The radius $R_h$ and mass $M_h$ of a black hole are related by $R_h = 2 GM_h/c^2$ , where c is the speed of light. Assume that the gravitational acceleration $a_g$ of an object at a distance $r_o = 1.001 R_h$ from the center of a black hole is given (it is, for large black holes). In terms of $M_h$ , find $a_g$ at $r_o$ .

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It is given that the radius $R_{h}$ and mass $M_{h}$ of a black hole are related by,

\begin{align} R_{h}=\frac{2 G M_{h} }{ c^{2}}\end{align}

The acceleration of gravity of an object at a distance of $r_{o}=1.001 R_{h}$ from the center of a black hole is given by,

a_{g}=\frac{G M}{r^{2}}=\frac{G M_{h}}{\left(1.001 R_{h}\right)^{2}}

substitute from (1) to get,

a_{g}=\frac{G M_{h}}{(1.001)^{2}\left(2 G M_{h} / c^{2}\right)^{2}}=\frac{c^{4}}{(2.002)^{2} G} \frac{1}{M_{h}}

substitute with the givens to get,

\begin{align*} a_{g}&=\frac{(3.0 \times 10^{8} \mathrm{~m/s})^{4}}{(2.002)^{2} (6.67 \times 10^{-11} \mathrm{~m^{3} / s^{2} \cdot kg})} \frac{1}{M_{h}}\\ &=\frac{\left(3.02 \times 10^{43} \mathrm{~kg \cdot m/s^{2}}\right)}{M_{h}} \end{align*}

\begin{align}\boxed{a_{g}=\frac{\left(3.02 \times 10^{43} \mathrm{~kg \cdot m/s^{2}}\right)}{M_{h}}}\end{align}

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