In 1939, Albert Einstein published a paper entitled "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses." In it, he considers the problem of whether it is possible to reach a spacetime metric containing singularities in a real physical system – that is, starting from actual gravitating masses.
Consider a system of many small objects, with equal masses, which move under the influence of the gravitational field of the entire system. Furthermore suppose that the particles move in concentric, randomly oriented circular orbits, so that the overall gravitational field is approximately static and spherically symmetric. Then the spacetime metric is
$$ds^2 = -a(r)(dx^2 + dy^2 + dz^2) + b(r)\, dt^2 ~$$
where $a(r)$ and $b(r)$ are functions of the radius $r^2 = x^2 + y^2 + z^2$. By substituting this into the Einstein field equation
$$G_{\mu\nu} = R_{\mu\nu} - \frac12 g_{\mu\nu}R + \kappa T_{\mu\nu} = 0$$
one can obtain differential equations for $a(r)$ and $b(r)$:
$$\alpha'' + \frac{\alpha'}{r} + \frac14 \alpha'^2 - \frac1{r^2} + \kappa mn a^{-1/2} \left( \frac{\alpha'^2}{\frac32 \alpha'^2 - \frac2{r^2}} \right)^{1/2} = 0$$
$$\beta' = \frac1{\alpha'} \left( \frac2{r^2} - \frac12 \alpha'^2 \right)$$
where $\alpha = \ln (r^2 a)$; $\beta = \ln b$; $m$ is the mass of each particle; and $n$ is the particle density.
At first glance, these equations do not look very tractable. However, the idealized limiting case where the gravitating particles are concentrated within an infinitesimally thin spherical shell of radius $r_0 \pm \Delta$ is relatively simple. Einstein solves this case and shows that:
$$r_0 > \frac\mu2 (2 + \sqrt3)$$
where $\mu/2 = \frac12 \left( \frac{\kappa}{8\pi} mN \right)$ is the Schwarzschild radius, with $mN$ being the total mass of the system.
Since this lower bound is above the Schwarzschild radius, Einstein reasons, a system of masses in circular orbits cannot form a black hole. He also generalizes this result to the case of continuous particle density and obtains a similar lower bound. Physically, as $r_0$ decreases towards the bound (that is, as the system of masses becomes smaller and smaller), the kinetic energy of the system goes to infinity. Intuitively, one would expect that the same reasoning should apply to other systems as well, though the fully general case is not addressed rigorously in the paper.
Einstein concludes:
The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths, it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.
Today, we know that black holes exist; hence, the argument above must be wrong. But where was Einstein's mistake? Was it the assumption of stable circular orbits?
I saw this question: Einstein and the existence of Black Holes However, the question does not discuss the actual argument made by Einstein himself, and the currently accepted answer simply states that Einstein's arguments are no more than heuristics and intuition.
