This is a spin off from another thread:(adsbygoogle = window.adsbygoogle || []).push({});

First there are a couple of mathpages http://mathpages.com/rr/s8-04/8-04.htm and http://mathpages.com/rr/s8-03/8-03.htm that discuss the refractive index model and highlights the differences. TrickyDicky said: ↑.... To do this, we shall change the independent variable in Eq. (4.67) from the afﬁne parameter λ to the coordinate time t by using the relation 0=dt2 +gαβ/g00dxα dxβ. (4.75) ...The Fermat principle is now equivalent to the statement that such a gravitational ﬁeld acts like a medium with a refractive index n(x)=f(x)/ |g00(x)|. In addition to the bending of light, such an effective refractive index will also lead to a time delay in the propagation of light rays. This delay, called Shapiro time-delay has been observationally veriﬁed."

The first obvious objection is that the 'medium' must have a refractive index that is independent of frequency, because there is no rainbow effect. In nature all known refractive materials are frequency dependent so the metric can't be interpreted in terms of any normal material. Then again, a vacuum can't be treated as a normal material as it does not cause any friction on objects passing through it.

The second objection is that the refractive index cannot be isotropic. In the radial direction, the refractive index has to be ##N_r = 1/(1=2m/r)## and in the horizontal direction the refractive index has to be ##N_x = 1/\sqrt{1-2m/r}##. This objection is not as strong as the previous one as many materials have non isotropic refractive index. See http://en.wikipedia.org/wiki/Double_refraction. Some artificial composite materials called meta-materials have refractive properties never seen in nature, such as negative refractive indexes.

Mathpages uses an isotropic refractive index of ##N = 1/(1=2m/r)## to approximate the Schwarzschild metric and finds some differences, mainly that the deflection is not quite right and that the photon orbit is at r=4m rather than the usual 3m.

Mathpages does not explore the non isotropic model in any detail and I was curious what that would look like, starting with this simplified model with parallel thin layers:

The incident ray passes from layer r to layer r' with the refractive index and velocity of the ray broken down into orthogonal components. Here ##\gamma(r)## and ##\gamma(r')## mean ##1/\sqrt{1-2m/r}## and ##1/\sqrt{1-2m/r'}## respectively. Everything is measured in the coordinates of the observer at infinity. Layer r' is lower in the gravitational field than layer r and has a higher refractive index. By simple trigonometry it is easy to determine that:

[tex]\frac{\tan(\theta)}{\tan(\theta')} = \frac{V_x/V_r}{V_x'/V_r'} = \frac{\gamma(r)}{\gamma(r')} = \frac{N_x}{N_x'}[/tex]

This is an interesting, yet still surprisingly simple version of Snell's law.

Mathpages mentions that the optical model does not take account of the additional curvature due to the curvature of spacetime. If we take the above simplified model with flat parallel layers and adapt it to the form of concentric spherical shells, it seems almost certain that their will be additional curvature of the incident ray due to the curvature of the refractive index shells, over and above the the parallel model, but whether that is enough to duplicate what is actually observed remains to be analysed.

P.S. The point of this exercise is to try and produce a mathematical model that may simplify some calculations or provide insight.

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# Schwarzschild metric in terms of refractive index

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