2022 | *Improvement on the rough solutions of 3D compressible Euler equations* | Huali Zhang, Lars AnderssonLink zum Preprint

### Improvement on the rough solutions of 3D compressible Euler equations

#### Autoren: Huali Zhang, Lars Andersson
(2022)

Wang proved the local existence and uniqueness of solutions for 3D isentropic compressible Euler equations by vector-field approach if the initial velocity \( v_0\), logarithmic density \(\rho_0\) and specific vorticity \(w_0\) satisfy \( (v_0, \rho_0, w_0) \in H^s\times H^s \times H^{s_0} (2<s_0<s)\), and the author proposed a conjecture that the local solutions of 3D isentropic, compressible Euler equations is well posed if \( (v_0, \rho_0,w_0) \in H^{2+}\times H^{2+}\times H^{\frac{3}{2}+}\). Inspired by the pioneer work, we study the local existence, uniqueness, and continuous dependence of rough solutions of 3D compressible Euler equations with dynamic vorticity and entropy.

We firstly prove the local well-posedness of solutions via Smith-Tataru's approach when the initial velocity \(v_0\), logarithmic density \(\rho_0\), entropy \(h_0\), and specific vorticity \(w_0\) satisfy \( (v_0, \rho_0,h_0,w_0) \in H^s \times H^s \times H^{s_0+1}\times H^{s_0}(2<s_0<s)\). So this recovers Wang's result. This result has three distinct features: (i) a Strichartz estimate of linear wave equations endowed with a acoustic metric, (ii) a continuous dependence on initial data with low regularity, (iii) a type of Strichartz estimates of solutions with a regularity of the velocity 2+, density 2+, entropy \(\frak52+\) and specific vorticity \(\frak32+\). Secondly, based on these key Strichartz estimates, we obtain the existence and uniqueness of solutions of 3D compressible Euler equations if the initial data satisfies \( (v_0, \rho_0, h_0, w_0) \in H^{\frac52} \times H^{\frac52} \times H^{\frac52+}\times H^{\frac32+}\). Furthermore, we prove the existence and uniqueness of solutions of 3D isentropic compressible Euler equations if \((v_0, \rho_0, w_0) \in H^{2+} \times H^{2+} \times H^{2}\).

2022 | *Mode analysis for the linearized Einstein equations on the Kerr metric : the large \(\mathfrak{a}\) case* | Lars Andersson, Dietrich Häfner, Bernard F. WhitingLink zum Preprint

### Mode analysis for the linearized Einstein equations on the Kerr metric : the large \(\mathfrak{a}\) case

#### Autoren: Lars Andersson, Dietrich Häfner, Bernard F. Whiting
(2022)

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 \(e^{-i t_\ast \sigma} \tilde h (r,\theta,\varphi)\) where \(t_\ast\) 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 \(\tilde h\) has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy. No mode solutions exist for \(\mathfrak{J} \sigma \geq 0\), \(\sigma \neq 0\). 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 7−dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by the third author and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime due to Aksteiner et al.

2021 | *Cauchy Problem for Incompressible Neo-Hookean materials* | Lars Andersson, Lev KapitanskiZeitschrift: to appear in Archive for Rational Mechanics and AnalysisLink zum Preprint

### Cauchy Problem for Incompressible Neo-Hookean materials

#### Autoren: Lars Andersson, Lev Kapitanski
(2021)

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension *d*≥2. We are here interested primarily in the low regularity case, *s*≤*s**c**r**i**t*=*d*/2+1. For *d*=2,3, we prove existence and uniqueness for *s*0<*s*≤*s**c**r**i**t*, and we can prove well-posedness, but for a smaller range, *s*1<*s*≤*s**c**r**i**t*,

if *d*=2if *d*=3,*s*0=74,*s*1=74+65√−78,*s*0=2,*s*1=1+32−−√

We consider the initial deformations of the form *x*(0,*ξ*)=*A**ξ*+*φ*(*ξ*), where *A* is a constant *S**L*(*d*,R) matrix, and *φ*∈*H**s*+1. For the full range (in *s*) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of *φ*.

A key observation in the proof is that the equations of evolution for the vorticities decomposes 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**c**r**i**t* is necessary.

Zeitschrift:

to appear in Archive for Rational Mechanics and Analysis

2021 | *Gravitational Instantons and special geometry* | Steffen Aksteiner, Lars AnderssonLink zum Preprint

### Gravitational Instantons and special geometry

#### Autoren: Steffen Aksteiner, Lars Andersson
(2021)

In this paper we show that the Chen-Teo gravitational instanton [14], which provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture, is Hermitian and non-Kaehler. It follows that all known examples of gravitational instantons are Hermitian.

2021 | *On the rough solutions of 3D compressible Euler equations: an alternative proof* | Huali Zhang, Lars AnderssonLink zum Preprint

### On the rough solutions of 3D compressible Euler equations: an alternative proof

#### Autoren: Huali Zhang, Lars Andersson
(2021)

The well-posedness of Cauchy problem of 3D compressible Euler equations is studied. By using Smith-Tataru's approach \cite{ST}, we prove the local existence, uniqueness and stability of solutions for Cauchy problem of 3D compressible Euler equations, where the initial data of velocity, density, specific vorticity *v*,*ρ*∈*H**s*,*ϖ*∈*H**s*0(2<*s*0<*s*). It's an alternative and simplified proof of the result given by Q. Wang in \cite{WQEuler}.

2021 | *Nonlinear radiation gauge for near Kerr spacetimes* | Lars Andersson, Thomas Bäckdahl, Pieter Blue, Siyuan MaLink zum Preprint

### Nonlinear radiation gauge for near Kerr spacetimes

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue, Siyuan Ma
(2021)

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 [DOI:10.1103/PhysRevD.11.2042] on the Kerr background. It has been shown by Price, Shankar and Whiting [arXiv:gr-qc/0611070] that IRG/ORG are consistent gauges for the linearized vacuum Einstein equation on Petrov type II backgrounds. In [arXiv:1903.03859] 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.

2021 | *Helicity and spin conservation in Maxwell theory and Linearized Gravity* | Lars Andersson, Sajad Aghapour, Reebhu BhattacharyyaZeitschrift: General Relativity and GravitationSeiten: Article number 102Band: 53Link zur Publikation
,
Link zum Preprint

### Helicity and spin conservation in Maxwell theory and Linearized Gravity

#### Autoren: Lars Andersson, Sajad Aghapour, Reebhu Bhattacharyya
(2021)

A duality symmetric formulation of linearized gravity has been introduced by Barnett and used to show the conservation of helicity. However, the relation between helicity and spin as well as the separate conservation of the spin and orbital parts of angular momentum, which is known to hold in Maxwell theory, was not considered. These conservation laws are known to follow from the conservation of the so-called helicity array, an analog of the zilch tensor, which includes helicity, spin, and spin-flux or infra-zilch. In the present paper we prove the conservation of spin and orbital angular momentum for linearized gravity on Minkowski space, and construct the analog of the helicity array for linearized gravity.

Zeitschrift:

General Relativity and Gravitation

Seiten:

Article number 102

2021 | *On the geometry of Petrov type II spacetimes* | Steffen Aksteiner, Lars Andersson, Bernardo Araneda, Bernard WhitingZeitschrift: Classical and Quantum GravitySeiten: 135023Band: 38Link zur Publikation
,
Link zum Preprint

### On the geometry of Petrov type II spacetimes

#### Autoren: Steffen Aksteiner, Lars Andersson, Bernardo Araneda, Bernard Whiting
(2021)

In general, geometries of Petrov type II do not admit symmetries in terms of Killing vectors or spinors. We introduce a weaker form of Killing equations which do admit solutions. In particular, there is an analog of the Penrose-Walker Killing spinor. Some of its properties, including associated conservation laws, are discussed. Perturbations of Petrov type II Einstein geometries in terms of a complex scalar Debye potential yield complex solutions to the linearized Einstein equations. The complex linearized Weyl tensor is shown to be half Petrov type N. The remaining curvature component on the algebraically special side is reduced to a first order differential operator acting on the potential.

Zeitschrift:

Classical and Quantum Gravity

2021 | *Helicity, Spin, and Infra-zilch of Light: a Lorentz Covariant Formulation* | Sajad Aghapour, Lars Andersson, Kjell RosquistZeitschrift: Ann. Phys.Seiten: 168535Band: 432Link zur Publikation
,
Link zum Preprint

### Helicity, Spin, and Infra-zilch of Light: a Lorentz Covariant Formulation

#### Autoren: Sajad Aghapour, Lars Andersson, Kjell Rosquist
(2021)

The spin part of angular momentum of the electromagnetic field is known since the 1990's to be a separately conserved quantity. Cameron et al. introduced the helicity array, a non-covariant analog of Lipkin's zilch tensor, which expresses the hierarchy of conservation laws including helicity, spin, as well as the spin-flux or infra-zilch. In this paper, a novel conserved Lorentz covariant tensor, termed the helicity tensor, is introduced. The conservation laws arising from the helicity array can be obtained from the helicity tensor. The Lorentz covariance of the helicity tensor is in contrast to previous formulations of the helicity hierarchy of conservation laws, which required the non-Lorentz covariant transverse gauge. The helicity tensor is shown to arise as a Noether current for a variational symmetry of a duality-symmetric Lagrangian for Maxwell theory.

2021 | *Compatibility complex for black hole spacetimes* | Steffen Aksteiner, Lars Andersson, Thomas Bäckdahl, Igor Khavkine, Bernard WhitingZeitschrift: Communications in Mathematical PhysicsLink zur Publikation
,
Link zum Preprint

### Compatibility complex for black hole spacetimes

#### Autoren: Steffen Aksteiner, Lars Andersson, Thomas Bäckdahl, Igor Khavkine, Bernard Whiting
(2021)

The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (S.A, T.B.) in (arXiv:1803.05341) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from (arXiv:1803.05341) with the first compatibility operator from that complex.

Zeitschrift:

Communications in Mathematical Physics

2021 | *Nilpotent symmetries as a mechanism for Grand Unification* | Lars Andersson, Andras Laszo, Blazej RubaZeitschrift: Journal of High Energy PhysicsVerlag: SpringerLink zur Publikation
,
Link zum Preprint

### Nilpotent symmetries as a mechanism for Grand Unification

#### Autoren: Lars Andersson, Andras Laszo, Blazej Ruba
(2021)

In the classic Coleman--Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact- \(U(1)\)- component, a Lorentz- \(SL(2,\mathbb{C})\) -component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman--Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.

Zeitschrift:

Journal of High Energy Physics

2021 | *Propagation of polarized gravitational waves* | Lars Andersson, Jérémie Joudioux, Marius A. Oancea, Ayush RajZeitschrift: Phys. Rev. DSeiten: 044053Band: 103Link zur Publikation
,
Link zum Preprint

### Propagation of polarized gravitational waves

#### Autoren: Lars Andersson, Jérémie Joudioux, Marius A. Oancea, Ayush Raj
(2021)

The propagation of high-frequency gravitational waves can be analyzed using the geometrical optics approximation. In the case of large but finite frequencies, the geometrical optics approximation is no longer accurate, and polarization-dependent corrections at first order in wavelength modify the propagation of gravitational waves via a spin-orbit coupling mechanism. We present a covariant derivation from first principles of effective ray equations describing the propagation of polarized gravitational waves, up to first-order terms in wavelength, on arbitrary spacetime backgrounds. The effective ray equations describe a gravitational spin Hall effect for gravitational waves and are of the same form as those describing the gravitational spin Hall effect of light, derived from Maxwell's equations.

2020 | *Gravitational spin Hall effect of light* | Marius A. Oancea, Jérémie Joudioux, I. Y. Dodin, D. E. Ruiz, Claudio F. Paganini, Lars AnderssonZeitschrift: Phys. Rev. DSeiten: 024075Band: 102Link zur Publikation
,
Link zum Preprint

### Gravitational spin Hall effect of light

#### Autoren: Marius A. Oancea, Jérémie Joudioux, I. Y. Dodin, D. E. Ruiz, Claudio F. Paganini, Lars Andersson
(2020)

The propagation of electromagnetic waves in vacuum is often described within the geometrical optics approximation, which predicts that wave rays follow null geodesics. However, this model is valid only in the limit of infinitely high frequencies. At large but finite frequencies, diffraction can still be negligible, but the ray dynamics becomes affected by the evolution of the wave polarization. Hence, rays can deviate from null geodesics, which is known as the gravitational spin Hall effect of light. In the literature, this effect has been calculated ad hoc for a number of special cases, but no general description has been proposed. Here, we present a covariant Wentzel-Kramers-Brillouin analysis from first principles for the propagation of light in arbitrary curved spacetimes. We obtain polarization-dependent ray equations describing the gravitational spin Hall effect of light. We also present numerical examples of polarization-dependent ray dynamics in the Schwarzschild spacetime, and the magnitude of the effect is briefly discussed. The analysis reported here is analogous to that of the spin Hall effect of light in inhomogeneous media, which has been experimentally verified.

2020 | *Global stability of spacetimes with supersymmetric compactifications* | Lars Andersson, Pieter Blue, Zoe Wyatt, Shing-Tung YauLink zum Preprint

### Global stability of spacetimes with supersymmetric compactifications

#### Autoren: Lars Andersson, Pieter Blue, Zoe Wyatt, Shing-Tung Yau
(2020)

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.

2020 | *The Zilch Electromagnetic Conservation Law in Variational Characteristic Form* | Sajad Aghapour, Lars Andersson, Kjell RosquistZeitschrift: J.Math.Phys.Seiten: 122902Band: 61Link zur Publikation
,
Link zum Preprint

### The Zilch Electromagnetic Conservation Law in Variational Characteristic Form

#### Autoren: Sajad Aghapour, Lars Andersson, Kjell Rosquist
(2020)

In this paper we consider the zilch conservation laws for Maxwell theory and demonstrate that in the duality-symmetric version of Maxwell theory, the zilch arises as a Noether current for a variational symmetry of the duality symmetric Lagrangian which we identify through an application of the reverse of the Noether theorem. A variational symmetry leaves Lagrangian invariant up to a total divergence, without restricting to solutions of the field equations. This fact was previously known only for the so-called chirality current, i.e. the 00-component of zilch.

2020 | *Nonlinear stability of the Milne model with matter* | Lars Andersson, David FajmanZeitschrift: Commun. Math. Phys.Seiten: 261–298Band: 378Link zur Publikation
,
Link zum Preprint

### Nonlinear stability of the Milne model with matter

#### Autoren: Lars Andersson, David Fajman
(2020)

We show that any 3+1-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein-Vlasov system. For the analysis of the Einstein equations we use the constant-mean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric L^2-estimates based on the Sasaki-metric. The resulting estimates on the energy momentum tensor are then upgraded by employing the natural continuity equation for the energy density.

Zeitschrift:

Commun. Math. Phys.

2020 | *Morawetz estimate for linearized gravity in Schwarzschild* | Lars Andersson, Pieter Blue, Jinhua WangZeitschrift: Annales Henri PoincaréVerlag: SpringerSeiten: 1–53Link zur Publikation
,
Link zum Preprint

### Morawetz estimate for linearized gravity in Schwarzschild

#### Autoren: Lars Andersson, Pieter Blue, Jinhua Wang
(2020)

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the r^{p} hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerilli variables decay as t^{-3/2} in fixed regions of r.

Zeitschrift:

Annales Henri Poincaré

2019 | *Variational principle for the Einstein-Vlasov equations* | Lars Andersson, Mikołaj KorzyńskiLink zum Preprint

### Variational principle for the Einstein-Vlasov equations

#### Autoren: Lars Andersson, Mikołaj Korzyński
(2019)

The Einstein-Vlasov equations govern Einstein spacetimes filled with matter which interacts only via gravitation. The matter, described by a distribution function on phase space, evolves under the collisionless Boltzmann equation, corresponding to the free geodesic motion of the particles, while the source of the gravitational field is given by the stress-energy tensor defined in terms of momenta of the distribution function. As no variational derivation of the Einstein-Vlasov system appears to exist in the literature, we here set out to fill this gap. In our approach we treat the matter as a generalized type of fluid, flowing in the tangent bundle instead of the spacetime. We present the actions for the Einstein-Vlasov system in both the Lagrangian and Eulerian pictures.

2019 | *Stability for linearized gravity on the Kerr spacetime* | Lars Andersson, Thomas Bäckdahl, Pieter Blue, Siyuan MaLink zum Preprint

### Stability for linearized gravity on the Kerr spacetime

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue, Siyuan Ma
(2019)

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the Kerr black hole exterior. 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 arXiv:1708.07385. 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.

The proof relies on three key steps. First, there are energy decay estimates for the Teukolsky equation, proved by applying weighted multiplier estimates to a system of spin-weighted wave equations derived from the Teukolsky equation, and making use of the pigeonhole principle for the resulting hierarchy of weighted energy estimates. Second, working in the outgoing radiation gauge, the linearized Einstein equations are written as a system of transport equations, driven by one of the Teukolsky scalars. Third, expansions for the relevant curvature, connection, and metric components can be made near null infinity. An analysis of the dynamics on future null infinity, together with the Teukolsky Starobinsky Identity plays an important role in the argument.

2019 | *An overview of the gravitational spin Hall effect* | Marius A. Oancea, Claudio F. Paganini, Jérémie Joudioux, Lars AnderssonLink zum Preprint

### An overview of the gravitational spin Hall effect

#### Autoren: Marius A. Oancea, Claudio F. Paganini, Jérémie Joudioux, Lars Andersson
(2019)

In General Relativity, the propagation of electromagnetic waves is usually described by the vacuum Maxwell's equations on a fixed curved background. In the limit of infinitely high frequencies, electromagnetic waves can be localized as point particles, following null geodesics. However, at finite frequencies, electromagnetic waves can no longer be treated as point particles following null geodesics, and the spin angular momentum of light comes into play, via the spin-curvature coupling. We will refer to this effect as the gravitational spin Hall effect of light. Here, we review a series of theoretical results related to the gravitational spin Hall effect of light, and we compare the predictions of different models. The analogy with the spin Hall effect in Optics is also explored, since in this field the effect is well understood, both theoretically and experimentally.

2019 | *On the asymptotic behavior of static perfect fluids* | Lars Andersson, Annegret Y. BurtscherZeitschrift: Annales Henri PoincaréVerlag: SpringerSeiten: 813-857Band: 20, no. 3Link zur Publikation
,
Link zum Preprint

### On the asymptotic behavior of static perfect fluids

#### Autoren: Lars Andersson, Annegret Y. Burtscher
(2019)

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear and polytropic-type equations of state. In order to capture the asymptotic behavior we introduce a notion of scaled quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In particular, these spacetimes are asymptotically simple.

Zeitschrift:

Annales Henri Poincaré

2019 | *New identities for linearized gravity on the Kerr spacetime* | Steffen Aksteiner, Lars Andersson, Thomas BäckdahlZeitschrift: Physical Review DVerlag: American Physical SocietySeiten: 044043Band: 99Link zur Publikation
,
Link zum Preprint

### New identities for linearized gravity on the Kerr spacetime

#### Autoren: Steffen Aksteiner, Lars Andersson, Thomas Bäckdahl
(2019)

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor M_{ab} which solves the linearized Einstein equations. The identity makes this manifest by relating M_{ab} to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of M_{ab} gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of nontrivial gauge invariants for linearized gravity.

Zeitschrift:

Physical Review D

Verlag:

American Physical Society

2018 | *On the geometry and topology of initial data sets with horizons* | Lars Andersson, Mattias Dahl, Gregory J. Galloway, Daniel PollackZeitschrift: Asian J. Math.Verlag: International PressSeiten: 863-881Band: 22, no. 5Link zur Publikation
,
Link zum Preprint

### On the geometry and topology of initial data sets with horizons

#### Autoren: Lars Andersson, Mattias Dahl, Gregory J. Galloway, Daniel Pollack
(2018)

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M,g,K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M\∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established by Eichmair, Galloway, and Pollack in dimension n = 3, to higher dimensions.

Zeitschrift:

Asian J. Math.

Verlag:

International Press

2018 | *Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime* | Lars Andersson, Pieter Blue, Jeremie JoudiouxZeitschrift: Commun. Part. Diff. Eq.Verlag: Taylor & FrancisSeiten: 47-65Band: 43, no. 1Link zur Publikation
,
Link zum Preprint

### Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

#### Autoren: Lars Andersson, Pieter Blue, Jeremie Joudioux
(2018)

This paper proves the existence of a bounded energy and integrated energy decay for solutions of the massless Vlasov equation in the exterior of a very slowly rotating Kerr spacetime. This combines methods previously developed to prove similar results for the wave equation on the exterior of a very slowly rotating Kerr spacetime with recent work applying the vector-field method to the relativistic Vlasov equation.

Zeitschrift:

Commun. Part. Diff. Eq.

2018 | *Wave and Dirac equations on manifolds* | Lars Andersson, Christian BärVerlag: de GruyterBuchtitel: Space - Time - MatterSeiten: 324-348Link zur Publikation
,
Link zum Preprint

### Wave and Dirac equations on manifolds

#### Autoren: Lars Andersson, Christian Bär
(2018)

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.

Buchtitel:

Space - Time - Matter

2018 | *Geometry of black hole spacetimes* | Lars Andersson, Thomas Bäckdahl, Pieter BlueReihe: London Mathematical Society Lecture Note SeriesVerlag: Cambridge University PressBuchtitel: T. Daudé, D. Häfner, J.-P. Nicolas (eds.): Asymptotic Analysis in General RelativitySeiten: 9-85Band: 443Link zur Publikation
,
Link zum Preprint

### Geometry of black hole spacetimes

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue
(2018)

These notes, based on lectures given at the summer school on Asymptotic Analysis in General Relativity, collect material on the Einstein equations, the geometry of black hole spacetimes, and the analysis of fields on black hole backgrounds. The Kerr model of a rotating black hole in vacuum is expected to be unique and stable. The problem of proving these fundamental facts provides the background for the material presented in these notes.

Among the many topics which are relevant for the uniqueness and stability problems are the theory of fields on black hole spacetimes, in particular for gravitational perturbations of the Kerr black hole, and more generally, the study of nonlinear field equations in the presence of trapping. The study of these questions requires tools from several different fields, including Lorentzian geometry, hyperbolic differential equations and spin geometry, which are all relevant to the black hole stability problem.

Reihe:

London Mathematical Society Lecture Note Series

Verlag:

Cambridge University Press

Buchtitel:

T. Daudé, D. Häfner, J.-P. Nicolas (eds.): Asymptotic Analysis in General Relativity

2017 | *Mode stability on the real axis* | Lars Andersson, Siyuan Ma, Claudio Paganini, Bernard F. WhitingZeitschrift: J. Math. Phys.Verlag: AIP PublishingSeiten: 072501Band: 58, no. 7Link zur Publikation
,
Link zum Preprint

### Mode stability on the real axis

#### Autoren: Lars Andersson, Siyuan Ma, Claudio Paganini, Bernard F. Whiting
(2017)

A generalization of the mode stability result of Whiting (1989) for the Teukolsky equation is proved for the case of real frequencies. The main result of the paper states that a separated solution of the Teukolsky equation governing massless test fields on the Kerr spacetime, which is purely outgoing at infinity, and purely ingoing at the horizon, must vanish. This has the consequence, that for real frequencies, there are linearly independent fundamental solutions of the radial Teukolsky equation which are purely ingoing at the horizon, and purely outgoing at infinity, respectively. This fact yields a representation formula for solutions of the inhomogenous Teukolsky equation.

Zeitschrift:

J. Math. Phys.

2017 | *Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System* | Lars Andersson, Nishanth Gudapati, Jeremie SzeftelZeitschrift: Ann. PDEVerlag: SpringerSeiten: Art. 13Band: 3, no. 2Link zur Publikation
,
Link zum Preprint

### Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

#### Autoren: Lars Andersson, Nishanth Gudapati, Jeremie Szeftel
(2017)

In this paper we consider the equivariant 2+1 dimensional Einstein-wave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.

2017 | *A new tensorial conservation law for Maxwell fields on the Kerr background* | Lars Andersson, Thomas Bäckdahl, Pieter BlueZeitschrift: J. Differential Geom.Verlag: Project EuclidSeiten: 163-176Band: 105, no. 2Link zur Publikation
,
Link zum Preprint

### A new tensorial conservation law for Maxwell fields on the Kerr background

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue
(2017)

A new, conserved, symmetric tensor field for a source-free Maxwell test field on a four-dimensional spacetime with a conformal Killing–Yano tensor, satisfying a certain compatibility condition, is introduced. In particular, this construction works for the Kerr spacetime.

Zeitschrift:

J. Differential Geom.

2016 | *Dynamical compact elastic bodies in general relativity* | Lars Andersson, Todd A. Oliynyk, Bernd G. SchmidtZeitschrift: Archive for Rational Mechanics and AnalysisVerlag: SpringerSeiten: 849-887Band: 220, no. 2Link zur Publikation
,
Link zum Preprint

### Dynamical compact elastic bodies in general relativity

#### Autoren: Lars Andersson, Todd A. Oliynyk, Bernd G. Schmidt
(2016)

We prove the local existence of solutions to the Einstein-Elastic equations that represent self-gravitating, relativistic elastic bodies with compact support.

Zeitschrift:

Archive for Rational Mechanics and Analysis

2016 | *Decay of solutions to the Maxwell equation on the Schwarzschild background* | Lars Andersson, Thomas Bäckdahl, Pieter BlueZeitschrift: Class. Quantum Grav.Verlag: IOP PublishingSeiten: 085010Band: 33, no. 8Link zur Publikation
,
Link zum Preprint

### Decay of solutions to the Maxwell equation on the Schwarzschild background

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue
(2016)

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab}(\partial_t)^ b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.

Zeitschrift:

Class. Quantum Grav.

2016 | *Gauge-invariant perturbations of Schwarzschild spacetime* | Abhay G. Shaw, Bernard F. Whiting, Steffen Aksteiner, Lars Andersson, Thomas BäckdahlLink zum Preprint

### Gauge-invariant perturbations of Schwarzschild spacetime

#### Autoren: Abhay G. Shaw, Bernard F. Whiting, Steffen Aksteiner, Lars Andersson, Thomas Bäckdahl
(2016)

We study perturbations of Schwarzschild spacetime in a coordinate-free, covariant form. The GHP formulation, having the advantage of not only being covariant but also tetrad-rotation invariant, is used to write down the previously known odd- and even-parity gauge-invariants and the equations they satisfy. Additionally, in the even-parity sector, a new invariant and the second order hyperbolic equation it satisfies are presented. Chandrasekhar's work on transformations of solutions for perturbation equations on Schwarzschild spacetime is translated into the GHP form, i.e., solutions for the equations of the even- and odd-parity invariants are written in terms of one another, and the extreme Weyl scalars; and solutions for the equations of these latter invariants are also written in terms of one another. Recently, further gauge invariants previously used by Steven Detweiler have been described. His method is translated into GHP form and his basic invariants are presented here. We also show how these invariants can be written in terms of curvature invariants.

2015 | *Self-gravitating elastic bodies* | Lars AnderssonReihe: Fundamental Theories of PhysicsVerlag: SpringerBuchtitel: D. Puetzfeld, C. Lämmerzahl, B. Schutz (eds.): Equations of Motion in Relativistic GravitySeiten: 543-559Band: Volume 179Link zur Publikation
,
Link zum Preprint

### Self-gravitating elastic bodies

#### Autoren: Lars Andersson
(2015)

Extended objects in GR are often modelled using distributional solutions of the Einstein equations with point-like sources, or as the limit of infinitesimally small "test" objects. In this note, I will consider models of finite self-gravitating extended objects, which make it possible to give a rigorous treatment of the initial value problem for (finite) extended objects.

Reihe:

Fundamental Theories of Physics

Buchtitel:

D. Puetzfeld, C. Lämmerzahl, B. Schutz (eds.): Equations of Motion in Relativistic Gravity

2015 | *Hidden symmetries and decay for the wave equation on the Kerr spacetime* | Lars Andersson, Pieter BlueZeitschrift: Annals of MathematicsVerlag: Princeton UniversitySeiten: 787-853Band: 182, no. 3Link zur Publikation
,
Link zum Preprint

### Hidden symmetries and decay for the wave equation on the Kerr spacetime

#### Autoren: Lars Andersson, Pieter Blue
(2015)

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

Zeitschrift:

Annals of Mathematics

Verlag:

Princeton University

2015 | *Spin geometry and conservation laws in the Kerr spacetime* | Lars Andersson, Thomas Bäckdahl, Pieter BlueReihe: Surveys in Differential GeometryVerlag: International Press of BostonBuchtitel: L. Bieri, S.-T. Yau (eds.): One hundred years of general relativitySeiten: 183-226Band: Volume 20Link zur Publikation
,
Link zum Preprint

### Spin geometry and conservation laws in the Kerr spacetime

#### Autoren: Lars Andersson, Thomas Bäckdahl, Pieter Blue
(2015)

In this paper we will review some facts, both classical and recent, concerning the geometry and analysis of the Kerr and related black hole spacetimes. This includes the analysis of test fields on these spacetimes. Central to our analysis is the existence of a valence $(2,0)$ Killing spinor, which we use to construct symmetry operators and conserved currents as well as a new energy momentum tensor for the Maxwell test fields on a class of spacetimes containing the Kerr spacetime. We then outline how this new energy momentum tensor can be used to obtain decay estimated for Maxwell test fields. An important motivation for this work is the black hole stability problem, where fields with non-zero spin present interesting new challenges. The main tool in the analysis is the 2-spinor calculus, and for completeness we introduce its main features.

Reihe:

Surveys in Differential Geometry

Verlag:

International Press of Boston

Buchtitel:

L. Bieri, S.-T. Yau (eds.): One hundred years of general relativity

2015 | *Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior* | Lars Andersson, Pieter BlueZeitschrift: J. Hyperbolic Differ. Equ.Verlag: World ScientificSeiten: 689-743Band: 12, no. 4Link zur Publikation
,
Link zum Preprint

### Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior

#### Autoren: Lars Andersson, Pieter Blue
(2015)

We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant $t$. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localised energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in $t$. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.

Zeitschrift:

J. Hyperbolic Differ. Equ.

2015 | *The Merger of Small and Large Black Holes* | P. Mösta, L. Andersson, J. Metzger, B. Szilágyi, J. WinicourZeitschrift: Class. Quantum Grav.Verlag: IOP PublishingSeiten: 235003Band: 32, no. 23Link zur Publikation
,
Link zum Preprint

### The Merger of Small and Large Black Holes

#### Autoren: P. Mösta, L. Andersson, J. Metzger, B. Szilágyi, J. Winicour
(2015)

We present simulations of binary black holes mergers in which, after the common outer horizon has formed, the marginally outer trapped surfaces (MOTSs) corresponding to the individual black holes continue to approach and eventually penetrate each other. This has very interesting consequences according to recent results in the theory of MOTSs. Uniqueness and stability theorems imply that two MOTSs which touch with a common outer normal must be identical. This suggests a possible dramatic consequence of the collision between a small and large black hole. If the penetration were to continue to completion then the two MOTSs would have to coalesce, by some combination of the small one growing and the big one shrinking. Here we explore the relationship between theory and numerical simulations, in which a small black hole has halfway penetrated a large one.

Zeitschrift:

Class. Quantum Grav.

2015 | *Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?* | T. Buchert, M. Carfora, G.F.R. Ellis, E.W. Kolb, M.A.H. MacCallum, J.J. Ostrowski, S. Räsänen, B.F. Roukema, L. Andersson, A.A. Coley, D.L. WiltshireZeitschrift: Class. Quantum Grav.Verlag: IOP PublishingSeiten: 215021Band: 32, no. 21Link zur Publikation
,
Link zum Preprint

### Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?

#### Autoren: T. Buchert, M. Carfora, G.F.R. Ellis, E.W. Kolb, M.A.H. MacCallum, J.J. Ostrowski, S. Räsänen, B.F. Roukema, L. Andersson, A.A. Coley, D.L. Wiltshire
(2015)

No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. Specifically, we demonstrate, referring to their recent review paper, that (i) their two-dimensional example used to illustrate the fitting problem differs from the actual problem in important respects, and it assumes what is to be proven; (ii) the proof of the trace-free property of backreaction is unphysical and the theorem about it fails to be a mathematically general statement; (iii) the scheme that underlies the trace-free theorem does not involve averaging and therefore does not capture crucial non-local effects; (iv) their arguments are to a large extent coordinate-dependent, and (v) many of their criticisms of backreaction frameworks do not apply to the published definitions of these frameworks. It is therefore incorrect to infer that Green and Wald have proven a general result that addresses the essential physical questions of backreaction in cosmology.

Zeitschrift:

Class. Quantum Grav.