TheCosmologist

But not pseudoscientific. I guess speculation, its wording, is a matter of relative perspective since the variability of light is synonymous mathematically to relativity, both special and general cases. However, I will accept it isn't mainstream, as in the sense that it is the most "popular view," however, it is garnering a large following. He actually makes very good points about the inaccuracies that are purported and as a matter of science neglecting historical significant models which had been superceded by a popularity contest rather than the rigour of scientific scrutiny at full throttle.

The Smallest Black Hole Say a black hole's Schwarzchild radius is equal to the Planck length then a horizontal formula can be established as [math]A = 4\pi \ell^2[/math] [1]. I've tried to find an analogue of this set up online but can't find any reading material on it. I'll expand further on this at the end and ask a few questions. We start with the equation for the Schwarzschild radius: [math]R_s = \frac{2Gm}{c^2}[/math] where G is the gravitational constant, m is the mass of the black hole, and c is the speed of light. If the black hole has a Schwarzschild radius equal to the Planck length, we have: [math]\frac{2Gm}{c^2} = \ell_P[/math] Solving for the mass, we get: [math]m = \frac{\ell_P c^2}{2G}[/math] Next, we use the equation for the radius of the photon sphere: [math]R_p = \frac{3}{2}R_s[/math] Substituting in the expression for [math]R_s[/math], we have: [math]R_p = \frac{3}{2}\cdot\frac{2Gm}{c^2} = \frac{3}{2}\cdot\frac{2G}{c^2}\cdot\frac{\ell_P c^2}{2G} = \frac{3}{2}\ell_P[/math] So the radius of the photon sphere in this case is indeed 1.5 times larger than the Planck length, as expected. Then a standard formula would be [math]R_p > \frac{3}{2}R_s = \frac{2Gm}{c^2} = \frac{3}{2}R_s[/math] Going back now to, [math]A = 4\pi \ell^2[/math] Surely this would be the smallest area that is computable within physics, since physics breaks down below the Planck scales? I've been trying to visualise such a small object and the immense curvature it should possess as posed by general relativity. I took my sights to using the spacetime uncertainty, [math]\Delta x\ c\ \Delta t \geq \ell^2_P[/math] we can state that this equation be taken to the Planck domain as the shortest interval or length: [math]ds^2 = g_{tt}\ \Delta x\ c\Delta t \geq \ell^2_P[/math] And of course, the equation can undergo a curve in the pseudo Reimannian manifold, which is akin to a curve between two Planck regions, [math]ds^2 = g\ \Delta x\ c\Delta t \geq \ell^2_P[/math] Using the spacetime uncertainty, [math]\Delta x\ c\ \Delta t \geq \ell^2_P[/math] we can state that this equation be taken to the Planck domain as the shortest interval or length: [math]ds^2 = g_{tt}\ \Delta x\ c\Delta t \geq \ell^2_P[/math] And of course, the equation can undergo a curve in the pseudo Reimannian manifold, which is akin to a curve between two Planck regions, [math]ds^2 = g\ \Delta x\ c\Delta t \geq \ell^2_P[/math] The metric can be rewritten as [math]ds^2 = g\ \Delta x\ c\Delta t = g(\Delta \mathbf{q}_1 \Delta \mathbf{q}_2) \geq \ell^2_P[/math] Where [math]\mathbf{q}[/math] is the infinitesimal displacement by generalised coordinates which acts on the integral as [math]\mathbf{q}(\lambda_1) [/math] and [math]\mathbf{q}(\lambda_2)[/math] Which gives the standard curve equation: [math]ds = \int_{\lambda_1}^{\lambda_2} d\lambda \sqrt{|ds^2|}[/math] Considerations: It is theoretically possible for the photon ring in the micro black hole case to be comprised of virtual particles instead of the usual on-shell photon sphere for macroscopic cases of the same phenomenon.

Gravitational Index of Refraction Using the surface gravity entropy of the form: [math]\mathbf{S} \leq k_B\frac{2 \pi\ c\ E}{\hbar \kappa}[/math] Plugging in the dimensions of the surface gravity will yield a ratio which we can surreptiously take as the index of refraction which can be both expressed in terms of electromagnetic or gravity models. For example: [math]\mathbf{S} \leq k_B\frac{2 \pi\ c\ E\ t}{\hbar\ v} = k_B\frac{2 \pi\ n\ E\ t}{ \hbar}[/math] Where [math]n = \frac{c}{v}[/math]. Such ideas in which gravity can be treated as a special limit of some optical theory has long impressed me. In the context of the free-space wavenumber [math]k_0[/math] compared to the usual wave number, a relationship to the index of refraction can be made such that, demonstrated quickly with [math]c = 1[/math], [math]k = k_0\sqrt{\mu_0\ \nu_0} = k_0n[/math] There's two ways we can implement this into the entropy equation - the first way is to simply rearrange this formula for the index of refraction and plug in directly - so firstly, rearranging, after restoring the units of the speed of light [math]\frac{k}{k_0} = c \cdot \sqrt{\mu_0\ \nu_0} = n[/math] Restoring the units is quite simple when you know [math]\sqrt{\mu_0\ \nu_0}= \frac{1}{c}[/math] This is arguably the most recognized form of the index of refraction and was the starter to the gravitational aether theory for light being able to escape the prism of a black hole. I won't make a big mention on this model but I'll leave it in the air for later analysis. This relationship yields: [math]\mathbf{S} \leq k_B\frac{2 \pi\ n\ E\ t}{ \hbar}[/math] [math]= k_B \cdot \frac{k}{k_0}(\frac{2 \pi\ E\ t}{ \hbar})[/math] [math]= k_Bc\ \sqrt{\mu_0 \nu_0} \cdot (\frac{2 \pi\ E\ R }{ \hbar})[/math] Where in the last expression, we absorbed the factor of c back into the time component, as as always we can construct the uncertainty principle from the parts in the numerator if we so chose. The second way is to state the uncertainty [math]\Delta E\ \Delta R = c\Delta p\ \Delta x[/math] We won't be implementing the uncertainty relationship in this particular exercise, the previous relationship was just to point out dimensional grounds between [math]dE\ dx[/math] and [math]c dp\ dt[/math]. Using that we can shift between two different representations of the non commuting variables, [math]\mathbf{S} \leq k_B\frac{2 \pi\ n\ E\ t}{\hbar} = k_B\frac{2 \pi\ n\ p\ ct}{\hbar}[/math] The point is that in quantum mechanics we interpret the wave number as being a measure of the momentum of a particle, with the rule that [math]p = \hbar k[/math] so that relation tells us that [math]p \approx \frac{h}{\Delta x}[/math] This allows us to write, [math]\mathbf{S} \leq k_B\frac{2 \pi\ n\ E\ t}{\hbar}[/math] [math]= k_B\frac{2 \pi\ nk_0\ \hbar\ c t}{\hbar}[/math] [math]= 2 \pi\ k_B\ n k_0\ x[/math] By making the free wave number as a coefficient on the index of refraction allows us to write it alternatively as [math]= 2k_B \pi\ c\ \sqrt{\mu_0\ \nu_0}\ k_0 x[/math] By reminding ourselves of [math]k = k_0c\ \sqrt{\mu_0\ \nu_0} = k_0n[/math] While it's very interesting to consider this electromagnetic interpretation, the more satisfying case maybe one which suits the gravitational index of refraction and in fact, there's something quite elegant when deriving it from the following line element: [math]ds^2 = -g_{tt} c^2dt^2 + g_{rr}dr^2 + r^2d\Omega^2[/math] Some definitions are required, such as for light, it moves along the null geodesic [math]ds^2 = 0[/math] The velocity is simply [math]v = \frac{dR}{dt}[/math] And the metrics satisfy [math]g_{tt} = g^{rr}[/math] Solving for the velocity, we find for Schwarzchild geometry that [math]v = \frac{dR}{dt} = c \cdot g_{tt} = c \cdot (1 - \frac{2Gm}{Rc^2})[/math] Where [math]g_{tt} = (1 - \frac{2Gm}{Rc^2})[/math] Just a little algebra required now to find: [math]n = \frac{c}{c \cdot (1 - \frac{2Gm}{Rc^2})}[/math] [math]n = (1 - \frac{2Gm}{Rc^2})^{-1}[/math] This means that the entropy equation takes the form now of [math]\mathbf{S} \leq k_B\frac{2 \pi\ E\ t}{ \hbar} \cdot (1 - \frac{2Gm}{Rc^2})^{-1}[/math] [math]= k_B\frac{2 \pi\ E\ t}{ \hbar} \cdot \frac{1}{(1 - \frac{2Gm}{Rc^2})}[/math] Which is a new formula that literally takes into question the refractive index of the localized geometry. And of course the uncertainty enters naturally as [math]\mathbf{S} \leq k_B\frac{2 \pi\ \Delta E\ \Delta t}{ \hbar} \cdot (1 - \frac{2Gm}{Rc^2})^{-1}[/math] [math]= k_B\frac{2 \pi\ \Delta E\ \Delta t}{ \hbar} \cdot \frac{1}{(1 - \frac{2Gm}{Rc^2})}[/math] Planck Acceleration For A Black Hole In a gravitational field it was shown that the relativistic formula of an accelerated charged system is given by a power formula of (in which I modified it slightly in modern notation) as [math]P = \frac{2}{3} \frac{Q^2}{c^3} \cdot \frac{a^2}{(\frac{g_{tt}(R)}{g_{tt}(S)})^2}[/math] [math]= \frac{2}{3} \frac{Q^2}{c^3} \cdot \frac{a^2}{(1 + z)^2}[/math] Where [math](1 + z)^2[/math] is the redshift ans is equivalent to [math](\frac{g_{tt}(R)}{g_{tt}(S)})^2[/math] Which is further equivalent to the ratio of index of refractions as you will notice by the notation from the first part linking it to [math]g_{tt} = (1 - \frac{2Gm}{Rc^2})[/math] I constructed the same idea this time for the Larmor formula combined with the Black hole inequality. And it should be noted, the way the equations are presented here are heavily based on a model, which I call the Krafty black hole, but is really original work by L. Motz et al. See reference 3. Before we had constructed a power equation for an accelerated charged black hole under a novel approach from the inequality giving, while this time we take the maximum acceleration possible, [math]P = \frac{2}{3}\frac{(\frac{mRa^2_{max}}{c})}{(1 + z)^2}[/math] [math]\leq - [\frac{Gm^2}{c} + \frac{Q^2}{c^3}] \cdot \frac{a^2_P}{(1 + z)^2}[/math] The maximal Acceleration it can undergo then is [math]a^2 = \frac{m^2_P c^6}{\hbar^2} = \frac{c^7}{G\hbar}[/math] From using this I obtain: [math]P = \frac{2}{3}\frac{m^2_Pc^5R}{\hbar^2} \cdot \frac{1}{(1 + z)^2}[math] [math]\leq - [\frac{Gm^2}{c} + \frac{Q^2}{c^3}] \cdot \frac{a_{max}}{(1 + z)^2}[/math] And [math]P = \frac{2}{3}\frac{mRc^6}{G\hbar} \cdot \frac{1}{(1 + z)^2}[/math] And [math]P = \frac{2c^4}{3G}\frac{Jc}{\hbar} \cdot \frac{1}{(1 + z)^2}[/math] The last result is a bit remarkable revealing [math]Jc \approx Q^2[/math] and [math]\frac{Jc}{\hbar}[/math] can be seen as the Von Klitzing constant for conductors, with [math]\frac{c^4}{G}[/math] as the upper limit of gravitation. I made a post recently in which black holes have been in literature modelled with conducting surfaces. Kip Thorne was one such physicist who showed and even calculated the conductive surface of a charged black hole. Gravitational Corrections Gravitational corrections on the potential can be made, relatively simply enough. Setting [math]G=c=1[/math] (natural unit system) and this time concentrating on a black hole solution which takes into account rotation within the inequality, I present to first approximation: [math]P = \frac{2}{3}\frac{a^2}{1 - \Delta \phi}(Q +\frac{J}{m})^2 \leq \frac{2}{3}mr_g \cdot \frac{a^2}{(1 - \frac{m}{r_1r_2}(r_2 - r_1))^2}[/math] [math]= \frac{2}{3}\frac{r_g}{m}\frac{1}{(1 - \frac{m}{r_1r_2}(r_2 - r_1))^2}(\frac{d\mathbf{p_{\mu}}}{d\tau}\frac{d\mathbf{p^{\mu}}}{d\tau})[/math] This latter equality is a generalized formula under relativity for the Larmor radiation, which includes bold-p for the four momentum and tau as the proper time. With [math]\Delta \phi = \phi_1 - \phi_2 = \frac{m}{r_1r_2}(r_2 - r_1)[/math] Also we can see how the inner product gives [math]\frac{d\mathbf{p_{\mu}}}{d\tau}\frac{d\mathbf{p^{\mu}}}{d\tau}= \beta^2(\frac{d{p_{\mu}}}{d\tau})^2 - (\frac{d\mathbf{p^{\mu}}}{d\tau})^2[/math] So that in the limit of [math]\beta^2 << 1[/math] it reproduces the non-relativistic limit. See ref 4. Referenced https://www.google.com/url?sa=t&source=web&rct=j&url=https://vixra.org/pdf/1903.0407v2.pdf&ved=2ahUKEwjDu_iUnKX9AhXtTEEAHWHVDWkQFnoECFwQAQ&usg=AOvVaw3nHrIxGEqQuVWveAgu8ald https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.researchgate.net/publication/258707167_Gravitational_wave_derived_from_fluid_mechanics_applied_on_the_permittivity_and_the_permeability_of_free_space&ved=2ahUKEwi2q-HpnaX9AhUO87sIHSctChYQFnoECAkQAQ&usg=AOvVaw0RjnV9u-qwEw6ZJBpUgh0x https://www.google.com/url?sa=t&source=web&rct=j&url=https://worldscientific.com/doi/abs/10.1142/S0217732398000887&ved=2ahUKEwi6u4DFoqX9AhXHQEEAHfTvCYMQFnoECA0QAQ&usg=AOvVaw2dFDMKN12k9bOKLOS1Z16- Larmor formula - Wikipedia Calculation of the Universal Gravitational Constant, of the Hubble Constant, and of the Average CMB Temperature Another good one on reference 5. This latter paper is a great one. One implication of a lightspeed which was spatially variable, is that light could potential escape black holes, albeit it would take a very long time to do so. And if you're not convinced, these ideas of a variable speed of light was taken seriously by a number of great Physicists. In one such case, some physicists stumbled upon Einstein's relativity before him, by implying that the speed of light was variable in gravitational field. A. Unzicker explains this best in a series of videos which can be found on YouTube, to which I shall provide some links to below. I don't know what happened to mathtex in this case. I'll just have to redo it soon.

I thought you'd appreciate the time that I took to rebuke the statements you have made about nothing being "new" here as I finally came to realise what your issue is. So please tell me if anything I've said in the forthcoming post, is incorrect in any way to the way your mindset works on this issue On The Elastic Paper: Part II A Rebuttal To Accusation and Further Insights When I proposed this new theory, a critique, and not a very insightful one, couldn't ascertain why my formulas where new. Here I point out a historical reference to the Dirac equation - Dirac knew about the mass energy equivalence in its relativistic form, equally, from Einstein’s own special theory. He was also aware of how to make energy and momentum operators. When upgrading observables to their operator equivalences, we call this process a second quantization method. Now, like all theoretical scientists today, we “stand on the shoulders of giants” as Einstein was modest to admit. The point of Dirac, was to meld quantum theory in a “wedding” with relativity. We do not say, like I was accused of, of taking credit of already known formula's. Indeed, the formulas I constructed are but an extension not dissimilar to the process Dirac took, except mine concentrated on general relativity. Would we say he “stole the work” and then accuse of nothing being “new” about the Dirac equation…? Of course not. Further Insights to Build a Mental Picture Now I'll provide a deeper insight into one of my last formulae, namely the higher curvature corrections and higher powers of the Planck constant which revealed equally higher powers of the gravitational fine structure. [math]\delta A = c\int\ d^4x\ \frac{g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \mathbf{A}\ k\ \int dk[/math] [math]+ \int \frac{dk}{k}(\mathbf{B}^2 \alpha^{-1}_G\ \mathbf{R}^2 + \mathbf{C}^3\alpha^{-2}_G \ \mathbf{R}^{ik}\mathbf{R}_{ik}[/math] [math]+\mathbf{D}^4 \alpha^{-3}_G\ \mathbf{R}^{ikjm}\mathbf{R}_{ikjm} )]\frac{8\pi G}{c^4}[/math] The higher powers of the Planck constant could mean we are dealing with more virtual particles given in some differential volume element, which take the form in curved spacetime as [math]dV = \sqrt{g}\ dxdydz[/math] It would be naive to think we are dealing with a single virtual particle in all cases, moreover, further folly to think that the prediction made in the first paper about how curvature can produce fluctuations, to neglect a notion that higher power curvature corrections produce a flurry of energetic virtual particles in a given space to be devoid from the theoretical model presented. Indeed, during the first instant of our universe when it nucleosynthesized into existence, there was present the strongest gravitational forces that have ever existed in our universe. This is what I call the “birthplace of real fluctuations,” as the gravitational field again, was able to boost a reasonable quantity of the off-shell particles into long-lived real on shell matter, the same observable “stuff” that we see in present cosmology which encompasses anything between 1–4% of all spacetime volume. It must have been a very fast process which does owe some merit to inflation - if it has been a slow process during the initial expansion phase, much more observable matter would have been present today and the observed density would be of orders much larger. There's around [math]3 \times 10^{80}[/math] particles roughly speaking in our universe, give or take a power of ten. If the expansion phase had been much slower, the magnitudes could have been tens upon hundreds of powers larger. So rapid initial expansion seems to concur why spacetime appears so dilute in this respect also. The Singularity Sakharov mentioned the initial singularity in his paper - for many reasons, many cosmological models now avoid singular solutions to spacetime. One quantum law which should forbid singularities for example, is that you cannot squeeze a particle into a region of spacetime that is smaller than its own wavelength, meaning the initial phase was not [math]R =0[/math]. * just as a side note, even Hawking and Penrose abandoned their own singularity theorems as some historical reference. An important feature of the aforementioned equation is that it contains the wave number of the fluctuations. As mentioned earlier, it would be naive to think we are dealing with a unit spacetime cell, with one fluctuation, the equation deals with higher powers of curvature closely tied to the element volume - it’s basically telling us that any region of spacetime larger than a discrete cell can have two, and even more fluctuations in a given increasing volume that we integrate over and is the true frequency divided by the speed of the wave and thus equal to the number of waves in a unit distance. This is why melding Sakharovs equation and the action is important as it deals with distances as it is an action principle. Further, Bosons can fall into the same energy states, whilst the Fermions cannot - so the physics can be as exotic as the model allows for given fluctuations in a region of spacetime. In previous investigations I made on the Bekenstein entropy and other entropy equations, I discovered that non-commutation was something seemingly written into the structure of black holes. Don't worry, we won't digress from the topic too much, I just want to introduce quickly why the wave number is interesting for any calculation when done properly. [math]\mathbf{S} \leq k_B\frac{2 \pi mc^2R}{4 \hbar c} = k_B\frac{2 \pi\ c\ mR}{4 \hbar}[/math] This encompasses the deBroglie wavelength as [math]\mathbf{S} \leq k_B\frac{2 \pi mc^2R}{4 \hbar c} = k_B\frac{2 \pi\ c\ R}{4 \lambda}[/math] It just so happens we can replace [math]\frac{2\pi\ R}{\lambda}[/math] Where the wave number is [math]k = \frac{2\pi}{\lambda}[/math] With [math]\Delta k \Delta x \geq \frac{1}{2}[/math] Which is an analogue of an uncertainty principle obtained by Fourier, well before Heisenberg was on the picture, that a picture stated that a superposition of waves could not have both a small size and also a small number of frequencies. It makes the entropy appear as [math]\mathbf{S} \leq k_B\frac{\Delta R}{4 \Delta \lambda} = k_B\frac{ \Delta k \Delta x}{4}[/math] This is where the Fourier transform part cones in... The momentum of the system is expressed as [math]p = (\frac{h}{2\pi})k[/math] By multiplying [math]\Delta k \Delta x \geq \frac{1}{2}[/math] By [math]\frac{h}{2\pi}[/math] allows us to obtain [math](\Delta k \frac{h}{2\pi}) \Delta x = \Delta p \Delta x \geq \frac{h}{2\pi} = \frac{\hbar}{2}[/math] To be fair, the first part really addresses a rebuttal of your statements about nothing being new here, the rest was just showcase.

you're very strange. I am very much a fleshy human being.

Curious statement, in what sense of the matter am I relatable to a chatbot? Sorry what? I am confused now! You do realise the equations in the OP are based on the melding of field theory with the gravitational action, and has not been don3 before. What part am I trying to take credit for except which was my own contribution of the melding of the two? I've been very specific claiming this.

Yes it is new You seem seem a bit confused over this matter. It's very much new as no one has ever wrote the action in terms of field theory like this. That's what constitutes the equation to be "new."

It's new because the gravitational action has never been written to encorporate the fluctuations of the ground state fields. It's called the elastic action because Sakharov coined the term metrical elasticity in his paper which is recited. Bold R is the curvature scalar, bold k is simply a constant which is found in the action. All bold notation has been throughly explained in the OP. I have made a mention of Sakharov including the original Langrangian. My conclusions are the following: fluctuations can and do effect the background of space and has strange analogy to the Casimir effect (as mentioned by Sakharov) but more curiously we reach a chicken and egg problem. What came first, curvature or fluctuations? Can curvature of the manifold produce the fluctuations? Moreover when the curvature is extremely significant, these virtual off shell particles can become real on shell particles by virtue of what is called a "gravitational boost." It's not a well-known subject, but gravity can affect the way these particles are created, and is very similar to how Hawking radiation becomes long-lived fluctuations from the curvature decay of black holes. As mentioned, this effect of curvature on fluctuations, and fluctuations on curvature holds vital and important questions about the nature of quantum gravity.

I am the sole author of this work, all rights are reserved to me. Elastic Action: A Wedding of Quantum Field Theory with the General Relativiatic Action This tackles a question on how Sakharov's ground state field for virtual particles enters the gravitational action. In it I conclude that maybe the cosmological constant is in fact a renormalization constant which is only set to zero for flat Euclidean spacetime. The jury us still out, but most respected astrophysicist tend to agree that while spacetime looks quite flat, it probably isn't exactly flat, it's just a very good approximation and his equations on a cosmological scale would predict a small curve like we expect. Let's identify variables [math]A[/math] - action [math]g[/math] - metric [math]x[/math] - variable spatial coordinate [math]c[/math] - speed of light [math]\mathbf{R}[/math] - Ricci curvature scalar [math]\hbar[/math] - Planck constant, reduced [math]G[/math] - Newtons constant [math]k[/math] - wave number * We will use [math]\mathbf{k}[/math] as a constant [math]\frac{8πG}{c^4}[/math] which is the upper value of the gravitational constant In the style of Sakharov, we'd like to write a Langrangian of the ground state fluctuations which has a contribution of geometry by off-shell virtual particles. It's a rare paper to find, but his original ideas can be found here, (a link can be found in references to his paper), https://www.atticusrarebooks.com/pages/books/719/a-d-sakharov-andrei/vacuum-quantum-fluctuations-in-curved-space-and-the-theory-of-gravitation-in-soviet-physics His original ideas can be taken as a precursor to Bogoliubov transformations used to describe how gravity jiggles these off-shell particles at the horizon of supermassive Black holes, owing to their name as Hawking radiation. We then use Sakharov's curvature corrected Langrangian [math]\mathcal{L} = \mathbf{R}\ \hbar c\ k\ \int dk + \mathbf{R}^2\ \hbar c\ \int \frac{dk}{k} + C[/math] [math]= \mathbf{R}\ \mathbf{A}\ k\ \int dk + \mathbf{R}^2\ \mathbf{B}\ \int \frac{dk}{k} + C[/math] [math]= \mathbf{R}\ \mathbf{A}\ k\ \int dk + \mathbf{R}^2\ \mathbf{B}\ \int d\log k + C[/math] Where [math]\int \frac{dk}{k} \approx 137[/math] (the inverse fine structure). C is a renormalization constant set to zero for flat Euclidean spacetime - loosely it could be seen as the cosmological constant. At no point in spacetime is the cosmological constant really zero, so this argues for a microscopic description of curvature. This isn't meant to mean that flat spacetime is devoid of fluctuations however - modern field theory assures us that there is no such thing as empty space as fluctuations are being produced constantly, both on curved backgrounds and in flat spacetime. To meld field theory with the gravitational action, we first identify the action as [math]A = c \int\ d^4x\ \frac{\sqrt{-g}}{\mathbf{k}}[\mathbf{R} +…][/math] *the presence of the speed of light as a coefficient of the integrand is because of the dimensiones that requires an energy multiplied by a time, which is also the units of action, ie one can use [math]dx = cdt[/math]. In the Landau and Lifshiftz theory of fields, the action becomes [math]A = \frac{c^3}{8\pi G}\ \int\ d^4x\ \sqrt{-g}[\mathbf{R} +…][/math] This is just another way of writing [math]A = c\ \int\ d^4x\ \frac{\sqrt{-g}}{\mathbf{k}}[\mathbf{R} +…][/math] Let's show quickly why. Using [math]\frac{G}{c^2} = \frac{\ell}{m}[/math] Then we can crunch the dimensions down easily on, [math]A = \frac{c^3}{8\pi G}\ \int\ d^4x\ \sqrt{-g}[\mathbf{R} +…][/math] We invert the length to mass ratio and plug in, which leaves a coefficient of c on the mass term, and the RHS absorbs the inverse lengths to produce the mass density, and we absorb one factor of c from the four dimensional volume element [math]\frac{mc}{\ell} (\ell^4) \cdot \frac{1}{\ell^2} \cdot dxdydz\ cdt = \rho c^2 dt \cdot dxdydz\ dt[/math] Which has units of energy times time, which are the dimensions of action. Plugging in the nodes from the first formula, (which is the background curvature correction of off shell virtual particles discovered by Sakharov), we unify quantum field theory with gravity in the following way, and name it the "elastic action," and we drop the constant of integration for brevity, [math]A = c\ \int\ d^4x\ \frac{\sqrt{-g}}{\mathbf{k}}[\mathbf{R}\ \hbar c\ k\ \int dk[/math] [math]+ \mathbf{R}^2\ \hbar c\ \int \frac{dk}{k}]\frac{16\pi G}{c^4}[/math] It may look funny that we have had to introduce yet another [math]\frac{16\pi G}{c^4}[/math] but this is very much required as a coefficient that "undoes" what we did mathematically inside of the square parenthesis. I'll show why in the footnotes. This quantum correction is applied to the theory of relativity, melding the two nicely. In short it says the gravitational action involves the ground state zero point energy fields. First Varied Action Using the formulation set above, we now can apply the varied action. Notice the variation does not apply to the constants or as usual, to do this is a straightforward process where I will use Einstein summation. [math]\delta A = c\ \int d^4x\ \frac{\sqrt{-g}}{\mathbf{k}}[\delta g^{\mu \nu} \mathbf{R}_{\mu \nu}\ \mathbf{A}\ k\ \int dk[/math] [math]+ \delta g \mathbf{R}^{\mu \nu}\mathbf{R}_{\mu \nu}\ \mathbf{B}\ \int \frac{dk}{k}]\frac{16\pi G}{c^4}[/math] This is the varied action, we simply split the corrected Reimann curvature R^(μv)R_(μv) like so, as it acts like the fourth inverse power over the wavelengths. Notes: To understand why we had to correct the dimensions for a new coefficient [math]\frac{16\pi G}{c^4}[/math] plays to functional roles. The constants 16 \pi removes the constants found in [math]\mathbf{k}[/math] and [math]\frac{G}{c^4}[/math] is the (only) reasonable gravitational coefficient that removes the dimensions of the Sakharov terms. We just want to focus on how this altered the dimensions and how we must fix those dimensions. Focusing on this, [math]\mathbf{R}\ \hbar c\ k[/math] We know what is, and so it's dimensions are still inverse length squared. The k is called the wave number and has inverse unit of length. All-in-all, we have dimensions of charge squared divided by a length, giving an energy, further with another inverse length cubed, giving the appropriate dimensions of energy density. What we "put in" those brackets, must be undone, and there's a straight-forward way to do it. We don't need to "undo" what we have in since it already features in the action, but we will concentrate on [math]\hbar c\ k\ \int dk = \frac{\hbar c}{\ell^2}[/math] We understand that the following dimensions must hold true: [math]\frac{G}{c^2} \equiv \frac{\ell}{m}[/math] Since dimensionally-speaking [math]\frac{\hbar c}{\ell^2} = \frac{Gm^2}{\ell^2}[/math] Then we can decompose it in the following way: [math]\frac{Gm}{\ell^2} = \frac{Gm}{\ell}\frac{c^2}{G} = \frac{m}{\ell} \cdot c^2 = \frac{c^4}{G}[/math] Interesting isn't it? It seems then the solution has been found. In order to "undo" what we did, it requires a correction coefficient of the upper limit of gravity as [math]\frac{c^4}{G}[/math] (by taking its inverse). Why its inverse? Simply because if [math]\frac{\hbar c}{\ell^2} = \frac{Gm^2}{\ell^2} = \frac{c^4}{G}[/math] Then we must invert to remove these unwanted dimensions. Second Varied Action Starting from the Lagrangian density we derived earlier: [math]\mathcal{L} = c\ \frac{\sqrt{-g}}{\mathbf{k}} \left[\mathbf{R} \hbar c\ k \int dk + \mathbf{R}^2 \hbar c\ \int \frac{dk}{k} \right] \frac{16 \pi G}{c^4}[/math] To compute the variation of the action, we need to vary the metric tensor g_{\mu\nu} and integrate the variation over all spacetime coordinates: [math]\delta A = \int d^4x\ \delta\left(\mathcal{L}\sqrt{-g}\right)[/math] Using the product rule of variation, we have: [math]\delta A = \int d^4x\ \left[\delta\mathcal{L}\sqrt{-g} + \mathcal{L}\delta(\sqrt{-g})\right][/math] We can simplify the second term using the variation of the determinant of the metric tensor: [math]\delta(\sqrt{-g}) = \frac{1}{2}\sqrt{-g}\ g^{\mu\nu}\delta g_{\mu\nu}[/math] Substituting this expression and our variation of the Lagrangian density into the equation for the variation of the action, we obtain: [math]\delta A = c \int d^4x\ \frac{\sqrt{-g}}{\mathbf{k}} [\delta g^{\mu \nu}\ \mathbf{R}_{\mu \nu}\ \mathbf{A}\ k \int dk[/math] [math]+ \delta g \mathbf{R}^{\mu \nu} \mathbf{R}_{\mu \nu}\ \mathbf{B}\ \int \frac{dk}{k}]\frac{16 \pi G}{c^4}[/math] where we have defined [math]\mathbf{A} = \hbar c[/math] and [math]\mathbf{B} = \hbar c[/math]. Note that the variation of the determinant of the metric tensor has been expressed in terms of the variation of the metric tensor [math]g_{\mu\nu}[/math], using the inverse metric [math]g^{\mu\nu}[/math] and the Ricci tensor [math]\mathbf{R}_{\mu\nu}[/math]. Substituting our expression for [math]\delta(\sqrt{-g})[/math] gives: [math]\delta A = -\frac{c}{2}\ \int\ d^4x\ \frac{\sqrt{-g}\ g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \hbar c\ k\ \int dk[/math] [math]+ \mathbf{R}^2\ \hbar c\ \int \frac{dk}{k}]\frac{16\pi G}{c^4}[/math] We can simplify this expression by using the fact that [math]\sqrt{-g}\ g^{\mu\nu} = -2g^{\mu\nu}[/math], which gives: [math]\delta A = c\int\ d^4x\ \frac{g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \mathbf{A}\ k\ \int dk[/math] [math]+ \mathbf{R}^2\ \mathbf{B}\ \int \frac{dk}{k}]\frac{8\pi G}{c^4}[/math] Higher Powers Sakharov concludes the higher powers are taken like so: [math]\int \frac{dk}{k}(\mathbf{B}\ \mathbf{R}^2 + \mathbf{C}\ \mathbf{R}^{ik}\mathbf{R}_{ik} +\mathbf{D}\ \mathbf{R}^{ikjm}\mathbf{R}_{ikjm} … + higher\ powers)[/math] Where [math]\int \frac{dk}{k} \approx 137[/math] [math]\delta A = c\int\ d^4x\ \frac{g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \mathbf{A}\ k\ \int dk + \int \frac{dk}{k}(\mathbf{B}\ \mathbf{R}^2 + \mathbf{C}\ \mathbf{R}^{ik}\mathbf{R}_{ik} +\mathbf{D}\ \mathbf{R}^{ikjm}\mathbf{R}_{ikjm} )]\frac{8\pi G}{c^4}[/math] Higher Powers of Fluctuations [math]\mathbf{R}\ \hbar c\ k \int dk[/math], taking higher powers of [math]\hbar c[/math] requires that the dimensions are scaled appropriately… Say the higher powers don't just affect the curvature, but affects higher powers of [math]\hbar c = (A, B,C,D) \approx 1[/math] So taking higher powers of [math]\hbar c[/math] in gives [math]\delta A = c\int\ d^4x\ \frac{g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \mathbf{A}\ k\ \int dk + \int \frac{dk}{k}(\mathbf{B}^2\frac{1}{Gm^2}\ \mathbf{R}^2[/math] [math]+ \mathbf{C}^3 \frac{1}{G^2m^4}\ \mathbf{R}^{ik}\mathbf{R}_{ik} +\mathbf{D}^4 \frac{1}{G^6m^8}\ \mathbf{R}^{ikjm}\mathbf{R}_{ikjm} )]\frac{8\pi G}{c^4}[/math] Would be the same as writing [math]\delta A = c\int\ d^4x\ \frac{g^{\mu\nu}\delta g_{\mu\nu}}{\mathbf{k}}[\mathbf{R}\ \mathbf{A}\ k\ \int dk + \int d\log k(\mathbf{B}^2 \alpha^{-1}_G\ \mathbf{R}^2[/math] [math]+ \mathbf{C}^3\alpha^{-2}_G \ \mathbf{R}^{ik}\mathbf{R}_{ik} +\mathbf{D}^4 \alpha^{-3}_G\ \mathbf{R}^{ikjm}\mathbf{R}_{ikjm} )]\frac{8\pi G}{c^4}[/math] Where we use the gravitational fine structure to normalise the higher dimensions, [math]\mathbf{R}^2 \hbar^2 c^2(\frac{1}{Gm^2}) k \int dk = \mathbf{R}^2 \hbar c\ \alpha^{-1}_G k \int dk[/math] While the fine structure constant and Newton's gravitational constant are both fundamental constants of nature, they describe completely different physical phenomena. Therefore, it is not accurate to say that higher powers of the fine structure constant are equivalent to correcting gravity at higher powers. However, there are some theories, such as string theory, that suggest a connection between the values of fundamental constants and the properties of space-time, including the strength of gravity. In these theories, it is possible that changes in the fine structure constant could lead to changes in the properties of space-time and therefore, to corrections in the theory of gravity. But this is a highly speculative area of research and is not yet well understood. Taking higher powers of the fine structure constant can reveal higher order corrections to physical phenomena that cannot be accounted for by classical or first-order quantum mechanical calculations. In particular, higher-order quantum corrections are important in understanding the behavior of subatomic particles and the interactions between them. For example, the anomalous magnetic moment of the electron can be calculated with higher and higher accuracy by taking into account higher order QED corrections involving higher powers of the fine structure constant. The electron's anomalous magnetic moment has been measured experimentally to extremely high precision, and the agreement between theory and experiment is a remarkable demonstration of the power of higher-order quantum corrections. Similarly, in quantum chromodynamics (QCD), the theory that describes the strong nuclear force, higher order corrections involving higher powers of the strong coupling constant (the analog of the fine structure constant for the strong force) are important for understanding the properties of hadrons (particles made of quarks), and for calculating the scattering amplitudes of quarks and gluons. In general, higher order corrections involving higher powers of dimensionless coupling constants are important for understanding the behavior of quantum field theories at energies far beyond the scales of current experiments. Such corrections can also provide clues to the existence of new physics beyond the Standard Model of particle physics. References http://ayuba.fr/pdf/sakharov_qvf.pdf