Gravitational Index of Refraction 

Fascinating topic, and beautifully illustrated. Thanks for making this!

I actually recall someone model a black hole in Blender by putting spheres of greater refractive index around it

Grant's @3blue1brown video on refractive index is one of the best I've seen. His microscopic view of individual synchronized oscillations really hits home with how light is observed at the macro level. This takes his work up a major notch. Thank you!

Incredible. The very way of thinking is new to me, like suddenly being teleported to a new realm. Thank you sir.

1:47 made me laugh. "ACHTUNG: Schematics are Not to Scale." So hilariously funny! Subscribed.

My preferred way of understanding gravitational lensing is in terms of Huygens' wavefront hypothesis. I will discuss that in two stages: - First in terms of an early exploratory theory by Einstein (1907), that already had curvature of time, but not yet curvature of space. - Second in terms of the fully fledged GR. Einstein's 1907 explorator theory proposed that deeper in a gravitational well a smaller amount of proper time elapses. So: according to that 1907 exploratory theory: deeper in a gravitational well the locally measured speed of light will be slower. For celestial objects moving at non-relativistic velocity: this curvature-of-time theory reproduces newtonian gravity. Also, the 1907 exploratory theory was already sufficient to account for the (much later conducted) Pound-Rebka experiment. My understanding is that Einstein explored what the effect would be on a Huygens wavefront grazing the Sun. The wavefront would undergo a slight turn, in accordance with the difference in speed of light as a function of radial distance. Einstein arrived at a value of something like 0.8 arc-sec, about half the value that the fully fledged theory predicts. In the years after 1907 Einstein sussed out that the theory needed curvature of space too. One of the clues to that was the Ehrenfest paradox; for a rotating disk the ratio of radius and circumference is not exactly 2pi; there is something non-euclidean going on. In terms of the fully fledged GR: around a source of spacetime curvature the ratio of radius and circumference is not exactly 2pi. As a condition to be satisfied by the theory: the radius/circumference difference is to be such that it precisely matches the gravitational time dilation, such that at any distance to the source of spacetime curvature the same speed of light obtains. That means that for a Huygens wavefront grazing the Sun there is a double whammy. Even when not counting a gravitational time dilation effect: there is a space curvature effect that result in a turning of the orientation of the wavefront. The overall effect is a deflection of 1.75 arc-sec. For light the curvature-of-time aspect of spacetime curvature has a comparatively small effect, because light is moving so fast. There is not enough time; the curvature-of-time effect has very little opportunity to make a difference. It is only for light that the aspect of curvature of space contributes a significant proportion of the total effect. By contrast: for the planets of the solar system the contribution of the curvature-of-space aspect is extremely small; the motion can almost entirely be accounted for in terms of the curvature-of-time aspect. The precession of the perihelion of Mercury correlates with the curvature-of-space aspect, that gives an indication how small that contribution is. I want to emphasize that I am totally onboard with the idea of thinking in terms of index of refraction of a gradient index lens; the spacetime curvature is acting as a gradient index lens.

Very good video! My instinctive explanation for why the light seems to slow down when going towards the source of the gravity is that due to the space-time curvature there's just more space than there ought to be if the local spacetime was flat, that means the light must take longer to get to the destination which looks to outside observers as if it was slower

There's a nice journal article floating around the internet "F = ma for Optics" by James Evans and Mark Rosenquist that relates potential energy to index of refraction. The potential ~ -n^2 / 2. They go through several elementary examples such as a plane dielectric interface and the case with cylindrical symmetry.

We don't use refractive indices in gravitational physics as the refractive index is a function of the coordinates, i.e. every point in space has a different value for n(r). This is in contrast to some medium, e.g. flint glass, that has a constant refractive index of say n=1.58 for some choice of wavelength (also note another difference in that the gravitational field is not dispersive).

Fascinating presentation and theories. I can't help getting the feeling that you're holding back, based on your last statement. I think you are really on to something bigger. Go for it! Best of luck.

The simulations appear beautiful as they do very well illustrate the intuitive sense we have of how gravitational lensing would distort light, allowing for the problems depicting something this vast on a small screen, and I strongly believe they are useful. Nice work!

You are certainly one of the best persons to explain complex thins in simple words. After Richard Feynman of course. Not even getting into quantum mechanics. Even Huygens-Fresnel simulations are mind blowing for one that tries to visualise it. Thank you so much for the sharing of your work Sir.

Wow what an amazing video. The animations, explanations, and humour are all on point!

I was delighted to see I might have been nearly right with some of my speculations and I even nearly understood about 50% of what was said. Thank you very much.

Years have I waited for someone to explain as I've frailly understood the fundamental behaviors of wave theory. Your insight also reinforces the idea of resonant effects concerning the apparent observed phenomena of light slowing through a Bose Einstein condensate from laser synchronization of atomic state to and from the chaotic environment we live from day to day. Light as a particle cannot explain that. Thanks for filling in some of the missing key components so many confuse and for expounding my own thought. If I can understand it so clearly, it's not that difficult. You bring order to chaos.

I loved the visualizations so much! ❤ I'm right now working on my physics degree final project about the bending of light around black holes. The idea of the gravitational index of refraction is not new, but it isn't very well known. This is the first video I see on the topic.

great video again! it is always fun to see a semi-familiar concept through a different... "lens" 😸

Another mind blowing video. Thanks for all the effort you go to. It is appreciated here on the Isle of Wight!

Браво!! Я очень давно ждал такого красивого представления о зависимостях скорости времени к сути материй и пространств! Благодарю! Bravo!! I have been waiting for such a beautiful idea of ​​the dependence of the speed of time on the essence of matter and space for a very long time! Thank you

Thank you, your clear thinking is contagious.

Absolutely beautiful video once again! I never thought about light following the curvature of spacetime from the perspective of the refractive index.

WOW! That was very cool, and actually easy to follow. Thank you so much for such a refreshing insight into reality.

Great work! Thank you for this extraordinary theory and simulations. Appreciate!

Thank you for making these videos.

Very enlightening and thought provoking. Thanks.

The "refractive index" of space not only depends on the gravitational potential, but depends on the impact parameter of the light rays. For example, if you want to simulate how light bends around a black hole by replacing the black hole with a spherical glass of variating index n(r), you will find this doesn't work. You will need n=n(r,a) with a the impact parameter, so as to match black hold deviation angles as fonction of a.

Brilliant! Simplest, most intuitive, and usefully correct explanation of general relativity I've ever seen (way better than the ball on a stretchy surface)! Glad I watched to the end before commenting- half way through I was going to say "you're much closer than you think" 😅

Defining a (derived) parameter called gravitational index of refraction makes perfect sense... once you realize that what general relativity (and special relativity, as well) actually describes (as "curvature of spacetime") is really how spacetime *density* changes in the presence of mass (energy). You simply start with the assumption that spacetime is discrete, and made of "spatial cells" (or 4D hyperspehrical nodes)... not of constant, but rather of variable sizes (which is the same as saying that Planck length [and Planck time as well] is not constant, but variable), where size of a cell is inversely proportional to the mass/energy existing in its surroundings, and directly proportional to the distance(s) from that mass/energy. In other words, the greater the mass/energy, and the smaller the distance from that mass/energy, the smaller the size of the spatial cell. This model of variable Planck lengths gives exactly the same results as general relativity, but explains much more intuitively why light (seemingly) slows down as it approaches mass/energy. What really happens in that scenario is that photons propagate through space at constant speed (which is the [constant] speed of light for all observers in all frames of reference), and since spacetime around a massive object is denser, light approaching that massive object has to take more (discrete) *steps* than light going around it. Light (that is, a photon) still moves at exactly the same speed (defined as c = Planck length / Planck time, where both length and "time" [which is just another spatial dimension] change by exactly the same factor) through spacetime (regardless of spacetime's local density), and it is really the *transformations* (those equations from special relativity being such transformations) between frames of reference (for an observer close to the massive object and an observer far away from it) that give the *illusion* of light (and time) moving slower the closer it is to a massive object. I could go into more detail, and explain *why* spatial cells shrink in the presence of mass/energy, but the gist of it is that it has to do with *constraining infinite potential* of Dirac delta function (which contains all frequencies [between 0 and infinity]), and one way to do that is by limiting the (infinite) number of (discrete) frequencies by bounding the spacetime in which those frequencies can *physically* exist. That is, only frequencies that have periods which are whole numbers of (hyperspherical) cell's diameter can physically exist within the cell, and the smaller the cell's size, the less frequencies can physically exist inside of it. Note that there will still be an infinite number of possible frequencies within a smaller cell, but that (infinite) number will be smaller than the (also infinite) number of frequencies possible within a larger cell. I mentioned all of this once, in another channel, but (once again) another way to accomplish this constraining (of infinite potentials) is by increasing the "depth" of the "potential well" that these spatial cells are really acting as, while leaving the size of all the cells constant (exactly the same) regardless of the presence or absence of any mass/energy. Such a (hypothetical) universe would have, more-or-less, the same properties (electromagnetic, strong and weak nuclear forces) as this universe, but it wouldn't have any gravity (or, rather, any gravitational effects) in it. So... yeah, gravitational index of refraction is one way to describe the phenomenon of gravity. Another way would be to define a different (also derived) parameter that we could call 'spatial compression rate' (Scr), or something to that effect, which would describe how the size of spatial cells shrinks in the presence of mass/energy, and this parameter would (obviously) have the (normalized) value of 1 at infinite distance from any mass/energy, and some minimum value at the distance of... not zero (since spacetime is discrete, distance of zero can never be reached), but something really, really small. It should be possible to calculate the smallest possible value for Scr in a given universe, as the smallest possible value corresponds to the size (Planck Length) of the cells on the surface of a black hole which has the size (total mass) of that universe itself (where that black hole has uniform energy distribution/density across its whole surface, and is also non-rotating). We could then multiply Scr with the Planck length (and the Planck time, as both must change by the same factor to maintain the hyprespherical nature of spatial cells, and also the constant speed of light) of empty space (with no mass/energy in it) to get exactly the same "curvature" of spacetime (in the presence of mass/energy) as general relativity predicts. ... and that's where a simulation can come in very handy, indeed, because a simulation can search through the whole space of all possible (candidate) functions for Scr (including [infinite] power tower functions), and find solution that fits best all the observations (both cosmological observations and observations from quantum/gravitational experiments) in virtually no time. P.S. This model (of variable Planck lengths) also explains how a black hole that looks like it's only a couple of miles in diameter (when looked at from the outside) can contain a whole universe that (apparently) has the size of (dozens of) billions of light years across. For example, the (local) value of Scr for *this black hole universe* (when observed from the outside) would, therefore, be something of the order of 10^-23. Do be mindful, though, that this is Scr value (of this black hole) that's *only applicable* in the universe *outside* of this black hole, that is, the universe inside of which this black hole exists. A black hole (or any mass, for that matter) existing *inside* of this black hole universe (there is, apparently, a deep nesting of black holes taking place in this... "omniverse") would have a completely different (purely universe-local) value of Scr... unless one figures out the exact function for *global* Scr (one that would be applicable to the whole "omniverse"), but that would be "slightly" unrealistic to expect if one can't even move outside of this black hole universe (at will), much less between all of them.

Fascinating and thought provoking, great viewing !.....cheers.

Thank you for your efforts. Wonderful work, clear explanations. Thanks from Germany.

Fascinating presentation.

Excellent as always!

This is mind blowingly awesome.

I was S O Terrible in doing Lens in my college Physics classes... This sort-of helps....so I will view this video, again.

Indeed, the usual illustrations for the curved spacetime of General Relativity are displaying gravitational potential rather than actual spacetime curvature, which is very ironic, because gravitationnal potential is a purely Newtonian concept.

It's all one universe, everything has to be connected somehow. Great work illustrating that.

Interesting topic, thanks for this!

Beautiful... (in my personal view), I always wondered how information was encoded on a blackhole. Thank you for sharing.

I ever thought you have very nice intuitions in many aspects of optics and this goes over all that, since maybe you "touched" the missing link in quantum gravity theory, FANTASTIC !!!

Thanks for the nice clear thinking.

Simple and elegant!!! 👏🏻

3Blue1Brown is fantastic! Unbelievably good!

Excellent. I remember refraction as the reason for light to bend around a gravitational body because light has no mass and no time.

fantastic explanation! thank you sir!

A beautiful and clear illustration. My first reaction was that gravitational potential is reversely proportional to the square of the radius, and not simply reversly proportional to the radius, but I may fool myself. The derivative of a square is linear after all. We usually say that light entering a gravitational well maintains it's speed, but gets a smaller wavelength. That can only be true if your illustration for an external observer is correct. This compression effect is most obvious with a black hole. Objects falling into it ends up as stickers fastened to the surface of the black hole, where time is infinitely compressed. When I watch your videos I can feel that there are Nobel prize worthy discoveries hiding behind the scenery. These illustrations are powerful tools to understand the behavior of waves well enough that we can begin to make practical inventions based on them. Maybe we can create telescope mirrors and lenses that compensate for their own flaws? Maybe it can even help us resolve the current cosmological crisis?

Very good, Generally gravitational potential is a good representation in today's practice. I myself derived few good equations for mass and the index. You have carefully avoided the negative (-) value for reflective index but now in high intensity laser this is no more negligible. We all wish good from you.

6:38 analogy between classical mechanics and variable refractive index was actually very popular idea for school physics olympiads around 2 decades ago. Indeed, if you look at toy problem of particle with kinetic energy K hitting border between two volumes in which it's potential energy differs by U then resulting formula looks precisely like Snell's law with refraction index ratio of \sqrt{1 - U/K}. Analogy can be further extended from here to arbitrary potential fields, and gravity is especially convenient since you can factor out particle mass. Although, these physics problems usually worked other way around by reducing some weird optics to mechanics. 14:15 light doesn't slow down, but wavelength decreases as one would expect. 14:49 speed of light isn't the most important constant since it's more-or-less fully determined by our system of measurement choice :)

SO WELL EXPLAINED ty

Absolutely fantastic video. It would be very interesting to map a large section of space, the great attractor, and farther things, so show how it all looks across a whole gravitationally bound area, AND those that are not bound, so how expansion v gravitational lensing work. (expansion not causing relativistic effects) I've watched it 4 times to really try to understand the whole idea - but first, the very idea of saying gravitational index, so succinctly, is brilliant. Scripting is brilliant, right as I am saying "ah but what about..." - you're right there answering the questions and leading us on in thinking. I've often argued on physics forums that we're already within the limits of visible relativistic effects - relative to the great attractor we're traveling at 600 km/s - which would be detectable real-world (world?) motion, with relativistic effects. Event the smallest motion is relativistic (there is no threshold, but i've heard 4% bandied around as detectable, but 0.002 could be on sensitive stuff) anyway, that's the late night grasp I have on this, amazing stuff and love the direction, BEST CHANNEL ON RU-vid!

Nice work!

your videos taught me to think of light as waves

I love how other people have the same correlations as I do lol... literally a couple weeks ago I was attempting to simulate black holes in a renderer with a gradient-controlled index of refraction with mildly good results. So cool to see a video on this exact idea... Hopefully this might give me some insight to make my simulations a bit more accurate

It was established more than a century ago that gravitational lensing occurs as light passes close to the surface of the Sun from some distant object. During a solar eclipse it was observed that objects that should have been occluded by the Sun/Moon became visible slightly earlier than was mathematically predicted if light always travelled in a straight line. So gravitational lensing is an established fact. It is also apparent that light passing close to any massive object in space must also be subjected to some gravitational distortion or lensing and this effect will become more apparent looking further out towards the visible limits of the Universe, simply because of the increasing likelihood of light originating so far away encountering some massive object on its’ way to Earth. Is there a universal coefficient of diffraction that can be applied as a general principle to light travelling from progressively further distances from Earth? Sadly the answer has to be no because the density of matter distribution in space and therefore the gravitational fields are not constant, although over a sufficiently large distance perhaps a rough average value could be assigned. While this would not be directly related to gravitational lensing, it might provide the means for a correction factor that would allow us to make sense of some of the more controversial results being delivered by the latest space telescopes.

I thought it was very good to be honest, but I love the science, so most videos I find to be great. Thanks again. Peace ✌️ 😎.

This approach looks like a straightforward way to model gravitational lensing around a galaxy cluster using brute force Finite Element Analysis calculations on a computer.

This video is quite good. I would be interested in seeing the details nailed down for a real cluster as best as possible instead of just swept under a rug. The concept looks very interesting. 👍

Sometimes I see outrageous sciency video titles and know right away that's just bait. Not this time around.

Interesting simulation. As an optics expert, you should be able to calculate the type of optical system capable of forming a proper image from the gravitational lens of the sun in order to create the biggest telescope ever. I remember reading that such telescope should be capable of seeing car around hypothetical planets around Alpha Centauri.

The difference between identical events that unfold serially or in parallel can be thought of as ocean waves on an ideal beach. From a line perpendicular to the shore, the waves are in series, but from a line parallel to the shore, the waves are unfolding simultaneously, or parallel. Something tells me this orthogonal relationship, and the similar orthogonal relation of electric and magnetic fields, has lovely secrets to tell, that a higher magnitude ordering principle is hiding beyond this orthogonal relation.

Very very cool. And thank you (as always) for your efforts into pedagogy. Does speed of light not just determine how clocks tick..? Eg, in a vacuum, the time a photon needs to travel 299,792,458m defines a local second..?

great video!!!

I like Samantha Critoforotetti Cameo and bravery at display 😂😊

Loved it!

Fantastico! 🤗

Very interesting, thanks.

Very nice! Such gravitational bending through Fermat principle was proposed by Robert Dicke as "Variable speed of light"

Would this explain the 'apparent' increase in the angular size of galaxies 7.7 Billion Light years from Earth? A phenomena that required the introduction of 'Dark energy'? So if the 'vacuum' has a refractive index higher than 1 (which it clearly does) we should be able to calculate & show it's the result of magnification and not Doppler redshift?

This is good. I remember doing derivations on these where how gravitational lens effect the refraction for JEE Advance physics problems. Were snells law constant becomes a function of gravitation potential

Wonderful.

Fascinating! How about the variable of wavelength on refraction? Would that also be analogous with "gravitational refraction?" And how would that relate to Red Shift?

Very very interesting ....

I am curious about your perspective on photons. I was looking at the differences between near field and far field EM interactions, and I noticed that photons appear to act quite a bit like soliton waves. Could that be what they are, or are there other explanations?

nice video kudos

14:04 Since the light is 'bent' by the object, wouldn't it bend toward the leading edge of the object, an upstream 'acceleration' gradient?

At 12:40 - Detectors placed at the top and bottom of this image on the paths highlighted, would ‘see’ the emitter in two ways and therefore in two positions. So for every gravitational lensing we observe there are at least two ghost images which are observable from elsewhere. In which case, our observations can be subjected to the same ghost image artefacts. How many observed galaxies are duplicates?

Amazing. But could an optical refractive index gradient induce light to orbit the massive object, in the same way a black hole can do so at its Schwarzschild boundary? How would you describe that gradient?

I've thought about this so much! My supposition is that around the photon sphere of a blackhole, there should be a rainbow either side, or a blackholebow if you will. Thanks for this video!

The thing is, the light still *does* move in a straight line - what happens is the very concept of a straight line *itself* is changed. That's what it means to curve spacetime, straight lines now do things you wouldn't expect from flat, uncurved space, such as diverge then reconverge. So there's no need to slow down the light going by, even in a pseudo-sense like with a lens, because there's no refraction going on, just the straight lines it was already going in happening to converge somewhere

Some years ago I watched a vid' that seemed to resurrect the theory of Luminiferous aether. I commented then, about the possibility of space having a refractive index and, if it did, how would that result in the way we interpret red shift and other data collected from spectra. Thank you for this vid'. It explains my thoughts in more detail than I ever could and has also convinced me that I am not a complete lunatic. 😎👍

This approach to the line of inquiry about refraction and electromagnetic waves is a prudent one. Nobody can write off gravitational lensing effects, so it’s a good way to anchor the pathway to more speculative hypothesis you may have in mind to explore. The realization that the rules or “formula” of physics or constants may change for each instance of a series within a nested stack of discontinuities, seems to be the view that keeps marking its appearance in my research travels, and resonates with similar ideas. One similar idea is the extended set of Riemannian manifolds, where seemingly linear euclidian space and related local action is subsumed within a larger non-linear space or manifold, which fools the investigator into thinking the local space is all there is, because the subsuming geometry is not always measurable or determinable from the linear perspective, because it passes through a singularity. If a system has singularities, there is a higher principle at work, and when “captured” can linearize them, such as Gauss remapping the square root of negative numbers onto the complex plane. The poles are the areas where greatest rate of change is occurring. The related issue of qualative change, as opposed to quantitative change, is important when looking at non-linear models. In the former, you sometimes have to throw out the system formula because the major rules change for each newly added principle, giving such a system an unpredictable nature sometimes, the mathematicians worst nightmare. I still suspect there is a non-linear aspect that electrodynamics overlooked, that only shows up to play at extremely high frequencies, to form particles. That idea still lingers. Perhaps it would be like a refraction shock-wave that morphs into a closed torus, but only at very high frequencies. Some kind of reflective bubble must form to make a particle, or that essentially is the particle, the interplay between a wave and space time at extreme frequencies, a self-generated resonant cavity composed of spacetime itself. Looking forward to see the path lit. Fine work!

Absolutely excellent video.Btw where can I find the puppet of samantha cristoforetti?😅

There are blind spots. Places where we should be able to see the emisor, but the lensing effect moves all light away. It is like the radar blindspots in submarines because of the salt gradient in the sea. Remember this from a video series regarding submarines of Smarter Everyday channel

Gravitational refraction or diffraction through plasma atmosphere surrounding stars?

This reminds me of one time I found a site giving the equation for the equivalent index of a black hole. If I ever get around to learning blender I want to make an accurate black hole render.

good job

I had a similar thought experiment (i.e., bridging relativity into microscopic optics) during my Ph.D. years. My research was on applied optics and imaging, so I didn't have much time nor expertise to explore further. It's good to see a well-explained video on this topic. On the other hand, I'm interested in knowing if this concept can be used to explain diffraction (or unify refraction and diffraction into a single general concept). I know that we can use Huygens' principle to describe how diffraction works. However, the explanation of why it happens doesn't really satisfy me, especially at the aperture boundary. What if the light was diffracted due to space being bent at the aperture boundary? If that were the case, would different materials give different diffraction patterns (because they have different masses)? We know that's not true because the diffraction pattern only depends on distance, wavelength, and aperture size. The material of the aperture does not matter. Or does it? I don't know. Has it been rigorously tested/measured? I was looking for an analogy by exploring how water waves diffract. If I'm not mistaken, the math for water diffraction also uses a similar equation by Sommerfeld (i.e., optics). Water has viscosity (which I assume plays a part in water diffraction). But light has no such properties. I'm pretty sure I'm 99.9% wrong here. But if someone can shed some light on this (pun intended), it would be much appreciated.

By the way, what is the 'k' constant? I was able to get everything else properly setup, but I dont have k

Can you reconstruct images shifted by gravitational lensing? |Eg, can you effectively look behind stuff?

The fact that even the most extreme vacuum conditions in deep space contain at least a million atoms per cubic meter, combined with the functionally inconceivable distances and timescales involved, surely points to something like an "effective" refractive index for space. My question is how different wavelengths of EM radiation would respond; we know X-Rays don't refract the way 400-700nm visible light does for example on entering glass and so this adds another fascinating dimension. The time seems ripe however for an overthrow of the old "certainties" concerning gravitational lenses, Doppler-shift etc.

1:30 you need to publish a paper about types of "gravitational aberrations" ...classify them, name them, etc.

Dude, your videos are always brilliant. You would have made the best physics teacher had you taken a different career path 😅

Hey there, I have a question. Could you demonstrate what the double slit experiment would look like, if one slit was double the width - or even a variable fraction - of the other. Would appreciate an answer :)

Really cool. Take the equation for time dilation as a function of radius from the mass and see if it matches up with your 1-k*Vg equation.

What happens if you try to use snell's law with the bending GR predicts?

12:49 what you’re showing is an viewer with the mass and the light source. But it seems the viewer would also have a good view of the light source on either side of the galaxy too. Later in the video you mention the effects wouldn’t be as pronounced in reality, but I can imaging these patterns could create some intriguing artefacts that would be difficult for astronomers to resolve.

The refractive index is more accurately modeled by n=e^(-V_g/c^2). Source is the Wikipedia page on gravitational time dilation. I'm also a physicist. Regarding 18:50, I tried to see if I could reproduce GR with a generalized version of Lorentz Ether Theory(LET). It worked great for light, but unfortunately failed horrifically for literally everything else. There was nothing to set the scale of particles, and therefore no way to dictate the volume of a region of spacetime. As such, a clock could shrink by n and run n times faster. In order to fix this, a scale parameter had to be added. At this point though, I was just representing the metric tensor in an exceedingly cumbersome way with no benefits, so I deemed this attempt a dead end. It was like that VSL stuff you see sometimes on RU-vid, but with frame dragging. Not sure if those guys have gotten that far yet(it really isn't that far, pretty trivial).

Hey, so this is the first time I've seen a video on this. I've had this propagating in my head for a while, because I'm a stereoscopic artist and the things that can be seen when treating gravitational lensing copies as stereo pairs... are absolutely fascinating. But nobody seems to be researching it or even willing to believe anything COULD be seen this way. I'm sharing this because stereo-lensing makes subtleties obvious, and I think these subtleties infer a long chain of this refraction index that might help you refine your equation. Are you familiar with the work of Viktor Toth, and his lens bridging simulations?

Actually the most general form of the dielectric 'constant' is a fourth rank tensor just like the Riemann tensor and it has the same symmetry properties.

The speed of light slows down when it passes through glass, based on the reference frame outside the glass. So what happens when the reference frame moves inside the glass, does the speed of light outside the glass, become faster than "c" ?

circa 11:00, if you look at the Schwarzschild metric, n=c/(dr/dt) goes as (1-1/r) with r in units of Schwarzschild radius, so you picked the right form.