In General relativity, energy formula of a body thrown straight up to the infinity is

$\large {E=\frac{mc^2}{\sqrt{1-R_S/R}}}$

As we know relativistic energy formula is

$\large {E=\frac{mc^2}{\sqrt{1-v^2/c^2}}}$

So

$\large {\frac{mc^2}{\sqrt{1-v_e^2/c^2}}=\frac{mc^2}{\sqrt{1-R_S/R}}}$

and hence escape velocity equation in General relativity is

${\large {v_e^2=c^2\frac{R_S}{R}}}$

where $R_S=2GM/c^2$ - Schwarzschild radius of black hole, and $R>R_S$

It's easy to derive that

${\large {v_e=c\sqrt{\frac{R_S}{R}}}=\sqrt{\frac {2GM}{R}}}$

So escape velocity formula in General relativity and Newton gravity is the same.

In General relativity, energy formula of a body thrown straight up to the infinity is

$\large {E=\frac{mc^2}{\sqrt{1-R_S/R}}}$

As we know relativistic energy formula is

$\large {E=\frac{mc^2}{\sqrt{1-v^2/c^2}}}$

So

$\large {\frac{mc^2}{\sqrt{1-v_e^2/c^2}}=\frac{mc^2}{\sqrt{1-R_S/R}}}$

hence escape velocity equation in General relativity is

${\large {v_e^2=c^2\frac{R_S}{R}}}$

where $R_S=2GM/c^2$ - Schwarzschild radius of a black hole, and $R>R_S$

It's easy to derive that

${\large {v_e=c\sqrt{\frac{R_S}{R}}}=\sqrt{\frac {2GM}{R}}}$

So escape velocity formula in General relativity and Newton gravity is the same.