There is no more important model of a solid of revolution as a model of hyperbolic geometry than the pseudosphere, shown above. The pseudosphere is an open surface of infinite extent having negative constant curvature that, surprisingly, has a finite volume. However, when the pseudosphere is interpreted in terms of an indefinite Lorentzian metric, incongruties become apparent.
When attempting to interpret spacetime, as described by a Lorentzian metric, in terms of the pseudosphere, circles wrapping around the bugle correspond to fixed positions that are a measure of proper time, or the time that is recorded by the observer's wrist watch. In contrast, lines directed along the surface represent coordinate time so the distance is the product of the speed of light and the interval of time elapsed. This obviously corresponds to light trajectories--and not material trajectories--since nothing travels faster than or even at the speed of light.
How does a pseudosphere, or any solid of revolution, come to be interpreted by a Lorentzian metric? The Lorentzian metric for the pseudosphere is
ds^2 = cosh^2(r)dt^2 - sinh^2(r)dr^2 (1)
where s is proper time in units c=1, t coordinate time, and r is Euclidean distance. Since the metric (1) is indefinite, we may rearrange it into a definite metric by making coordinate time the dependent variable, thus obtaining what is referred to as an Epstein chart
dt^2= sech^2(r) ds^2 + tanh^2(r)dr^2. (2)
To see that this has negative constant curvature set tanh(r)dr=du, so that cosh(r)=e^u. Then (2) can be written in polar form as
dt^2= du^2 + e^(-2u) ds^2. (3)
It is clear from (3) that the polar plot (u,s), that u behaves as a radial coordinate and the proper time as the azimuthal angle. Thus, the proper time (when multiplied by c) is distance.
If we set h=e^(-u). then the Gaussian curvature is K=-h"/h=-1. Thus, we may avail ourselves of the Killing-Hopf theorem, also known as Cartan's theorem that a complete connected Riemann manifold of constant curvature is isometric to a sphere, Euclidean space, or a hyperbolic space by a group acting discontinuously. Obviously, the most famous of these is the Poincare conjecture that any 3D finite space which is closed and connected is homeomorphic to a 3D sphere.
But what has (3) to do with a Lorentzian metric? for it requires coordinate, or lab, time intervals to be interpreted as "distance", and proper time as rotations. Consider the exterior Schwarzschild problem with Lorentzian metric
ds^2 = A(r) dt^2 - B(r) dr^2 (4)
where
A(r) = 1- 2GM/r = 1/B(r) (5)
still in units c=1.
Converting (4) into an Epstein chart results in
dt^2 = dr^2/A^2(r) + ds^2/A(r). (4)
Because of the choice (5), this is not a metric of constant curvature. However, if we were to consider the case of constant density instead of constant mass, thereby transferring our attention to the Schwarzschild inner solution,
A(r)= 1- G rho r^2 (5)
where 1/sqrt(G rho) is the Newtonian free-fall time with rho the constant density of the star,
we would get a metric of constant negative curvature.
The metric (4) would then become the Beltrami metric--with all its implications. In view of (5) this completely connected Riemann space would be isomorphic to a saddle, or a hyperbolic space of constant, negative curvature. The Beltrami metric (4) has been enshrined into the Poincare' disc model of hyperbolic geometry, which, according to geometers, is the true analogue of the sphere for constant negative curvature, and not the pseudosphere. In particular, it predicts that geodesics diverge from one another instead of bunching together as in the case of positive, constant curvature. This seems surprising as gravity is an attractive phenomenon where we would expect the opposite behavior. This is but one more nail in the coffin of considering an indefinite metric for a completely static gravitational interaction.
These implications are that:
(1) the coordinate time, t, is distance. But distance of what? For even r does not measure radial distance in the external Schwarzschild solution;
(2) the coordinate time, s, is the azimuthal angle. But of what? Why should proper time be periodic?
All this goes to show that an indefinite Lorentzian metric has no business in dealing with a static scalar gravitational problem, or any other static problem for that matter. It's like trying to fit your foot into the wrong size shoe. Time(s) does (do) NOT enter into the Schwarzschild problem. The way in which the Schwarzschild solution was derived does not even allow for a central mass of size M or density rho.