Einstein and Gravitation — Question 7

Question 7

Using half the relativistic light deflection biases the inferred mass

For light passing a spherical source in the weak-field limit, use the general-relativistic deflection $\alpha=4GM/(bc^2)$, where $b$ is impact parameter. An observed deflection is $4.0\times10^{-6}$ rad at $b=10^9\,\mathrm m$. Use $c=3\times10^8\,\mathrm{m/s}$ and $G=6.67\times10^{-11}$ SI.

1. Infer the source mass.

2. If an analyst instead used $\alpha=2GM/(bc^2)$, how would the inferred mass be biased? Predict the actual deflection at twice $b$.

Question 7 - Solution

Using half the relativistic light deflection biases the inferred mass

(a) Derivation and calculation

Rearrange the supplied weak-field deflection law: $$\alpha=\frac{4GM}{bc^2}\quad\Longrightarrow\quad M=\frac{\alpha bc^2}{4G}.$$ $$\boxed{M=\frac{(4.0\times10^{-6})(10^9)(3\times10^8)^2} {4(6.67\times10^{-11})} =1.34933\times10^{30}\ \mathrm{kg}}.$$

(b) Result and interpretation

Using half the correct coefficient gives $$M_{\mathrm{wrong}}=\frac{\alpha bc^2}{2G}=\boxed{2M}.$$ For the actual source at twice the impact parameter, $$\boxed{\alpha(2b)=\frac{4GM}{2bc^2}=\frac{\alpha(b)}2 =2.0\times10^{-6}\ \mathrm{rad}}.$$ Inverse-impact-parameter scaling alone cannot distinguish formulas with different normalization.

Physical check

The compactness at closest approach is $GM/(bc^2)=\alpha/4=10^{-6}\ll1$, consistent with the weak-field model.