Fetching the paper…
Reading the bibliography…
These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds.
K. Schwarzschild Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie Sitzungsber. d. Preuss. Akad. d. Wissenschaften 1
1916
Earlier work this paper cites.
K. Schwarzschild Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie Sitzungsber. d. Preuss. Akad. d. Wissenschaften, Berlin (1916)
1916
Earlier work this paper cites.
A. Einstein Über Gravitationswellen Königlich Preussiche Akademie der Wissenschaften Berlin Sitzungberichte (1918) 154-167
1918
Earlier work this paper cites.
F. Kottler Über die physikalischen Grundlagen der Einsteinschen Gravitationstheorie , Ann. Phys. 56
1918
Earlier work this paper cites.
E. Noether Invariante Variationsprobleme Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse (1918) 235–257
1918
Earlier work this paper cites.
H. Weyl Raum, Zeit, Materie Springer, Berlin, 1919
1919
Earlier work this paper cites.
H. Weyl Über die statischen kugelsymmetrischen Lösungen von Einsteins “kosmologischen” Gravitationsgleichungen , Phys. Z. 20
1919
Earlier work this paper cites.
T. de Donder La gravifique Einsteinienne Annales de l’Observatoire Royal de Belgique, Brussels, 1921
1921
Earlier work this paper cites.
G. Lemaitre L’Univers en Expansion Publication du Laboratoire d’Astronomie et de Géodésie de l’Université de Louvain 9
1932
Earlier work this paper cites.
J. Schauder Das Anfangswertproblem einer quasilinearen hyperbolischen Differential- gleichung zweiter Ordnung in beliebiger Anzahl yon urtabhängigen Veränderliehen Fundam. Math. 24
1935
Earlier work this paper cites.
K. Stellmacher Zum Anfangswertproblem der Gravitationsgleichungen Math. Ann. 115
1938
Earlier work this paper cites.
A. Einstein On a stationary system with spherical symmetry consisting of many gravitating masses. Ann. Math. 40
1939
Earlier work this paper cites.
J. R. Oppenheimer and H. Snyder On continued gravitational contraction Phys. Rev. 56
1939
Earlier work this paper cites.
J. L. Synge The gravitational field of a particle Proc. Roy. Irish Acad. 53
1950
Earlier work this paper cites.
Y. Choquet-Bruhat Théoreme d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires Acta Math. 88
1952
Earlier work this paper cites.
J. Leray Hyperbolic differential equations The Institute for Advanced Study, Princeton, N. J., 1953
1953
Earlier work this paper cites.
J. L. Synge Relativity: the special theory North-Holland Publishing Co., Amsterdam, 1956
1956
Earlier work this paper cites.
T. Regge and J. Wheeler Stability of a Schwarzschild singularity Phys. Rev. 108
1957
Earlier work this paper cites.
H. Buchdahl General relativistic fluid spheres Phys. Rev. 116
1959
Earlier work this paper cites.
M. Kruskal Maximal extension of Schwarzschild metric Phys. Rev. 119
1960
Earlier work this paper cites.
C. S. Morawetz The limiting amplitude principle Comm. Pure Appl. Math. 15
1962
Earlier work this paper cites.
R. Kerr Gravitational field of a spinning mass as an example of algebraically special metrics Phys. Rev. Lett. 11
1963
Earlier work this paper cites.
V. Fock The theory of space, time and gravitation The Macmillan Co., New York, 1964
1964
Earlier work this paper cites.
B. Carter Global structure of the Kerr family of gravitational fields Phys. Rev. 174
1968
Earlier work this paper cites.
B. Carter Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations Comm. Math. Phys. 10
1968
Earlier work this paper cites.
Y. Choquet-Bruhat and R. Geroch Global aspects of the Cauchy problem in general relativity Comm. Math. Phys. 14
1969
Earlier work this paper cites.
R. Penrose Gravitational collapse: the role of general relativity Rev. del Nuovo Cimento 1
1969
Earlier work this paper cites.
D. Christodoulou Reversible and irreversible transformations in black-hole physics Phys. Rev. Lett. 25
1970
Earlier work this paper cites.
M. Walker and R. Penrose On quadratic first integrals of the geodesic equations for type 22 {22} spacetimes Comm. Math. Phys. 18
1970
Earlier work this paper cites.
B. Carter Black hole equilibrium states , in Black Holes (Les Houches Lectures), edited B. S. DeWitt and C. DeWitt (Gordon and Breach, New York, 1972)
1972
Earlier work this paper cites.
R. Price Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations Phys. Rev. D (3) 5
1972
Earlier work this paper cites.
J. M. Bardeen, B. Carter, and S. W. Hawking The four laws of black hole mechanics Commun. Math. Phys. 31
1973
Earlier work this paper cites.
S. W. Hawking and G. F. R. Ellis The large scale structure of space-time Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973
1973
Earlier work this paper cites.
C. W.Misner, K. S. Thorne, and J. A. Wheeler Gravitation W. H. Freeman and Co., San Francisco, Calif., 1973
1973
Earlier work this paper cites.
J. D. Bekenstein Generalized second law of thermodynamics in black hole physics , Phys. Rev. D 9
1974
Earlier work this paper cites.
1976
Earlier work this paper cites.
G. W. Gibbons and S. W. Hawking Cosmological event horizons, thermodynamics, and particle creation Phys. Rev. D. 15
1977
Earlier work this paper cites.
R. Penrose Singularities and time asymmetry In “General Relativity–an Einstein Survey” S. Hawking, W. Israel ed., Cambridge University Press, Cambridge, 1979
1979
Earlier work this paper cites.
R. Schoen and S.-T. Yau On the proof of the positive mass conjecture in general relativity Comm. Math. Phys. 16
1979
Earlier work this paper cites.
R. M. Wald Note on the stability of the Schwarzschild metric J. Math. Phys. 20
1979
Earlier work this paper cites.
S. Detweiler Klein-Gordon equation and rotating black holes Phys. Rev. D 22
1980
Earlier work this paper cites.
V. Moncrief and D. Eardley The global existence problem and cosmic censorship in general relativity Gen. Rel. Grav. 13
1981
Earlier work this paper cites.
R. Schoen and S.-T. Yau Proof of the positive mass theorem. II. Comm. Math. Phys. 79
1981
Earlier work this paper cites.
E. Witten A new proof of the positive energy theorem Comm. Math. Phys. 80
1981
Earlier work this paper cites.
P. Breitenlohner and D. Z. Freedman Stability in gauged extended supergravity Ann. Phys. 144
1982
Cited alongside, same era.
D. Christodoulou Violation of cosmic censorship in the gravitational collapse of a dust cloud Comm. Math. Phys. 93
1984
Cited alongside, same era.
R. Wald General relativity University of Chicago Press, Chicago, 1984
1984
Cited alongside, same era.
J. Dimock Scattering for the Wave equation on the Schwarzschild metric , Gen. Rel. Grav. 17
1985
Cited alongside, same era.
S. Klainerman Uniform decay estimates and the Lorentz invariance of the classical wave equation Comm. Pure Appl. Math. 38
1985
Cited alongside, same era.
R. Bartnik The mass of an asymptotically flat manifold Comm. Pure Appl. Math. 39
D. Christodoulou The global initial value problem in general relativity 9th Marcel Grosmann Meeting (Rome 2000), 44–54, World Sci. Publishing, 2002
2002
Later among the works it cites.
P. Chruściel Black holes Lect. Notes Phys. 604
2002
Later among the works it cites.
P. Blue and A. Soffer Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates , Adv. Differential Equations 8
2003
Later among the works it cites.
D. Christodoulou Mathematical Problems of General Relativity Theory , ETH-Zürich lecture notes, 2003
2003
Later among the works it cites.
M. Dafermos Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations Ann. of Math. 158
2003
Later among the works it cites.
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
1986
Cited alongside, same era.
H. Friedrich The existence of n n -geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure Comm. Math. Phys. 107
1986
Cited alongside, same era.
S. Klainerman The null condition and global existence to nonlinear wave equations Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), 293–326, Lectures in Appl. Math. 23
1986
Cited alongside, same era.
R. C. Myers and M. J. Perry Black holes In higher dimensional space-times Ann. Phys. (N.Y.) 172
1986
Cited alongside, same era.
D. Christodoulou A mathematical theory of gravitational collapse Comm. Math. Phys. 109
1987
Cited alongside, same era.
J. Dimock and B. Kay Classical and quantum scattering theory for linear scalar field on the Schwarzschild metric I Ann. Phys., 175
1987
Cited alongside, same era.
B. Kay and R. Wald Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2 2 -sphere Classical Quantum Gravity 4
1987
Cited alongside, same era.
F. Finster, N. Kamran, J. Smoller, S.-T. Yau The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry Adv. Theor. Math. Phys. 7
2003
Later among the works it cites.
D. Häfner Sur la théorie de la diffusion pour l’équation de Klein-Gordon dans la métrique de Kerr Dissertationes Mathematicae 421
2003
Later among the works it cites.
L. Andersson and V. Moncrief Future complete vacuum spacetimes In: The Einstein equations and the large slace behavior of gravitational fields, 299–330, Birkhäuser, Basel, 2004
2004
Later among the works it cites.
V. Cardoso and O. J. C. Dias Small Kerr-anti-de Sitter black holes are unstable Phys. Rev. D 70
2004
Later among the works it cites.
D. Häfner and J.-P. Nicolas Scattering of massless Dirac fields by a Kerr black hole Rev. Math. Phys. 16
2004
Later among the works it cites.
R. Beig and J. M. Heinzle CMC-Slicings of Kottler-Schwarzschild-de Sitter Cosmologies , Comm. Math. Phys., 260
2005
Later among the works it cites.
P. Bizon, T. Chmaj, B. Schmidt Critical behavior in vacuum gravitational collapse in 4 + 1 4+1 dimensions Phys. Rev. Lett. 95
2005
Later among the works it cites.
Z. W. Chong, G. W. Gibbons, H. Lu, and C. N. Pope Separability and Killing tensors in Kerr-Taub-NUT-de Sitter metrics in higher dimensions Phys. Lett. B609
2005
Later among the works it cites.
M. Dafermos The interior of charged black holes and the problem of uniqueness in general relativity Comm. Pure App. Math. 58
2005
Later among the works it cites.
M. Dafermos Spherically symmetric spacetimes with a trapped surface Class. Quantum Grav. 22
2005
Later among the works it cites.
M. Dafermos and I. Rodnianski A proof of Price’s law for the collapse of a self-gravitating scalar field , Invent. Math. 162
2005
Later among the works it cites.
M. Dafermos and I. Rodnianski Small-amplitude nonlinear waves on a black hole background J. Math. Pures Appl. 84
2005
Later among the works it cites.
S. Klainerman and I. Rodnianski Rough solutions of the Einstein vacuum equations Ann. of Math. 161
2005
Later among the works it cites.
A. Rendall Theorems on existence and global dynamics for the Einstein equations Living Rev. Relativity 8
2005
Later among the works it cites.
P. Bizon, B. Schmidt How to bypass Birkhoff through extra dimensions: a simple framework for investigating the gravitational collapse in vacuum Internat. J. Modern Phys. D 15
2006
Later among the works it cites.
P. Blue and J. Sterbenz Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space Comm. Math. Phys. 268
2006
Later among the works it cites.
Y. Choquet-Bruhat, P.T. Chruściel, and J. Loizelet Global solutions of the Einstein-Maxwell equations in higher dimensions Classical and Quantum Gravity 23
2006
Later among the works it cites.
M. Dafermos and G. Holzegel On the nonlinear stability of higher dimensional triaxial Bianchi-IX black holes Adv. Theor. Math. Phys. 10
2006
Later among the works it cites.
F. Finster, N. Kamran, J. Smoller, S. T. Yau Decay of solutions of the wave equation in Kerr geometry Comm. Math. Phys. 264
2006
Later among the works it cites.
F. Finster and H. Schmid Spectral estimates and non-selfadjoint perturbations of spheroidal wave operators J. Reine Angew. Math. 601
2006
Later among the works it cites.
H. Andréasson On the Buchdahl inequality for spherically symmetric static shells Comm. Math. Phys. 274
2007
Later among the works it cites.
L. Bieri An extension of the stability theorem of the Minkowski space in general relativity , Thesis, ETH, 2007
2007
Later among the works it cites.
J.-F. Bony and D. Häfner Decay and non-decay of the local energy for the wave equation in the de Sitter-Schwarzschild metric , preprint 2007
2007
Later among the works it cites.
Y. Choquet-Bruhat, J. Isenberg, D. Pollack The constraint equations for the Einstein-scalar field system on compact manifolds Classical Quantum Gravity 24
2007
Later among the works it cites.
D. Christodoulou The formation of shocks in 3-dimensional fluids EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007
2007
Later among the works it cites.
C. Gundlach and J. M. Martín-García Critical Phenomena in Gravitational Collapse Living Rev. Relativity 10
2007
Later among the works it cites.
S. Klainerman and I. Rodnianski A Kirchoff-Sobolev parametrix for the wave equation and applications J. Hyperbolic Differ. Equ. 4
2007
Later among the works it cites.
J. Metcalfe Strichartz estimates on Schwarzschild space-times Oberwolfach Reports 44
2007
Later among the works it cites.
A. Vasy The wave equation on asymptotically de Sitter-like spaces Oberwolfach Reports 41
2007
Later among the works it cites.
S. Alinhac Energy multipliers for perturbations of Schwarzschild metric preprint, 2008
2008
Closest in time.
A. Bachelot The Dirac system on the anti-de Sitter universe Commun. Math. Phys. 283
2008
Closest in time.
P. Chruściel and D. Pollack Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends Annales Henri Poincaré 9
2008
Closest in time.
R. Emparan and H. Reall Black Holes in Higher Dimensions Living Rev. Relativity 11
2008
Closest in time.
G. Holzegel The stability problem for black holes Ph. D. Dissertation, University of Cambridge, 2008
2008
Closest in time.
D. Kubiznak Hidden symmetries of higher-dimensional rotating black holes Ph.D. Thesis, University of Alberta, September 2008
2008
Closest in time.
A. Rendall Partial differential equations in general relativity Oxford Graduate Texts in Mathematics 16
2008
Closest in time.
H. Ringström Future stability of the Einstein-non linear scalar field system Invent. Math. 173
2008
Closest in time.
H. Nussbaumer and L. Bieri Discovering the Expanding Universe Cam. Univ. Press, to appear 2009
2009
Closest in time.