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

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)

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

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

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

Precession of Mercury’s perihelion

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)

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

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

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

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

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

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

Illustrated London News of November 22, 1919

Illustrated London News of November 22, 1919

New York Times, 10 Nov. 1919

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

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