Dual Laws and The Inner and Outer Schwarzschild Solutions

It is quite remarkable that the inner and outer Schwarzschild solutions manifest the same dual laws as the laws of attraction an elliptical orbit when the source is located at the origin of the ellipse and when it is located at one of the two foci. That is, in the case where the source is located at the center of the ellipse the law of attract is Hooke's whereas when the source is located at the focus it is an inverse square law.

Moreover, the inner solution would show the same gravitational repulsion for velocities greater than c divided by the square root of 3 that the outer solution manifests. But that is a mere illusion since switching from coordinate time to local time eliminates entirely the negative repulsion. Considering the dual laws of Hooke and Newton, the "fictitious" time ds=dt/r is replaced by its complementary time (1-a/r), where a=2GM/c^2 is a constant in the Schwarzschild outer solution while the "fictitious" time ds=r^2dt, which would pertain Kepler's aerial theorem, is replaced by dt'=(1-br^2)dt in the inner solution where b=G(rho)/c^2, where (rho) is the constant mass density, analogous to the constant mass M in the outer solution.

The metric reads


in the radial case, where B=1/A. This condition is equivalent to the condition that the eigenvalues of the Ricci tensor all be zero. In the outer solution, B=1-a/r, while in the inner solution it is B=1-br^2. Again we see the duality of the linear Hooke law and the inverse square law appear in these complementary "fictitious" times.

Using coordinate time, t, instead of local, or fictitious time, s, the equation of motions for outer and inner solutions are

a=-(GM/r^2){1-2GM/c^2r -3v^2/B}



where a is the acceleration and v is the velocity. Neglecting the second term on the right hand side of both expressions, because it is small relative to the other terms, the acceleration becomes positive in the first case for v>c/sqrt(3), and the acceleration becomes negative in the second case under exactly the same condition. In the former, the inverse square law becomes positive, while in the latter it is Hooker's law which becomes negative, indicating instability or repulsion in both cases.

Instead dividing the metric by ds^2 and solving for the square of the velocity gives


where E=Bdt/ds, is the total energy. From the expressions for B, it is clear that Newton's law of inverse square results in the outer solution and Hooke's law for the inner solution.

The relation between dual laws and conformality of the metric was first alluded to by Edward Kasner in 1909. Due to the inaccessibility of the work published in 1913, it was rediscovered by numerous people afterwards. The relation to the conformality of the metrics has gone unnoticed until now though.

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