The Mars-Simon tensor (MST), which, e.g., plays a crucial role to provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, satisfies a Bianchi-like equation. In this paper, we analyze this equation in close analogy to the Bianchi equation, in particular it will be shown that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a space-like (i.e., for > 0). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at , for which existence and uniqueness results are derived.
REFERENCES
1.
Alexakis
, S.
, Ionescu
, A. D.
, and Klainerman
, S.
, “Uniqueness of smooth stationary black holes in vacuum: Small perturbations of the Kerr spaces
,” Commun. Math. Phys.
299
, 89
–127
(2010
).2.
Ames
, E.
, Beyer
, F.
, Isenberg
, J.
, and LeFloch
, P. G.
, “Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes
,” Ann. Henri Poincaré
14
(6
), 1445
–1523
(2013
).3.
Ames
, E.
, Beyer
, F.
, Isenberg
, J.
, and LeFloch
, P. G.
, “Quasilinear symmetric hyperbolic Fuchsian systems in several space dimensions
,” in Complex Analysis and Dynamical Systems V
, edited by Agranovsky
, M.
, Ben-Artzi
, M.
, Galloway
, G. J.
, Karp
, L.
, Maz’ya
, V.
, Reich
, S.
, Shoikhet
, D.
, Weinstein
, G.
, and Zalcman
, L.
(American Mathematical Society
, Providence, RI
, 2013
).4.
Ames
, E.
, Beyer
, F.
, Isenberg
, J.
, and LeFloch
, P. G.
, “A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges
,” J. Geom. Phys.
121
, 42
–71
(2017
).5.
Andersson
, L.
and Rendall
, A. D.
, “Quiescent cosmological singularities
,” Commun. Math. Phys.
218
(3
), 479
–511
(2001
).6.
Beig
, R.
and Chruściel
, P. T.
, “Shielding linearised gravity
,” Phys. Rev. D
95
, 064063
(2017
).7.
Beyer
, F.
and LeFloch
, P. G.
, “Second-order hyperbolic Fuchsian systems and applications
,” Classical Quantum Gravity
27
(24
), 245012
(2010
).8.
Beyer
, F.
and LeFloch
, P. G.
, “Second-order hyperbolic Fuchsian systems: Asymptotic behavior of geodesics in Gowdy spacetimes
,” Phys. Rev. D
84
(8
), 084036
(2011
).9.
Beyer
, F.
and Hennig
, J.
, “Smooth Gowdy-symmetric generalized Taub-NUT solutions
,” Classical Quantum Gravity
29
, 245017
(2012
).10.
Beyer
, F.
and LeFloch
, P. G.
, “Self-gravitating fluid flows with Gowdy symmetry near cosmological singularities
,” Commun. Partial Differ. Equations
42
(8
), 1199
–1248
(2017
).11.
Buchdahl
, H. A.
, “On the compatibility of relativistic wave equations for particles of higher spin in the presence of a gravitational field
,” Nuovo Cimento
10
, 96
–103
(1958
).12.
Choquet-Bruhat
, Y.
and Isenberg
, J.
, “Half polarized U(1)-symmetric vacuum spacetimes with AVTD behavior
,” J. Geom. Phys.
56
(8
), 1199
–1214
(2006
).13.
Chruściel
, P. T.
, “The geometry of black holes
,” lecture notes, 2015
, http://homepage.univie.ac.at/piotr.chrusciel/teaching/BlackHoles/BlackHolesViennaJanuary2015.pdf.14.
Claudel
, C. M.
and Newman
, K. P.
, “The Cauchy problem for quasi-linear hyperbolic evolution problems with a singularity in the time
,” Proc. R. Soc. A
454
(1972
), 1073
–1107
(1998
).15.
Damour
, T.
, Henneaux
, M.
, Rendall
, A. D.
, and Weaver
, M.
, “Kasner-like behaviour for subcritical Einstein-matter systems
,” Ann. Henri Poincaré
3
(6
), 1049
–1111
(2002
).16.
Friedrich
, H.
, “Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant
,” J. Geom. Phys.
3
, 101
–117
(1986
).17.
Friedrich
, H.
, “Conformal Einstein evolution
,” in The Conformal Structure of Space-time: Geometry, Analysis, Numerics
, edited by Frauendiener
, J.
and Friedrich
, H.
(Springer
, Berlin, Heidelberg
, 2002
), pp. 1
–50
.18.
Friedrich
, H.
and Rendall
, A. D.
, “The Cauchy problem for the Einstein equations
,” in Einstein’s Field Equations and Their Physical Implications
(Springer Berlin Heidelberg
, Berlin, Heidelberg
, 2000
), pp. 127
–223
.19.
Henneaux
, M.
, Persson
, D.
, and Spindel
, P.
, “Spacelike singularities and hidden symmetries of gravity
,” Living Rev. Relativ.
11
(1
), 1
(2008
).20.
Hintz
, P.
and Vasy
, A.
, “The global non-linear stability of the Kerr-de Sitter family of black holes
,” e-print arXiv:1606.04014 [math.DG] (2016
).21.
Ionescu
, A. D.
and Klainerman
, S.
, “On the uniqueness of smooth, stationary black holes in vacuum
,” Inventiones Math.
175
, 35
–102
(2009
).22.
Isenberg
, J.
and Kichenassamy
, S.
, “Asymptotic behavior in polarized T2-symmetric vacuum space–times
,” J. Math. Phys.
40
(1
), 340
(1999
).23.
Isenberg
, J.
and Moncrief
, V.
, “Asymptotic behaviour in polarized and half-polarized U(1) symmetric vacuum spacetimes
,” Classical Quantum Gravity
19
(21
), 5361
(2002
).24.
Israel
, W.
, Differential Forms in General Relativity
, Communications of the Dublin Institute for Advanced Studies: Series A Vol. 19 (Dublin Institute for Advanced Studies
, 1970
), pp. 1
–100
.25.
Mars
, M.
, “A spacetime characterization of the Kerr metric
,” Classical Quantum Gravity
16
, 2507
–2523
(1999
).26.
Mars
, M.
, “Uniqueness properties of the Kerr metric
,” Classical Quantum Gravity
17
, 3353
–3373
(2000
).27.
Mars
, M.
, Paetz
, T.-T.
, Senovilla
, J. M. M.
, and Simon
, W.
, “Characterization of (asymptotically) Kerr-de Sitter-like spacetimes at null infinity
,” Classical Quantum Gravity
33
, 155001
(2016
).28.
Mars
, M.
and Senovilla
, J. M. M.
, “A spacetime characterization of the Kerr-NUT-(A)de Sitter and related metrics
,” Ann. Henri Poincaré
16
, 1509
–1550
(2015
).29.
Mars
, M.
and Senovilla
, J. M. M.
, “Spacetime characterizations of Λ-vacuum metrics with a null Killing 2-form
,” Classical Quantum Gravity
33
, 195004
(2016
).30.
Paetz
, T.-T.
, “KIDs prefer special cones
,” Classical Quantum Gravity
31
, 085007
(2014
).31.
Paetz
, T.-T.
, “Killing initial data on space-like conformal boundaries
,” J. Geom. Phys.
106
, 51
–69
(2016
).32.
Paetz
, T.-T.
, “Algorithmic characterization results for the Kerr-NUT-(A)dS space-time. I. A space-time approach
,” J. Math. Phys.
58
, 042501
(2017
).33.
Penrose
, R.
, “Asymptotic properties of fields and space-time
,” Phys. Rev. Lett.
10
, 66
–68
(1963
).34.
Penrose
, R.
, “Zero rest-mass fields including gravitation: Asymptotic behavior
,” Proc. R. Soc. A
284
, 159
–203
(1965
).35.
Rendall
, A. D.
, “Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
,” Classical Quantum Gravity
17
(16
), 3305
–3316
(2000
).36.
Ringström
, H.
, The Cauchy Problem in General Relativity
, ESI Lectures in Mathematics and Physics (European Mathematical Society
, Zürich, Switzerland
, 2009
).37.
Simon
, W.
, “Characterizations of the Kerr metric
,” Gen. Relativ. Gravitation
16
, 465
–476
(1984
).38.
Wald
, R. M.
, General Relativity
(The University of Chicago Press
, Chicago, London
, 1984
).39.
Wald
, R. M.
, “Spin-two fields and general covariance
,” Phys. Rev. D
33
, 3613
–3625
(1986
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.