At the Schwarzschild radius, the escape velocity is the speed of light for all objects. However, I'm looking for an equation that will calculate the escape velocity as the radius of the object increases from its Schwarzschild radius, while keeping its mass constant.
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$\begingroup$ Are you interested in the classical escape velocity equation of Newton's dynamic or are you interested in a General Relativity focused answer? $\endgroup$– NoumenoCommented Jul 23, 2020 at 18:56
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$\begingroup$ Newton would be sufficient - thank you. $\endgroup$– TivityCommented Jul 24, 2020 at 0:22
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1 Answer
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As follows from energy conservation, the gravitational time dilation near a Schwarzschild object is equal to the velocity time dilation at the escape speed:
$$\dfrac{\tau}{t}=\sqrt{1-\dfrac{r_s}{r}}=\sqrt{1-\dfrac{v^2}{c^2}}$$
$$v=c\sqrt{\dfrac{r_s}{r}}$$
$$v=\sqrt{\dfrac{2GM}{r}}$$
Where $c$ is the speed of light, $G$ is the gravitational constant, $M$ is the mass of the object, and $r_s$ is the Schwarzschild radius. Due to energy conservation, this result is the same for the Newtonian gravity.
answered Jul 24, 2020 at 8:13