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Introduction to the General Relativity course

Introduction to the General Relativity course

First lecture of the course "General Relativity With Astrophysical Applications", taught by Prof. Rodrigo Nemmen (USP).

• Practical information
• Outline of the course
• When is GR relevant?
• Relativistic astrophysical phenomena

https://rodrigonemmen.com/teaching/relatividade-geral-e-aplicacoes-astrofisicas/

Rodrigo Nemmen

August 05, 2019
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  1. Relatividade Geral
    e Aplicações
    Astrofísicas
    AGA0319
    Rodrigo Nemmen

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  2. Apresentação do curso

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

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

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

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

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  9. Google Classroom

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

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  11. Desligar o seu smartphone, tablet, laptop etc

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  12. Paving the ground for general
    relativity

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  13. 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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  14. 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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  15. 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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  16. 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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  17. Spacetime
    x, y, z, t
    4D space:

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

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

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

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  21. The Elegant Universe. Nova / PBS

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  22. “Gravity is geometry”

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  23. 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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  24. 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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  25. 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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  26. A general relativity primer
    Einstein’s field equation
    Stress-energy
    Ricci curvature Metric
    Ricci
    scalar

    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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  27. 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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  28. 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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  29. 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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  30. 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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  31. Phenomena for which general
    relativity is important

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  32. 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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  33. 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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  34. 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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  35. 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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  36. \ ' 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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  37. \ ' 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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  38. \ ' 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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  39. \ ' 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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  40. 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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  41. Event Horizon Telescope:
    The first black hole image

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  42. Event Horizon Telescope antennas

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

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

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  45. Gravitational wave observatories
    with LIGO/Virgo

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

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

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  48. \ ' 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

    View full-size slide

  49. \ ' 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

    View full-size slide

  50. \ ' 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

    View full-size slide