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Register a new userConformal geodesics on gravitational instantons
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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.
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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 prove uniqueness and existence theorems for four-dimensional asymptotically flat, Ricci-flat, gravitational instantons with a torus symmetry. In particular, we prove that such instantons are uniquely characterised by their rod structure, which is data that encodes the fixed point sets of the torus action. Furthermore, we establish that for every admissible rod structure there exists an instanton that is smooth up to possible conical singularities at the axes of symmetry. The proofs involve adapting the methods that are used to establish black hole uniqueness theorems, to a harmonic map formulation of Ricci-flat metrics with torus symmetry, where the target space is directly related to the metric (rather than auxiliary potentials). We also give an elementary proof of the nonexistence of asymptotically flat toric half-flat instantons. Finally, we derive a general set of identities that relate asymptotic invariants such as the mass to the rod structure.
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 $mathbb{CP}^2$, Eguchi--Hanson and anti-self-dual Taub-NUT we obtain upper and lower bounds on the embedding class.
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
We study non-Einstein Bach-flat gravitational instanton solutions that can be regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions of Einstein gravity to conformal gravity. These solutions include non-Einstein spaces which are either asymptotically locally flat spacetimes (ALF) or asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with different asymptotic conditions exist: we find geometries that present a weakened AlAdS asymptotia, exhibiting the typical low decaying mode of conformal gravity. This permits to identify the simple Neumann boundary condition that, as it happens in the asymptotically AdS sector, selects the Einstein solution out of the solutions of conformal gravity. All the geometries present non-vanishing Hirzebruch signature and Euler characteristic, being single-centered instantons. We compute the topological charges as well as the Noether charges of the Taub-NUT/Bolt and Eguchi-Hanson spacetimes, which happen to be finite. This enables us to study the thermodynamic properties of these geometries.
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