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## Homework Statement

The radius R

_{h}and mass M of a black hole are related by R

_{h}= [itex]\frac{2GM}{c

^{2}}[/itex], where c is the speed of light. Assume that the gravitational acceleration ag of an object at a distance r

_{o}= (1 + ε)R

_{h}from the center of a black hole is given by a

_{g}= [itex]\frac{GM}{r

^{2}}[/itex], where ε is a small positive number. (This formula is valid for certain values of ε. Exactly how large or small ε must be depends on the size of the black hole.) (a) What is a

_{g}at r

_{o}for a mass M black hole, to first order in ε? (b) If an astronaut with a height of H is at r

_{o}with her feet toward this black hole, what is the difference in gravitational acceleration between her head and her feet? Assume H << R

_{h}. Express your answers in terms of M, G, H, c, and ε.

## Homework Equations

a

_{g}= [itex]\frac{GM}{r^{2}_{g}}[/itex]

R

_{h}= [itex]\frac{2GM}{c^{2}}[/itex]

r[itex]_{0}[/itex] = (1+[itex]\epsilon[/itex])R[itex]_{h}[/itex]

## The Attempt at a Solution

a[itex]_{0}[/itex] = [itex]\frac{GM}{r^{2}_{0}}[/itex]

= [itex]\frac{GM}{[(1+\epsilon)^{2}R_{h}]^{2}}[/itex]

= [itex]\frac{GM}{(1+\epsilon)^{2}(\frac{2GM}{c^{2}}^{2}}[/itex]

=[itex]\frac{c^{4}}{4(1+\epsilon)^{2}GM}[/itex]

part b)

a

_{g}= [itex]\frac{GM}{r^{2}_{g}}[/itex]

differentiating gives da

_{g}= [itex]\frac{-2GM}{r^{3}}[/itex]dr

= [itex]\frac{-2GM}{r^{3}}[/itex]dr

=[itex]\frac{-2GM}{[(1+\epsilon)R_{h}]^{3}}[/itex]

=[itex]\frac{-2GMH}{(1+\epsilon)^{3}\frac{2GM}{c^{2}}^{3}}[/itex]

=[itex]\frac{-Hc^{6}}{4(1+\epsilon)^{3}GM^{2}}[/itex]

I'm not sure why my answers are wrong, but my hint says

The gravitational acceleration depends on the mass of the black hole and the distance from its center. Here the distance is a not fixed value but a certain multiple of the radius Rh. Use Taylor series or the binomial expansion to determine the leading order dependence on ε.

Which doesn't help me. I'd appreciate any hints/explanations :)