Scalar tachyonic instabilities in gravitational backgrounds: Existence and growth rate

TL;DR

This paper studies tachyonic scalar field instabilities in Reissner-Nordström-deSitter black hole backgrounds, finding that the critical mass for instability remains zero, but the growth rate is reduced, especially near extremal conditions, due to the cosmological horizon.

Contribution

It demonstrates that the critical mass for tachyonic instability remains zero in RN-dS backgrounds, and analyzes how the growth rate is affected by the spacetime parameters.

Findings

Critical mass for instability remains zero in RN-dS backgrounds.

Growth rate of tachyonic instabilities is reduced compared to flat space.

Near extremal SdS black holes exhibit maximal delay in instability growth.

Abstract

It is well known that the Klein Gordon (KG) equation $\Box \Phi + m^2\Phi=0$ has tachyonic unstable modes on large scales ($k^2<\vert m \vert^2$) for $m^2<m_{cr}^2=0$ in a flat Minkowski spacetime with maximum growth rate $\Omega_{F}(m)= \vert m \vert$ achieved at $k=0$. We investigate these instabilities in a Reissner-Nordstr\"om-deSitter (RN-dS) background spacetime with mass $M$, charge $Q$, cosmological constant $\Lambda>0$ and multiple horizons. By solving the KG equation in the range between the event and cosmological horizons, using tortoise coordinates $r_*$, we identify the bound states of the emerging Schrodinger-like Regge-Wheeler equation corresponding to instabilities. We find that the critical value $m_{cr}$ such that for $m^2<m_{cr}^2$ bound states and instabilities appear, remains equal to the flat space value $m_{cr}=0$ for all values of background metric parameters despite the locally negative nature of the Regge-Wheeler potential for $m=0$. However, the growth rate $\Omega$ of tachyonic instabilities for $m^2<0$ gets significantly reduced compared to the flat case for all parameter values of the background metric ($\Omega(Q/M,M^2 \Lambda, mM)< \vert m \vert$). This increased lifetime of tachyonic instabilities is maximal in the case of a near extreme Schwarzschild-deSitter (SdS) black hole where $Q=0$ and the cosmological horizon is nearly equal to the event horizon ($\xi \equiv 9M^2 \Lambda \simeq 1$). The physical reason for this delay of instability growth appears to be the existence of a cosmological horizon that tends to narrow the negative range of the Regge-Wheeler potential in tortoise coordinates.