José Natário
Position
Full Professor
Member of the Center for Mathematical Analysis, Geometry and Dynamical Systems
Member of the Portuguese Relativity and Gravitation Society
Organizer of the Mathematical Relativity Seminar
Academic Degrees
Agregação em Matemática, 2016, Instituto Superior Técnico
D.Phil. (Mathematical Sciences), 2000, University of Oxford
Mestre em Matemática Aplicada, 1996, Instituto Superior Técnico
Licenciado em Matemática Aplicada e Computação, 1994, Instituto Superior Técnico
Research Interests
General Relativity, Mathematical Physics, Geometry
An Introduction to Mathematical Relativity
An Introduction to Riemannian Geometry, with Leonor Godinho
General Relativity Without Calculus
The Lorentz group, Mobius transformations and the sky of a rapidly moving observer, appendix to Geometric Mechanics, by Waldyr Muniz Oliva
QuasiMaxwell form of Einstein's equations, appendix to Geometric Mechanics, by Waldyr Muniz Oliva
Mathematical Relativity
[62] Spherically symmetric elastic bodies in general relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), 2023.
[61] Scale invariant elastic stars in General Relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), 2023.
[60] Message in a bottle: energy extraction from bouncing geometries (joint with Vitor Cardoso, João L. Costa and Zhen Zhong), Phys. Rev. D 108 (2023) 024071.
[59] Reference frames in General Relativity and the galactic rotation curves (joint with Filipe Costa, Francisco Frutos Alfaro and Michael Soffel), Phys. Rev. D 108 (2023) 044056.
[58] Electromagnetic radiation reaction and energy extraction from black holes: the tail cannot be ignored (joint with João Sieiro dos Santos and Vitor Cardoso), Phys. Rev. D 107 (2023) 064046.
[57] Curvature and dynamical spacetimes: can we peer into the quantum regime? (joint with Vitor Cardoso, David Hilditch, Krinio Marouda and Ulrich Sperhake), Class. Quantum Grav. 40 (2023) 065008.
[56] Compactness bounds in General Relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), Phys. Rev. D 106 (2022) L041502.
[55] Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes (joint with Flavio Rossetti), J. Math. Phys. 64 (2023) 032504.
[54] Framedragging: meaning, myths, and misconceptions (joint with Filipe Costa), Universe 7 (2021) 388.
[53] Compact elastic objects in general relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), Phys. Rev. D 105 (2022) 044025.
[52] Extrinsic black hole uniqueness in pure Lovelock gravity (joint with Levi Lima and Frederico Girão), Bull. Braz. Math. Soc. 53 (2022) 721739.
[51] Addendum to “Strong cosmic censorship: The nonlinear story” (joint with Raimon Luna, Miguel Zilhão, Vitor Cardoso and João L. Costa), Phys. Rev. D 103 (2021) 104043.
[50] Poynting vector, superPoynting vector, and principal observers in electromagnetism and general relativity (joint with Filipe Costa and Lode Wylleman), Class. Quantum Grav. 38 (2021) 165009.
[49] Riemannian manifolds dual to static spacetimes (joint with Carolina Figueiredo), Gen. Rel. Grav. 52 (2020) 84.
[48] Proof of the weak cosmic censorship conjecture for several extremal black holes (joint with Jéssica Gonçalves), Gen. Rel. Grav. 52 (2020) 94.
[47] Linear relativistic thermoelastic rod (joint with Anne Franzen), J. Hyperbolic Differ. Equ. 17 (2020) 863882.
[46] Gravitomagnetism in the Lewis cylindrical metrics (joint with Filipe Costa and Nilton Santos), Class. Quantum Grav. 38 (2021) 055003.
[45] Rigid elastic solids in relativity, Class. Quantum Grav. 37 (2020) 235006.
[44] Decay of solutions to the KleinGordon equation on some expanding cosmological spacetimes (joint with Amol Sasane), Ann. Henri Poincaré 23 (2022) 23452389.
[43] Test fields cannot destroy extremal de Sitter black holes (joint with Rodrigo Vicente), Gen. Rel. Grav. 52 (2020) 5.
[42] Elastic shocks in relativistic rigid rods and balls (joint with João L. Costa), Proc. Royal Soc. Lond. A 475 (2019) 20180858.
[41] Strong Cosmic Censorship: the nonlinear story (joint with Raimon Luna, Miguel Zilhão, Vitor Cardoso and João L. Costa), Phys. Rev. D 99 (2019) 064014.
[40] Solutions of the wave equation bounded at the Big Bang (joint with Pedro M. Girão and Jorge Drumond Silva), Class. Quantum Grav. 36 (2019) 075016.
[39] Decay of solutions of the wave equation in expanding cosmological spacetimes (joint with João L. Costa and Pedro Oliveira), J. Hyperbolic Differ. Equ. 16 (2019) 3558.
[38] Cosmic nohair in spherically symmetric black hole spacetimes (joint with João L. Costa and Pedro Oliveira), Ann. Henri Poincaré 20 (2019) 30593090.
[37] Rotating elastic string loops in flat and black hole spacetimes: stability, cosmic censorship and the Penrose process (joint with Leonel Queimada and Rodrigo Vicente), Class. Quantum Grav. 35 (2018) 075003.
[36] On the occurrence of mass inflation for the EinsteinMaxwellscalar field system with a cosmological constant and an exponential Price law (joint with João L. Costa, Pedro M. Girão and Jorge Drumond Silva), Commun. Math. Phys. 361 (2018) 289341.
[35] On a remarkable electromagnetic field in the Einstein Universe (joint with Jarosław Kopiński), Gen. Rel. Grav. 49 (2017) 81.
[34] Gravitomagnetism and the significance of the curvature scalar invariants (joint with Filipe Costa and Lode Wylleman), Phys. Rev. D 104 (2021) 084081.
[33] Test fields cannot destroy extremal black holes (joint with Leonel Queimada and Rodrigo Vicente), Class. Quantum Grav. 33 (2016) 175002.
[32] The Coriolis field (joint with Filipe Costa), Am. J. Phys. 84 (2016) 388395.
[31] Center of mass, spin supplementary conditions, and the momentum of spinning particles (joint with Filipe Costa), in D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, Springer, 2015, pp. 215258.
[30] On the global uniqueness for the EinsteinMaxwellscalar field system with a cosmological constant. Part 3: Mass inflation and extendibility of the solutions (joint with João L. Costa, Pedro M. Girão and Jorge Drumond Silva), Ann. PDE (2017) 3: 8.
[29] On the global uniqueness for the EinsteinMaxwellscalar field system with a cosmological constant. Part 2: Structure of the solutions and stability of the Cauchy horizon (joint with João L. Costa, Pedro M. Girão and Jorge Drumond Silva), Commun. Math. Phys. 339 (2015) 903947.
[28] On the global uniqueness for the EinsteinMaxwellscalar field system with a cosmological constant. Part 1: Well posedness and breakdown criterion (joint with João L. Costa, Pedro M. Girão and Jorge Drumond Silva), Class. Quantum Grav. 32 (2015) 015017.
[27] Relativistic elasticity of rigid rods and strings, Gen. Rel. Grav. 46 (2014) 1816.
[26] A Minkowskitype inequality for convex surfaces in the hyperbolic 3space, Differ. Geom. Appl. 41 (2015) 102109.
[25] Optimal escape from circular orbits around black holes, Gen. Rel. Grav. 45 (2013) 811818.
[24] Spacetime dynamics of spinning particles  exact gravitoelectromagnetic analogies (joint with Filipe Costa and Miguel Zilhão), Phys. Rev. D 93 (2016) 104006.
[23] Gravitoelectromagnetic analogies (joint with Filipe Costa), Gen. Rel. Grav. 46 (2014) 1792.
[22] The problem of a selfgravitating scalar field with positive cosmological constant (joint with João L. Costa e Artur Alho), Ann. Henri Poincaré 14 (2013) 10771107.
[21] Mathisson's helical motions for a spinning particle  are they unphysical? (joint with Filipe Costa, Carlos Herdeiro and Miguel Zilhão), Phys. Rev. D 85 (2012) 024001.
[20] Spherical linear waves in de Sitter spacetime (joint with João L. Costa e Artur Alho), J. Math. Phys. 53 (2012) 052501.
[19] Optimal time travel in the Gödel universe, Gen. Rel. Grav. 44 (2012) 855874.
[18] The rocket problem in general relativity (joint with Pedro G. Henriques), J. Optim. Theory Appl. 154 (2012) 500524.
[17] An elementary derivation of the Montgomery phase formula for the Euler top, J. Geom. Mech. 2 (2010) 113118.
[16] Formation of higherdimensional topological black holes (joint with Filipe Mena and Paul Tod), Ann. Henri Poincaré 10 (2010) 13591376.
[15] PainlevéGullstrand coordinates for the Kerr solution, Gen. Rel. Grav. 41 (2009) 25792586.
[14] Matching stationary spacetimes (joint with Filipe Mena), J. Geom. Phys. 59 (2009) 448458.
[13] Avoiding closed timelike curves with a collapsing rotating null dust shell (joint with Filipe Mena and Paul Tod), Class. Quantum Grav. 25 (2008) 045016.
[12] Greybody factors for dDimensional black holes (joint with Troels Harmark and Ricardo Schiappa), Adv. Theor. Math. Phys. 14 (2010) 727794.
[11] Gravitational collapse to toroidal and higher genus asymptotically AdS black holes (joint with Filipe Mena and Paul Tod), Adv. Theor. Math. Phys. 12 (2008) 11631181.
[10] Tangent Euler top in general relativity, Commun. Math. Phys. 281 (2008) 387400.
[9] QuasiMaxwell interpretation of the spincurvature coupling, Gen. Rel. Grav. 39 (2007) 14771487.
[8] Relativity and singularities  a short introduction for mathematicians, Resenhas IMEUSP 6 (2005) 309335.
[7] Newtonian limits of warp drive spacetimes, Gen. Rel. Grav. 38 (2006) 475484.
[6] Homogeneous cosmologies from the quasiMaxwell formalism (joint with João L. Costa), J. Math. Phys. 46 (2005) 082501.
[5] On the classification of asymptotic quasinormal frequencies for ddimensional black holes and quantum gravity (joint with Ricardo Schiappa), Adv. Theor. Math. Phys. 8 (2004) 10011131.
[4] Asymptotic quasinormal frequencies for black holes in nonasymptotically flat spacetimes (joint with Vitor Cardoso and Ricardo Schiappa), J. Math. Phys. 45 (2004) 46984713.
[3] Linking, Legendrian linking and causality (joint with Paul Tod), Proc. London Math. Soc. (3) 88 (2004) 251272.
[2] Linking and causality in (2+1)dimensional static spacetimes, Class. Quantum Grav. 19 (2002) 31153126.
[1] Warp drive with zero expansion, Class. Quantum Grav. 19 (2002) 11571165.
Relativistic elasticity and compactness bounds (November 2022)
Compact elastic objects in general relativity (September 2021)
Decay of solutions of the wave equation in expanding cosmological spacetimes (August 2021)
Penrose and the 2020 Physics Nobel Prize (December 2020)
Solutions of the wave equation bounded at the Big Bang (July 2019)
Cosmic nohair in spherically symmetric black hole spacetimes (July 2019)
Mathematical Relativity and the Cosmic Censorship Conjecture (November 2018)
Strong cosmic censorship in spherical symmetry (September 2017)
An elementary derivation of the Montgomery phase formula for the Euler top (September 2016)
Relativistic elasticity of rigid rods and strings (July 2014)
Gravitoelectromagnetic analogies (April 2014)
Bernoulli's elastic curves, rockets, and time travel (July 2013)
Optimal time travel in the Gödel universe (January 2013)
Formation of Higherdimensional Topological Black Holes (April 2010)
Asymptotic Quasinormal Frequencies for dDimensional Black Holes (February 2009)
Greybody Factors for dDimensional Black Holes (December 2008)
Avoiding closed timelike curves with a collapsing rotating null dust shell (September 2008)
An Overview of Mathematical General Relativity (March 2005)
Warp Drive (April 2003)
SpanishPortuguese Relativity Meeting 2016 (September 2016)
100 anos de Relatividade Geral (July 2015)
Mira Fernandes Lectures on Mathematical Relativity (June 2013)
A Geometria da Relatividade (July 2010)
Mathematical Relativity in Lisbon (June 2009)
A Geometria da Relatividade (July 2005)
The Hidden Geometry (February 2021)
Perguntas de Matemática (July 2019)
Matemática e Ficção Científica (May 2019)
A Geometria Escondida (February 2018)
A Relatividade do Tempo (March 2015)
O GPS e a Teoria da Relatividade (February 2015)
A Linguagem Secreta do Universo (October 2012)
Bilhares Matemáticos (March 2009)
Penrose e o Prémio Nobel da Física de 2020 (March 2021)
Teorema de Penrose: o teorema que valeu um Prémio Nobel (January 2021)
O GPS e a Teoria da Relatividade (May 2013)
A Linguagem Secreta do Universo (October 2012)
O Teorema de GaussBonnet (March 2012)
Como Rodam as Coisas (February 2011)
Espaçotempo de Minkowski: a Física como Geometria (June 2010)
O que é a curvatura de uma superfície? (January 2010)
Algebraic and Geometric Methods in Engineering and Physics
Semester 1 — 2023/2024
Semester 1 — 2022/2023
Semester 1 — 2020/2021
Advanced Topics in Geometry
Semester 2 — 2019/2020
Calculus II
Semester 2 — 2021/2022
Semester 2 — 2019/2020
Semester 2 — 2018/2019
Semester 2 — 2012/2013
Semester 1 — 2010/2011
Semester 1 — 2009/2010
Semester 2 — 2008/2009
Semester 2 — 2007/2008
Complex Analysis and Differential Equations
Semester 2 — 2016/2017
Differential Geometry
Semester 1 — 2013/2014
Semester 1 — 2005/2006
Semester 1 — 2004/2005
Differential Geometry of Curves and Surfaces
Semester 1 — 2023/2024
Geometric Mechanics
Semester 1 — 2022/2023
Semester 1 — 2020/2021
Semester 1 — 2019/2020
Semester 1 — 2018/2019
Semester 1 — 2016/2017
Semester 1 — 2015/2016
Semester 1 — 2012/2013
Semester 1 — 2010/2011
Semester 2 — 2002/2003
Semester 2 — 2001/2002
Semester 2 — 2000/2001
Lie Groups and Lie Algebras
Semester 1 — 2012/2013
Semester 1 — 2008/2009
Mathematical Analysis III
Semester 2 — 2005/2006
Semester 1 — 2003/2004
Semester 1 — 2002/2003
Semester 1 — 2001/2002
Semester 1 — 2000/2001
Mathematical Analysis IV
Semester 2 — 2000/2001
Mathematical Relativity
Semester 2 — 2021/2022
Semester 2 — 2014/2015
Semester 2 — 2011/2012
Semester 2 — 2009/2010
Riemannian Geometry
Semester 1 — 2011/2012
Semester 1 — 2006/2007
Semester 2 — 2003/2004
PostDoc
Anne Franzen (started in 2016)
Filipe Costa (started in 2013)
Artur Alho (started in 2013)
PhD
Miguel Duarte (20182022), Asymptotics in Generalized Harmonic Gauge
Pedro Oliveira (20142018), Cosmic NoHair for Spherically Symmetric Black Hole Spacetimes
Artur Alho (20082012), On the future stability of cosmological solutions of the Einsteinnonlinear scalar field system
MSc
João Silva (2022/2023), The Yamabe Problem
Flavio Rossetti (2020/2021), The Wave Equation in Cosmological Spacetimes
Gonçalo Fonseca (2020/2021), On the 3dimensional Geroch Conjecture and the Positive
Mass Theorem
Amol Sasane (2018/2019), Decay of solutions to the KleinGordon equation on some expanding cosmological spacetimes
Pedro Oliveira (2012/2013), MinOo's Conjecture: Positive Mass Theorem in the de Sitter Universe
Sílvia Guerra (2006/2007), Integrabilidade do Fluxo Geodésico num Elipsóide
João Costa (2002/2003), Cosmologias em Espaçotempos Estacionários
Undergraduate Theses
Luís Machado (2021/2022), Construction of Penrose Diagrams for Static Spherically Symmetric Spacetimes
Patrícia Engrácia (2002/2003), Universo de Gödel
Undergraduate Research Scholarships
Paulo Mourão (2019/2020)
Leonel Queimada (2016/2017)
Rodrigo Vicente (2015/2016)
Course Projects
Sara Sal Santos (2018/2019), Viagens no tempo em Relatividade Geral
Renata Amado (2017/2018), Paradoxo de Olbers  Porque é que o céu à noite é escuro?
Laura Melo (2016/2017), Distorções Relativistas
Ana Branca (2015/2016), Buracos Negros: uma leve abordagem
Manuel Santos (2015/2016), Classificação de Modelos Cosmológicos Homogéneos
Pedro Peres (2012/2013), Viagens no tempo no Universo de Gödel
Pedro Filipe (2012/2013), Determinação da parametrização e curvatura de uma superfície em R^{3} com a métrica medida por observadores estacionários no espaçotempo de Schwarzschild
Rui Pinto (2008/2009), A Conjectura de Censura Cósmica de A a B
Gulbenkian program "Novos Talentos em Matemática"
Carolina Figueiredo (2018/2019), Riemannian manifolds dual to static spacetimes
João Barata (2016/2017), Two dimensional Tangent Euler Top: classical and relativistic dynamics
Leonel Queimada (2015/2016), From Relativistic Elasticity to Cosmic Censorship
António Lages (2014/2015), Relevant Aspects of Riemannian Geometry
Francisco Lopes (2011/2012), Geodésicas Nulas em Relatividade Geral
Ana Rita Nunes (2006/2007), A Forma do Universo
Ana Knopfly (2002/2003), Problema Restrito dos 3 Corpos
Ida Griffith (2001/2002), Transformações de Lorentz e de Möbius
Gulbenkian program "Novos Talentos em Física"
Joana Yao (2022/2023), Study of Static Spatially Compact Universes
