Gravitational Corrections On The Ricci Flow

[Dubbelosix]

 

 

The Einstein equations, namely the Einstein tensor is related to the curvature scalar as

 

G^μν = 1/2g^μν R - R^μν

 

Note the almost identical nature found in the flow equation

 

∂_0 g^μν = - 1/2 R^μν + 1/n (R g^μv)

 

[1] - If this was to follow a geometric power series would be an interesting approach to look at but from this I can see a distinct possible series just from looking at it numerically,

 

1/2 > 1 > 2

 

It starts at one half, then to the whole number, meaning it has spanned its own number in one jump. The second jump gets to the whole number 2, meaning it has doubled the jump it had done previously. Continuing that pattern:

 

1/2 > 1 > 2 > 4 > 8 > 16... +

 

So that the corrections arise, with the second form in terms of an identity matrix M, following another identity matrix in higher arrranges

 

∂_0 g^μν = - 1/2 R^μν + M(1/1) (R g^μv) + M(2/2) (R g^μv) +.....

 

Higher powers then gives

 

M(4/2) (R g^μv) + M(8/2) (R g^μv)

 

An interesting thing to note that 8/2 naturally is... The identity matrix of g^μv g_μv = 4. So even the matrix in this higher power can be rewritten as

 

∂_0 g^μν = - 1/2 R^μν + M(g^μv g_μv) (R g^μv)

 

(2)

 

But you cannot do away with the correction of - 1/2 R^μν because this is foolish when you understand that the first term and the second term always define the four dimensional statements of reality when it gives rise to curvature. This is why we cannot do away with time, whether you class is it as simply innate to the changes of things, because physics has often oversimplified itself, such as the holographic principle, as inspiring as it has been since Dr. d'Hooft introduced it. Though by no means, we are to take previous attempts to oversimplify physics as wrong, as it can show us more about where we have gone wrong. This is how science propagates.

 

The corrections simply means that there are higher powers to gravitational corrections, naturally, but being a pseudo force we need to be careful how we interpret these higher powers as it is a product of the metric itself but not by any specific particle alone since all particles contribute to gravity through the stress energy. The cannot be stressed enough from my own statements and a few scientists even before me. What still interests me is that Einstein knew his equations where for pseudo fields but he was romanced it seems from the statements of particles and fields that he forget his own first principles, and those principles being rooted in how we understand pseudo fields and why we do not quantize them. The higher corrections on the Ricci flow however can play a type of antigravity as found in the study of the quark star, which was then introduced as a cosmological Principle, not so much as it had to be rated to the quark star, though such approaches seems natural to avoid singularities, but it seems the whole picture is not fully understood anyway because singularities have romanced even some of the greatest scientists who took mathematics far too seriously. Mathematics is just a language to explain physical effects, not all mathematics is even observable. Limits in nature however, do exist but the limits are not absolute. I believe this dichotomy needs to be broken before actual progress can be made in the direction we should have believed in. The antigravity however, is not an antigravity at all, but has to do with stresses and pressures inside the system, just as I hold today, that the laws of physics do not vanish inside of black holes (1), but are a type of condensate, just as a pre big bang model is also a condensate model. The reason why I believe this strongly was from a justification of a thought experiment in which larger condensates give off less radiation than smaller ones, which would unify Hawking radiation into the gravitational system. The system is not bounded entirely, photons can escape from black holes, meaning we will have to resort to a new model. The best so far I have read if the gravitational aether in which photons do experience length contractions, but the photon cannot just simply reach a zero speed limit. This even defies quantum mechanics and creates a number of problems. If photons can escape then it would solve the information paradox literally over night.

 

Footnotes

 

(1) -... No more in fact, that physics vanishes inside an atom, for if the wave nature did not exist, the electons would deplete its energy very fast by falling towards the centre. The centre of an atom is no more a singularity in this case either, but these principles can explain why for instance, in-falling matter inside a black hole is stabalized in a similar way by using wave mechanics. This is why the quark star idea should be taken a bit more seriously. The tendancy of classical mechanics allowing something to crash to a centre neither means it has fallen towards a singularity, its just that the atom would radiate much quicker than what classical mechanics had predicted. This is why there are no ultimately stable black holes but in theory you can make a stable black hole so long as it is being observed or interacted, in the most exact way possible. Though even then, I argue the atom will eventually evaporate due to similar effects known as the anti zeno effect. Until we can absolutely stablize a particle so it cannot radiate in the way we expect, can we rule out absolute refrigerators in science. It just seems it is highly unlikely.

 

(2) - An interesting thing to note is that numeracy is just the study of numbers, all equations boil down to numbers. It's a misunderstanding of many today about how important numeracy is, with some even classifying it as a pseudoscience. Numeracy is just the study of numbers and how they are to be interpreted, as strange as that may sound. But it's not all that strange if the study of numbers can tell us something about the underlying principles of physics and how the universe works.

 

Extra questions for the footnotes, "is it possible the negative values will boil down to a minimal length in physics? And to add to this, does it mean that a Planck length should be taken more seriously than the Planck mass as some fundamental system, can you have one without the other? And to add to that, the higher powers must eventually reach the Planck upper limit of the gravitational force? " I ask this question because the Planck mass is just a quantity, but it does not define the volumetric meaning, a particle with a Planck mass truly would be a micro black hole. The Planck mass is almost well defined, but it depends on approximations. The human eyelash is approximately made of a Planck mass, and this has always interested me for these questions posted today, because great scientists have taken the Planck particle as something fundamental to nature when really it seems the mass charge is fundamental (ie. As shown from Weinberg mass formula which can predict a wide range of particles on the standard model, but since we have not detected any particle with a Planck mass, it seems Arun and Sivaram may be correct in thinking it is too fundamental even for natural fields. If there is a range of Gravitational corrections in this sense, this absolute value of upper limit appears to be only a special correction case found in Einsteins (G/c^4) approach and those who tried to help unify the physics....

[Dubbelosix]

The four dimensionality of the flow was shown from my previous work as the following.

 

∂_0 g_μν = - k (2 ∇ T_μν - ∂_0 g_μν T)

 

 

Defining ∇ as a three space gradient operator then it can be more compactly written as

 

∂_0 g_μν(R, t) = - 2 k □² T_μν

 

= - 2k (∂²_x T + ∂²_y T + ∂²_z T - ∂²_0 T)

 

The box operator as related to the Ricci curvature is given in literature as

 

g^μv R_μv = g^μv ∇_μ∇_v

 

= g^μv ∇_μ (∂_v) = g^μv (∂_μ∂_v + Γ^σ_μv ∂_σ)

 

The basic result for the flow in curved space is the tensor relationships

 

∇_0 g_μv(R, t) = - 2 R_μv = - 2(∂_μ∂_v + Γ^σ_μv ∂_σ)

 

= - 2(∂_μ + Γ_μ)(∂_v + Γ_v)

 

= - 2(∂_μ∂_v + ∂_μΓ_v + Γ_μ ∂_v + Γ_μ Γ_v)

 

In rotating curved spaces capable of making space flow in a particular direction involves not setting Γ_μ Γ_v = 0 which involves dynamic torsion in the metric. There is no special reason in general relativity to set this zero in any realistic model of spacetime preserving the Poincare symmetries. The connections may even apply to its own uncertainty relationship between space and time.

Edited February 20, 2020 by Dubbelosix

[Dubbelosix]

The Ricci flow is the heat equation of the Riemannian Manifold. Following those principles we have

 

G_μν = R_μν - 1/2 g_μν (-kT)

 

∂_0 g_μν(R ) = - k (2 ∂_0 T_μν - ∂_0 g T)

 

This is still under the description of T_μν meaning the equation can further be written as

 

F = 2A^μv (R_μν + D/2 R g_μν + ½ g_μν T)

 

Let's plug all the necessary numerical coefficients in for curiosity, by changing it slightly under the similar representation of

 

F = 2A^μv (R_μν + D/2 R g_μν) + ½ (g_μν g^μνT)

 

(under a new defined Einstein proportionality [variable]) since the upper limit is only a special case, not a true general case.

Edited February 16, 2020 by Dubbelosix