Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 3

For the Komar integral of a five-dimensional Reissner--Nordstrom metric, use the one-form metrically dual to the stationary Killing vector field: $k^{♭}=-f(r)dt$. Its exterior derivative is $d{k}^{♭}=f^{\prime }(r)dt\wedge dr$. The stated orientation gives volume formBecause the squared norm of $dt\wedge dr$ is $g^{tt}{g}^{rr}=-1$, the defining wedge identity for the Hodge star operator givesThus the Komar integral over the positively oriented angular three-sphere issince the unit three-sphere has volume $2{\pi }^2$. Write $s=r_{+}^2+r_{-}^2$, $q=r_{+}^2{r}_{-}^2$. Then $f=1-s/r^2+q/r^4$ and $r^3{f}^{\prime }=2s-4q/r^2$. With the normalization specified for the Komar mass,The orientation makes this positive. The problem's coefficient $1/(16\pi )$ must be retained: it is not the usual dimension-adjusted five-dimensional ADM-normalized Komar coefficient. With that conventional normalization the ADM mass would instead be $3\pi s/8$.

For radial null geodesics, the metric gives $dt/dr=\pm f^{-1}$ in the exterior. Define the tortoise coordinate byAn outgoing future radial ray has $dt/dr=f^{-1}$, so $u=t-r_{\star }$ is constant; an ingoing ray has $dt/dr=-f^{-1}$, so $v=t+r_{\star }$ is constant. These describe the null trajectories independently of the choice of affine parameter.

For the tortoise coordinate of a five-dimensional charged black hole, with $a=r_{+}$, $b=r_{-}$ and $\Delta =a^2-b^2$, partial fractions giveThus a useful explicit tortoise coordinate isThe logarithms signal failure of the static coordinates at the horizons, rather than a curvature singularity there.

Since $v=t+r_{\star }$, $dt=dv-f^{-1}dr$. Substitution into the radial metric cancels the $d{r}^2/f$ terms. The ingoing Eddington-Finkelstein coordinates giveThe radial block has determinant $-1$, even when $f=0$. Its coefficients are analytic for every $r>0$, so this form extends through $r=r_{+}$ and also through $r=r_{-}$ to $0<r<r_{+}$. Ordinary angular coordinate singularities are removed by angular charts on the three-sphere. The genuine endpoint $r=0$ is not included in this extension.

The field $n=-{\partial }_r$ is an ingoing null tangent. It is future-directed in the original exterior, since its inner product with the future stationary Killing vector field is $-1$, and it remains future-directed by continuation. This fixes the time orientation used in the next part.

For an asymptotically flat spacetime with a chosen exterior component of future null infinity, the black-hole region isnamely the events from which no future-directed causal curve can reach that infinity. Here ${\mathcal{I}}^{+}$ is the infinity of the original $r>r_{+}$ exterior.

Let a future causal tangent in the ingoing Eddington-Finkelstein coordinates have components $(\dot{v},\dot{r},\dot{\Omega })$. Its inner product with the future null field $n=-{\partial }_r$ is $-\dot{v}$, so $\dot{v}\ge 0$. The causal inequality isIf $\dot{v}>0$, this impliesIn the band $r_{-}<r<r_{+}$, $f<0$, so $\dot{r}<0$. If $\dot{v}=0$, the causal inequality forces $\dot{\Omega }=0$, and future direction means the remaining radial tangent is a nonnegative multiple of $n$, again giving $\dot{r}\le 0$. This is causal trapping between two spherical horizons: $r$ is nonincreasing along every future causal curve in the between-horizon band. No event there can escape outward. An event with $0<r\le r_{-}$ would also have to cross this band outward to reach the original exterior, which is impossible. Thus all $0<r<r_{+}$ events in this extension lie in the black-hole region.

Conversely, from any $r>r_{+}$ point, the outgoing radial null ray $u=\mathrm{constant}$ satisfies $dr/dv=f/2>0$. It reaches arbitrarily large $r$ and then the original future null infinity. Therefore the exterior does not intersect the black-hole region, and $r=r_{+}$ is its event horizon. This conclusion is relative to the selected asymptotic end; the maximal charged-black-hole extension can have other asymptotic ends.

In the ingoing Eddington-Finkelstein coordinates, the stationary Killing vector field is $k={\partial }_v$ with norm $k^2=-f$. The normal vector to a level set of $r$ isAt $r=r_{+}$ it equals $k^a$, which is null there. Thus this null hypersurface is a Killing horizon of $k$. Its surface gravity is fixed by ${\nabla }_a(k^2)=-2\kappa k_a$ on the horizon. Since $k_a=(dr{)}_a$ there and ${\nabla }_a(k^2)=-f^{\prime }(r_{+})(dr{)}_a$,The positive sign follows from the future outer-horizon generator, normalized by the static time at infinity. It vanishes as the two radii approach the extremal limit.

The inner horizon in the nonextreme Reissner-Nordstrom spacetime diagram is a Cauchy horizon. Radiation from exterior perturbations can approach it at arbitrarily late advanced time and undergo unbounded blueshift. Here its positive blueshift scale is ${\kappa }_{-}=(r_{+}^2-r_{-}^2)/r_{-}^3$. The inner-horizon blueshift coordinate behaves as $V_{-}\propto -e^{-{\kappa }_{-}v}$, soEven a decaying power-law tail can therefore generate divergent local flux. With counterstreaming radiation, backreaction produces mass inflation and curvature growth, rather than the smooth inner horizon of the exact solution.

This supports the strong cosmic censorship conjecture: a generically perturbed maximal Cauchy development is expected to lose the smooth extension across its Cauchy horizon, restoring predictability in the appropriate regularity class. The exact smooth inner horizon is unstable, not a robust failure of deterministic evolution. Curvature divergence supports an obstruction to a $C^2$ metric extension; it does not by itself exclude every continuous metric extension. The regularity class is part of the conjecture, and the blueshift argument is evidence, not a general theorem proving it.