Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 57 / 4 / i

A positive-radius horizon occurs at a zero of V. Set $u=r^2>0$; multiplying by $r^4$ gives $u^2-2Mu+Q^2=0$. Thus real roots require $M^2-Q^2\ge 0$. With $M>0$, the horizon condition and radii areFor $0<|Q|<M$ both roots are positive, giving an outer event horizon and an inner Cauchy horizon. For $|Q|=M$ they coincide at $r_h=\sqrt{M}$. For $Q=0$ only $r_{+}=\sqrt{2M}$ is a horizon: the formal root $u=0$ was introduced by multiplication and is the singular origin, where the original expression for V is undefined. For $|Q|>M$ there is no horizon and the central singularity is naked. Each positive root is regular in ingoing coordinates $v=t+r_{\ast }$, where the radial metric is $-Vd{v}^2+2dvdr$.