"Bernard Lavenda
The correct metrics of constant curvature are given in Robertson-Noonan eqns (14.12), Relativity and Cosmology. Hyperbolic distance is arctanh, not arcsinh as in Landau and Lifshitz (111.12), Classical Theory of Fields. The latter is the circumference of a hyperbolic circle of radius r, i.e. 2 pi R sinh r/R, where R is the absolute constant of the geometry (31.5) Busemann and Kelly, Projective Geometry and Projective Metrics. It was even known to Einstein that the ratio of the circumference of a rotating disc to the radius was greater than 2 pi. In the RW metric at constant r, you would get the ratio equal to 2 pi for k=-1 (constant negative curvature). So RW is wrong and is not to be found in the Robertson-Noonan monograph except in the forward by Fowler.
The Schwarzschild solution is inconsistent. The outer solution has non-constant curvature while the inner solution has constant curvature. There is no matching condition on the boundary. It goes from M/r->rho r^2, i.e., from constant mass M to constant density rho. In Moller, eqn (79) of Ch. XI, the \Lambda term is negative, cf. (98) and the spatial geometry of the inner solution is given by (104) with 2m missing. This I interpret as going from one of constant mass to one of constant density. Moreover, the inner solution (104) does not agree with the Riemann metric (14-10) in Roberston-Noonan. The historical evolution is Riemann->Beltrami->Poincare. These are known metrics of constant curvature, and I showed that it is the Beltrami metric that characterizes a uniformly rotating disc. One of the major results is that you cannot have local observers at the (inertial) center and on the periphery communicating their time differences and thereby knowing their positions on the disc. This is what Einstein contended who did not realize that as you move about on the disc your rulers and clocks will shrink or enlarge with you so that you have no way of knowing where you are in reference to someone else. It will also take you an infinite time to reach the boundary of the disc. Any theory of relativity must reduce to these metrics in the constant curvature limit so that there was no reason for looking for new metrics that would be solutions to Einstein's equation in the constant curvature limit. "
The correct metrics of constant curvature are given in Robertson-Noonan eqns (14.12), Relativity and Cosmology. Hyperbolic distance is arctanh, not arcsinh as in Landau and Lifshitz (111.12), Classical Theory of Fields. The latter is the circumference of a hyperbolic circle of radius r, i.e. 2 pi R sinh r/R, where R is the absolute constant of the geometry (31.5) Busemann and Kelly, Projective Geometry and Projective Metrics. It was even known to Einstein that the ratio of the circumference of a rotating disc to the radius was greater than 2 pi. In the RW metric at constant r, you would get the ratio equal to 2 pi for k=-1 (constant negative curvature). So RW is wrong and is not to be found in the Robertson-Noonan monograph except in the forward by Fowler.
The Schwarzschild solution is inconsistent. The outer solution has non-constant curvature while the inner solution has constant curvature. There is no matching condition on the boundary. It goes from M/r->rho r^2, i.e., from constant mass M to constant density rho. In Moller, eqn (79) of Ch. XI, the \Lambda term is negative, cf. (98) and the spatial geometry of the inner solution is given by (104) with 2m missing. This I interpret as going from one of constant mass to one of constant density. Moreover, the inner solution (104) does not agree with the Riemann metric (14-10) in Roberston-Noonan. The historical evolution is Riemann->Beltrami->Poincare. These are known metrics of constant curvature, and I showed that it is the Beltrami metric that characterizes a uniformly rotating disc. One of the major results is that you cannot have local observers at the (inertial) center and on the periphery communicating their time differences and thereby knowing their positions on the disc. This is what Einstein contended who did not realize that as you move about on the disc your rulers and clocks will shrink or enlarge with you so that you have no way of knowing where you are in reference to someone else. It will also take you an infinite time to reach the boundary of the disc. Any theory of relativity must reduce to these metrics in the constant curvature limit so that there was no reason for looking for new metrics that would be solutions to Einstein's equation in the constant curvature limit. "
The Schwarzschild solution is inconsistent. The outer solution has non-constant curvature while the inner solution has constant curvature. There is no matching condition on the boundary. It goes from M/r->rho r^2, i.e., from constant mass M to constant density rho. In Moller, eqn (79) of Ch. XI, the \Lambda term is negative, cf. (98) and the spatial geometry of the inner solution is given by (104) with 2m missing. This I interpret as going from one of constant mass to one of constant density. Moreover, the inner solution (104) does not agree with the Riemann metric (14-10) in Roberston-Noonan. The historical evolution is Riemann->Beltrami->Poincare. These are known metrics of constant curvature, and I showed that it is the Beltrami metric that characterizes a uniformly rotating disc. One of the major results is that you cannot have local observers at the (inertial) center and on the periphery communicating their time differences and thereby knowing their positions on the disc. This is what Einstein contended who did not realize that as you move about on the disc your rulers and clocks will shrink or enlarge with you so that you have no way of knowing where you are in reference to someone else. It will also take you an infinite time to reach the boundary of the disc. Any theory of relativity must reduce to these metrics in the constant curvature limit so that there was no reason for looking for new metrics that would be solutions to Einstein's equation in the constant curvature limit. "
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