Mordred

A refractive index requires scatterrings. Those scatterrings and the degree of change in angles will involve the wavelength. If I look at an Einstein Ring I will get distrortions that do not depend on wavelength. I have been involved in doing those tests studying the CMB as my speciality is early universe dynamics. In order to get a telescope even the Hubble telescope on must often use a gravitational lens to extend the distance to get deep field imaging. A simple analogy for time dilation would be signal propogation delay. Take a digital signal in electronics you can delay that signal through electrical cross talk. Now extend that analogy to the 18 coupling constants of the standard model of particle physics.

No I am a Professional Cosmologist with degrees in particle physics. I can prove any refractive index treatment of spacetime wrong. I can do the same with any treatment of spacetime as a medium wrong. Lol all I have to do is point out previous research papers. I have even been hired to research refractive indexes by a survey camera manufacturer. That grant paid my income for a year. I have also done spectronomy research on a couple of gravity wells.

No forget refractive index. The correct application is Principle of least action via the Langrangian. Which is completely different from refractive index. How can you have a refractive index when the mean average number density of particles amounts to 5 protons per cubic metre in interstellar space ?

What curves is the principle of least action which relates the potential of the field to the kinetic energy of the particle. If you really want to understand curvature then you need to study the Principle of least action with the geodesic equations. (Do not mistake a field as a medium) a field is an abstract descriptive of values or mathematical objects under a geometry descriptive.) Of course it does. Spacetime has no refractive index... Particles behave in accordance to how they couple with a field or other fields. The couplings is what lead to the mass terms. Mass is resistance to inertia change.

No light doesn't require a medium to travel. This is where it differs from sound waves. It was this very thought that led to Eather theories. The Michelson and Morley experiment is one of the numerous tests that proved the medium view incorrect.

Spacetime isn't a medium. Any treatment of spacetime as a medium will be easily proven inaccurate to observational evidence. As to the first part. The geodesic equations do not involve electrons. A good example is light curves in the FLRW metric. (Universe geometry) however after the CMB there are no free electrons and this no Compton scatterring. How one describes mass must also work in cosmology applications as well as near massive bodies. GR and the FLRW are fully compatible theories that do not depend on any medium or specific particle composition.

Correct and never examined the effect of frequency in accordance to Snell's law. Let alone the one way two effects of a medium on light as per the M and M experiments. (Lol a little side note I had the opportunity to prove a peer reviewed article wrong on applying Snell's law to describe gravitational lensing to a PH.D in astrophysics. He pulled his article off Arxiv)

Spacetime curvature does not involve any refractive index. The mass term cannot be described under refractive index. Or radiation pressure (pressure has directional components).

The statement of the photon gravitating to a higher refractive index in the above will not work for gravitational lensing. Different frequencies of light respond differently in a medium. (Prism being one example) A Gravitational lens doesn't have the same effect. Ie a spectrograph looking at a gravitational lens will not see the same prismatic effect. Spacetime curvature doesn't depend on frequency. (If you try to treat spacetime as a medium you will invariably get the wrong answers) I could easily falsify any medium association.

No I am describing a rank two tensor under GR ( though just the starting steps to understand a rank two tensor). I hadn't gotten into components of a vector. (A special rank two tensor would be the Dyad.. The reason you need a rank two tensor describe gravity is that you a gradient to describe gravity.

Well that would certainly involve a lot of antisymmetry relations. Acceleration caused a rotation due to rapidity. Torsion would give antisymmetry to the metric tensor. Ie to describe torsion using the metric tensor you would have to specify a direction of rotation. What you actually need is a covector vector and a vector. The covariant vector is the column vectors while the vector is the row vectors. Using the two vectors above will preserve invatiance under coordinate transformations. Gravity itself is a form of flux of the energy momentum stress tensor. With the Minkowskii tensor you have already made a coordinate choice (cartesian) so you can use the inner product of two vectors. Which will return a scalar value [math]\mu\cdot\nu=s[/math] the Minkowskii tensor is orthogonal all orthogonal groups are symmetric and commute. [math]\mu\cdot\nu=\nu\cdot\mu[/math] However this would not be invariant under coordinate transformation so the column vector would use a covector.

Somehow I don't want to know. 😖

Every physics theory has competitive theories this is true of inflation. The CMB itself is supportive evidence through big bang nucleosynthesis of the BB. One of the difficult things to explain is how the supercooling due to rapid expansion and reheating due to the inflation slow roll leads to the metalicity values measured at z=1090. When you get right down to it the percentages match those predicted by inflation and quite frankly limit the range of viable inflation models. So it really doesn't matter what one believes. The only thing that matters is what observational evidence tells us. If you want a listing of viable inflation models that match observational evidence (though the last update was 2013.) See here https://arxiv.org/abs/1303.3787 The opening section explains the criteria. Personally I'm a fan of a single scalar field with a low kinetic term Higgs inflation. However my opinion doesn't mean it's factual. Lol at least that one is still viable according to Planck datasets.

It describes the probability wavefunctions. An oscillator isn't restricted to object motion but can also be used to describe any repeating varying value. A sinusoidal waveform in electronics is a good example.

With the uncertainty principle one cannot accurately measure the position and momentum of a particle at the same time. Measuring one observable P or Q will interfere with the other. Also the more accurately you measure one the less accurate you can determine the other. Both observable's will have a probability amplitude.

I like your reference four paper, there is several Langrangians in that paper I will latex later on to have a handy copy of them. I also like what you did with the overbrace and underbrace. I don't see any problems thus far

Looks good thus far the reference 7 page 11 equation 47 has the covariant derivative of the graviton propogator, as your employing the same tensors you should be be good.

See here for further detail you will also find the one loop vacuum polarization propogator handy https://arxiv.org/abs/1504.00894 Lol this article does mention the k term

Here is one of the better papers for the graviton spin 2. Note equation [math]g_{\mu\nu}\rightarrow \eta_{\mu\nu}+k h_{\mu\nu}[/math] https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/gr-qc/0607045&ved=2ahUKEwjCzqGC1PXmAhW1oFsKHT-HAs4QFjAAegQIBBAB&usg=AOvVaw30-hmokcbjp_amGXWvZtet The article provides the general spin Compton scattering for the other spin statistics as well as spin 2. My recommendation is to start with the linearized Einstein Hilbert action. See the following Doctorate thesis. (It's a common methodology for modelling the graviton) https://www.google.com/url?sa=t&source=web&rct=j&url=https://academiccommons.columbia.edu/download/fedora_content/download/ac:201929/content/GarciaSaenz_columbia_0054D_13501.pdf&ved=2ahUKEwi90_f23vXmAhX6JTQIHd0cCU8QFjACegQIAhAB&usg=AOvVaw22sjBJZaLwriZmm2fuX0wt By using the Einstein Hilbert action your already working in quanta Ie quanta of action and thus can make the correlations to the creation and annihilation operators for the Feymann path integrals. Though the difficulty will be avoiding infinite one loop corrections. Divergence that ruins renormalization Edit almost forgot in the above equation [math]k^2=16\pi G [/math] I'm not sure any of the above articles mention that. The above formula is a fairly standard equation in numerous papers for massless spin 2 propogators.

Much better for the Jacobian in spherical weak field limit. However It looks to me your graviton application is spin 1 dipolar. You need quaternion relations spin 2 (quadrupolar). To match up to gravity wave data. ( Though I will have to study your equations for reference 6 further) Let dig up some good graviton modelling and the relevant GR spin 2 application. Keep in mind in order to properly model the gravitons you will need it's wavefunction for transerve and longitudinal component. (In gauge treatments you must be renormalizable. )

Lol I always seem to get caught by spell check lol. Correction made thanks for the catch

Here is one example Schwartzchild metric Vacuum solution [latex]T_{ab}=0[/latex] which corresponds to an unaccelerated freefall frame [latex]G_{ab}=dx^adx^b[/latex] if [latex]ds^2> 0[/latex] =spacelike propertime= [latex]\sqrt{ds^2}[/latex] [latex]ds^2<0[/latex] timelike =[latex]\sqrt{-ds^2}[/latex] [latex] ds^2=0[/latex] null=lightcone spherical polar coordinates [latex](x^0,x^1,x^2,x^3)=(\tau,r,\theta,\phi)[/latex] [latex] G_{\alpha\beta} =\begin{pmatrix}-1+\frac{2M}{r}& 0 & 0& 0 \\ 0 &1+\frac{2M}{r}^{-1}& 0 & 0 \\0 & 0& r^2 & 0 \\0 & 0 &0& r^2sin^2\theta\end{pmatrix}[/latex] line element [latex]ds^2=-(1-\frac{2M}{r}dt)^2+(1-\frac{2M}{r})^{-1}+dr^2+r^2(d \phi^2 sin^2\phi d\theta^2)[/latex]

Lol correction applied

[math]g_{\alpha\beta}[/math] is the metric tensor the indices run (0,1,2,3). The form will vary according to the spacetime being modelled it can have either or both the covariant and contravariant terms accordingly to the Einstein summation convention. In the above its specifying covariant.

[math]\mu\cdot\nu=\nu\cdot\mu=\eta [/math] is the inner product symmetry relations for the Minkowskii metric tensor. You identified it as the GR symmetry matrix expression. You need the first order partials for the Jacobian matrix while I don't have polar coordinate form handy you can look here for the Minkowskii form though you will have to switch the signature. https://en.m.wikipedia.org/wiki/Four-gradient The one forms mentioned are invariant under coordinate change