Question

The condition for a uniform spherical mass $m$ of radius $r$ to be a black hole is [$G=$gravitational constant, $g=$ acceleration due to gravity].

• A. ${\left[\frac{2Gm}{r}\right]}^{1/2}\le c$
• B. ${\left[\frac{2gm}{r}\right]}^{1/2}=c$
• C. ${\left[\frac{2Gm}{r}\right]}^{1/2}\ge c$
• D. ${\left[\frac{gm}{r}\right]}^{1/2}\ge c$

Answer 1

Answer: (C)

${\left[\frac{2Gm}{r}\right]}^{1/2}\ge c$

For a uniform spherical mass to be a black hole, total energy of the "particle + spherical mass" system $\le 0$
Let the mass of particle be $m_1$ moving with its maximum speed $c$.
$\therefore$ $\frac{1}{2}{m}_1{c}^2+(-\frac{Gm{m}_1}{r})\le 0$
$⟹c\le [\frac{2Gm}{r}]$