We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are: compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.

In general relativity, the dynamics of spinning particles is governed by the Mathisson-Papapetrou-Dixon equations, which are most commonly applied to massive bodies, but the framework also works in the massless case. Such massless versions naturally arise, for example, in the description of energy centroids of high-frequency wave packets. In this work, we consider massless spinning particles in spacetimes with hidden symmetries and we derive the generalized conservation laws associated with conformal Killing-Yano tensors. We then show that the spin Hall equations, a particular case of the Mathisson-Papapetrou-Dixon equations restricted to massless particles with longitudinal angular momentum, are completely integrable in a large class of type D spacetimes. Additionally, we also show that for massive spinning particles, the generalized Carter constant associated with Killing-Yano tensors is conserved independently of the choice of spin supplementary condition.

We prove infinitesimal rigidity and integrability of the moduli space for Hermitian gravitational instantons. Together with the recent proof by Biquard, Gauduchon, and LeBrun of local rigidity for Hermitian instantons, this completes the picture of the moduli space of Hermitian gravitational instantons, both for the compact and non-compact cases. An important step in the proof is to show that provided certain boundary conditions hold, a curve of Riemannian metrics passing through a Hermitian non-Kähler Einstein metric is conformally Kähler to second perturbative order. This uses ideas of Wu and LeBrun.

Uniqueness results for asymptotically locally flat and asymptotically flat S^1-symmetric gravitational instantons are proved using a divergence identity of the type used in uniqueness proofs for static black holes, combined with results derived from the G-signature theorem. Our results include a proof of the S^1-symmetric version of the Euclidean Black Hole Uniqueness conjecture, a uniqueness result for the Taub-bolt family of instantons, as well as a proof that an ALF S^1-symmetric instanton with the topology of the Chen-Teo family of instantons is Hermitian.

We develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with arbitrary connections. We construct the associated star product and compute its semiclassical expansion up to third order in the expansion parameter. A central feature of our approach is a one-to-one correspondence between formally self-adjoint symbols and formally self-adjoint operators, extending known results from flat space to curved geometries. In addition, we analyze the Moyal equation satisfied by the Wigner function in this setting and provide explicit computations of Weyl symbols for several physically significant operators, including the Dirac, Maxwell, linearized Yang-Mills, and linearized Einstein operators. Our results lay the foundation for future developments in quantum field theory on curved spacetimes, semiclassical analysis, and chiral kinetic theory.

We consider the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike 2-sphere Sigma approaches an apparent horizon from the exterior. We find that if the apparent horizon Sigma cannot be isometrically embedded into R-3 , the Wang-Yau quasi-local energy diverges at the horizon. Further, in this situation, the optimal embedding equation does not admit any solution that is smooth up to the horizon.

The main result of this paper is that infinitesimal rigidity of the moduli space for Hermitian gravitational instantons holds. An important step in the proof is to show that, provided certain boundary conditions hold, a curve of Riemannian metrics passing through a Hermitian non-Kähler Einstein metric is conformally Kähler to second perturbative order. This uses ideas of Wu and LeBrun. Hermitian non-Kähler Einstein 4-manifolds have a quasi-locally conserved charge, which is shown to correspond to a parameter of the moduli space. This charge is evaluated for all explicitly known examples. It follows from the proof of our main theorem that infinitesimal Einstein deformations admit a closed 2-form that measures the perturbation in the moduli parameter.

In this note, we prove the Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons which are either asymptotically locally flat (ALF) and Ricci-flat, or compact and Einstein with positive cosmological constant. We show that the Teukolsky equation on any such manifold is a positive definite operator. We also discuss the compatibility of the results with the existence of negative modes associated to variational instabilities.

We review Wang-Yau quasi-local definitions along the line of gravitational Hamiltonian. This makes clear the connection and difference between Wang-Yau definition and Brown-York or even global ADM definition. We make a brief comment on admissibility condition in Wang-Yau quasi-lcoal mass. We extend the positivity proof for Wang-Yau quasi-local energy to allow possible presence of strictly stable apparent horizons through establishing solvability of Dirac equation in certain 3-manifolds that possess cylindrical ends, as in the case of Jang's graph blowing up at marginally outer trapped surfaces.

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the domain of outer communication of the Kerr black hole spacetime. The estimates are valid for the full subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky Master Equation holds. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity.

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1-form wave operator on the Kerr metric in the large a case. By mode solutions we mean solutions of the form e(-it*sigma)h(-)(r, theta, phi) where t(*) is a suitable time variable. The corresponding Fourier-transformed 1-form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and hQ has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of H & auml;fner et al. (2021). No mode solutions exist for (sic)sigma >= 0,sigma not equal 0. For sigma = 0 mode solutions are Coulomb solutions for the 1-form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and B & auml;ckdahl (2018) and Aksteiner et al. (2021).

In this paper, we study some of the properties of the G -* 0 limit of the Kerr-Newman solution of Einstein-Maxwell equations. Carter noted the near equality between the gyromagnetic ratio, or g-factor, of the Kerr-Newman solution and of the electron. This observation is a consequence of the multipole structure of the Kerr-Newman field. We discuss additional coincidences between the Kerr-Newman multipole structure and the properties of the electron. In contrast to the Coulomb field, this spinning Maxwell field has a finite Lagrangian. Moreover, by evaluating the Lagrangian for the superposition of two such KerrNewman electromagnetic fields on a flat background, we are able to find their interaction potential. This yields a correction to the Coulomb interaction due to the spin of the field.

The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci-flat family of metrics on CP2/S-1, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. In this paper we show that the Chen-Teo instanton is Hermitian and non-Kahler. It follows that all known examples of gravitational instantons are Hermitian.

A quasi-local mass, typically defined as an integral over a spacelike 2-surface Σ, should encode information about the gravitational field within a finite, extended region bounded by Σ. Therefore, in attempts to quantize gravity, one may consider an infinite dimensional space of 2-surfaces instead of an infinite dimensional space of 4-dimensional Lorentzian spacetimes. However, existing definitions for quasilocal mass only applies to surfaces outside an horizon whose mean curvature vector is spacelike. In this paper, we propose an extension of the Wang-Yau quasi-local energy/mass to surfaces with timelike mean curvature vector, including in particular trapped surfaces. We adopt the same canonical gauge as in the Wang-Yau quasi-local energy but allow the pulled back "killing vector" to the physical spacetime to be spacelike. We define the new quasi-local energy along the Hamiltonian formulation of the Wang-Yau quasi-local energy. The new definition yields a positive definite surface energy density and a new divergence free current. Calculations for coordinate spheres in Kerr family spacetime are shown. In the spherical symmetric case, our definition reduces to a previous definition .

Identifying a general quasi-local notion of energy-momentum and angular momentum would be an important advance in general relativity with potentially important consequences for mathematical and astrophysical studies in general relativity. In this paper we study a promising approach to this problem first proposed by Wang and Yau in 2009 based on isometric embeddings of closed surfaces in Minkowski space. We study the properties of the Wang-Yau quasi-local mass in high accuracy numerical simulations of the head-on collisions of two non-spinning black holes within full general relativity. We discuss the behavior of the Wang-Yau quasi-local mass on constant expansion surfaces and we compare its behavior with the irreducible mass. We investigate the time evolution of the Wang-Yau Quasi-local mass in numerical examples. In addition we discuss mathematical subtleties in defining the Wang-Yau mass for marginally trapped surfaces.

In many areas of physics, the propagation of wave packets carrying intrinsic angular momentum is generally influenced by spin-orbit interactions. This is the main mechanism behind spin Hall effects, which result in wave packets following spin-dependent trajectories. Spin Hall effects have been observed in several experiments for electrons in condensed matter systems and for light propagating in inhomogeneous optical media. Similar effects have also been predicted for wave packets propagating in inhomogeneous gravitational fields. We give a brief introduction to gravitational spin Hall effects, emphasizing the analogies with the spin Hall effect of light in optics. Furthermore, we review the most promising astrophysical avenues that could lead to experimental observations of the gravitational spin Hall effect.

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

This paper proves the stability, with respect to the evolution determined by the vacuum Einstein equations, of the Cartesian product of high-dimensional Minkowski space with a compact, Ricci-flat Riemannian manifold that admits a spin structure and a nonzero parallel spinor. Such a product includes the example of Calabi-Yau and other special holonomy compactifications, which play a central role in supergravity and string theory. The stability proved in this paper provides a counter example to an instability argument by Penrose.

In this paper, we introduce and explore the properties of a new gauge choice for the vacuum Einstein equation inspired by the ingoing and outgoing radiation gauges (IRG, ORG) for the linearized vacuum Einstein equation introduced by Chrzanowski in his work on metric reconstruction (Chrzanowski in Phys Rev D 11:2042-2062, 1975) on the Kerr background. It has been shown by Price et al. (Class Quantum Gravity 24:2367-2388, 2007) that the IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In (Andersson et al. Stability for linearized gravity on the Kerr spacetime, 2019), the ORG was used in proving linearized stability for the Kerr spacetime, and the new non-linear radiation gauge introduced here is a direct generalization of that gauge condition, and is intended to be used to study the stability of Kerr black holes under the evolution generated by the vacuum Einstein equation.

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity $(\mathbf{v}_0, \rho_0, \mathbf{w}_0) \in H^{2+} \times H^{2+} \times H^{2}$, improving on the regularity conditions of \cite{WQEuler}. The continuous dependence on initial data for rough solutions of the compressible Euler system is new, even with the same regularity conditions as in \cite{WQEuler}. In addition, we prove new local well-posedness results for the 3D compressible Euler system with entropy.