A Brans-Dicke universe has

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

\omega_{\rm BD}=8

, giving

\gamma_{\rm PPN}=(1+\omega_{\rm BD})/(2+\omega_{\rm BD})

and

\beta_{\rm PPN}=1

. A radar signal travels from Earth to a transponder on a Mars orbiter while Mars is almost directly behind the Sun, so the ray grazes the solar limb. Take the Earth-Sun distance as

r_1=1.496\times10^{11}\,\text{m}

, the Mars-Sun distance as

r_2=2.280\times10^{11}\,\text{m}

, the impact parameter as the solar radius

b=R_\odot=6.96\times10^{8}\,\text{m}

, and

GM_\odot/c^2=1.48\,\text{km}

. The excess round-trip coordinate time (the Shapiro delay), generalised to the PPN framework, is

\Delta t_{\rm S}=\frac{2GM_\odot}{c^3}(1+\gamma_{\rm PPN})\ln\!\left(\frac{4r_1r_2}{b^2}\right).

((a)) Compute

\Delta t_{\rm S}

predicted by general relativity (

\gamma_{\rm PPN}=1

), in microseconds. ((b)) Compute

\Delta t_{\rm S}

predicted by this Brans-Dicke theory, and give its difference from the general-relativistic value in microseconds. ((c)) Universality of free fall has been confirmed to a few parts in

10^{15}

. Explain why this Brans-Dicke theory, whose Shapiro delay differs measurably from general relativity's, is not already excluded by that result: state which assumption used to build the Einstein field equations in the Newtonian limit it drops, and what matter still couples to.

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