'Naked' Singularities Exist: the Schwarzschild Metric Doesn't Apply

Newton was well-aware of the fact that an inverse cubic orbit implies a bound orbit reaches the origin "by an infinite number of spiral revolutions..." This he first mentioned to Hooke in a letter addressed to him on the 13th of December 1679. General relativity, basing itself on the outer Schwarzschild solution, claims that "naked" singularities, like the inverse cubic law,  don't exist. But, has anyone ever asked to what central force law the Schwarzschild metric corresponds? It certainly will not due to claim asymptotic flatness, and in the weak field limit, claim that a Newtonian potential exists. 

Moller, in his Theory of Relativity dissects Einstein's expression for the deflection of light by a massive body into the sum of a light ray passing through a medium with a nonuniform index of refraction,

$$n_S^2=\frac{c^2}{1-\mathcal{R}/r},$$

and a second contribution coming from the non-Euclidean nature of the metric,

$$\frac{\dot{r}^2}{1-\mathcal{R}/r}+r^2\dot{\varphi}^2=c^2.$$

A general property of uniform deflection is that the medium possess singularities. The index of refraction should not diverge more strongly than $1/r$ as $r\rightarrow 0$. This is clearly  not case in Schwarzschild's case. So in order to rationalize the result, Penrose came up with the assumption that Nature does not like "naked" singularities. 

Apart from the fact that Schwarzschild's inner solution has been swept under the carpet, an index of refraction does not meet the requirements of uniform deflection. In fact, optics clearly shows that "naked" singularities exist. Hence, there must be something wrong with Schwarzschild's outer metric which is bounded by the Schwarzschild radius, $\mathcal{R}$. In fact, we will show that it violates the conservation of energy.

 

It has long been known (1911, known as Bohlin's principle) that there is a certain duality between Hooke\s law, described by the differential equation

$$\frac{d^2z}{dt^2}+z=0,$$

and Newton's law

$$\frac{d^2w}{d\tau^2}+\frac{4Ew}{|w|^3}=0,$$

where $2E$ is the total energy of the oscillator

$$\left|\frac{dz}{dt}\right|^2+z^2=2E.$$

We also require that Kepler's law of areas be obeyed: the planets sweep out equal areas in equal times, with the sun at one of the foci of the ellipse. This means for $z$ that

$$|z^2|\frac{d\varphi}{dt}=L,$$

while for $w$, it implies

$$|w|^2\frac{d\varphi}{d\tau}=L.$$ This means that the two times, $t$, and $\tau$, must be related by

$$\frac{d\tau}{dt}=\frac{|w|^2}{|z|^2}.$$

For the conformal transformation,

$$w=z^2,$$

$\tau$-time will increase more rapidly that $t$-time by

$$\frac{d\tau}{dt}=|z|^2.$$

The duality can be expressed by the Maupertuis-Jacobi (MJ) principle:

$$E-U(z)=\left|\frac{dw}{dz}\right|^2\left(E^{\prime}-U^{\prime}(w)\right),$$

which bears a striking similarity to Fermat's optical principe when we define the index of refraction as

$$n^2=E-U.$$

Let $w=z^{\alpha}$ be any conformal mapping. Then, such a mapping transforms the field with potential $U(z)=\left|\frac{dw}{dz}\right|^2$ into trajectories of motion of a field with potential $U^{\prime}(w)=\left|\frac{dz}{dw}\right|^2$. That is, the duality principle is expressed by the conditions

$$U(z)U^{\prime}(w)=EE^{\prime}=-1.$$

For $\alpha=2$, the conformal mapping transforms trajectories with potential $U(z)=4|z|^2$ into those with potential $-1/4|w|$. The former is the potential of an harmonic oscillator, while the latter is Newton's gravitational potential. Thus, the  Maupertuis-Jacobi principle is satisfied identically

$$1-4|z|^2=4|z|^2\left(-1+\frac{1}{4|w|}\right).$$

Now, the same principle is also satisfied by a linear repulsion law:

$$\frac{d^2z}{dt^2}-z=0.$$

The same mapping now transforms trajectories with potential $U(z)=-4|z|^2$ into those with potential $U^{\prime}=+1/4|w|$. In other words, repulsion appears as negative gravitation. Again, the MJ principle,

$$-1+4|z|^2=4|z|^2\left(1-\frac{1}{4|w|}\right),$$

is satisfied identically.

The attractive case contains turning points, such as those that exist in scattering theory. Denoting $4|z|$ by $r$, the left-hand side of the MJ principle:

$$n_L^2=1-r^2,$$

while, the right-hand side gives an index of refraction

$$n_E^2=\frac{1}{r}-1,$$

where $r$ denotes $4|w|$. The former is the Luneburg index of refraction while the latter is known as Easton's index of refraction.  It satisfies the condition for uniform deflection, diverging no greater than $1/r$ as $r\rightarrow 0$.Eaton's index of refraction profile is shown below for $r<1$. Rays come from infinity and return to infinity in the same direction, 

cat's eye

it a perfect cat's eye. Unlike the Schwarzschild index of refraction, $n_S$, $n_E$ has a naked singularity at $r=0$. 

$n_E^2$ also happens to be Newton's index of refraction for an inverse square central force law. It will account for any conic orbit, regardless whether the origin is at the center or a focus since the two are related by the squaring the former. It is commonly believed that optical orbits are the zero-energy limit of mechanical ones. The barrier is completely illusory as Maxwell's fish-eye has demonstrated: Newtonian elliptic orbits with an index of refraction $n_E^2$ are stereographic projections of  curves on the surface of a sphere representing Maxwell's fish-eye.

In the case of repulsion, we define the indices of refraction as the negative inverses of attraction. On the left-hand side we have an index of refraction.

$$n^2=\frac{1}{r^2-1},$$

while on the right-hand side of the MJ principle the index of refraction is

$$n^2_S=\frac{1}{1-1/r}.$$

This would precisely the Schwarzschild index of refraction given above. What was gravitational attraction now has become gravitational repulsion. In comparison with the Eaton lens, where the index of refraction tends to infinity as $r\rightarrow 0$ as $r^{-1/2}$, Schwarzschild's index of refraction tends to $\infty$ as $r\rightarrow 1$! The barrier, or horizon, which has been put into place has nothing to do with a scattering process, regardless of whether it be electromagnetic or gravitational. It is as if gravity has become repulsive, in the sense discussed above.

General relativity was so impressed with the (outer) Schwarzschild solution that they invented a "Cosmic Censorship" principle to hide 'naked' singularities. Optics has no such qualms, and the Eaton lens manifest the general feature of a uniformly deflecting lens as we have discussed above. Without singularities there can be no uniform deflection. Then what is the significance of  using the (outer) Schwarzschild metric as a prototype metric for discussing the deflection of light rays by a  massive body? For small $r$, the so-called turning parameter $\rho\equiv nr$ must tend to zero, implying that $n$ must diverge more slowly than $1/r$.

This is certainly not satisfied by the Schwarzschild index of reflection. In fact, there is absolutely no justification for using the Schwarzschild metric in the calculation of the angle of deflection of light rays, other than it was the only one on the market. And instead of attracting light rays, the massive body repulses them according to the Schwarzschild metric. This is manifested by the fact that the scattering angle which the general relativists calculate is positive, meaning that light rays bend outward in violation of Snell's law which says that when light rays traverse a medium with an index of refraction greater than $1$ they must bend inward.