Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 1 / b

FactorIntroduce $v=t+r_{\ast }$ with $d{r}_{\ast }/dr=f^{-1}$. Since $dt=dv-dr/f$,These are the charged analogue of ingoing Eddington-Finkelstein coordinates. The $(v,r)$ block has determinant $-1$, so the metric and its inverse are nonsingular at $r=r_{+}$. For $M\ge Q$, $r_{+}$ is real and the metric extends analytically through it. The original potential becomes $A=-Qdv/r^2+Qdr/(r^{2}f)$; the radial term is locally exact, and a gauge transformation gives the horizon-regular potential $A=-Qdv/r^2$.

The stationary Killing vector field is $k={\partial }_t={\partial }_v$, with $k^2=-f$. The normal to $r=\text{constant}$ has squared norm $g^{rr}=f$, so $r=r_{+}$ is a null hypersurface; there $k$ is both tangent and normal. It is therefore a Killing horizon.

For a static metric of this form, the surface gravity is $\kappa =f^{\prime }(r_{+})/2$. Since $r_{+}^2{r}_{-}^2=Q^2$ and $r_{+}^2+r_{-}^2=2M$,For the extremal black hole $M=Q$, the horizons coincide and $\kappa =0$.

The zeros of $f$ give three causal structures.

Thus the cases are subextremal, extremal, and overextremal according as $M>Q$, $M=Q$, and $M<Q$.

For this charged scalar field, $\sqrt{-g}=r^3\sin \theta \cos \theta$. On $e^{-i\omega t}$, the gauge-covariant derivative of a charged scalar field supplies the shifted frequencyThe wave equation becomesSeparation of variables with constant $\lambda$ givesandThe angular equation is the axisymmetric spherical harmonic equation on the three-sphere; regular solutions have $\lambda =\ell (\ell +2)$ with $\ell =0,2,4,\dots$, the even values selected by the absence of $\varphi$ and $\psi$ dependence.