Q:

What equation can I use to calculate escape velocity as the radius increases from the Schwarzschild radius?

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.

A:

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.

