We use the isometric embedding of the spatial horizon of fast rotating Kerr black hole in a hyperbolic space to compute the quasi-local mass of the horizon for any value of the spin parameter $j=J/m^2$. The mass is monotonically decreasing from twice
the ADM mass at $j=0$ to $1.76569m$ at $j=sqrt{3}/2$. It then monotonicaly increases to a maximum around $j=0.99907$, and finally decreases to $2.01966m$ for $j=1$ which corresponds to the extreme Kerr black hole.
Conformal geodesics are solutions to a system of third order of equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third--order
conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth order ODEs arising from this Lagrangian, and show that some of its integral curves are spirals.
We construct the normal forms of null-Kahler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified set
ting) appear in the Bridgeland stability conditions of the moduli spaces of Calabi-Yau three-folds. Using twistor methods we show that, in dimension four - where there is a connection with dispersionless integrability - the cohomogeneity-one anti-self-dual null-Kahler metrics are generically characterised by solutions to Painleve I or Painleve II ODEs.
We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity $mathcal{I}^-$ contains the whole of the future conformal infinity $mathcal{I}
^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the connection with scattering theory in the presence of positive mass.
We construct isometric and conformally isometric embeddings of some gravitational instantons in $mathbb{R}^8$ and $mathbb{R}^7$. In particular we show that the embedding class of the Einstein--Maxwell instanton due to Burns is equal to $3$. For $math
bb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.
We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(infty)$ Toda equation. We give several examples of new, explicit solutions of the Toda equ
ation, and construct their mini--twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle $T^*N$ of any projective structure $(N, [ abla])$. We show that there is a canonical Einstein of metric on an $R^*$--bundle over $T^*N$, with a connection whose curvature is the pull--back of the natural symplectic structure from $T^*N$.
We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kahler four--manifold the conformal geodesic equations reduce to geodesic equations of
a charged particle moving in a constant self--dual magnetic field. In the case of the anti--self--dual Taub NUT instanton we integrate these equations completely by separating the Hamilton--Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four--manifold which is not a symmetric space. In the case of the Eguchi--Hanson we find all conformal geodesics which lie on the three--dimensional orbits of the isometry group. In the non--hyper--Kahler case of the Fubini--Study metric on $CP^2$ we use the first integrals arising from the conformal Killing--Yano tensors to recover the known complete integrability of conformal geodesics.
We find necessary and sufficient conditions for existence of a locally isometric embedding of a vacuum space-time into a conformally-flat 5-space. We explicitly construct such embeddings for any spherically symmetric Lorentzian metric in $3+1$ dimens
ions as a hypersurface in $R^{4, 1}$. For the Schwarzschild metric the embedding is global, and extends through the horizon all the way to the $r=0$ singularity. We discuss the asymptotic properties of the embedding in the context of Penroses theorem on Schwarzschild causality. We finally show that the Hawking temperature of the Schwarzschild metric agrees with the Unruh temperature measured by an observer moving along hyperbolae in $R^{4, 1}$.
We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metri
c projective compactification. We construct a para--$c$--projective compactification of neutral signature Einstein metrics canonically defined on certain rank--$n$ affine bundles $M$ over $n$-dimensional manifolds endowed with projective structures.
We construct a new class of solutions to the dispersionless hyper--CR equation, and show how any solution to this equation gives rise to a supersymmetric Einstein--Maxwell cosmological space--time in $(3+1)$--dimensions.
