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    [**Abstract**]{}

    We study the absorption probability and Hawking radiation spectra of a phantom scalar field in the Kerr black hole spacetime. We find that the presence of the negative kinetic energy terms modifies the standard results in the greybody factor, super-radiance and Hawking radiation. Comparing with the usual scalar particle, the phantom scalar emission is enhanced in the black hole spacetime.
author:
- Songbai Chen
- Jiliang Jing
title: '[**Phantom scalar emission in the Kerr black hole spacetime**]{}'
---

=0.8 cm

Introduction
============

Our Universe is assumed to be filled with dark energy because it can explain the accelerated expansion of the Universe, which is strongly supported by many cosmological observations[@1b; @1b0]. Dark energy is an exotic energy component with negative pressure and constitutes about $72\%$ of present total cosmic energy. The leading interpretation of such a dark energy is the cosmological constant with an equation of state $\omega_x=-1$ (for a classic review see [@1ax], for a recent nice review see [@1ar0], and for a recent discussion see [@1ar1; @1b1; @1ar2]). The energy density of this dark energy is associated with quantum vacuum [@1ax; @1ar0; @1ar1; @1ar2; @1b1; @1as]. Although this explanation is consistent with observational data, it is plagued with the so-called coincidence problem namely, “why are the vacuum and matter energy densities of precisely the same order today?". Therefore the dynamical scalar fields, such as quintessence [@2a], k-essence [@3a] and phantom field [@4a], are proposed as possible alternatives of dark energy.

Comparing with other dynamical scalar fields, the phantom field model is more interesting because that it has a negative kinetic energy and the super negative equation of state $\omega_x<-1$. Although the null energy condition is violated, this dark energy model is not ruled out by recent precise observational data involving CMB, Hubble Space Telescope and type Ia Supernova [@5a]. The dynamical evolution of the phantom field in the cosmology has been investigated in the last years [@6a1; @6a11; @6a2; @6a3; @6a4; @6a5; @6a6; @6a7]. It shows that the energy density increases with the time and approaches to infinity in a finite time [@6a1]. This implies in the standard Einstein cosmology the flat universe dominated by phantom energy will blow up incessantly and arrive at a future singularity finally named big rip which has such a strong exclusive force that anything in the universe including the large galaxies will be torn up. Recently, many efforts have been working to avoid the big rip [@7b]. It has argued that this future singularity could be vanished in the universe if one considers the effects from loop quantum gravity [@lc3; @lg4; @lg6; @lg7].

The presence of negative kinetic energy results in many exotic properties of phantom field in the black hole spacetime. E. Babichev *et al* [@Eph] considered the phantom energy accretion of black hole and find that the mass of the black hole is decreased. This can be explained by that the kinetic energy of the phantom field is negative which yields the super negative equation of state $\omega_x<-1$. The decrease of mass of black hole in the phantom energy accretion will lead to the ratio between charge and mass of black hole could be larger than $1$ ($\frac{Q}{M}>1$ ) and there may exists a naked singularity [@Eph1], which implies that the cosmological censorship is violated. The negative kinetic energy also yields that the dynamical evolution of phantom scalar perturbation possesses some unique characteristics in the black hole spacetime [@sjp]. One of is that it grows with an exponential rate in the late-time evolution rather than decays as the usual scalar perturbations. These new results will excite more efforts to be devoted to the study of phantom energy in the background of a black hole. In this paper we will focus on the Hawking radiation of the phantom scalar particles in the Kerr black hole spacetime and see what effect of the negative kinetic energy on the power and angular momentum emission spectra of the Hawking radiation.

Phantom scalar emission in the Kerr black hole spacetime
========================================================

In the curve spacetime, the action of the phantom scalar field with the negative kinetic energy term is $$\begin{aligned}
S=\int d^4x \sqrt{-g}\bigg[-\frac{R}{16\pi
G}-\frac{1}{2}\partial_{\mu}\psi\partial^{\mu}\psi+V(\psi)\bigg].\end{aligned}$$ Here we take metric signature $(+ - - -)$ and the potential $V(\psi)=-\frac{1}{2}\mu^2\psi^2$, where $\mu$ is the mass of the scalar field. Varying the action with respect to $\psi$, we obtain the Klein-Gordon equation for a phantom scalar field in the curve spacetime $$\begin{aligned}
\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})
\psi-\mu^2\psi=0.\label{WE}\end{aligned}$$ The presence of negative kinetic energy leads to the sign of the mass term $\mu^2$ is negative in the wave equation, which will yield the peculiar properties of Hawking radiation of the phantom scalar particle in the black hole spacetime.

The well-known Kerr metric in the Boyer-Lindquist coordinate is $$\begin{aligned}
ds^2&=&\bigg(1-\frac{2Mr}{\Sigma}\bigg)dt^2+\frac{4Mra\sin^2{\theta}}{\Sigma}dtd\phi-\frac{\Sigma}{\Delta}dr^2
-\Sigma
d\theta^2-\bigg(r^2+a^2+\frac{2Mra^2\sin^2{\theta}}{\Sigma}\bigg)\sin^2{\theta}d\phi^2,\label{m1}\end{aligned}$$ with $$\begin{aligned}
\Delta=r^2-2Mr+a^2,\;\;\;\;\;\;\;\;\;\;\;\;\Sigma=r^2+a^2\cos^2{\theta},\end{aligned}$$ where $M$ is the mass and $a$ is the angular momentum of the black hole. Equation (\[WE\]) is separable in terms of the spheroidal harmonics $\psi(t,r,\theta,\phi)=e^{i(-\omega
t+m\phi)}R(r)S(\theta)$. The angular and the radial functions $S(\theta)$, $R(r)$ obey to $$\begin{aligned}
\frac{1}{\sin{\theta}}\frac{d}{d\theta}\bigg[\sin{\theta} \frac{d
S(\theta)}{d \theta}\bigg]
+\bigg[(\omega^2+\mu^2)a^2\cos^2{\theta}-\frac{m^2}{\sin^2{\theta}}+\lambda\bigg]S(\theta)=0,\label{angd}\end{aligned}$$ and $$\begin{aligned}
\frac{d}{dr}\bigg[\Delta\frac{d R(r)}{dr}\bigg]
+\bigg\{\frac{[(r^2+a^2)\omega-ma]^2}{\Delta}+\mu^2r^2-E_{lm}\bigg\}R(r)=0,\label{radial}\end{aligned}$$ respectively. Where $\lambda$ is the eigenvalues and the function $E_{lm}=\lambda+\omega^2a^2-2ma\omega$. In order to calculate the absorption probability $|\mathcal{A}_{lm}|^2$ and the luminosity of Hawking radiation for a phantom scalar particle, we must solve the radial equation (\[radial\]) above. Following the standard matching techniques [@Haw3; @Kanti; @Kan1; @Kan2; @Kan3; @Kan4; @Kan5; @Kan6; @Haw4; @Haw5], we can create a smooth analytical solution of the equation (\[radial\]) in the low-energy and low-angular momentum limit. Near the horizon ($r\simeq r_+$) regime and at infinity, it has the form $$\begin{aligned}
&&r\rightarrow r_+,~~~~~~
R(r=r_+)=A^{(r=r_+)}_{in}e^{-(\omega-m\Omega_H)r_*}+A^{(r=r_+)}_{out}e^{(\omega-m\Omega_H)r_*},\\
&&r\rightarrow \infty,~~~~~~
R(r=\infty)=A^{(r=\infty)}_{in}e^{-\sqrt{(\omega^2+\mu^2)r}}+A^{(r=\infty)}_{out}e^{\sqrt{(\omega^2+\mu^2)r}},\end{aligned}$$ respectively. Unlike the usual scalar particle, we find that for the phantom particle with an arbitrary value of $\mu$ the solution above denotes an incoming and an outgoing spherical waves at large distance from the black hole. From this solution, we can calculate the absorption probability $$\begin{aligned}
|\mathcal{A}_{lm}|^2=1-\bigg|\frac{A^{(r=\infty)}_{out}}{A^{(r=\infty)}_{in}}\bigg|^2.\label{GFA}\end{aligned}$$ The power and and angular momentum emission spectra of phantom scalar particle is then written as $$\begin{aligned}
\frac{d^2E}{dtd\omega}&=&\sum_{l,m}
\frac{\omega^2}{e^{\;(\omega-m\Omega_H)/T_{H}}-1}\frac{N_l|\mathcal{A}_{lm}|^2}{2\pi\sqrt{\omega^2+\mu^2}},\label{H1}\\
\frac{d^2J}{dtd\omega}&=&\sum_{l,m}
\frac{m\omega}{e^{\;(\omega-m\Omega_H)/T_{H}}-1}\frac{N_l|\mathcal{A}_{lm}|^2}{2\pi\sqrt{\omega^2+\mu^2}},\end{aligned}$$ where $T_H$ is the Hawking temperature of Kerr black hole. These equations can be integrated numerically.

Here we present the numerical results about the absorption probability $|\mathcal{A}_{lm}|^2$ and the Hawking radiation of a phantom scalar field in the background of a Kerr black hole.

In Fig.(1), we fix $a=0.2$ and examine the dependence of the absorption probability of phantom scalar particle on its mass $\mu$ for the first partial waves ($l=0, \;m=0$) in the background of a Kerr black hole. Comparing with the usual scalar particle, the absorption probability of the phantom increase rather than decrease as the mass $\mu$ increases. This is not surprising because that in the wave equation of phantom field (\[WE\]) the sign of the mass term is negative get rise to the absorption probability has the form $|\mathcal{A}_{00}|^2\sim 4\omega \sqrt{\omega^2+\mu^2}(r^2_+-a^2)$ in the low-energy and low-angular momentum limit. For other values of $l$, we also find that the absorption probability of phantom scalar particle increase with the mass $\mu$.

![Variety of the absorption probability $|\mathcal{A}_{lm}|^2$ of a scalar particle propagating in the Kerr black hole with its mass $\mu$ (the left panel is for a usual scalar particle and the right panel is for a phantom one) for fixed $l=0$, $m=0$ and $a=0.2$. We set $M=1$.](usA00-u.eps "fig:"){width="8.0cm"}![Variety of the absorption probability $|\mathcal{A}_{lm}|^2$ of a scalar particle propagating in the Kerr black hole with its mass $\mu$ (the left panel is for a usual scalar particle and the right panel is for a phantom one) for fixed $l=0$, $m=0$ and $a=0.2$. We set $M=1$.](A00-u.eps "fig:"){width="8.0cm"}

\[fig1\]

![The dependence of the super-radiance on the mass $\mu$ (the left panel is for a usual scalar particle and the right panel is for a phantom one) in the Kerr black hole for fixed $l=1$, $m=1$ and $a=0.2$. We set $M=1$.](ussuper-u.eps "fig:"){width="8.0cm"}![The dependence of the super-radiance on the mass $\mu$ (the left panel is for a usual scalar particle and the right panel is for a phantom one) in the Kerr black hole for fixed $l=1$, $m=1$ and $a=0.2$. We set $M=1$.](super-u.eps "fig:"){width="8.0cm"}

\[fig2\]

![Variety of the luminosity of Hawking radiation $L$ of scalar particles propagating with $\mu$ in the Kerr black hole, for $l=0, m=0$ and different $a$. We set $M=1$. The left panel for a usual scalar particle and the right panel for a phantom particle.](Lusxs.eps "fig:"){width="8.0cm"} ![Variety of the luminosity of Hawking radiation $L$ of scalar particles propagating with $\mu$ in the Kerr black hole, for $l=0, m=0$ and different $a$. We set $M=1$. The left panel for a usual scalar particle and the right panel for a phantom particle.](Lphxs.eps "fig:"){width="8.0cm"}

\[fig3\]

![The relative luminosity of Hawking radiation between phantom and usual massive scalar particles (with the same mass as the phantom field) for different values of $\mu$ in the Kerr black hole. We set $M=1$.](Lph-luxs.eps){width="8.0cm"}

\[fig4\]

In Fig.(2), we plot the absorption probability of phantom particle for the mode ($l=1,m=1$) and find that the super-radiance possesses some peculiar properties in this case. First, for phantom particle the range of $\omega$ for the super-radiance to happen is independent of $\mu$, which means that the phantom scalar particle can be radiated away without any constraint of the particle’s energy. This can be explained by that the wave-function of phantom particle at the spatial infinity has the form $e^{i\sqrt{\omega^2+\mu^2}\;r_*}$. It is different from that of the usual scalar particles. It is well known that the usual scalar particle with the energy $\omega^2<\mu^2$ cannot be radiated away. These particles are is forced to be trapped in the system and the amplified wave can accumulate in the potential well, which will yield black hole bomb in which the mass $\mu$ of a usual scalar particle plays the role of a natural mirror. While in the super-radiance of the phantom scalar particle, it can not yield black hole bomb since all of phantom scalar particles can be radiated away and their wave functions are not in the bounded-state. Second, the magnitude of the super-radiance of phantom particle increases with the mass $\mu$. While for usual scalar particle, it is decreases. The new properties of super-radiance imply that the phantom field will enhance Hawking radiation of the black hole.

![Power emission spectra of usual scalar particle (the left panel) and of the phantom scalar one (the right panel) for different values of $\mu$ in the Kerr black hole. We set $M=1$.](hs1.eps "fig:"){width="8.0cm"} ![Power emission spectra of usual scalar particle (the left panel) and of the phantom scalar one (the right panel) for different values of $\mu$ in the Kerr black hole. We set $M=1$.](hp1.eps "fig:"){width="8.0cm"}

\[fig5\]

![Angular momentum spectra of usual scalar particle (the left panel) and of the phantom scalar one (the right panel) for different values of $\mu$ in the Kerr black hole. We set $M=1$.](hs2.eps "fig:"){width="8.0cm"} ![Angular momentum spectra of usual scalar particle (the left panel) and of the phantom scalar one (the right panel) for different values of $\mu$ in the Kerr black hole. We set $M=1$.](hp2.eps "fig:"){width="8.0cm"}

\[fig6\]

Let us now to study the luminosity of the Hawking radiation of phantom particles in the Kerr black hole spacetimes. We first focus on the first partial waves ($l=0$, $m=0$), which plays a dominant role in the greybody factor. The luminosity of the Hawking radiation for the phantom scalar particle can be expressed as $$\begin{aligned}
L&=&\int^{\infty}_0
\frac{|\mathcal{A}_{00}|^2}{\sqrt{\omega^2+\mu^2}}\frac{\omega^2}{e^{\;(\omega-m\Omega_H)/T_{H}}-1}\frac{d\omega}{2\pi}.
\label{LHK}\end{aligned}$$ Here the low bound of the integral is zero rather than $\mu$ since the emission of the phantom scalar particle is not constrained by the value of the particle mass $\mu$. In generally, this integral cannot be computed analytically. In figure (3), we present some numerical results. For the fixed $\mu$, the Hawking radiation of the phantom scalar particles decreases as the angular momentum a increases. This is the same as that of the usual scalar particles in the Kerr black hole. But for the fixed $a$, with the mass $\mu$ increase the Hawking radiation of black hole increases for the phantom scalar field and decreases for the usual scalar field. Therefore the presence of negative kinetic leads to that the emission of phantom field increases and black hole evaporates more quickly. Figure (4) also tell us that with the increase of the mass $\mu$ the difference between the emission of phantom and usual scalar particle increase rapidly. When $\mu=0.1$, the luminosity of Hawking radiation of phantom field is almost 1.5 multiples of that of the usual scalar one for all $a$. Thus for the larger $\mu$, the Hawking radiation of black hole is dominated by phantom scalar field. In Figs.(5) and (6), we summed up to $l=4$ modes in calculating the energy and angular momentum emission rates. We find that the mass $\mu$ of phantom scalar particle enhance the the power and angular momentum emission spectra. Our results are consistent with that obtained in the phantom energy accretion of black hole where the black hole mass is decreased. This leads to increase of the temperature of the black hole and enhance the luminosity of the Hawking radiation.

summary
=======

In this paper, we have studied the greybody factor and Hawking radiation for a phantom scalar field in the background of Kerr black hole in the low-energy and low-angular momentum approximation. We have found that the absorption probability and Hawking radiation contain the imprint of the phantom scalar field. The effects of $\mu$ on the super-radiance and the luminosity of the Hawking radiation for phantom scalar field are different from that for the usual scalar field since phantom scalar particle possesses a negative kinetic energy. Our results may help us to detect whether our universe is filled with phantom field or not. It would be of interest to generalize our study to other black hole spacetimes, such as stringy black holes etc. Work in this direction will be reported in the future.

This work was partially supported by the National Natural Science Foundation of China under Grant No.10875041 and the construct program of key disciplines in Hunan Province. J. L. Jing’s work was partially supported by the National Natural Science Foundation of China under Grant No.10675045, No.10875040 and No.10935013; 973 Program Grant No. 2010CB833004 and the Hunan Provincial Natural Science Foundation of China under Grant No.08JJ3010.

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