The assertion regarding the Schwarzschild radius suggests the event horizon in the Schwarzschild solution is coincident with the central singularity, implying that the minimum value of \( r \) (which he denotes as \( u \)) is \(\frac{2GM}{c^2}\).

However, this is a misunderstanding of the standard interpretation of the Schwarzschild metric.

In general relativity, the Schwarzschild radius

\( r_s = \frac{2GM}{c^2} \) marks the location of the event horizon of a non-rotating black hole. The term \( r \) in the Schwarzschild metric is a radial coordinate, not the radial distance from the center, but it is not the same as the proper distance from the singularity. The proper distance becomes infinite as one approaches the event horizon from the outside.

The Schwarzschild metric is:

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

Here, \( r \) is the Schwarzschild radial coordinate, and \( d\Omega^2 \) represents the angular part of the metric. At \( r = r_s \), the metric coefficient \(\left(1 - \frac{2GM}{r}\right) \) goes to zero, leading to a coordinate singularity, which is interpreted as the event horizon.

therefore, the minimum value of \( r \) in the Schwarzschild solution corresponds to the event horizon, \( r_s \). The suggestion that there is no event horizon and that it is coincident with the central singularity is incorrect based on standard interpretations of the Schwarzschild metric in general relativity.

Harri Shore , the invariant line element is ds^2=(1-1/r)dt^2-dr^2/(1-1/r)-r^2df^2-r^2sin^2fdh^2 where i have normalized c and 2GM/c^2 to unity and denote colatitude by f and longitude by h. r,f and h are mere coordinates with no direct physical significance, in particular we should not conflate their meanings in the nonEuclidean spacetime of the Schwarzschild metric with their meanings in a Euclidean manifold. suppose we confine our calculations to a time slice, dt = 0 or t = constant, then we are dealing with a three dimensional space in a given reference frame, the rest frame of the black hole. the spatial distance differential in this space is given by dS=sqrt(-ds^2)--that is the standard interpretation of the significance of ds^2 for a space-like separation of coordinates (dt=0). S = integral dS along some curve, r = R(p), f = F(p), h = H(p) where R,F and H are some functions of an arbitrary parameter, p. Let us choose to measure the distance, S, along a radial curve, R = p, F = f0 = constant, H = h0 = constant, then S is the path integral S =integral dr/sqrt(1-1/r) from the origin at S = 0 (whatever value of the coordinate r corresponds to S =0) to some arbitrary value of the coordinate, r, along the path described parametrically by r = p, f = f0, h = h0. The physical singularity responsible for the Schwarzschild gravitational field presumably sits at the origin, S = 0. rewriting the integral, it is S=integral rdr/sqrt(r(r-1)), which is an elementary quadrature that can be looked up in standard tables of integrals. it is S = sqrt(r(r-1))+arctanh((r-1)/r), which is a function that returns as output a given value of S for a given value of input, the coordinate r. to state the matter with less prolixity, the coordinate r describes a point along the radial curve (defined parametrically by r = p, f = f0, h = h0) at spatial distance, S, from the origin, where the physical Schwarzschild singularity presumably sits. Substituting r = 1 (i.e., restoring normalizations, r = 2GM/c^2, the Schwarzschild parameter) in this function, we return a value S = 0: r = 2GM/c^2 corresponds to (or is equivalent to, or--with less prolixity--IS the origin at S = 0, the seat of the physical singularity. Just as no value of S less than 0 exists, no value of r less than 2GM/c^2 exists; there is no "inside" of the black hole where sqrt(1-1/r) is imaginary and there are irregularities in space and time. i am fully aware that this is at variance with standard interpretations of the schwarzschild metric, but my claim is precisely that the standard interpretations are inconsistent, or to state the case with more bluntness, they are dead WRONG! stuart boehmer