Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Induced Brownian motion by the Friedmann–Robertson–Walker spacetime in the presence of a cosmic string


In this paper, we investigate the quantum Brownian motion of a massive scalar point particle induced by the FRW spacetime in the presence of a linear topological defect named cosmic string. In addition, we also consider a flat boundary orthogonal to the defect to analyze its effect on the particle’s motion. For both cases, we found exact expressions for the renormalized mean square deviation of the particle velocity, the quantity that measures the induced Brownian motion, and obtain asymptotic expressions when the point particle is near and far away from the cosmic string and the boundary. Furthermore, in both cases, there appear divergencies in the mean square deviation of the particle velocity coming from a time interval that correspond to a round trip of the massive point particle between its position and the cosmic string/flat boundary. The number of divergencies depends upon the radial position of the particle and the parameter associated with the cosmic string in the case without boundary, and upon this parameter and the radial and z positions of the particle in the case with boundary.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    V.M. Mostepanenko, N.N. Trunov, The Casimir effect and its applications (Clarendon, Oxford, 1997), p. 199

  2. 2.

    M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, Advances in the Casimir effect, vol. 145 (OUP, Oxford, 2009)

  3. 3.

    K.A. Milton, The Casimir effect: Physical manifestations of zero-point energy (World Scientific, River Edge, 2001), p. 301

  4. 4.

    L.H. Ford, Gravitons and light cone fluctuations. Phys. Rev. D 51, 1692–1700 (1995). [arXiv: gr-qc/9410047]

  5. 5.

    H.F. Mota, E.R. Bezerra de Mello, C.H.G. Bessa, V.B. Bezerra, Light-cone fluctuations in the cosmic string spacetime. Phys. Rev. D 94, 024039 (2016). [arXiv:1606.0177]

  6. 6.

    G.H.S. Camargo, V.A. De Lorenci, C.C.H. Ribeiro, F.F. Rodrigues, M.M. Silva, Vacuum fluctuations of a scalar field near a reflecting boundary and their effects on the motion of a test particle. JHEP 07, 173 (2018). [arXiv:1709.1039]

  7. 7.

    G. Gour, L. Sriramkumar, Will small particles exhibit Brownian motion in the quantum vacuum? Found. Phys. 29, 1917–1949 (1999). [arXiv: quant-ph/9808032]

  8. 8.

    L.H. Ford, Stochastic spacetime and Brownian motion of test particles. Int. J. Theor. Phys. 44, 1753–1768 (2005). [arXiv: gr-qc/0501081]

  9. 9.

    C.H.G. Bessa, V.B. Bezerra, L.H. Ford, Brownian Motion in Robertson–Walker space-times from electromagnetic vacuum fluctuations. J. Math. Phys. 50, 062501 (2009). [arXiv:0804.1360]

  10. 10.

    H.-W. Yu, J. Chen, Brownian motion of a charged test particle in vacuum between two conducting plates. Phys. Rev. D 70, 125006 (2004). [arXiv: quant-ph/0412010]

  11. 11.

    H.-W. Yu, L.H. Ford, Vacuum fluctuations and Brownian motion of a charged test particle near a reflecting boundary. Phys. Rev. D 70, 065009 (2004). [arXiv: quant-ph/0406122]

  12. 12.

    V.A. De Lorenci, E.S. Moreira Jr., M.M. Silva, Quantum Brownian motion near a point-like reflecting boundary. Phys. Rev. D 90(2), 027702 (2014). [arXiv:1404.3115]

  13. 13.

    L.M. Burko, Selfforce on static charges in Schwarzschild space-time. Class. Quant. Grav. 17, 227–250 (2000). [arXiv: gr-qc/9911042]

  14. 14.

    A.G. Wiseman, The Selfforce on a static scalar test charge outside a Schwarzschild black hole. Phys. Rev. D 61, 084014 (2000). [arXiv: gr-qc/0001025]

  15. 15.

    E.R. Bezerra de Mello, A.A. Saharian, Scalar self-energy for a charged particle in global monopole spacetime with a spherical boundary. Class. Quant. Grav. 29, 135007 (2012). [arXiv:1201.1770]

  16. 16.

    L.E. Reichl, A modern course in statistical physics, 4th edn. (Wiley-VCH, Weinheim, 2016)

  17. 17.

    Das, A., Dalui, S., Chowdhury, C., Majhi, B.R.: Conformal vacuum and fluctuation-dissipation in de-Sitter Universe and Black Hole Spacetimes. arXiv:1902.0373

  18. 18.

    A. Adhikari, K. Bhattacharya, C. Chowdhury, B.R. Majhi, Fluctuation-dissipation relation in accelerated frames. Phys. Rev. D 97, 045003 (2018). [arXiv:1707.0133]

  19. 19.

    Planck Collaboration, P.A.R. Ade et al., Planck 2015 results—XVIII. Background geometry and topology of the Universe, Astron. Astrophys.594, A18 (2016) [arXiv:1502.0159]

  20. 20.

    S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008)

  21. 21.

    M. Hindmarsh, T. Kibble, Cosmic strings. Rept. Prog. Phys. 58, 477–562 (1995). [arXiv: hep-ph/9411342]

  22. 22.

    A. Vilenkin, E.P.S. Shellard, Cosmic strings and other topological defects. Cambridge monographs on mathematical physics (Cambridge Univ. Press, Cambridge, 1994)

  23. 23.

    E.J. Copeland, L. Pogosian, T. Vachaspati, Seeking string theory in the cosmos. Class. Quant. Grav. 28, 204009 (2011). [arXiv:1105.0207]

  24. 24.

    M. Hindmarsh, Signals of inflationary models with cosmic strings. Prog. Theor. Phys. Suppl. 190, 197–228 (2011). [arXiv:1106.0391]

  25. 25.

    H.F. Santana Mota, M. Hindmarsh, Big-bang nucleosynthesis and gamma-ray constraints on cosmic strings with a large Higgs condensate. Phys. Rev. D 91(4), 043001 (2015). [arXiv:1407.3599]

  26. 26.

    V.A. Gasilov, V.I. Masliankin, M.I. Khlopov, Gas-dynamic effects of cosmic strings. Astrofizika 23, 191–201 (1985)

  27. 27.

    N.D. Birrell, P.C.W. Davies, Quantum fields in curved space (Cambridge University Press, Cambridge, 1984)

  28. 28.

    H.F. Mota, E.R. Bezerra de Mello, K. Bakke, Scalar Casimir effect in a high-dimensional cosmic dispiration spacetime. Int. J. Mod. Phys. D 27(12), 1850107 (2018). [arXiv:1704.0186]

  29. 29.

    A. G. Smith, On the evolution of cosmic strings, in The formation and evolution of cosmic strings : proceedings of a workshop supported by the SERC and held in Cambridge, 3-7 July, 1989 (Gibbons, G.W., Hawking, S.W., Vachaspati, T. eds.), (Cambridge), p. 263, Cambridge University Press (1990)

  30. 30.

    E. Poisson, A. Pound, I. Vega, The Motion of point particles in curved spacetime. Living Rev. Rel. 14, 7 (2011). [arXiv:1102.0529]

  31. 31.

    T.C. Quinn, Axiomatic approach to radiation reaction of scalar point particles in curved space-time. Phys. Rev. D 62, 064029 (2000). [arXiv: gr-qc/0005030]

  32. 32.

    C.H.G. Bessa, V.B. Bezerra, E.R. Bezerra de Mello, H.F. Mota, Quantum Brownian motion in an analog Friedmann-Robertson-Walker geometry. Phys. Rev. D 95(8), 085020 (2017). [arXiv:1703.0652]

  33. 33.

    Planck Collaboration, P.A.R. Ade et al., Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, Astron. Astrophys. 571, A25, (2014) [arXiv:1303.5085]

  34. 34.

    P. Jain, S. Weinfurtner, M. Visser, C.W. Gardiner, Analogue model of a FRW universe in Bose–Einstein condensates: Application of the classical field method. Phys. Rev. A 76, 033616 (2007). [arXiv:0705.2077]

  35. 35.

    E. Bezerra de Mello, A. Saharian, Vacuum polarization by a flat boundary in cosmic string spacetime. Class. Quant. Grav. 28, 145008 (2011). [arXiv:1103.2550]

  36. 36.

    E.R. Bezerra de Mello, V.B. Bezerra, A.A. Saharian, H.H. Harutyunyan, Vacuum currents induced by a magnetic flux around a cosmic string with finite core. Phys. Rev. D 91(6), 064034 (2015). [arXiv:1411.1258]

Download references


H.F.S.M and E.R.B.M are partially supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil) under grants 305379/2017-8 and 313137/2014-5, respectively.

Author information

Correspondence to Herondy Francisco Santana Mota.

Appendix A: Wightman function

Appendix A: Wightman function

Let us here derive some important expressions which are necessary for the results obtained in the body of the text. The positive-energy Wightman function resulting from the solution (6), obeying Dirichlet boundary condition, can be written in the form [5, 35]:

$$\begin{aligned} W^{(\mathrm \pm )}(x,x') = \pm \frac{p}{8\pi ^2\rho \rho '}\int _{0}^{\infty }\mathrm{d}we^{-\frac{\zeta _{({\mp })}}{2\rho \rho '}w}\sum _{n=-\infty }^{\infty }e^{inp\Delta \phi }I_{p|n|}(w), \end{aligned}$$

where the ‘plus’ sign in \(W^{(\mathrm \pm )}(x,x')\) indicates the Wightman function due to only the cosmic string spacetime and the ‘minus’ sign indicates the Wightman function due to only a flat boundary [35] and:

$$\begin{aligned} \zeta _{({\mp })} = - \Delta \eta ^2 + (z{\mp } z')^2 + \rho ^2 + \rho '^2 . \end{aligned}$$

Therefore, the total renormalized Wightman function associated with a massless scalar field in the cosmic string spacetime obeying the Dirichlet boundary condition on the flat boundary is given by:

$$\begin{aligned} W_{\text {r}}(x,x') = W_{\text {r}}^{(+)}(x,x') + W^{(-)}(x,x'), \end{aligned}$$

where \(W_{\text {r}}^{(+)}(x,x') \) is the renormalized Wightman function purely due to the cosmic string topology, i.e., the Wightman function with the Minkowski spacetime divergent contribution already subtracted. Also, \(W^{(-)}(x,x')\) is the Wightman function induced by the plain boundary. Note that the Neumann boundary condition would only change the plus sign in Eq. (A3) by the minus sign.

Moreover, in Refs. [5, 36], a summation formula to perform the sum in n present in Eq. (A1) was derived and is given by:

$$\begin{aligned} \sum _{n=-\infty }^{\infty }e^{ipn\Delta \phi }I_{p|n|}(w)= & {} \frac{e^w}{p} + \frac{2}{p}\mathop {{\sum }'}\limits _{n=1}^{[q/2]}e^{w\cos \left( \frac{2\pi n}{p}-\Delta \phi \right) }\nonumber \\&- \frac{1}{2\pi }\sum _{j=+,-}\int _{0}^{\infty }d\xi \frac{\sin \left[ p\left( j\Delta \phi + \pi \right) \right] e^{-w\cosh (\xi )}}{[\cosh (p\xi ) - \cos (jp\Delta \phi + p\pi )]}, \end{aligned}$$

where [p / 2] represents the integer part of p / 2, and the prime on the sign of summation means that in the case p is an integer number, the term \(n=p/2\) should be taken with the coefficient 1 / 2. Note that, if \(p<2\) the summation contribution should be omitted. Thereby, using (A4), the Wightman function is obtained as [5, 35]:

$$\begin{aligned} W^{(\mathrm \pm )}(x,x')= & {} \pm \frac{1}{4\pi ^2}\frac{1}{\sigma _0^{({\mp })}} \pm \frac{1}{2\pi ^2}\mathop {{\sum }'}\limits _{n=1}^{[p/2]}\frac{1}{\sigma _n^{({\mp })}}\nonumber \\&{\mp }\frac{q}{8\pi ^3}\sum _{j=+,-}\int _{0}^{\infty }d\xi \frac{\sin \left[ p\left( j\Delta \phi + \pi \right) \right] }{[\cosh (p\xi ) - \cos (jp\Delta \phi + p\pi )]}\frac{1}{\sigma _\xi ^{({\mp })}}, \end{aligned}$$


$$\begin{aligned} \begin{aligned} \sigma _0^{({\mp })}&= \zeta _{({\mp })} - 2\rho \rho '\cos (\Delta \phi ),\\ \sigma _n^{({\mp })}&= \zeta _{({\mp })} - 2\rho \rho '\cos \left( \frac{2\pi n}{p}-\Delta \phi \right) ,\\ \sigma _{\xi }^{({\mp })}&= \zeta _{({\mp })} + 2\rho \rho '\cosh (\xi ). \end{aligned} \end{aligned}$$

Another important relation to calculate the azimuthal component of the dispersion in the particle velocity is obtained by taking the derivatives in \(\phi \) and \(\phi '\) of Eq. (A1); that is:

$$\begin{aligned} \partial _{\phi }\partial _{\phi '}W^{(\mathrm \pm )}(x,x') = \pm \frac{p^3}{8\pi ^2\rho \rho '}\int _{0}^{\infty }dwe^{-\frac{\zeta _{({\mp })}}{2\rho \rho '}w}\sum _{n=-\infty }^{\infty }n^2I_{p|n|}(w). \end{aligned}$$

Note that, after taking the derivatives in the azimuthal coordinates, we have taken the limit \(\phi '\rightarrow \phi \). Next, we can make use of the modified Bessel differential equation:

$$\begin{aligned} \sum _{n=-\infty }^{\infty }n^2I_{p|n|}(w) = \frac{1}{p^2}\left( w^2\partial _w^2 + w\partial _w - w^2\right) \sum _{n=-\infty }^{\infty }I_{p|n|}(w), \end{aligned}$$

to further work out the expression (A7). Thus:

$$\begin{aligned} \partial _{\phi }\partial _{\phi '}W^{(\mathrm \pm )}(x,x') = \pm \frac{p}{8\pi ^2\rho \rho '}\int _{0}^{\infty }dwe^{-\frac{\zeta _{({\mp })}}{2\rho \rho '}w}\left( w^2\partial _w^2 + w\partial _w - w^2\right) \sum _{n=-\infty }^{\infty }I_{p|n|}(w).\nonumber \\ \end{aligned}$$

We can again use the summation formula (A4) for \(\Delta \phi =0\) to get:

$$\begin{aligned}&\partial _{\phi }\partial _{\phi '}W^{(\mathrm \pm )}(x,x') \nonumber \\&\quad = \pm \frac{\rho \rho '}{2\pi ^2}\left[ \mathop {{\sum }'}\limits _{n=-[q/2]}^{[p/2]}M_n^{({\mp })}(x,x') - \frac{p}{\pi }\int _{0}^{\infty }d\xi \frac{\sin (p\pi )M_{\xi }^{({\mp })}(x,x')}{[\cosh (p\xi ) - \cos (p\pi )]}\right] , \end{aligned}$$


$$\begin{aligned} M_{\gamma }^{({\mp })}(x,x') = 16s_{\gamma }^2(s_{\gamma }^2 - 1)\frac{\rho \rho '}{[\sigma _{\gamma }^{({\mp })}]^3} + (1 - 2s_{\gamma }^2)\frac{1}{[\sigma _{\gamma }^{({\mp })}]^2}. \end{aligned}$$

Note that\(\gamma \) stands for n in the first term and \(\xi \) in the second term of Eq. (A10) and \(s_n=\sin (n\pi /p)\) and \(s_{\xi }=\cosh (\xi /2)\).

We want now to consider the integral in \(\eta \) of the functions \(M_{\gamma }^{({\mp })}(x,x')\) necessary for the calculation of the \(\phi \) component of the velocity dispersion; that is:

$$\begin{aligned} \int _{0}^{\eta }\mathrm{d}\eta _1\int _{0}^{\eta }\mathrm{d}\eta _2M_{\gamma }^{(+)}(x,x) = \frac{1}{\eta ^2c^4}\left[ -\frac{(s_{\gamma }^2 - 1)}{R^2s_{\gamma }^2(1-R^2s_{\gamma }^2)} + \frac{(s_{\gamma }^2 - 2)}{4R^3s_{\gamma }^3}\ln \left( \frac{Rs_{\gamma } + 1}{Rs_{\gamma } - 1}\right) ^2\right] \nonumber \\ \end{aligned}$$


$$\begin{aligned}&\int _{0}^{\eta }\mathrm{d}\eta _1\int _{0}^{\eta }\mathrm{d}\eta _2M_{\gamma }^{(-)}(x,x)\nonumber \\&\quad = \frac{1}{4\eta ^2c^4(R^2s_{\gamma }^2+Z^2)^{\frac{5}{2}}}\left[ \frac{4R^2s_{\gamma }^2(s_{\gamma }^2-1)\sqrt{R^2s_{\gamma }^2 +Z^2}}{(R^2s_{\gamma }^2 + Z^2 -1)}\right. \nonumber \\&\qquad \left. + [(1-2s_{\gamma }^2)Z^2 + (s_{\gamma }^2-2)R^2s_{\gamma }^2]\ln \left( \frac{\sqrt{R^2s_{\gamma }^2 +Z^2} + 1}{\sqrt{R^2s_{\gamma }^2 +Z^2} - 1}\right) ^2\right] . \end{aligned}$$

Therefore, the expressions (A10)–(A13) are all necessary to calculate Eqs. (26) and (43).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Santana Mota, H.F., Bezerra de Mello, E.R. Induced Brownian motion by the Friedmann–Robertson–Walker spacetime in the presence of a cosmic string. Eur. Phys. J. Plus 135, 12 (2020).

Download citation