Yes. The cleanest way to derive this is to start with Newtonian gravitational collapse, then show where Newtonian gravity fails and replace it with general relativity, obtaining the Schwarzschild radius and the condition for a black hole.

One important correction first: an isolated Sun like ours will not naturally collapse into a black hole at the end of its life. Its mass is too small; it will lose its outer layers and leave a white dwarf. But mathematically, if the Sun’s entire mass could somehow be compressed sufficiently, it would become a black hole.

1. Start with the Sun as a self-gravitating sphere

Take a spherical star of mass (M) and radius (R).

For the Sun,

\[M_\odot \approx 1.989\times10^{30}\ {\rm kg},\] \[R_\odot \approx 6.957\times10^8\ {\rm m}.\]

Consider a small mass (dm) at radius (r). The gravitational acceleration produced by the mass inside (r) is

\[g(r)=\frac{Gm(r)}{r^2}.\]

For a spherical object, the shell theorem tells us that the exterior shells do not contribute to the gravitational force at (r).

If the density is approximately uniform,

\[m(r)=\frac{4\pi}{3}\rho r^3.\]

Therefore

\[g(r)=\frac{4\pi G\rho}{3}r.\]

The gravitational force becomes stronger as the material moves inward relative to the outer layers, driving collapse when pressure can no longer support the star.

2. Gravitational potential energy

The gravitational binding energy of a uniform sphere is

\[U=-\frac{3GM^2}{5R}.\]

This equation is extremely important.

As the star contracts,

\[R\rightarrow R/2,\]

we get

\[U\rightarrow 2U.\]

So gravitational energy is released as the radius decreases.

More generally,

\[\boxed{ U\sim-\frac{GM^2}{R} }\]

and therefore

\[|U|\rightarrow\infty \qquad\text{as}\qquad R\rightarrow0\]

in the Newtonian approximation.

Of course, this divergence tells us that Newtonian gravity has ceased to be an adequate description. We need general relativity.

3. What prevents a normal star from collapsing?

A star reaches equilibrium through approximately

\[\boxed{ \text{gravity} \leftrightarrow \text{pressure} }\]

or, in Newtonian form, the equation of hydrostatic equilibrium:

\[\boxed{ \frac{dP}{dr} = -\frac{Gm(r)\rho(r)}{r^2} }\]

with

\[\frac{dm}{dr}=4\pi r^2\rho.\]

Thus the structure of a star is determined by

\[\boxed{ \begin{aligned} \frac{dm}{dr}&=4\pi r^2\rho,\\ \frac{dP}{dr}&=-\frac{Gm\rho}{r^2},\\ P&=P(\rho,T,\ldots). \end{aligned}}\]

The last equation is the equation of state.

During the Sun’s normal life, thermal pressure generated by nuclear fusion helps support it.

Eventually, however, different forms of pressure can become important:

thermal pressure,
electron degeneracy pressure,
neutron degeneracy pressure,
nuclear interactions.

A black hole forms when none of the available pressure mechanisms can produce a stable configuration before the collapsing material passes inside its gravitational radius.

4. The key relativistic condition

Now consider the escape velocity from the surface of an object:

\[v_{\rm esc}=\sqrt{\frac{2GM}{R}}.\]

\[v_{\rm esc}=c,\]

then

\[c=\sqrt{\frac{2GM}{R}}.\]

Squaring,

\[c^2=\frac{2GM}{R}.\]

Therefore

\[\boxed{ R=\frac{2GM}{c^2} }\]

This is the Schwarzschild radius:

\[\boxed{ r_s=\frac{2GM}{c^2}. }\]

This Newtonian escape-velocity derivation happens to give the same radius obtained exactly from the Schwarzschild solution of Einstein’s equations.

5. How small would the Sun have to become?

Put

\[M=M_\odot.\]

Then

\[r_s= \frac{2GM_\odot}{c^2}.\]

Using

\[G=6.67430\times10^{-11}\ {\rm m^3kg^{-1}s^{-2}},\] \[M_\odot=1.989\times10^{30}\ {\rm kg},\] \[c=2.998\times10^8\ {\rm m/s},\]

we obtain approximately

\[\boxed{ r_s\approx2.95\ {\rm km}. }\]

So a mass equal to the Sun’s mass would have to be compressed to a radius of roughly

\[\boxed{3\ {\rm km}}\]

to form a Schwarzschild black hole.

Compare:

\[R_\odot\approx696,000\ {\rm km}.\]

Thus the required compression is enormous:

\[\frac{R_\odot}{r_s} \approx \frac{696000}{2.95} \approx236,000.\]

The Sun would have to shrink by roughly 236,000 times in radius.

6. General relativity gives the real explanation

The previous argument is useful, but the fundamental derivation comes from Einstein’s field equations:

\[\boxed{ G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}. }\]

The left-hand side describes spacetime curvature.

The right-hand side describes matter and energy.

For a nonrotating, spherically symmetric object outside the matter,

\[T_{\mu\nu}=0.\]

Einstein’s equations then have the Schwarzschild solution:

\[\boxed{ ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2d\Omega^2 }\]

where

\[d\Omega^2=d\theta^2+\sin^2\theta\,d\phi^2.\]

The crucial quantity is

\[f(r)=1-\frac{2GM}{rc^2}.\]

\[f(r)=0,\]

we have

\[1-\frac{2GM}{rc^2}=0.\]

Consequently,

\[\boxed{ r=\frac{2GM}{c^2}=r_s. }\]

This is the event horizon of the Schwarzschild black hole.

7. Why this creates a black hole

The Schwarzschild metric contains the factor

\[1-\frac{r_s}{r}.\]

Outside the horizon,

\[r>r_s,\]

\[1-\frac{r_s}{r}>0.\]

Inside,

\[r<r_s,\]

and the causal structure changes dramatically.

The important statement is not merely that the escape velocity exceeds (c).

Rather:

\[\boxed{ \text{inside }r_s,\quad \text{all future-directed causal paths lead toward smaller }r. }\]

That includes paths followed by light.

So the defining property of the event horizon is

\[\boxed{ \text{nothing inside the horizon can communicate with the external universe.} }\]

8. What happens during gravitational collapse?

Imagine the Sun somehow loses its ability to support itself.

Its radius evolves approximately as

\[R(t)\downarrow.\]

Initially,

\[R\gg r_s.\]

Gravity is strong, but the object is not a black hole.

Eventually,

\[R(t)=r_s.\]

At this point the surface crosses the Schwarzschild radius.

Afterward,

\[R(t)<r_s.\]

The horizon has formed.

A useful conceptual picture is

\[\boxed{ R>r_s \quad\longrightarrow\quad R=r_s \quad\longrightarrow\quad R<r_s }\]

and therefore

\[\boxed{ \text{star} \rightarrow \text{collapse} \rightarrow \text{event horizon} \rightarrow \text{black hole}. }\]

9. But what actually causes the collapse?

This is where stellar astrophysics becomes important.

Suppose pressure is (P). Hydrostatic equilibrium requires approximately

\[\frac{dP}{dr} \sim -\frac{GM\rho}{R^2}.\]

As (R) becomes smaller,

\[\frac{GM\rho}{R^2}\]

becomes enormous.

The required pressure therefore increases.

But pressure itself contributes to the gravitational field in general relativity.

This leads to the relativistic version of hydrostatic equilibrium: the Tolman–Oppenheimer–Volkoff equation.

For a spherical relativistic star,

\[\boxed{ \frac{dP}{dr} = -\frac{ G \left(\rho+\frac{P}{c^2}\right) \left( m(r)+\frac{4\pi r^3P}{c^2} \right) }{ r^2 \left( 1-\frac{2Gm(r)}{rc^2} \right) }. }\]

with

\[\boxed{ \frac{dm}{dr}=4\pi r^2\rho. }\]

Notice something profound.

Newtonian gravity gives approximately

\[\frac{dP}{dr} \sim -\frac{Gm\rho}{r^2}.\]

GR introduces two major corrections:

\[\boxed{ \rho\rightarrow \rho+\frac{P}{c^2} }\]

and

\[\boxed{ m\rightarrow m+\frac{4\pi r^3P}{c^2}. }\]

Thus pressure itself gravitates.

This becomes increasingly important during collapse.

10. The denominator reveals the black-hole instability

Look at the TOV equation again:

\[\frac{dP}{dr} = -\frac{ G \left(\rho+\frac{P}{c^2}\right) \left( m+\frac{4\pi r^3P}{c^2} \right) }{ r^2 \left( 1-\frac{2Gm}{rc^2} \right) }.\]

The denominator contains

\[\boxed{ 1-\frac{2Gm(r)}{rc^2}. }\]

\[\frac{2Gm(r)}{rc^2}\rightarrow1,\]

the required pressure gradient becomes extremely large.

\[r=\frac{2Gm}{c^2},\]

the static stellar description breaks down.

This is the relativistic mathematical signature of the formation of a trapped region/event horizon.

11. Why the actual Sun doesn’t become a black hole

This is crucial.

The Sun’s mass is

\[M_\odot\approx1.989\times10^{30}\ {\rm kg}.\]

Its Schwarzschild radius is only

\[r_s\approx2.95\ {\rm km}.\]

But the Sun cannot simply collapse to (3) km.

When the Sun eventually exhausts its nuclear fuel, its evolution leads through later stellar stages and ultimately to a white dwarf of roughly solar mass scale, rather than a black hole.

Electron degeneracy pressure can support a white dwarf.

The maximum mass for a cold, idealized electron-degenerate white dwarf is approximately the Chandrasekhar mass:

\[\boxed{ M_{\rm Ch}\approx1.4M_\odot. }\]

A star substantially more massive than this can undergo further collapse, depending on its evolutionary history, losing mass along the way.

Eventually, sufficiently massive stellar remnants can form neutron stars or black holes.

12. The deeper condition: compactness

A particularly elegant way to characterize gravitational collapse is through the dimensionless compactness parameter

\[\boxed{ \mathcal C=\frac{GM}{Rc^2}. }\]

For the Sun,

\[\mathcal C_\odot = \frac{GM_\odot}{R_\odot c^2} \approx2.1\times10^{-6}.\]

Very weak relativistic gravity.

For a Schwarzschild black hole,

\[R=\frac{2GM}{c^2},\]

\[\boxed{ \mathcal C=\frac12. }\]

Thus the Sun would have to increase its compactness from roughly

\[2.1\times10^{-6}\]

\[0.5.\]

That’s a factor of roughly

\[\boxed{2.4\times10^5}.\]

13. The whole derivation in one chain

The essential mathematical chain is

\[\boxed{ \text{mass} \rightarrow \text{gravity} \rightarrow \text{loss of pressure support} \rightarrow R\downarrow }\]

followed by

\[\boxed{ \frac{2GM}{Rc^2}\rightarrow1. }\]

Einstein’s equations give

\[G_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu},\]

whose spherical vacuum solution is

\[ds^2= -\left(1-\frac{2GM}{rc^2}\right)c^2dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 +r^2d\Omega^2.\]

The horizon occurs at

\[1-\frac{2GM}{rc^2}=0,\]

hence

\[\boxed{ r_h=r_s=\frac{2GM}{c^2}. }\]

For the Sun:

\[\boxed{ r_s(M_\odot)\approx2.95\ {\rm km}. }\]

Therefore:

\[\boxed{ \text{If the Sun's entire mass were compressed inside } \sim3\text{ km,} }\]

it would be a Schwarzschild black hole.

There is an even more interesting next step: we can derive the collapse dynamically from Einstein’s equations, using the Oppenheimer–Snyder model of a homogeneous spherical dust star, and explicitly derive the trajectory (R(\tau)), the formation of the apparent horizon, the event horizon, and finally the (r=0) singularity. That gives a genuine mathematical model of a star collapsing into a black hole rather than just the static Schwarzschild-radius argument.