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Gravitational collapse with tachyon field and barotropic fluid

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Abstract

A particular class of space-time, with a tachyon field, \(\phi \), and a barotropic fluid constituting the matter content, is considered herein as a model for gravitational collapse. For simplicity, the tachyon potential is assumed to be of inverse square form i.e., \(V(\phi )\sim \phi ^{-2}\). Our purpose, by making use of the specific kinematical features of the tachyon, which are rather different from a standard scalar field, is to establish the several types of asymptotic behavior that our matter content induces. Employing a dynamical system analysis, complemented by a thorough numerical study, we find classical solutions corresponding to a naked singularity or a black hole formation. In particular, there is a subset where the fluid and tachyon participate in an interesting tracking behaviour, depending sensitively on the initial conditions for the energy densities of the tachyon field and barotropic fluid. Two other classes of solutions are present, corresponding respectively, to either a tachyon or a barotropic fluid regime. Which of these emerges as dominant, will depend on the choice of the barotropic parameter, \(\gamma \). Furthermore, these collapsing scenarios both have as final state the formation of a black hole.

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Notes

  1. Notice that, since the interior space-time (4) is spatially homogeneous, it is obvious that, the energy-momentum tensor must also be spatially homogeneous.

  2. Using the energy density (11) and pressure (15), the weak energy condition can be written as follows

    $$\begin{aligned} \rho +p=\frac{nc}{3a^{n}}>0. \end{aligned}$$
    (12)

    The weak energy condition is satisfied throughout the collapse process (see also next section for more details regarding the issue of the energy conditions).

  3. We note that satisfying the weak energy condition implies that the null energy condition (NEC) is held as well.

  4. We recall that throughout this paper \(H<0\), i.e., \(\dot{a}<0\) is assumed.

  5. Or equivalently, the surface where the trajectories will be present can be written as \(y^{2}=\left(1-s\right)^{2}\left(1-x^{2}\right)\).

  6. In the absence of a barotropic fluid, the effective phase space is one-dimensional.

  7. The time at which the collapse reaches the singularity is finite. Thus the tachyon field at the singularity remains finite as \(\phi (t_{s})=t_{s}+\phi _{0}\).

  8. For the \(\phi <0\) branch, \(\phi _{0}\) is negative at the initial condition. Thus, the absolute value of the tachyon field starts to decrease from the initial configuration as \(\phi (t)=t-|\phi _{0}|\) until the singular point at time \(t_{s}<|\phi _{0}|\), where tachyon field reaches its minimum but non-zero value \(\phi _{s}\). This leads it uphill the potential until the singular epoch, where the potential becomes maximum but finite.

  9. On the other hand, for the \(\phi <0\) branch, the tachyon field increases from its initial condition as time evolves, proceeding to ever less negative values as \(\phi \rightarrow 0^{-}\). In this case, it proceeds downhill the tachyon potential till the system reaches \(a=0\) at \(t=t_{s}\), where the Hubble rate and hence the total energy density of the system diverge. This implies that the time at which the collapse system reaches the singularity is always \(t_{s}\le \phi _{0}\).

  10. For the \(\phi <0\) branch, the time dependence of the tachyon field is given by \(\phi (t)=\sqrt{\gamma }t+\phi _{0}\). Therefore the tachyon field starts its evolution from the initial configuration at \(\phi _{0}<0\), decreasing in time, going uphill the potential.

  11. The case of a standard scalar field was investigated in [42, 67].

  12. It is worthwhile to compare the result obtained herein for a homogeneous (tachyonic) collapsing matter field with the gravitational collapse of a k-essence scalar field with non-standard kinematic terms in [58], where the scalar field has a dependence to the radius \(r\) i.e., \(\phi =\phi (r)\). In both models, the collapsing systems lead to the black hole formation.

  13. Satisfying the SEC for the tachyon field demands that \(\rho _{\phi }+3p_{\phi }\ge 0\). Thus, the SEC holds if \(\dot{\phi }^{2}>2/3\).

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Acknowledgments

The authors would like to thank P. Joshi, A. Khaleghi, C. Kiefer, F. C. Mena, A. Vikman, and J. Ward for making useful suggestions and comments. They are also grateful to the referee for the useful comments and suggestions on the issue of energy conditions. YT is supported by the Portuguese Agency Fundação para a Ciência e Tecnologia through SFRH/BD/43709/2008. This research work was also supported by the grant CERN/FP/109351/2009 and CERN/FP/116373/2010.

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Authors and Affiliations

  1. Departamento de Física, Universidade da Beira Interior, 6200 , Covilhã, Portugal

    Yaser Tavakoli, João Marto & Paulo Vargas Moniz

  2. Department of Physics, Shahid Beheshti University, G. C., Evin, 19839 , Tehran, Iran

    Amir Hadi Ziaie

  3. CENTRA, IST, Av. Rovisco Pais, 1, Lisbon, Portugal

    Paulo Vargas Moniz

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  1. Yaser Tavakoli
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Correspondence to João Marto.

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Tavakoli, Y., Marto, J., Ziaie, A.H. et al. Gravitational collapse with tachyon field and barotropic fluid. Gen Relativ Gravit 45, 819–844 (2013). https://doi.org/10.1007/s10714-013-1503-3

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