= Solution
Given an <asymptotically flat spacetime> with a <conformal completion> and <future null infinity> $\mathscr I^+$, its <black hole> region is
$$
\boxed{\mathcal B=\mathcal M\setminus J^-(\mathscr I^+).}
$$
The <event horizon> is its boundary. For the <magnetically charged dilaton black hole>, first assume $0\le a=r_-<b=r_+$. The physical asymptotic component has $r>a$, and the ingoing chart extends regularly across $r=b$. Its future ingoing null direction is $-\partial_r$. For $a<r<b$,
$$
\nabla r=\partial_v+V\partial_r,\qquad g(\nabla r,\nabla r)=V<0,\qquad g(\nabla r,-\partial_r)=-1.
$$
Hence $\nabla r$ is future timelike. Every nonzero future causal tangent $X$ satisfies $X(r)=g(\nabla r,X)<0$. A future causal curve from this interior can neither cross back to $r=b$ nor reach the asymptotic end $r\to\infty$, so it cannot reach $\mathscr I^+$. It lies in $\mathcal B$.
This proof uses the future black-hole extension. The radial interval alone does not distinguish it from the past <white hole> interior of the maximal extension; that interior has the same interval and can send signals to infinity. Thus the statement needs this future-component qualification if the maximal spacetime is intended.
The <Kretschmann scalar> confirms that $r=a$ is a genuine <curvature singularity>: for $0<a<b$, its numerator at $r=a$ is $3a^4(a-b)^2$, and
$$
R_{abcd}R^{abcd}\sim\frac{3(a-b)^2}{4a^2(r-a)^4}.
$$
For $a=0$ it instead reduces to $12b^2/r^6$. Since $V<0$ on approaching the singularity from the interior, the singularity is spacelike. At $r=b>a$ the curvature is finite.
For the maximal extension, take $U=-e^{-u/(2b)}$, $W=e^{v/(2b)}$ in the right exterior and continue analytically across the horizons. Then
$$
UW=\left(1-\frac rb\right)e^{r/b},\qquad
ds^2=-\frac{4b^3}{r}e^{-r/b}dU\,dW+R^2d\Omega_2^2.
$$
The singularity is $UW=(1-a/b)e^{a/b}>0$ and the horizons are $U=0$ and $W=0$. Rescaling the null coordinates by the square root of this positive constant before arctangent compactification produces the usual qualitative <Penrose diagram> of <Schwarzschild spacetime>: two exteriors, a future black-hole region, a past white-hole region, and future and past spacelike singularities. The dashed lines below are the horizon branches; only the future interior is the black-hole region.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-311.png]
{title=Radial <effective potential>}
If $a>b$, the physical domain remains $r>a$, but now $V>0$ throughout it. The apparent zero $r=b$ is beyond the curvature singularity and is not an accessible horizon. The singularity is timelike, and outgoing radial null rays with $\dot r=E>0$ escape to infinity from arbitrarily near it. Thus
$$
\boxed{r_->r_+:\ \text{a timelike naked singularity, with no black-hole horizon}.}
$$
The excluded equality $r_-=r_+$ is a singular limiting case, not a regular horizon. These facts describe the <causal structure of the magnetic dilaton black hole>.
Back to article page
