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# 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. ## Rodrigo Nemmen

November 13, 2018

## Transcript

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

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)

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

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

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

6. Precession of Mercury’s perihelion

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)

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

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

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

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

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

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

14. Illustrated London News of November 22, 1919

15. Illustrated London News of November 22, 1919

16. New York Times, 10 Nov. 1919

17. LUZES TODAS
TORTAS NOS CÉUS
Cientistas inquietos pelos
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

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