Books
Book Chapters
Lecture Notes
Research Papers
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$José Natário$

José Natário

Phone	(351) 218 417 114

Email	jnatar@math.tecnico.ulisboa.pt

Office	Mathematics Building, 4th floor, room 4.29

Postal Address	Departamento de Matemática
	Instituto Superior Técnico
	Av. Rovisco Pais
	1049-001 Lisboa, Portugal

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

Mestrado em Matemática Aplicada, 1996, Instituto Superior Técnico

Licenciatura em Matemática Aplicada e Computação, 1994, Instituto Superior Técnico

Research Interests

General Relativity, Mathematical Physics, Geometry

Books

An Introduction to Mathematical Relativity

An Introduction to Riemannian Geometry, with Leonor Godinho

General Relativity Without Calculus

Book Chapters

The Lorentz group, Mobius transformations and the sky of a rapidly moving observer, appendix to Geometric Mechanics, by Waldyr Muniz Oliva

Quasi-Maxwell form of Einstein's equations, appendix to Geometric Mechanics, by Waldyr Muniz Oliva

Lecture Notes

Mathematical Relativity

Research Papers

[70] An exact solution describing a scalar counterpart to the Schwarzschild-Melvin Universe (joint with Vitor Cardoso), 2024.

[69] Floating orbits and energy extraction from magnetized Kerr black holes (joint with João Sieiro dos Santos and Vitor Cardoso), Phys. Rev. D Accepted Paper.

[68] Relativistic elastic membranes: rotating disks and Dyson spheres (joint with Paulo Mourão and Rodrigo Vicente), Class. Quantum Grav. Accepted Manuscript.

[67] Elastic rods and elastic spinning rings as gravitational wave detectors (joint with Amol Sasane and Rodrigo Vicente), 2024.

[66] Free-falling motion of an elastic rigid rod towards a Schwarzschild black hole (joint with Luís Machado and Jorge Drumond Silva), Class. Quantum Grav. 41 (2024) 215002.

[65] Radiation reaction in weakly magnetized black holes: can the tail term be ignored in the strong field regime? (joint with João Sieiro dos Santos and Vitor Cardoso), Phys. Rev. D 109 (2023) 124032.

[64] Comment on "The Tully-Fisher law and dark matter effects derived via modified symmetries" by Arraut Ivan (joint with Filipe Costa, Francisco Frutos Alfaro and Michael Soffel), EPL 146 (2024) 19002.

[63] Relativistic effects cannot explain galactic dynamics (joint with Filipe Costa), Phys. Rev. D 110 (2024) 064056.

[62] Spherically symmetric elastic bodies in general relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), Class. Quantum Grav. 41 (2024) 073002.

[61] Scale invariant elastic stars in General Relativity (joint with Artur Alho, Paolo Pani and Guilherme Raposo), Phys. Rev. D 109 (2024) 064037.

[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 term 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] Frame-dragging: 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) 721-739.

[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, super-Poynting 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) 863-882.

[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 Klein-Gordon equation on some expanding cosmological spacetimes (joint with Amol Sasane), Ann. Henri Poincaré 23 (2022) 2345-2389.

[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) 35-58.

[38] Cosmic no-hair in spherically symmetric black hole spacetimes (joint with João L. Costa and Pedro Oliveira), Ann. Henri Poincaré 20 (2019) 3059-3090.

[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 Einstein-Maxwell-scalar 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) 289-341.

[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) 388-395.

[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. 215-258.

[30] On the global uniqueness for the Einstein-Maxwell-scalar 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 Einstein-Maxwell-scalar 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) 903-947.

[28] On the global uniqueness for the Einstein-Maxwell-scalar 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 Minkowski-type inequality for convex surfaces in the hyperbolic 3-space, Differ. Geom. Appl. 41 (2015) 102-109.

[25] Optimal escape from circular orbits around black holes, Gen. Rel. Grav. 45 (2013) 811-818.

[24] Spacetime dynamics of spinning particles - exact gravito-electromagnetic analogies (joint with Filipe Costa and Miguel Zilhão), Phys. Rev. D 93 (2016) 104006.

[23] Gravito-electromagnetic analogies (joint with Filipe Costa), Gen. Rel. Grav. 46 (2014) 1792.

[22] The problem of a self-gravitating scalar field with positive cosmological constant (joint with João L. Costa e Artur Alho), Ann. Henri Poincaré 14 (2013) 1077-1107.

[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) 855-874.

[18] The rocket problem in general relativity (joint with Pedro G. Henriques), J. Optim. Theory Appl. 154 (2012) 500-524.

[17] An elementary derivation of the Montgomery phase formula for the Euler top, J. Geom. Mech. 2 (2010) 113-118.

[16] Formation of higher-dimensional topological black holes (joint with Filipe Mena and Paul Tod), Ann. Henri Poincaré 10 (2010) 1359-1376.

[15] Painlevé-Gullstrand coordinates for the Kerr solution, Gen. Rel. Grav. 41 (2009) 2579-2586.

[14] Matching stationary spacetimes (joint with Filipe Mena), J. Geom. Phys. 59 (2009) 448-458.

[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 d-Dimensional black holes (joint with Troels Harmark and Ricardo Schiappa), Adv. Theor. Math. Phys. 14 (2010) 727-794.

[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) 1163-1181.

[10] Tangent Euler top in general relativity, Commun. Math. Phys. 281 (2008) 387-400.

[9] Quasi-Maxwell interpretation of the spin-curvature coupling, Gen. Rel. Grav. 39 (2007) 1477-1487.

[8] Relativity and singularities - a short introduction for mathematicians, Resenhas IME-USP 6 (2005) 309-335.

[7] Newtonian limits of warp drive spacetimes, Gen. Rel. Grav. 38 (2006) 475-484.

[6] Homogeneous cosmologies from the quasi-Maxwell formalism (joint with João L. Costa), J. Math. Phys. 46 (2005) 082501.

[5] On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity (joint with Ricardo Schiappa), Adv. Theor. Math. Phys. 8 (2004) 1001-1131.

[4] Asymptotic quasinormal frequencies for black holes in non-asymptotically flat spacetimes (joint with Vitor Cardoso and Ricardo Schiappa), J. Math. Phys. 45 (2004) 4698-4713.

[3] Linking, Legendrian linking and causality (joint with Paul Tod), Proc. London Math. Soc. (3) 88 (2004) 251-272.

[2] Linking and causality in (2+1)-dimensional static spacetimes, Class. Quantum Grav. 19 (2002) 3115-3126.

[1] Warp drive with zero expansion, Class. Quantum Grav. 19 (2002) 1157-1165.

Talks

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 no-hair 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)

Gravito-electromagnetic analogies (April 2014)

Bernoulli's elastic curves, rockets, and time travel (July 2013)

Optimal time travel in the Gödel universe (January 2013)

Formation of Higher-dimensional Topological Black Holes (April 2010)

Asymptotic Quasi-normal Frequencies for d-Dimensional Black Holes (February 2009)

Greybody Factors for d-Dimensional 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)

Local Activities

Spanish-Portuguese 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)

Outreach Talks (mostly in Portuguese)

The mathematics of waves and black holes (March 2024)

The Hidden Geometry (February 2024)

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)

Outreach Articles (in Portuguese)

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 Gauss-Bonnet (March 2012)

Como Rodam as Coisas (February 2011)

Espaço-tempo de Minkowski: a Física como Geometria (June 2010)

O que é a curvatura de uma superfície? (January 2010)

Teaching (mostly in Portuguese)

Students (mostly in Portuguese)

PostDoc

Anne Franzen (2016-2020)

Filipe Costa (2013-2018)

Artur Alho (2013-2018)

PhD

João Santos (started in 2024)

Flavio Rossetti (2021-2024), Stability of the Cauchy Horizon for Cosmological Black Holes in Spherical Symmetry

Miguel Duarte (2018-2022), Asymptotics in Generalized Harmonic Gauge

Pedro Oliveira (2014-2018), Cosmic No-Hair for Spherically Symmetric Black Hole Spacetimes

Artur Alho (2008-2012), On the future stability of cosmological solutions of the Einstein-nonlinear scalar field system

MSc

Luís Machado (2023/2024), Boundedness and decay of solutions of the wave equation in Minkowski’s and Schwarzschild’s spacetimes

José Queirós (2022/2023), On the energy of gravitational waves in a de Sitter background

João Santos (2022/2023), Radiation reaction in magnetized black holes

João Silva (2022/2023), The Yamabe Problem

Flavio Rossetti (2020/2021), The Wave Equation in Cosmological Spacetimes

Gonçalo Fonseca (2020/2021), On the 3-dimensional Geroch Conjecture and the Positive Mass Theorem

Amol Sasane (2018/2019), Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes

Pedro Oliveira (2012/2013), Min-Oo'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

Filipe Santos (2023/2024), Teoremas de Singularidade de Hawking-Penrose

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

João Santos (2023/2024), Spinorial proof of the positive mass theorem

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 R³ com a métrica medida por observadores estacionários no espaço-tempo 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

José Natário

Position

Academic Degrees

Research Interests

Algebraic and Geometric Methods in Engineering and Physics

Advanced Topics in Geometry

Calculus II

Complex Analysis and Differential Equations

Differential Geometry

Differential Geometry of Curves and Surfaces

Geometric Mechanics

Lie Groups and Lie Algebras

Mathematical Analysis III

Mathematical Analysis IV