Solar System tests of general relativity

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

• Parametrized-Post-Newtonian metric (PPN)
• Precession of the perihelion of Mercury measurement
• Deflection of light by the Sun
• The 1919 total eclipse expedition and observations

Credit for the slides/figures belongs to Rodrigo Nemmen, unless otherwise stated.

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

Transcript

1. AGA0319
   Rodrigo Nemmen
   Solar System Tests of
   General Relativity
   Astrophysical Applications of GR I
   

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2. Comparing GR with other theories of gravity
   We need a way of testing GR with observations
   Specifically: quantify possible deviation from GR in data
   Standard way: parametrized-Post-Newtonian (PPN)
   

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3. Parametrized-Post-Newtonian (PPN)
   framework
   Begin with general
   static, spherically
   symmetric metric
   ds2 = − A(r)(cdt)2 + B(r)dr2 + r2(dθ2 + sin2 θdϕ2)
   A(r) = 1 −
   2GM
   c2r
   + ⋯ B(r) = 1 + ⋯
   Agreement with
   Newton requires
   Define γ and β
   A(r) = 1 −
   2GM
   c2r
   + 2(β − γ)
   (
   GM
   c2r )
   2
   + ⋯
   B(r) = 1 + 2γ
   (
   GM
   c2r )
   + ⋯
   If γ=1 and β=1 → recover general relativity
   PPN metric
   

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4. Parametrized-Post-Newtonian (PPN)
   framework
   ds2 = − A(r)(cdt)2 + B(r)dr2 + r2(dθ2 + sin2 θdϕ2)
   A(r) = 1 −
   2GM
   c2r
   + 2(β − γ)
   (
   GM
   c2r )
   2
   + ⋯
   B(r) = 1 + 2γ
   (
   GM
   c2r )
   + ⋯
   If γ=1 and β=1 → GR is correct
   PPN metric
   If γ≠1 and β≠1 → Einstein is wrong
   

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5. Relativistic effects in the PPN metric
   Δϕ
   def
   =
   (
   1 + γ
   2 ) (
   4GM
   c2b )
   Deflection angle of light
   ray passing by mass M
   Δϕ
   prec
   =
   1
   3
   (2 + 2γ − β)
   6πGM
   c2a(1 − ϵ2)
   Precession of perihelion
   of planet per orbit
   

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6. Precession of Mercury’s perihelion
   

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7. Precession of Mercury’s perihelion
   GR prediction Δφprec = 43″ / century Δϕ
   prec
   =
   6πG
   c2
   M
   a(1 − ϵ2)
   0.8 0.9 1.0 1.1 1.2
   38
   40
   42
   44
   46
   48
   γ
   Δφprec
   (″/century)
   

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8. Precession of Mercury’s perihelion
   GR prediction Δφprec = 43″ / century
   Observed precession = 5599.74″±0.65 / century
   Δϕ
   prec
   =
   6πG
   c2
   M
   a(1 − ϵ2)
   Effect Δφ (")
   Precession of equinoxes 5025.64
   Perturbation other planets 532
   Oblateness of Sun 0.03
   Total 5557
   Data - effects 43
   γ = 1.000 ± 0.002
   β = 1.000 ± 0.003
   PPN parameters
   

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9. Deflection of light by the Sun
   GR prediction Δφdef = 1.75″ Δϕ
   def
   =
   4GM
   c2b
   https://grupos.unican.es/glendama/Historical_intro.htm
   

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10. Deflection of light by the Sun
    GR prediction Δφdef = 1.75″ Δϕ
    def
    =
    4GM
    c2b
    https://grupos.unican.es/glendama/Historical_intro.htm
    

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11. Deflection of light by the Sun
    Δφ
    def = 1.75″
    https://medium.com/@GatotSoedarto/how-to-show-gravity-affects-light-ee9e8dfd33af
    

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12. Deflection of light by the Sun
    apparent position
    of stars in the sky
    

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13. Deflection of light by the Sun
    apparent position
    of stars in the sky
    M
    

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14. Illustrated London News of November 22, 1919
    

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15. Illustrated London News of November 22, 1919
    

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16. New York Times, 10 Nov. 1919
    

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17. LUZES TODAS
    TORTAS NOS CÉUS
    Cientistas inquietos pelos
    resultados das observações do
    eclipse
    Estrelas não estão onde
    deveriam estar, mas ninguém
    precisa se preocupar
    TEORIA DE EINSTEIN TRIUNFA
    UM LIVRO PARA DEZ SÁBIOS
    Ninguém mais no mundo
    consegue compreende-lo
    New York Times, 10 Nov. 1919
    

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18. Deflection of light by the Sun
    GR prediction Δφdef = 1.75″
    γ
    Δφdef (″)
    Δϕ
    def
    =
    4GM
    c2b
    0.8 0.9 1.0 1.1 1.2
    1.60
    1.65
    1.70
    1.75
    1.80
    1.85
    1.90
    β=1
    γ = 1.007 ± 0.009
    β = 1.000 ± 0.003
    PPN parameters
    

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