Proper time in General Relativity (GR) is a vestige of Special Relativity (SR). GR attributes proper time to a "test" particle, which has no affect nor is influenced by the gravitational field. Even after solving Einstein's equations in vacuum, Schwarzschild would still have found an expression for the proper time. It's bad enough to set an arbitrary integration constant equal to a central static mass, but the line element would also imply something traveling at a proper time in contrast to the laboratory time.
In 1917 and again in 1924, Hilbert address the problem of the seemingly illogical conclusion that once the speed (of what?) surpasses the speed of light divided by the square root of 3, the sign of the radial acceleration becomes negative. This would indicate gravitational repulsion, so what was attractive at low speeds has now become repulsive at high ones! Rather than question the validity of such a solution, people went ahead inventing anti-gravity machines, and trying to harness such energy for the propulsion of space projectiles. The long list of authors is impressive, and includes physicist turned science fiction writer, Robert Forward, Thorne, Hartle, and more recently McGruder III, who even succeeded in getting a letter into Nature. What folly!
The fundamental point is that GR requires energy and momentum conservation. It was clear that the Schwarzschild solution does not conserve the angular momentum in laboratory time. Moller, in his book The Theory of Relativity (1952), speaks of the "ambiguity" in defining a radial coordinate in a non-Euclidean setting and proceeds by using the Euclidean expression to determine the advance of the perihelion of Mercury. Yet, the term he throws out in the expression for the angular momentum is exactly the same order that he finds to the missing 43 arcseconds/century. How could he (or for that matter Einstein whom he follows) find the numerical result when the entire analysis is flawed? The answer is that the expression for the advance of the perihelion was given in 1898 by the German high school teacher, Paul Gerber. Moreover, it is well-known that Einstein was dissatisfied with his derivation (like his derivation of E=mc^2) and attempts several other derivations that turned out to be none better.
It was also known (see the historical review by Loinger and Marisco) that conservation of angular momentum is restored using the so-called proper time which differs from the laboratory term by the coefficient of the time component in the Schwarzschild metric. The derivative of proper time with respect to laboratory term is analogous to the square of an imaginary index of refraction for an Eaton lens when the Coulomb field is equated with the square of the speed of light.
Then using proper time, the square of the velocity was found to be twice the negative of the gravitational potential and differentiating it in time gave Newton's inverse law of attraction. This was done by Paul Drumaux in 1936, who was subsequently admonished for his conclusions: for both the negative repulsion and the event horizon of a black hole disappeared!
So if proper time is not "proper" than what is it? Dual laws have been known to exist since the beginning of the twentieth century. Hooke's law was shown to be the dual of Newton's inverse square law. In fact, the conformal transform transfers Hooke's position of the sun at the center of the ellipse to one of the foci of the ellipse, as Newton claimed. Obviously, the areas swept out were not the same but their ratios to their respective times were. This is none other than requiring Kepler's law of equal areas in equal times to be obeyed. And this precisely introduces a second time, what later Levi-Civita referred to as "fictitious" time. It turns out that this time is anything but "fictitious".
Newton's law contains a singular at the origin whereas Hooke's law does not. The former were referred to as "collision" orbits. They could be eliminated by considering the dual law. The procedure was referred to as the "regularization" of the orbit, and this necessitated Levi-Civita to introduce his "fictitious" time.
However, it was not until 1970 that Moser showed the equivalence of Hamilton flows on sphere, traversed in "fictitious" time, s, equal to the arc length transformation under inverse stereographic projection to trajectories of the Hamitonian flow of Kepler's function traversed in real (laboratory) time, t, on a specific level hypersurface where
where V(r) is the Newtonian gravitational potential. Parenthetically, this holds for all dual laws, not just the Newton-Hooke pair. In Schwarzschild's case, the above expression is replaced by
or Schwarzschild's B-coefficient.
The above derivative elicit an analogy with the square of the index of refraction. The gravitational potential creates another medium with a differing index of refraction than the original one light was propagating through. In Schwarzschild's case, the right hand side is the negative of the Eaton index of refraction where charge is replaced by the gravitational parameter, and the constant term is equated with the square of the speed of light. So instead of propagation we now have absorption in Schwarzschild's case.
It is rather ironic that the 1911 derivation of Einstein of the deflection of light uses precisely the above expression. This was pre-GR, where Einstein's relinquishes the constancy of the speed of light. It was off by a factor of "2" that Schwarzschild put in.
So Levi-Civita's "fictitious" time pertains to the arc lengths on the surface of a sphere whereas laboratory time pertains to arc length of Kepler's ellipses in the projected plane. Proper time does not exist, and neither does the Schwarzschild line element.