# A Roller Coaster Ride through Relativity/Appendix H

## The Gravitational Potential at the surface of a star

Gravitational Potential

The force of gravity on a mass m at a distance r away from a star of mass M is equal to:

$F = {GMm \over r^2}$

The work done in pulling the mass from the surface of the star out to infinity is therefore:

$W = \int_{r}^{\infty}{GMm \over r^2} dr$

This works out to be

$W = {GMm \over r}$

and therefore the gravitational potential difference Δφ between the surface and ∞ is

$\Delta \phi = {w \over m} = {GM \over R}$

Now by definition, the gravitational potential at ∞ is zero so the potential at the surface is negative, hence:

$\phi_s = - {GM \over R}$

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