Quantization of a scalar field in the Kerr spacetime

The interaction of a scalar field with the Kerr gravitational field is studied. The properties of the Klein-Gordon equation in the Kerr metric are reviewed, and a two-component formalism is developed. This formalism expresses the one-particle quantum theory for a massive scalar particle in the Kerr metric. A semiclassical analysis of the spontaneous emission of particles by a Kerr black hole is given. The quantization of a scalar field in the Kerr metric is developed, and a treatment of the spontaneous particle creation is given. The particular quantization given here leads to emission only into the classical superradiant modes and hence no emission by a Schwarzschild black hole. In the case in which $\omega M\ll 1$, where $M$ is the mass of the black hole and $\omega$ the frequency of a given mode, an explicit expression may be given for the rate at which particles are emitted into each mode. In the case in which the particle's mass is zero and $a\ll M$, $a=\mathrm{angular}\mathrm{momentum}\mathrm{per}\mathrm{unit}\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{black}\mathrm{hole}$, angular momentum per unit mass of the black hole, the total rate of loss of energy of the black hole is shown to be proportional to $\frac{a^6}{M^8}$. A discussion of the problem of the vacuum energy is given. It is shown that the energy of the vacuum state of a scalar field in a Kerr spacetime $a\ll M$ is the same as that for a Schwarzschild ($a=0\mathrm{}$) spacetime.