Suppose we have a material which has a very, very high refractive index, say 30-50. The critical angle for this material would be very small (1.5-2 degrees). Will the said object be even visible at all, given that most of the light that enters it would not be able to come out again, at least until a few thousand reflections? Can anyone imagine/describe what it would actually look like?
[Physics] A material with very high refractive index
refractionvisible-light
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First of all, atmospheric attenuation in the visible region is primarily due to scattering, not molecular absorption as in the infrared and microwave regions. This is perhaps not that important to your question, but a good thing to keep in mind. The ligth doesn't disappear, it only changes direction.
If you have the density profile of the atmosphere, let's denote it $\rho(h)$, where $h$ is the height above sea level, you can calulate what you call mass-thickness, (I will call it slant mass column and denote it $C$), using this integral:
$$ C=\int^{\infty}_{0}\frac{\rho(h)dh}{\sqrt{1-\left(\frac{\cos{\theta_0}}{1+h/R}\right)^2}} $$
or this one:
$$ C=\int^{\pi/2}_{\theta_0}\rho\left(R\left(\frac{\cos{\theta_0}}{\cos{\theta}}-1\right)\right)\frac{R\cos{\theta_0}}{\cos^2{\theta}}d\theta $$
where $\theta_0$ is the elevation angle of the sun and $R$ is the earth radius. These formulas are for an obeserver located at sea level.
This assumes that the light travels straight through the atmosphere. This assumption works reasonably well for high elevation angles, but for lower angles (think sunsets) you have to take atmospheric refraction into account and perhaps also some additional scattering effects.
Your digital camera experiment might work in theory but it will perhaps not be as easy as you would like. First of all you need to account for refraction when calculating the slant mass column as I explained above. Refraction is wavelength-dependent so you will have to do it for each color. Secondly, only Rayleigh scattering will be proportinal to $C$. You will also have Mie scattering due to aerosols. This will depend on the aerosol loadings in the atmosphere which will be variable. Mie scattering will also be more important for low elevation angles since a larger part of the path through the atmosphere will be in the lower parts, which have higher aerosol loadings.
The problem here isn't a simple algebra error, but rather an issue with the physics. A medium which at rest is isotropic no longer behaves as an isotropic medium when it is moving relativistically. Instead, it behaves as a nonreciprocal bianisotropic material.
In particular, the phase velocity of light at a particular frequency in a medium is no longer the same in all directions if the medium is moving. This effect can be detected even if the medium is moving at a non-relativistic speed, as in the Fizeau experiment.
Because of this, any attempt to determine one consistent number for what a refractive index turns into under a Lorentz transformation is doomed to failure, because the equivalent to the refractive index is no longer a single number in a frame in which the medium is moving. Instead, it's necessary to treat the refractive index (or really the relative permittivity, which is closely related) as a tensor.
Unfortunately, I'm unable to find a full tensor treatment of refraction in moving media online. I did find this paper on relativistic optics in moving media, but it uses Clifford algebra as an alternative to a tensor treatment, and unfortunately the paper is behind a paywall. However, the non-paywalled abstract for that paper is my source for the above claim that a medium that's isotropic at rest behaves as a nonreciprocal bianisotropic medium when moving.
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What makes diamond so attractive (at least to some who have bought into the marketing craze created by the de Beers cartel) is not just the refractive index but the dispersion (aka "fire"), which is among the highest of all gemstones. It's also important to the quality of a diamond to be as translucent as possible, especially for the larger stones and the cut is of extreme importance. A poorly cut stone with the wrong facet angles does not produce the sparkle that is considered a necessary quality criterion. In other words: a jewelry diamond is more than just its material properties, it's also an optimized optical system.
As a consequence for your hypothetical material it would be important to have extremely low absorption, otherwise there would be little if any light coming out. I don't know how dispersion would play out and what the ideal "cut" would have to look like. If anything that sounds like a non-trivial optics problem.