arXiv subjects
Publications and source records attributed to Maciej Dunajski.
The twistor correspondence provides a duality between curves in $S^4$ and ruled surfaces in $\mathbb{CP}^3$ in which the conformal properties of the former are reflected in the projective properties of the latter. We use this to characterise a $14$--dimensional family of homogeneous ruled surfaces in $\mathbb{CP}^3$ which correspond to loxodromes in $S^4$.
We demonstrate that all anti-self-dual Einstein metrics with non--zero cosmological constant $Λ$ locally arise from solutions of a single second order PDE introduced by Lipstein and Nagy. We show how this equation fits into the heavenly formalism of Plebański, and establish a Lax pair. Finally we show how Plebański's second heavenly equation arises in the limit as $Λ\rightarrow 0$.
We numerically study a formation of near extremal horizons from a gravitational collapse of radially symmetric gravitational waves in $4+1$ dimensions within the framework of pure Einstein gravity with positive cosmological constant. Evolution of a regular initial data with cosmological horizon leads to a formation of a black hole with mass exceeding $99\%$ of the extremal value corresponding to the black hole and cosmological horizons coinciding. We demonstrate how our results fit within the framework of characteristic gluing, and present some evidence that the third law of black hole thermodynamics may not hold in the cosmological context, where the extremality corresponds to the maximal mass of the Schwarzschild black hole in de--Sitter space.
We prove (Theorem 1.1.) that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained in \cite{DL23} for the extremal black hole horizons and completes the classification of compact quasi-Einstein 2-manifolds in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on (non-compact) 2-manifolds and on $S^2 \times S^1$.
We construct all axi-symmetric non-gradient $m$-quasi-Einstein structures on a two-sphere. This includes the spatial cross-section of the extreme Kerr black hole horizon corresponding to $m=2$, as well as a family of new regular metrics with $m\neq 2$ given in terms of hypergeometric functions. We also show that in the case $m=-1$ with vanishing cosmological constant the only orientable compact solution in dimension two is the flat torus, which proves that there are no compact surfaces with a metrisable affine connection with skew Ricci tensor.
Wojciech Kamiński has provided a non real-analytic counterexample to our claim in [1] that conformal geodesics cannot spiral. This erratum illustrates how the proof of Lemma 4.6 [1] (on which our claim was based) fails.
These notes are based on lecture courses I gave to third year mathematics students at Cambridge. They could form a basis of an elementary one--term lecture course on integrable systems covering the Arnold-Liouville theorem, inverse scattering transform, Hamiltonian methods in soliton theory and Lie point symmetries. No knowledge beyond basic calculus and ordinary differential equation is assumed.
We disprove the Euclidean Einstein--Maxwell Black Hole Uniqueness Conjecture, and thus demonstrate that the semi-classical properties of coupled gravitational and electromagnetic fields are more subtle than expected from Lorentzian general relativity, where the Kerr-Newman family of metrics yields the most general stationary and asymptotically flat black holes with a single event horizon. This is achieved by an explicit construction of a new three--parameter family of asymptotically flat Einstein--Maxwell instantons. These solutions are toric, regular, and free of conical and orbifold singularities on the manifold $M=\CP^2\setminus S^1$. In the case of vanishing charge, these instantons reduce to the Chen--Teo Ricci flat instantons.
We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non--linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $Λ^3(\cT)$, where ${\cT}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms. Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterisation of algebraically special Riemannian metrics of type $D$ in dimensions higher than four.
We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.
We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have a Lie algebra of Killing vector fields that contains $\mathfrak{sl}(2)\times \mathfrak{u}(1)$ in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the Kähler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector.
This is a review of gravitational instantons -- solutions to Riemannian Einstein or Einstein-Maxwell equations in four dimensions which yield complete metrics on non-compact four-manifolds, and which asymptotically `look like' flat space. The review focuses on examples, and is based on lectures given by the author at the Cracow School of Theoretical Physics held in Zakopane in June 2024.
Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form.
In https://pubs.ams.org/ebooks/pspum/103.2, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space $M$ of stability conditions of a $CY_3$ triangulated category. Given a non-degeneracy assumption, a feature of this structure is a complex hyper-Kähler metric with homothetic symmetry on the total space $X = TM$ of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the $A_2$ Joyce structure in https://link.springer.com/article/10.1007/s00208-021-02337-w, we obtain an explicit expression for a hyper-Kähler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrödinger equation with deformed polynomial oscillator potential of odd degree $2n+1$. The metric is defined on a total space $X$ of complex dimension $4n$ and fibres over a $2n$--dimensional manifold $M$ which can be identified with the unfolding of the $A_{2n}$-singularity. The hyper-Kähler structure is shown to be compatible with the natural symplectic structure on $M$ in the sense of admitting an affine symplectic fibration as defined in https://link.springer.com/article/10.1007/s11005-021-01388-z. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Plebański's heavenly equations that govern the hyper-Kähler condition. We introduce the notion of a projectable hyper-Lagrangian foliation and show that in dimension four such a foliation of $X$ leads to a linearisation of the heavenly equation. The hyper-Kähler metrics constructed here are shown to admit such a foliation.
We determine the frequency ratios $τ\equiv ω_z/ω_ρ$ for which the Hamiltonian system with a potential \[ V=\frac{1}{r}+\frac{1}{2}\Big({ω_ρ}^2(x^2+y^2)+{ω_z}^2 z^2\Big) \] is completely integrable. We relate this result to the existence of conformal Killing tensors of the associated Eisenhart metric on $\mathbb{R}^{1, 4}$. Finally we show that trajectories of a particle moving under the influence of the potential $V$ are not unparametrised geodesics of any Riemannian metric on $\mathbb{R}^3$.
Radial solutions to the elliptic sinh-Gordon and Tzitzeica equations can be interpreted as Abelian vortices on certain surfaces of revolution. These surfaces have a conical excess angle at infinity (in a way which makes them similar to Elizabethan ruff collars). While they can not be embedded in the Euclidean 3-space, we will show that they can be globally embedded in the hyperbolic space. The existence of these hyperbolic embeddings follows from the asymptotic analysis of a Painleve III ODE.
I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. At the non--relativistic level there is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes. This is the cosmological analogue of the Penrose effect, which can also be seen as a non--relativistic limit of the Unruh effect. The symmetries of the associated Schrödinger equation are generated by the Newton--Hooke algebra, which arises from a non--relativistic limit of a cosmological twistor space.
A quantum mechanical model for the Neon-20 nucleus is developed that allows for the splitting of a bipyramidal structure of five alpha-partices into an alpha-particle and an Oxygen-16 nucleus. The geometry of the configuration space is assumed to be a 3-dimensional spatial wormhole, and on the wormhole background there is an attractive short-range potential. This leads to a radial Schrödinger equation of the Heun form, which simplifies for threshold bound states to an associated Legendre equation that has explicit solutions. The energies of the true bound states for all spin/parities are numerically calculated, and match those of the well-established $K^π=0^+$, $K^π=0^-$, and certain higher rotational bands of Neon-20.