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Relatividade Geral e Aplicações Astrofísicas AGA0319 Rodrigo Nemmen

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Apresentação do curso

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Pré-requisitos • Curso introdutório de mecânica clássica: leis de conservação, problema da força central e mecânica Lagrangiana • Noções básicas de relatividade restrita (Física 4) e álgebra linear

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Website da disciplina Slides Datas das provas Divulgação das notas Datas quando não haverá aulas https://tinyurl.com/iag-gr

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Presença em aula Não é obrigatória exceto nos dias de exames. Presença e participação em aula serão levadas em conta nos casos de alunos com nota abaixo, mas perto do limiar de aprovação

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Exames em aula P1 11 de Setembro P2 9 de Outubro Psubs 4 de Dezembro

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Plantões de monitoria Quando: Terças-feiras, 11:00-12:30 Onde: Sala C302

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Livros-textos disponíveis para download: https://tinyurl.com/aga0319 (somente com email USP)

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Google Classroom

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1. classroom.google.com 2. App 3. A partir do gmail Como acessar:

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Desligar o seu smartphone, tablet, laptop etc

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Paving the ground for general relativity

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Leis da Leis da Relatividade Gravitação Q uântica Leis Quânticas Leis Newtonianas As leis físicas que governam o universo Planetas, estrelas, galáxias, aviões, carros, … Espaço e tempo curvos, expansão do universo, buracos negros, GPS, … Flutuações quânticas, lasers, LEDs, energia nuclear, química, … Big bang, singularidades, viagens no tempo (?), escala de Planck, energia escura (?), …

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Leis da Leis da Relatividade Gravitação Q uântica Leis Quânticas Leis Newtonianas As leis físicas que governam o universo Planetas, estrelas, galáxias, aviões, carros, … Espaço e tempo curvos, expansão do universo, buracos negros, GPS, … Flutuações quânticas, lasers, LEDs, energia nuclear, química, … Big bang, singularidades, viagens no tempo (?), escala de Planck, energia escura (?), …

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Leis da Leis da Relatividade Gravitação Q uântica Leis Quânticas Leis Newtonianas As leis físicas que governam o universo Planetas, estrelas, galáxias, aviões, carros, … Flutuações quânticas, lasers, LEDs, energia nuclear, química, … Big bang, singularidades, viagens no tempo (?), escala de Planck, energia escura (?), … Espaço e tempo curvos, expansão do universo, buracos negros, ondas gravitacionais, …

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Leis da Leis da Relatividade Gravitação Q uântica Leis Quânticas Leis Newtonianas As leis físicas que governam o universo Planetas, estrelas, galáxias, aviões, carros, … Flutuações quânticas, lasers, LEDs, energia nuclear, química, … Big bang, singularidades, viagens no tempo (?), escala de Planck, energia escura (?), … Espaço e tempo curvos, expansão do universo, buracos negros, ondas gravitacionais, …

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Spacetime x, y, z, t 4D space:

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Relatividade geral de Einstein: A gravidade corresponde a uma curvatura do espaço

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Gravity visualized: https://www.youtube.com/watch?v=MTY1Kje0yLg&list

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Gravity visualized: https://www.youtube.com/watch?v=MTY1Kje0yLg&list

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The Elegant Universe. Nova / PBS

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“Gravity is geometry”

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Lagrangian for standard model of particle physics http://www.symmetrymagazine.org/article/the-deconstructed-standard-model-equation gluon (strong force) W and Z bosons (weak force) weak interactions + Higgs Higgs ghosts Faddeev-Popov ghosts S = ∫ ℒ −gd4x δS δϕ = ∂ℒ ∂ϕ − ∂μ ( ∂ℒ ∂(∂μ ϕ) ) + ⋯ = 0 action Lagrange equations

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A general relativity primer Einstein’s field equation Stress-energy Ricci curvature Metric Ricci scalar Rμν − 1 2 gμν R = 8πG c4 Tμν

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A general relativity primer Einstein’s field equation Stress-energy Ricci curvature Metric Ricci scalar spacetime curvature 㱺 = constant × matter-energy Rμν − 1 2 gμν R = 8πG c4 Tμν

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A general relativity primer Einstein’s field equation Stress-energy Ricci curvature Metric Ricci scalar 㱺 For a free particle: Geodesic equation Newtonian analogue Poisson equation spacetime curvature = constant × matter-energy Rμν − 1 2 gμν R = 8πG c4 Tμν Solution to field equation gives Line element Metric

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561 D.3 Constructing Courses 1. Gravitational 2. Geometry 3. Space. Time, 4. Principles 5. Special PART Physics as Physics and Gravity of Special Relativistic in Newtonian Relativity Mecham‘cs Physics G . 7. The Description 6. Gravity as Spacetime 16. Gravitational 17. The Universe Waves Observed 18. Cosmological Models 9. The Geometry Outside a Sphen'cal Star PART 10. Solar System Tests of General Relativity 12. GraviIafional Collapse and Black Holes 14. A Little Rotation ll. Relativistic 13. Astrophysical 15. Rotatin'g Gravity In Black Holes Black Action Holes 20. A LittleMore Math 21. Curvature and the Em'stern‘ PART Equation Course structure Part I Part II Einstein equation Part IV Hartle General relativity basics Special relativity Spacetime explorations Part III

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561 D.3 Constructing Courses 1. Gravitational 2. Geometry 3. Space. Time, 4. Principles 5. Special PART Physics as Physics and Gravity of Special Relativistic in Newtonian Relativity Mecham‘cs Physics G . 7. The Description 6. Gravity as Spacetime 16. Gravitational 17. The Universe Waves Observed 18. Cosmological Models 9. The Geometry Outside a Sphen'cal Star PART 10. Solar System Tests of General Relativity 12. GraviIafional Collapse and Black Holes 14. A Little Rotation ll. Relativistic 13. Astrophysical 15. Rotatin'g Gravity In Black Holes Black Action Holes 20. A LittleMore Math 21. Curvature and the Em'stern‘ PART Equation Course structure Special relativity General relativity basics Part I Part II Einstein equation Part IV Hartle Spacetime explorations Part III

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All equations of motion are deterministic. No probabilities involved Once we specify the initial positions and velocities of particles, everything is determined! Evolving the gravitational field and matter dynamics for astrophysical situations can be challenging → General relativity is a classical theory Once initial conditions are given, the physical truth is perfectly determined Ψ(x, t) xμ, vμ

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Gravity is unscreened: there are no negative gravitational charges. It is not possible to shield the gravitational field Gravity is a long-range interaction. There is no characteristic length scale for gravitational interactions Gravity is the weakest of fundamental interactions between elementary particles. Some important properties of gravitational interaction These explain why gravity plays such a pivotal role in the universe Gravity governs large scale structure formation in the universe

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Phenomena for which general relativity is important

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vs. Newtonian gravity Given object of mass M and size R GR (general relativity) is important when When is general relativity important? GM Rc2 → 1

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vs. Newtonian gravity Given object of mass M and size R GR (general relativity) is important when \ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm Characteristic mass (g) Characteristic distance (cm) black hole interiors event horizons R s = 2GM c 2 When is general relativity important? GM Rc2 → 1

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vs. Newtonian gravity \ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm Characteristic mass (g) Characteristic distance (cm) black hole interiors event horizons R s = 2GM c 2 quantum gravity scale cosmological scales Given object of mass M and size R GR (general relativity) is important when GM Rc2 → 1 When is general relativity important?

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vs. Newtonian gravity \ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm Characteristic mass (g) Characteristic distance (cm) When is general relativity important? Given object of mass M and size R GR (general relativity) is important when GM Rc2 → 1 Hartle

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm vs. Newtonian gravity When is general relativity important? GM⊕ R⊕ c2 ∼ 10−9 Characteristic mass (g) Characteristic distance (cm) Earth Hartle

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm GM⊙ R⊙ c2 ∼ 10−6 Characteristic mass (g) Characteristic distance (cm) Sun Precession of perihelion of Mercury Bending of path of light rays passing near the Sun Solar System Hartle

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm GM Rc2 ∼ 0.1 Characteristic mass (g) Characteristic distance (cm) M ≲ 3M⊙ radius ~ 10 km (maximum mass) Credit: NASA's Goddard Space Flight Center/CI Lab Neutron stars Hartle

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm GM Rc2 = 0.5 Characteristic mass (g) Characteristic distance (cm) M ≳ 3M⊙ Black holes Hartle

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Two populations of black holes Supermassive 106-1010 solar masses one in every galactic nucleus 5-60 solar masses ~107 per galaxy Stellar

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Event Horizon Telescope: The first black hole image

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Event Horizon Telescope antennas

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https://www.youtube.com/watch?v=hebGhsNsjG0 Gravitational waves

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https://www.youtube.com/watch?v=hebGhsNsjG0 Gravitational waves

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Gravitational wave observatories with LIGO/Virgo

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Cosmology

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Quantum gravity: still a long way to go Credit: Greene

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Github Twitter Web E-mail Bitbucket Facebook Group figshare rodrigo.nemmen@iag.usp.br rodrigonemmen.com @nemmen rsnemmen facebook.com/rodrigonemmen nemmen blackholegroup.org bit.ly/2fax2cT

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Extra

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm black hole interiors event horizons R s = 2GM c 2

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm Hartle, w/ modifications by Nemmen black hole interiors event horizons R s = 2GM c 2

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\ ' Sun GPS orbit . neutron star primordial black hole evaporating today mass in grams 1010 I human universe at the end of inflation o laboratorymeasurement of Newton’ 3 G universe at the quantum gravity scale 10-10 I strand of DNA 10—20 probed by best I accelerators I hydrogen atom 10-30 10—30 10-20 10—10 1 101° 1020 1030 distance in cm Hartle, w/ modifications by Nemmen black hole interiors event horizons R s = 2GM c 2 quantum gravity scale dark matter? modified gravity? Milky Way