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Register a new userSelf-dual Gravitational Instantons in Conformal Gravity: Conserved Charges and Thermodynamics
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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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We show that there are a further infinite number of, previously unknown, supertranslation charges. These can be viewed as duals of the known BMS charges corresponding to supertranslations. In Newman-Penrose language, these new supertranslation charges roughly correspond to the imaginary part of the leading term in $psi_2$. We find these charges by dualising the Barnich-Brandt asymptotic charges and argue that this prescription gives rise to new bona fide charges at null infinity.
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.
We consider the consequences of the dual gravitational charges for the phase space of radiating modes, and find that they imply a new soft NUT theorem. In particular, we argue that the existence of these new charges removes the need for imposing boundary conditions at spacelike infinity that would otherwise preclude the existence of NUT charges.
Dual gravitational charges have been recently computed from the Holst term in tetrad variables using covariant phase space methods. We highlight that they originate from an exact 3-form in the tetrad symplectic potential that has no analogue in metric variables. Hence there exists a choice of the tetrad symplectic potential that sets the dual charges to zero. This observation relies on the ambiguity of the covariant phase space methods. To shed more light on the dual contributions, we use the Kosmann variation to compute (quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a formula that illustrates comprehensively why the dual contribution to the Hamiltonian charges: (i) vanishes for exact isometries and asymptotic symmetries at spatial infinity; (ii) persists for asymptotic symmetries at future null infinity, in addition to the usual BMS contribution. Finally, we point out that dual gravitational charges can be equally derived using the Barnich-Brandt prescription based on cohomological methods, and that the same considerations on asymptotic symmetries apply.
We provide a Hamiltonian derivation of recently discovered dual BMS charges. In order to do so, we work in the first order formalism and add to the usual Palatini action, the Holst term, which does not contribute to the equations of motion. We give a method for finding the leading order integrable dual charges `a la Wald-Zoupas and construct the corresponding charge algebra. We argue that in the presence of fermions, the relevant term that leads to dual charges is the topological Nieh-Yan term.
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