A numerical approach to particle creation in accelerating toy models

Abstract

The formation of black holes by the gravitational collapse of stars is known to spontaneously excite particle pairs out of the quantum vacuum. For the canonical vacuum state at past null infinity, the expected number of particles received at future null infinity can be obtained in full closed form at sufficiently late times. However, for intermediate times, or for more complicated astrophysical processes (e.g. binary black hole mergers), the problem is technically challenging and has not yet been resolved. We develop here a numerical approach to study scattering problems of massless quantum fields in asymptotically flat spacetimes, based on the hyperboloidal slice method used in numerical relativity and perturbation theory. This promising approach can reach both past and future null infinities, and therefore it has the potential to address the Hawking scattering problem more rigorously than evolution on the usual Cauchy slices. We test this approach with some dynamical toy models in Minkowski using effective potentials that mimic the effects of gravity, and compute the spectrum of particles received at future null infinity. We finally discuss future prospects for applying this framework in more relevant gravitational scenarios.

Figures (16)

• Figure 1: Penrose diagram illustrating the propagation of the modes in a Vaidya spacetime, similar to Figure 3.3 in doi:10.1142/p378.
• Figure 2: Penrose diagram of Minkowski spacetime depicting radiation propagating from $\mathrsfs{I}^-$ to $\mathrsfs{I}^+$, and the two foliations of spacetime using ingoing (constant $t_m$) and outgoing (constant $t_p$) hyperboloidal slices.
• Figure 3: Penrose diagram of Minkowksi spacetime showcasing a schematic of the relation between the information on outgoing and ingoing hyperboloidal slices. Information on the $t_p$ slice is present at multiple $t_m$ slices.
• Figure 4: Illustration of the flat bump function defined in (\ref{['Eq:BumpFunctionOutMode']}) with parameters $q=5$, $a=3$ and $b=3.2$.
• Figure 5: Frequency spectrum for the Bogoliubov coefficient $\beta_{\omega_0\omega}$ for fixed $\omega_0$ in the case of a vanishing potential barrier. The spectrum is computed from (\ref{['Eq:BetaNewExpansion']}) using a square window for the envelope of the out modes (up) and a bump function with $b-a=0.2$ for the envelope (down). The results found are compatible with zero, which indicates no particle creation, as expected.