Tweet
Tweet

Slide 1

Slide 1 text

Relatividade Geral e Aplicações Astrofísicas AGA0319 Rodrigo Nemmen

Slide 2

Slide 2 text

Apresentação do curso

Slide 3

Slide 3 text

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

Slide 4

Slide 4 text

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

Slide 5

Slide 5 text

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

Slide 6

Slide 6 text

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

Slide 7

Slide 7 text

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

Slide 8

Slide 8 text

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

Slide 9

Slide 9 text

Google Classroom

Slide 10

Slide 10 text

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

Slide 11

Slide 11 text

No content

Slide 12

Slide 12 text

No content

Slide 13

Slide 13 text

Desligar o seu smartphone, tablet, laptop etc

Slide 14

Slide 14 text

Paving the ground for general relativity

Slide 15

Slide 15 text

No content

Slide 16

Slide 16 text

No content

Slide 17

Slide 17 text

No content

Slide 18

Slide 18 text

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 (?), …

Slide 19

Slide 19 text

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 (?), …

Slide 20

Slide 20 text

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, …

Slide 21

Slide 21 text

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, …

Slide 22

Slide 22 text

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

Slide 23

Slide 23 text

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

Slide 24

Slide 24 text

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

Slide 25

Slide 25 text

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

Slide 26

Slide 26 text

The Elegant Universe. Nova / PBS

Slide 27

Slide 27 text

“Gravity is geometry”

Slide 28

Slide 28 text

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

Slide 29

Slide 29 text

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

Slide 30

Slide 30 text

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

Slide 31

Slide 31 text

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

Slide 32

Slide 32 text

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

Slide 33

Slide 33 text

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

Slide 34

Slide 34 text

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

Slide 35

Slide 35 text

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

Slide 36

Slide 36 text

Phenomena for which general relativity is important

Slide 37

Slide 37 text

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

Slide 38

Slide 38 text

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

Slide 39

Slide 39 text

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?

Slide 40

Slide 40 text

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

Slide 41

Slide 41 text

\ ' 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

Slide 42

Slide 42 text

\ ' 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

Slide 43

Slide 43 text

\ ' 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

Slide 44

Slide 44 text

\ ' 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

Slide 45

Slide 45 text

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

Slide 46

Slide 46 text

Event Horizon Telescope: The first black hole image

Slide 47

Slide 47 text

Event Horizon Telescope antennas

Slide 48

Slide 48 text

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

Slide 49

Slide 49 text

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

Slide 50

Slide 50 text

Gravitational wave observatories with LIGO/Virgo

Slide 51

Slide 51 text

Cosmology

Slide 52

Slide 52 text

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

Slide 53

Slide 53 text

Github Twitter Web E-mail Bitbucket Facebook Group figshare [email protected] rodrigonemmen.com @nemmen rsnemmen facebook.com/rodrigonemmen nemmen blackholegroup.org bit.ly/2fax2cT

Slide 54

Slide 54 text

Extra

Slide 55

Slide 55 text

\ ' 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

Slide 56

Slide 56 text

\ ' 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

Slide 57

Slide 57 text

\ ' 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