary seeds for structure formation [8].

While the absence of effects on rest particles could seem to limit the relevance of conical spacetimes, this aspect radically changes when tests involving charged particles are considered. It is well known that, in a curved background, the electric field of a rest point charge is not spherically symmetric and this leads to the existence of an electrostatic self-force on the charge; this is so because though one can always choose a freely falling frame were the Maxwell equations for flat space hold locally, they cannot globally admit the Coulomb solution. In particular, then, while a conical spacetime is locally flat, the existence of a deficit angle globally rules out the symmetric electric field solution for a point charge; hence an electrostatic self-force exists and a rest charge would develop an acceleration in such a background [9]. Besides, and this point is of central importance within our analysis, because the electric field probes the global aspects of a geometry it also allows to discern between two conical spaces which are identical at the position of the point charge but differ globally. For instance, as shown in Refs. [10, 11], one could use the self-force on a charge to identify different possible interior sources behind a shell supporting a conical geometry, or could even detect the existence of a thin-shell wormhole [12] connecting two conical spacetimes (see for instance Refs. [13, 14]), without the necessity of traveling across the wormhole throat. Note that wormholes of this class are supported by thin matter layers located at the throat; hence the exotic^{2} matter required for the existence of such geometries is confined at a finite radius and no direct evidence of it is available at the position of the observer using the charge as a test for the spacetime properties. Since the local curvature vanishes in these conical spacetimes, the self-force has its origin in the boundary conditions on the thin shell and infinity. The program of computing the electric self-force as a probe of the global properties of spacetime was continued in spherical thin-shell wormholes as well as in thin-shell closed universes (see for instance Refs. [18] and [19]).

The analysis of Refs. [10, 11] was recently extended [20] to address the problem of the self-force on a scalar point charge within the theory of a massless scalar field. Scalar fields are considered of interest in modern cosmology, both within the study of the early Universe and, after the first proposals relating the apparently necessary \(\Lambda \) term to a scalar field (see for instance Ref. [21]), also to understand the present day observations showing an accelerated expansion. Scalar fields with non-minimal coupling \(\xi \) to gravity have been included in inflationary models motivated in the early works [22, 23, 24] on the observation that the problem of an exceedingly small quartic self-coupling parameter, dictated by the amplitude of primordial scalar (density) perturbations [25], can be solved by using a non-minimally coupled inflaton with a large value of \(\xi \). From a different point of view, a fundamental aspect of the motivation for the study of non-minimally coupled scalar field is determining the different way in which it behaves in curved geometries, for example, with thin-shell matter sources; while in Maxwell theory the field of a charge is continuous across a non charged matter layer inducing a discontinuity of the extrinsic curvature, for a scalar charge the presence of an infinitely thin matter distribution generates what can be seen as new surface sources for the corresponding coupled scalar field. Thus one cannot, in principle, expect the scalar self-force to behave as the electrostatic self-force. The scalar self-force on a static particle is a problem recently addressed in a large class of spherically symmetric spacetimes [26, 27, 28, 29]. Much of this work was done in wormhole spacetimes, where some authors [30, 31, 32] noted that the self-force diverges for an infinite set of values of the curvature coupling \(\xi \). Bezerra and Khusnutdinov [33] analyzed the anomalous behavior through an analogous problem in scattering theory by defining an effective potential in a non-relativistic quantum mechanical problem and identifying the poles of the self-interaction with the bound states for the wave function. The study of Ref. [20] in cylindrical spacetimes was performed in terms of the coupling of a massless scalar field, and critical values of this constant for which the scalar field is unstable in the background configuration were identified. In addition, the aforementioned analogy proposed by Bezerra and Khusnutdinov was particularly useful in the latter work to understand the role of the coupling \(\xi = 1/4\), which cancels out the delta-like effective potential at the shell in cylindrically symmetric spacetimes, and for which the self-force changes its sign at the vicinities of the shell.

In the present work we generalize our previous results to the case of a massive scalar field. Such an extension is interesting for both formal and physical aspects. In particular, we well find that, for all other parameters fixed, a greater mass for the scalar field increases the domain of stability in terms of the coupling constant (see Sect. 3.3). From a physical point of view, quantitatively different results are to be expected. This can already be inferred from the behavior of the field of a point charge in the simplest situation of a flat spacetime (consider the field equation in Sect. 2 for the case \(R=0\), \(\sqrt{-g}=1\)): while for a massless field the solution is of the form 1 / *r*, for a field of mass *m* the solution takes the form \(e^{-mr}/r\). Thus, in comparison with the massless case, a more rapidly decaying field is expected, and consequently a weaker self-force. In what follows, the problem is studied analytically and numerically in terms of the coupling constant for different configurations of the background spacetime. The general formulation of the problem, as well as the treatment of the anomalous behavior pointed above, now for the massive case, are presented in Sect. 3 in the two types of geometries considered, that is, those associated to cylindrical shells joining an inner to an outer region, and wormholes of the thin-shell class. The self-interaction on scalar point charges is developed in Sect. 4, and the self-force is studied in Sects. 4.1 and 4.2 for each type of geometry. Throughout the article the geometrized unit system is used where \(c=G=1\).

## 2 Approach: massive scalar field coupled to curvature and scalar particle

*R*,

^{3}\(m_{0}\) over its world line \( \gamma \) and

*q*. The field equation for a massive scalar field coupled to gravity is derived requiring the action to be stationary under variations \(\delta \Phi (x)\). With a scalar charge

*q*at rest the inhomogeneous equation is:

*S*to be stationary under variations \(\delta y^{\alpha }(\tau )\) of the world line, i.e.

## 3 Green’s function for the massive scalar field in conical spacetimes

^{4}which yields the following radial equation

*n*-mode and

*k*-eigenvalue of the Fourier expansion. Integrating (14) over an infinitesimal radial interval around the position of the shell \(r=r_s\), we obtain

*z*line axis, or infinity, over the corresponding regions in Type I and Type II geometries.

### 3.1 Scalar Green’s function in Type I spacetimes

*G*in Type I spacetimes we define the \(\textit{internal}\) and \(\textit{external}\) functions as

*n*and

*k*. The boundary conditions at the axis and infinity are

### 3.2 Scalar Green’s function in Type II spacetimes

*G*as

*n*and

*k*. The boundary conditions are

### 3.3 Stability regions and resonant configurations

*k*appears in \(\mu = \sqrt{k^2 + m^2}\). This means that a Fourier integral in

*dk*for the \(n^{th}\) mode of the Green’s function presents a pole at \(k=k_p>0\) if \(\xi = \xi _p^{(n)}\). Nevertheless, the integral can be performed by splitting it at \(k_p \mp \epsilon \), with \(\epsilon \rightarrow 0\), canceling out the contributions of the lateral limits in the positive real line integral or, equivalently, using contour integrals for the complex-valued coefficient function to obtain the positive real half-line integral. On the other hand, a mode is divergent if the denominator of the coefficient in its integrand is null for \(k = 0\). Then, if a specific value of the coupling makes the \(n^{th}\) coefficient divergent at \(k=0\), the \(n^{th}\) mode in (13) is resonant and the massive coupled scalar field in the given conical thin-shell background has an unstable configuration. The critical coupling \(\xi _c^{(n)} = \xi _p^{(n)} \big |_{k=0}\) that generates a resonant \(n^{th}\) mode is:

*k*for the Fourier integral modes. Moreover, to analyze the dependence of the self-force in terms of the coupling constant we would like to go over values of \(\xi \) in an interval where we do not come upon resonant configurations. To find ranges of the coupling constant where no resonant configuration or poles are found, we will look for safety domains in the parameter space of the configuration.

Figure 1 shows the plot of \(\xi _{p}^{(n)}\) against \(r_{i}\mu \) for Type I spacetimes; in the example of Fig. 1a the trace of the extrinsic curvature jump \(\kappa \) is negative, and in Fig. 1b \(\kappa \) is positive. Each point in a curve \(\xi _{p}^{(n)}\) represents a divergence of the \(n^{th}\) coefficient for the corresponding values of \(\xi \) and \(r_i \mu \) at eigenvalues *k*. To analyze a particular configuration one must fix the product \(m r_i\), and look for the critical couplings at the intersection of the curves with the vertical line \({r_i \mu }|_{k=0} = m r_i\). To the right side of this line the curves represent values of the coupling for which poles appear in the corresponding *n* coefficient for some eigenvalue \(k>0\). In Fig. 1a a safety domain is painted in grey for a Type I geometry with \(\omega _i=1\), \(\omega _e=0.9\) (\(\kappa <0\)) and \(m r_i=1\). For this configuration, \(\xi _{c}^{(0)}|_{m r_i =1} \simeq -9.2\) is the greatest critical coupling, thus \(\xi > \xi _{c}^{(0)}|_{m r_i =1}\) is a stable range where no resonant configurations are encountered. There is no intersection of the curves \(\xi _{p}^{(n)}\) with the grey region; this ensures that while increasing the eigenvalue *k*, with fixed \(m r_i=1\), no poles appear for any coefficient *n* in the safety domain. Figure 1b shows, in grey, a safety domain for a Type I geometry with \(\omega _i=0.9\) and \(\omega _e=1\) (\(\kappa >0\)), where we chose again \(m r_i=1\). In this case \(\xi _{c}^{(0)}|_{m r_i =1} \simeq 8.6\) is the smallest critical coupling and \(\xi < \xi _{c}^{(0)}|_{m r_i =1}\) is the stable region for configurations with positive \(\kappa \) thin-shells.

## 4 Scalar charge self-interaction

### 4.1 Self-force in Type I geometries

*f*over the scalar particle in a Type I geometry with \( \omega = \omega _{i}\) in the interior region and \(\omega = \omega _{e}\) in the outer one is given by

Finally, Fig. 4 shows the dependance of the force on the mass by fixing different values of the product \(r_i m\). We observe that the self-force is attenuated with increasing \(r_i m\) at both sides of the shell. In every case the self-force decreases with increasing mass as expected from the exponential decay \(e^{-mr}/r\) in the general solution of the field in these conical geometries. Although this specific singular term is subtracted as seen from Eq. (50) in the regularization procedure, traces of the same dependance are found in the regular term \(G_{\omega }^{reg}\), where \(K_{1/2}(m D(r))/\sqrt{r} \sim e^{-mr}/r\). Of greater difficulty is interpreting exactly the same factor in the contribution \(G_{\xi }\) to the Green’s function. We note that in regions where \(\omega =1\), the derivative of the latter is the only term contributing to the self-force, and the same kind of decay and attenuation with increasing \(r_i m\) is observed, as shown in the inner region of Figs. 4a and b.

### 4.2 Self-force in Type II geometries

Figure 5a presents examples in a flat wormhole where \(r_0 m = 1\). For minimal coupling the self-force is repulsive from the thin-shell throat as it was in the previous cases (Type I spacetimes) with positive \(\kappa \). Negative values of the product \(\xi \kappa \) produce an increased repulsion, while a positive product \(\xi \kappa \) contributes with an attractive force towards the shell. At the specific value \(\xi = 1/4\) the force changes sign and an attractive force at the vicinities of the throat is observed for greater couplings. Figure 5b displays the self-force curves for different conical wormholes with \(r_0 m =0.1\) and \(\xi =0.1\) fixed. In all the cases the force is repulsive in the vicinity of the throat and becomes asymptotically attractive if \(\omega <1\). Figure 6 shows that, just as in Type I geometries, the self-force is attenuated as \(r_0 m\) increases.

## 5 Summary and conclusions

The arbitrarily coupled massive scalar field of a charged particle held at rest in conical thin-shell backgrounds was solved to characterize stable and resonant solutions in terms of the configuration parameters. We have determined a safety domain for the scalar Green’s function and have applied it in the calculation of the self-interaction on a scalar charge, obtained from a regularized field at the position of the particle. The background geometries are locally flat except at the cylindrical shell,^{5} so the field is coupled by \(\xi \) to the trace of the extrinsic curvature jump \(\kappa \). Two types of spacetimes were used to study the problem. Type I are conical geometries with the shell separating an interior from an exterior region characterized by negative or positive \(\kappa \) (ordinary or exotic matter, respectively, at the shell). Type II are cylindrical wormholes with a thin throat of positive \(\kappa \) separating different asymptotic regions.

Resonances and poles were identified in the Green’s function for the massive scalar field. Critical values \(\xi _c^{(n)}\) of the coupling constant were found for which the \(n^{th}\) mode of the Green’s function is resonant. For given spacetime parameters and mass *m* of the field, we determined a safety domain in terms of the coupling where no resonances or poles appear. The latter is the stability domain of the solutions and restricts the allowed values of the coupling constant for scalar field models in the considered backgrounds, i.e., Type I geometries with flat or conical asymptotics, and Type II wormhole spacetimes. As a general rule, \(\xi > \xi _c^{(0)}\) is a stable range if \(\kappa <0\), while \(\xi < \xi _c^{(0)}\) are stable if \(\kappa >0\). We found that the critical couplings for masses much greater than the reciprocal radius of the shell are \(\xi _c^{(n)} \sim m/\kappa \). Therefore, the stability regions can be enlarged, keeping the spacetime parameters fixed, by increasing the mass of the field. On the other hand, the stable range decreases for smaller masses. If \(m \rightarrow 0\) then \(\xi _c^{(0)} \rightarrow 0\), and the safety domain takes only positive values of \(\xi \) if \(\kappa <0\), or only negative values if \(\kappa >0\). This last result coincides with the critical couplings found on a previous work were we used a massless scalar field in conical backgrounds to test the self-force on a charged scalar particle [20]. Despite this, we note that in the mentioned article some part of the analysis was performed outside the safety domain.

Stable couplings within the safety domain were used to study the self-force problem for a static scalar charge in Type I and Type II geometries. The self-force is the only force on the particle in these locally flat spacetimes. Analyzing the problem in these simple backgrounds of the thin-shell type allows to isolate the effects of coupling to gravity at the shell and interprete how curvature affects the self-interaction directly from \(\kappa \). At minimal coupling the force is attractive to the shell in its vicinity if \(\kappa <0\), it is repulsive if \(\kappa >0\), and diverges as the particle gets closer to it. As a general feature for non-minimal coupling we have observed that a negative product \(\xi \kappa \) contributes with a repulsive force from the shell’s position, and a positive value of \(\xi \kappa \) contributes with an attraction towards the shell. The coupling \(\xi =1/4\), known for canceling out the delta-like effective potential at the shell for the radial solutions of the field [20], is manifested in the present work, i.e., with a negative (positive) \(\kappa \) the force is attractive (repulsive) as the particle approaches the shell if \(\xi <1/4\), and changes to a repulsion (an attraction) for \(\xi >1/4\). In all cases the absolute value of the self-force decreases with increasing mass as it was expected from the exponential decay \(e^{-mr}/r\) in the general solution of the field in conical geometries. The same behavior was obtained in [33] in the case of a massive scalar field coupled to a spherically symmetric wormhole spacetime with infinitely thin throat. In our cases, we observe that the field and force are localized at both sides of the shell either for Type I or Type II spacetimes.

## Footnotes

- 1.
The existence of small structures, as wiggles and kinks along the string, would modify this aspect, leading to \(g_{00}\ne 1\); see for instance Ref. [3].

- 2.
- 3.
The bare mass \(m_0\) must not be confused with a particular value of the field mass

*m*: for example, the latter could be zero if a massless scalar field is considered, while the point particle mass \(m_0\) is always non null. - 4.
The Kronecker delta \(\delta _{0,n}\) accounts for the 1 / 2 normalization factor of the zero mode with respect to the other base functions in the angular coordinate \(\theta \).

- 5.
And at the central axis in case of a conical interior.

## Notes

### Acknowledgements

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

## References

- 1.A. Vilenkin, E.P.S. Shellard,
*Cosmic Strings and Other Topological Defects*(Cambridge University Press, Cambridge, 1994)zbMATHGoogle Scholar - 2.A. Vilenkin, Phys. Rev. D
**23**, 4 (1981)CrossRefGoogle Scholar - 3.T. Vachaspati, A. Vilenkin, Phys. Rev. Lett.
**67**, 1057 (1991)ADSCrossRefGoogle Scholar - 4.W.A. Hiscock, Phys. Rev. D
**31**, 3288 (1985)ADSMathSciNetCrossRefGoogle Scholar - 5.A. Vilenkin, Phys. Rev. Lett.
**46**, 1169 (1981). [Erratum-ibid. 46, 1496 (1981)]ADSCrossRefGoogle Scholar - 6.N. Turok, R.H. Brandenberger, Phys. Rev. D
**33**, 2175 (1986)ADSCrossRefGoogle Scholar - 7.H. Sato, Prog. Theor. Phys.
**75**, 1342 (1986)ADSCrossRefGoogle Scholar - 8.R.J. Danos, R.H. Brandenberger, G. Holder, Phys. Rev. D
**82**, 023513 (2010)ADSCrossRefGoogle Scholar - 9.B. Linet, Phys. Rev. D
**33**, 1833 (1986)ADSCrossRefGoogle Scholar - 10.E. Rubín de Celis, O.P. Santillán, C. Simeone, Phys. Rev. D
**86**, 124009 (2012)ADSCrossRefGoogle Scholar - 11.E. Rubín de Celis, Eur. Phys. J. C
**76**, 92 (2016)ADSCrossRefGoogle Scholar - 12.M. Visser,
*Lorentzian Wormholes*(AIP Press, New York, 1996)Google Scholar - 13.E.F. Eiroa, C. Simeone, Phys. Rev. D
**70**, 4 (2004)CrossRefGoogle Scholar - 14.S. Danial Forghani, S. Habib Mazharimousavi, M. Halilsoy, (2018). arXiv:1807.05080 [gr-qc]
- 15.M.G. Richarte, C. Simeone, Phys. Rev. D
**76**, 087502 (2007). Erratum: Phys. Rev. D**77**, 089903 (2008)ADSCrossRefGoogle Scholar - 16.H. Maeda, M. Nozawa, Phys. Rev. D
**78**, 024005 (2008)ADSMathSciNetCrossRefGoogle Scholar - 17.C. Simeone, Phys. Rev. D
**83**, 087503 (2011)ADSCrossRefGoogle Scholar - 18.E. Rubín de Celis, O.P. Santillán, C. Simeone, Phys. Rev. D
**88**, 124012 (2013)ADSCrossRefGoogle Scholar - 19.K. Davidson, E. Poisson, Phys. Rev. D
**97**, 104030 (2018)ADSMathSciNetCrossRefGoogle Scholar - 20.M.C. Tomasini, E. Rubín de Celis, C. Simeone, Eur. Phys. J. C
**78**(2), 149 (2018)ADSCrossRefGoogle Scholar - 21.V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D
**9**, 373 (2000)ADSGoogle Scholar - 22.B.L. Spokoiny, Phys. Lett. B
**147**, 39 (1984)ADSCrossRefGoogle Scholar - 23.D.S. Salopek, J.R. Bond, J.M. Bardeen, Phys. Rev. D
**40**, 1753 (1989)ADSCrossRefGoogle Scholar - 24.R. Fakir, W.G. Unruh, Phys. Rev. D
**41**, 1783 (1990)ADSCrossRefGoogle Scholar - 25.S.W. Hawking, Phys. Lett.
**115B**, 295 (1982)ADSCrossRefGoogle Scholar - 26.E.R. Bezerra de Mello, A.A. Saharian, Class. Quant. Grav.
**29**, 135007 (2012)ADSCrossRefGoogle Scholar - 27.A.A. Popov, Phys. Lett. B
**693**, 180–183 (2010)ADSCrossRefGoogle Scholar - 28.A.A. Popov, O. Aslan, Int. J. Mod. Phys. A
**30**(22), 1550143 (2015)ADSCrossRefGoogle Scholar - 29.A.A. Popov, O. Aslan, Int. J. Geom. Meth. Mod. Phys.
**15**(03), 1850050 (2017)CrossRefGoogle Scholar - 30.P. Taylor, Phys. Rev. D
**87**, 2 (2013)Google Scholar - 31.P. Taylor, Phys. Rev. D
**90**, 2 (2014)Google Scholar - 32.P. Taylor, Phys. Rev. D
**95**(10), 109904(E) (2017)ADSCrossRefGoogle Scholar - 33.V.B. Bezerra, N.R. Khusnutdinov, Phys. Rev. D
**79**, 064012 (2009)ADSMathSciNetCrossRefGoogle Scholar - 34.E.F. Eiroa, E. Rubín de Celis, C. Simeone, Eur. Phys. J. C
**76**, 10 (2016)CrossRefGoogle Scholar - 35.E. Rubín de Celis, C. Simeone, Eur. Phys. J. Plus
**132**, 3 (2017)CrossRefGoogle Scholar - 36.E. Poisson, A. Pound, I. Vega, Liv. Rev. Relat.
**14**, 7 (2011). http://www.livingreviews.org/lrr-2011-7 - 37.S. Detweiler, B.F. Whiting, Phys. Rev. D
**67**, 024025 (2003)ADSCrossRefGoogle Scholar - 38.M.E.X. Guimarães, B. Linet, Commun. Math. Phys.
**165**, 297 (1994)ADSCrossRefGoogle Scholar

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