Stability and instability of Ellis and phantom wormholes: Are there ghosts?

It is concluded in the literature that the Ellis wormhole is unstable under small perturbations and would either decay to the Schwarzschild black hole or expand away to infinity. While this deterministic conclusion of instability is correct, we show that the Ellis wormhole reduces to the Schwarzschild black hole only when the Ellis solution parameter $\gamma$ assumes a complex value $-i$. We shall then reexamine the stability of Ellis and phantom wormholes from the viewpoint of local and asymptotic observers by using a completely different approach, viz., we adapt Tangherlini’s nondeterministic, prequantal statistical simulation about photon motion in the real optical medium to an effective medium reformulation of motions obtained via Hamilton’s optical-mechanical analogy in a gravity field. A crucial component of Tangherlini’s idea is the observed increase of momentum of the photons entering a real medium. We show that this fact has a heuristic parallel in the effective medium version of the Pound-Rebka experiment in gravity. Our conclusion is that there is a nonzero probability that Ellis and phantom wormholes could appear stable or unstable depending on the location of observers and on the values of $\gamma$, leading to the possibility of ghost wormholes (like ghost stars). The Schwarzschild horizon, however, would always certainly appear to be stable ($R=1$, $T=0$) to observers regardless of their location. Phantom wormholes of bounded mass in the extreme limit $a\to -1$ are also shown to be stable just as the Schwarzschild black hole is. We shall propose a thought experiment showing that our nondeterministic results could be numerically translated into observable deterministic signatures of ghost wormholes.

Figure 1

Plot of $\mathrm{\Phi }(r_{\mathrm{th}})$ and $r_{\mathrm{th}}(\gamma )$ vs $\gamma$. The larger the value of $\gamma$, the smaller the value of $r_{\mathrm{th}}$ and $\mathrm{\Phi }(r_{\mathrm{th}})$. One should choose $\alpha$ such that $\alpha <\mathrm{\Phi }(r_{\mathrm{th}})$ is needed for the reality of $N(r)$.

Figure 2

Reflection ($R$) and transmission ($T$) probabilities for the ingoing light pulse in the Ellis wormhole spacetime. We choose units such that $M=1$ and plot ($R$, $T$) at the throat, now only a function of $\gamma$ for the different locations of the observers [Eqs. (64)–(67)]. Asymptotic observers are denoted by the superscript a.o. and local observers by l.o. with a tilde for coefficients. It turns out that $R$ and $T$ reach fixed values at some value of $\gamma$, hence at some fixed size of the throat $r_{\mathrm{th}}$, where $R_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}\simeq 0.60$, $T_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}\simeq 0.40$, and ${\tilde{R}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}\simeq 0.20$ and ${\tilde{T}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}\simeq 0.80$, as shown in the figure. The values are in contrast to the Schwarzschild values $R=1$, $T=0$ at the horizon, which suggests that the horizon plays the role of a perfect mirror totally reflecting away the incoming pulse.

Figure 3

Reflection ($R$) and transmission ($T$) probabilities for ingoing de Broglie waves in the Ellis wormhole spacetime as perceived by the asymptotic observers. We choose units such that $M=1$ and values $\alpha ={\lambda }^{\prime }/{\lambda }_{\mathrm{C}}=0.01$, $\beta =0.2$ and 0.5. The plot ($R$, $T$) at the throat, now only a function of $\gamma$ [Eqs. (78) and (79)], shows that $R$ and $T$ reach fixed values at some value of $\gamma$, hence at some fixed size of the throat $r_{\mathrm{th}}$. Asymptotic observers are denoted by the superscript a.o., for whom ${\overline{R}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}\sim 0.76$, ${\overline{T}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}\sim 0.24$ when $\beta =0.2$. Reflection probability is reduced to ${\overline{R}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}=0.45$ when $\beta$ is increased, e.g., to 0.5, as shown in the figure. When $\beta \to 0$, stability is achieved: ${\overline{R}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}\to 1$, ${\overline{T}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}\to 0$.

Figure 4

Reflection ($R$) and transmission ($T$) probabilities for ingoing de Broglie waves in the Ellis wormhole spacetime perceived by the near-throat local observers. The plot ($R$, $T$) at the throat, now only a function of $\gamma$ [Eqs. (83) and (84)], shows de Broglie waves of high and low velocities $\beta =0.9$ and 0.01. The lesser the velocity, the higher the ${\widehat{R}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}$. Stability is achieved: ${\widehat{R}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}\to 1$,${\widehat{T}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}\to 0$ only in the limit $\beta \to 0$.

Figure 5

Reflection ($R$) and transmission ($T$) probabilities for de Broglie waves in the phantom wormhole spacetime ($-1<a<0$). The point $P$ corresponds to $R_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}={\tilde{R}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}=1$, ${\overline{R}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}={\widehat{R}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}=1$ and $Q$ corresponds to $T_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}={\tilde{T}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}=0$, ${\overline{T}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}={\widehat{T}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}=0$. See Eqs. (105) and (106). Likewise, $S$ corresponds to $R_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}={\tilde{R}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}=0.06$, ${\overline{R}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}={\widehat{R}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}=0.06$ and $U$ corresponds to $T_{\text{photon}}^{\mathrm{a}.\mathrm{o}.}={\tilde{T}}_{\text{photon}}^{\mathrm{l}.\mathrm{o}.}=0.94$, ${\overline{T}}_{\mathrm{dB}}^{\mathrm{a}.\mathrm{o}.}={\widehat{T}}_{\mathrm{dB}}^{\mathrm{l}.\mathrm{o}.}=0.94$. See Eqs. (107) and (108).