Larmor Radiation From Black Holes Under Weak Equivalence Principle and Gravitational Index of Refraction

TheCosmologist · February 21, 2023

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

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.

Edited February 21, 2023 by TheCosmologist

joigus · February 21, 2023

Oh, I see.

You mean the infamous narcissist heckler of Witten that says all of modern physics since Planck is nonsense:

https://youtu.be/k7EnQd-VGqU?t=27

I know what you're up to now.

Markus Hanke · February 21, 2023

!

Moderator Note

Thread moved to the ‘Speculations’ section, since this isn’t mainstream physics.

TheCosmologist · February 21, 2023

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.

7 hours ago, joigus said:

Oh, I see.

You mean the infamous narcissist heckler of Witten that says all of modern physics since Planck is nonsense:

https://youtu.be/k7EnQd-VGqU?t=27

I know what you're up to now.

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.

joigus · February 21, 2023

Actually, no. He's a well-known hackler of serious scientists in public talks. I know enough about him to know he's a crackpot magnet.

He claims that all of physics since Planck is wrong --I was patient enough to watch one of his videos or two. That's some time down the drain I'm not getting back.

He certainly doesn't understand the ideas behind renormalisation. I'm not saying quantum physics is problem-free, and there are no consistency issues. There are. But I see nothing of value in trying to substitute renormalisation strategies with WAG numerical games and numerical analysis. And all hand-waving. That's what he does.

Very similar to what you did by copying and pasting some formulae from a 1968 paper and pulling some numbers from a part of your anatomy, and substituting a logarithmically divergent integral by your wild guess.

Zero value from a scientific POV IMO.

TheCosmologist · February 21, 2023

3 hours ago, joigus said:

Actually, no. He's a well-known hackler of serious scientists in public talks. I know enough about him to know he's a crackpot magnet.

He claims that all of physics since Planck is wrong --I was patient enough to watch one of his videos or two. That's some time down the drain I'm not getting back.

He certainly doesn't understand the ideas behind renormalisation. I'm not saying quantum physics is problem-free, and there are no consistency issues. There are. But I see nothing of value in trying to substitute renormalisation strategies with WAG numerical games and numerical analysis. And all hand-waving. That's what he does.

Very similar to what you did by copying and pasting some formulae from a 1968 paper and pulling some numbers from a part of your anatomy, and substituting a logarithmically divergent integral by your wild guess.

Zero value from a scientific POV IMO.

Physics is a ruthless game. I don't know if I'd call him a true heckler, only passionate about directing physics back on a logical path.

Let me ask you a question, do you think string theory has actually provided true scientific reasoning towards unification? Read Lee Smolin, in fact, there's are dozens of scientists who are convincing in arguments, that string theory has been an ultimate failure.

So yes, I'd do the same if I was in his position as a populariser of scientific progression. When a theory isn't a theory, and is not testable or even falsifiable, then yes, we must heckle against the dogma. People like Unzicker is actually doing the world a favor, instead of being duped by scientists who are over-rated to the point that few challenge their belief system. Keep in mind, Witten has spent most of his career on string theory, which makes him bias. A true scientist doesn't overlook other possibilities based on the bias of their own time-consuming endevours.

Reference

The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next https://g.co/kgs/ySLGpD

BTW you have not addressed my rebuttal of your bizarre accusations that I had not created a new equation. Maybe you would be generous enough to find a counterargument. You know which thread I refer to.

joigus · February 21, 2023

29 minutes ago, TheCosmologist said:

Let me ask you a question, do you think string theory has actually provided true scientific reasoning towards unification? Read Lee Smolin, in fact, there's are dozens of scientists who are convincing in arguments, that string theory has been an ultimate failure.

Oh, come on. As Leonard Susskind put it, you can make a pretty good living today out of criticising every new idea that comes up. Beating string theory is beating a dead horse. It sells books too.

There are many other ideas besides string theory. What's his problem with Witten in particular? Do you know anything about string theory?

Your technique for renormalising a divergent logarithmic integral is making up its numerical value. Do I really need to say more?

exchemist · February 21, 2023

6 hours ago, TheCosmologist said:

Physics is a ruthless game. I don't know if I'd call him a true heckler, only passionate about directing physics back on a logical path.

Let me ask you a question, do you think string theory has actually provided true scientific reasoning towards unification? Read Lee Smolin, in fact, there's are dozens of scientists who are convincing in arguments, that string theory has been an ultimate failure.

So yes, I'd do the same if I was in his position as a populariser of scientific progression. When a theory isn't a theory, and is not testable or even falsifiable, then yes, we must heckle against the dogma. People like Unzicker is actually doing the world a favor, instead of being duped by scientists who are over-rated to the point that few challenge their belief system. Keep in mind, Witten has spent most of his career on string theory, which makes him bias. A true scientist doesn't overlook other possibilities based on the bias of their own time-consuming endevours.

Reference

The Trouble With Physics: The Rise of String Theory, The Fall of a Science, and What Comes Next https://g.co/kgs/ySLGpD

BTW you have not addressed my rebuttal of your bizarre accusations that I had not created a new equation. Maybe you would be generous enough to find a counterargument. You know which thread I refer to.

Maybe you can tell us how it is that somebody called Gareth Meredith wrote the paper at this link:

https://www.academia.edu/24475326/The_Larmor_Phenomenon_around_Quasars_as_an_Extension_to_Hawking_Radiation

From your comments on the other thread, I'm not to call you Gareth, apparently. 😁

TheCosmologist · February 22, 2023

No you're not to call me Gareth at all. We are not friends and therefore not on first term name calling, understand?

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Larmor Radiation From Black Holes Under Weak Equivalence Principle and Gravitational Index of Refraction

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