Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 2

Every metric coefficient is independent of $t$ and $\varphi$. The corresponding coordinate flows therefore preserve the metric, sosatisfy the Killing equation and are Killing vector fields. They generate stationarity and axial symmetry respectively.

Put $\mu =r_{+}(r_{+}^2+a^2)$. At large $r$, the one-form dual to $K$ hasOnly the $dr\wedge dt$ term contributes to the constant-$t,r$ Komar integral, andWith the stated orientation, the asymptotic Hodge star operator givesThe angular integral is the area of the unit four-sphere,ConsequentlyThis is the Komar mass of the Singly rotating six-dimensional Myers-Perry black hole in the units of the question.

Substituting the two differential coordinate transformations cancels every coefficient singular as ${\Delta }^{-1}$, so the ingoing Kerr-like coordinates are regular at $r=r_{+}$. The inverse metric applied to the normal covector $dr$ hasIts norm is $g^{rr}=\Delta /\Sigma$, which vanishes at $r=r_{+}$. Thus that level set is a null hypersurface. On it the raised normal is proportional toSince the transformed stationary and axial Killing fields are $K={\partial }_v$ and $m={\partial }_{\chi }$, the horizon is a Killing horizon generated by

For this horizon generator, direct evaluation of ${\xi }^b{\nabla }_b{\xi }^a=\kappa {\xi }^a$, or the radial derivative of $\Delta$, givesBecause $\Delta =r^2+a^2-\mu /r$ and $\mu =r_{+}(r_{+}^2+a^2)$,and thereforeThe induced horizon cross-section has volume elementUsing the unit-four-sphere integral from part b gives

On $\theta =\lambda =0$, the radial function $r\Delta =r^3+a^{2}r-\mu$ is strictly increasing and has the single positive root $r_{+}$. The curvature invariant diverges at $r=0$, and $g^{rr}=\Delta /\Sigma <0$ immediately inside the horizon, so this is a spacelike singularity. The maximally extended Penrose diagram therefore has the Schwarzschild form: two asymptotically flat exterior diamonds separated by future and past event horizons, with a spacelike future singularity above the black-hole regions and a spacelike past singularity below the white-hole regions. A collapse spacetime retains one exterior, the future horizon, and the future spacelike singularity.