Gravity, evaluated in the weak field of the solar system, behaves as a Brans-Dicke…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch Solar-system tests, the PPN framework and alternative theories

Problem

f(R)

gravity, evaluated in the weak field of the solar system, behaves as a Brans-Dicke theory with

\omega_{\rm BD}=0

, giving

\gamma_{\rm PPN}=1/2

and

\beta_{\rm PPN}=1

. A distant star's light grazes the limb of Jupiter, with mass

M_{\rm J}=1.90\times10^{27}\,\text{kg}

and equatorial radius

R_{\rm J}=7.15\times10^{7}\,\text{m}

. Take

G=6.674\times10^{-11}\,\text{N m}^2\,\text{kg}^{-2}

and

c=2.998\times10^{8}\,\text{m s}^{-1}

. The PPN-generalised deflection angle for a ray grazing a body of mass

M

at impact parameter

b

\Delta\phi=\frac{1+\gamma_{\rm PPN}}{2}\cdot\frac{4GM}{bc^2}.

((a)) Compute the deflection predicted by general relativity (

\gamma_{\rm PPN}=1

), in arcseconds. ((b)) Compute the deflection predicted by

f(R)

gravity under this assumption, in arcseconds. ((c)) Deriving the Newtonian limit of the Einstein field equations fixes the coefficient of the first-order term in

g_{00}

at exactly 1, with no free parameter there, while the coefficient in

g_{ij}

is free and becomes

\gamma_{\rm PPN}

. State which assumption of that derivation

f(R)

gravity drops, and explain why the leading

g_{00}

coefficient stays fixed at 1 even though the theory departs from general relativity.

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