Abstract
We study the emission of space-time waves produced by back-reaction effects during a collapse of a spherically symmetric universe with a time-dependent cosmological parameter, which is driven by a scalar field. As in a previous work, the final state avoids the final singularity due to the fact the co-moving relativistic observer never reaches the center, because the physical time evolution \(\mathrm{d}\tau =U_{0}\,\mathrm{d}x^0\), decelerates for a co-moving observer which falls with the collapse. The equation of state of the system depends on the rate of the collapse, but always is positive: \(0< \omega (p) < 0.25\).
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Notes
We can define the operator
$$\begin{aligned} \delta {\hat{x}}^{\alpha }(t,\vec {x}) = \frac{1}{(2\pi )^{3/2}} \int \mathrm{d}^3 k \, {\check{e}}^{\alpha } \left[ b_k \, {\hat{x}}_k(t,\vec {x}) + b^{\dagger }_k \, {\hat{x}}^*_k(t,\vec {x})\right] , \end{aligned}$$such that \(b^{\dagger }_k\) and \(b_k\) are the creation and destruction operators of space-time, such that \(\left\langle B \left| \left[ b_k,b^{\dagger }_{k'}\right] \right| B \right\rangle = \delta ^{(3)}(\vec {k}-\vec {k'})\) and \({\check{e}}^{\alpha }=\epsilon ^{\alpha }_{\,\,\,\,\beta \gamma \delta } {\check{e}}^{\beta } {\check{e}}^{\gamma }{\check{e}}^{\delta }\), where
in order to
We use the asymptotic expressions for the Bessel functions, for \(f(t) \gg 1\), which are
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Acknowledgements
This research was supported by the CONACyT-UDG Network Project No. 294625 “Agujeros Negros y Ondas Gravitatorias”. M. B. acknowledges CONICET, Argentina (PIP 11220150100072CO) and UNMdP (EXA852/18) for financial support.
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Hernández, J.M., Bellini, M. & Moreno, C. Space-time waves from a collapse with a time-dependent cosmological parameter. Eur. Phys. J. Plus 135, 207 (2020). https://doi.org/10.1140/epjp/s13360-020-00243-9
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DOI: https://doi.org/10.1140/epjp/s13360-020-00243-9