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Is Gravity Complex?


Dubbelosix

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Some straight-forward facts... no time messing about. 

 

1. Quantum gravity in the standard model lacks a universal time given here and what is famously known as the Wheeler deWitt equation, a complete 3 geometry of space pertaining the effects of gravity:

 

[math]\mathbf{H}|\Psi> = 0[/math]

 

But it suffers a not-too-well known problem if a quantum unification with the other forces to be feasible, that is, all fundamental fields have complexified terms, but it turns out the Wheeler deWitt equation is a ''real'' force... real, not in the sense it fundamental, but real in the sense it depends on no complex couplings. From the Wigner-inequality dependent curved Schrodinger equation revealed that there are such two fields

 

[math]\frac{\hbar^2}{2m^2}[\Gamma, \Gamma] =\ \frac{1}{2} <\dot{\psi}|\dot{\psi}>[/math]

 

And decomposing into bra-ket notation we get:

 

[math]|\dot{\psi}_{\mu}>\ = \frac{\hbar}{m}\Gamma_{\mu}[/math]

 

and it's conjugate

 

[math]<\dot{\psi}^{\mu}|\ = \frac{\hbar}{m}\Gamma^{\mu}[/math]

 
 
If time really does exist, do we need complexification? If so its an easy application:
 
 

[math]|\dot{\psi}_{\mu}>\ = \frac{i\hbar}{m}\Gamma_{\mu}[/math]

 

and it's conjugate

 

[math]<\dot{\psi}^{\mu}|\ = \frac{i\hbar}{m}\Gamma^{\mu}[/math]

 

This would mean gravity could be complexified but the real issue here is that gravity is not fundamental, as in say, electromagnetism. The irony that gravity is ''defined'' real as in non-complex, makes it a complex issue further realizing it might not even be a real force from the first principles of relativity. I feel it be likely gravity really is a pseudo force and doesn't have to follow the ordinary rules of quantum fields. 

Edited by Dubbelosix
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I have explained to a friend recently, if you want to understand gravity, forget extra dimensions, you don't work from the top to the bottom, you work from the bottom first, that way if one ever reaches a final theory it should in theory be simpler. Forget the complex additions of string theory, for it is dead. It's no longer a respected theory. 

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I found your thread on "Unifying Temperature Into Hilbert Space Through Geometrizing The Model". That's was not simple maths, but does explain why you think Verlindes ideas may be wrong. The laws of thermodynamics and entropy always apply even in space, do they not. A lot of your equations look very similar to Verlindes from memory, which they should as you are both trying to describe the same thing mathematically. This is the most recent paper Verlindes I can find at the moment. https://arxiv.org/pdf/1611.02269.pdf

Hossenfelder seems to support Verlindes ideas in this paper https://arxiv.org/pdf/1703.01415.pdf also in the following later paper ref reddhift https://arxiv.org/pdf/1803.08683.pdf she adds further credence to Verlindes theories.

 

I don't want to sound big headed but the math really isn't all that hard the Einstein notation is. 

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The Schrodinger equation question in my final exams was one I avoided :).  I have noted you do like your maths, which I can do, but makes my head hurt. A good way of learning for lots of individuals is to develop their own pet theories, as we all do when lacking a viable explanation for something. Current thought supported by a lot of theoretical dreamers all approaching from different angles, is that gravity operates via at least one extra dimension curving the underlying space time. Both time and space are emergent at the quantum level. Your set of equations don't appear to demonstrate this.

 

Taking from your first post " The irony that gravity is ''defined'' real as in non-complex, makes it a complex issue further realizing it might not even be a real force from the first principles of relativity" I am inclined to reword this statement to > The irony that gravity is ''defined'' real as in non-complex, makes it a simple issue further realizing it might not even be a real force from the first principles of QUANTUM MECHANICS".

 

My own pet theory is that science is not a religion, and it is very possible that Einsteins field equations are stretched beyond breaking point with dark matter. They like us operate in space time, and we could just be seeing reflections of the real world. Spooky action was something Einstein suggested for non locality but didn't believe in. It seems modern theories are taking the extra dimensions on board and getting results. 

 

I posted some interesting links above, you have not commented! all of which involve extra dimensions. 

 

Yes, you must start from bottom, not the top, the top was made by intelligent physicists who thought the knew how to unify physics. 

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  • 4 weeks later...

So I've talked myself out of the complex definition of the gravitational field, because it simply isn't needed and doesn't appear from first principles without a smudge factor. On my other thread I defined the equivalent set of terms for the opening post:

 

You will recall that we obtained the velocity squared formula. It might interest you to know that this formula is a covariant derivative away from  curve equation: It not only guides the particle using gravitational waves but it seems to contribute to the four velocity or more accurately, a four-acceleration

 

[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

And that four-acceleration is in fact implemented like:

 

[math]\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\partial \tau} = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]

 

 

To understand this last equation, it is best to view this part
 
[math]\frac{\partial \mathbf{U}^{\lambda}}{\partial \tau}[/math]
 
denoting flat space, and the second term is the correction, with
 
[math]\Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}[/math]
 
yielding what is known as the time operator:
 
[math]\frac{\mathbf{D}}{\mathbf{D}\tau} = \mathbf{U}^{\mu} \nabla_{\mu}[/math]
 
As always, the space derivatives are conventionally replaced by their connections. It can also be noticed from this equation:
 
[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]
 
That it can be separated to find a single solution for the velocity
 
[math]\frac{S(r,t)}{m}\Gamma_{\mu} = \mathbf{U}_{\mu}[/math]
 
Using
 
[math]\frac{\mathbf{D}}{\mathbf{D}\tau} = \mathbf{U}^{\mu} \nabla_{\mu}[/math]
 
we find
 
[math]\frac{S(r,t)}{m}[\nabla^{\mu},\Gamma_{\mu}] = \mathbf{U}_{\mu}\nabla^{\mu} =\frac{\mathbf{D}}{\mathbf{D}\tau} [/math]
 
The four acceleration is found as
 
[math]\frac{S(r,t)\mathbf{U}^{\lambda}}{m}[\nabla^{\mu},\Gamma_{\mu}] = \mathbf{U}^{\lambda}\mathbf{U}_{\mu}\nabla^{\mu} =\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\mathbf{D}\tau } = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]
 
We'll get back to this soon to see how it will define the curve for the Hilbert space. 
Edited by Dubbelosix
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So I've talked myself out of the complex definition of the gravitational field, because it simply isn't needed and doesn't appear from first principles without a smudge factor. On my other thread I defined the equivalent set of terms for the opening post:

 

You will recall that we obtained the velocity squared formula. It might interest you to know that this formula is a covariant derivative away from  curve equation: It not only guides the particle using gravitational waves but it seems to contribute to the four velocity or more accurately, a four-acceleration

 

[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

And that four-acceleration is in fact implemented like:

 

[math]\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\partial \tau} = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]

 

 

To understand this last equation, it is best to view this part
 
[math]\frac{\partial \mathbf{U}^{\lambda}}{\partial \tau}[/math]
 
denoting flat space, and the second term is the correction, with
 
[math]\Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}[/math]
 
yielding what is known as the time operator:
 
[math]\frac{\mathbf{D}}{\mathbf{D}\tau} = \mathbf{U}^{\mu} \nabla_{\mu}[/math]
 
As always, the space derivatives are conventionally replaced by their connections. It can also be noticed from this equation:
 
[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]
 
That it can be separated to find a single solution for the velocity
 
[math]\frac{S(r,t)}{m}\Gamma_{\mu} = \mathbf{U}_{\mu}[/math]
 
Using
 
[math]\frac{\mathbf{D}}{\mathbf{D}\tau} = \mathbf{U}^{\mu} \nabla_{\mu}[/math]
 
we find
 
[math]\frac{S(r,t)}{m}[\nabla^{\mu},\Gamma_{\mu}] = \mathbf{U}_{\mu}\nabla^{\mu} =\frac{\mathbf{D}}{\mathbf{D}\tau} [/math]
 
The four acceleration is found as
 
[math]\frac{S(r,t)\mathbf{U}^{\lambda}}{m}[\nabla^{\mu},\Gamma_{\mu}] = \mathbf{U}^{\lambda}\mathbf{U}_{\mu}\nabla^{\mu} =\frac{\mathbf{D}\mathbf{U}^{\lambda}}{\mathbf{D}\tau } = \frac{\partial \mathbf{U}^{\lambda}}{\partial \tau} + \Gamma^{\lambda}_{\mu \nu}\mathbf{U}^{\mu}\mathbf{U}^{\nu}[/math]
 
We'll get back to this soon to see how it will define the curve for the Hilbert space. 

 

 

 

So let's define the curve - we have defined it in another approach which served as the original toy model, which in the Hilbert space is defined by the wave function:

 

[math]\frac{ds}{dt} = \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]

 

We then retrieve the definition of the square of the velocity from the work cited above:

 

[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

This means it is the same thing as saying:

 

[math](\frac{ds}{dt})^2 = \frac{ds}{dt} \cdot \frac{ds}{dt} =\ <\dot{\psi}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

Distribution of the mass tensor could suffice from here and it would yield a similar result from a previous result approach concerning the mass tensor:

 

[math]\hat{H}^{\mu \nu} =\ <\dot{\psi}|M^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} M^{\mu \nu}[/math]

 

This can be considered as an approach to a quantum theory of gravity (which could easily remain classical) - the last equation has dimensions of energy. The stress energy tensor is related to Einstein's tensor as

 

[math]G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}[/math]

 

And the mass tensor is related to the stress energy as

 

[math]\int\ T^{\mu \nu}\ dV = M^{\mu \nu}c^2[/math]

 

So it's also true that we can define a mass density tensor through the stress energy relationship:

 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} \rho^{\mu \nu}[/math]

 

We can determine a wave equation for the propagation of the mass depends on the gravitational field:

 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} \Box \phi^{\mu \nu}[/math]

 

In which [math]\partial^{\mu}\partial_{\mu} = \Box[/math] is the d'Alembertian wave operator (which involves time derivatives as well as the usual space derivatives). The density tensor is

 

[math]\rho^{\mu \nu} = \Box \phi^{\mu \nu}[/math]

 

dimensions of

 

[math] \Box \phi = \frac{G \times mass}{volume} = G \rho[/math]

 

So we can see there is in fact an additional value of the gravitational constant which is attached to the mass density term. But for conventional terms, we can set [math]G = 1[/math] and happily satisfy the approach we are using. 

 

The partial derivatives can in normal form be replaced by the gravitational connections of the field: 

 

[math]\Box = g^{\mu \nu}\nabla_\mu\nabla_\nu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu)[/math]

 

The rules and crucial identities for the operator in curved space is

 
[math]\Box \phi = g^{\mu \nu}\nabla_{\mu}\nabla_{\nu} \phi = g^{\mu \nu}\nabla_{\mu}(\partial_{\nu}\phi)[/math]
 
[math]= g^{\mu \nu}(\partial_{\mu}\partial_{\nu} \phi - \Gamma^{\sigma}_{\mu \nu} \partial_{\sigma}\phi) = (\partial_{\mu}\partial^{\mu} - g^{\mu \nu}\Gamma^{\sigma}_{\mu \nu}\partial_{\sigma})\phi[/math]
 
This then also allows us to write:
 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ =  \frac{\phi^{\mu \nu}}{\sqrt{-g}}\partial_\mu \mathbf{U}^{\mu}(\sqrt{-g}\partial^{\mu}\mathbf{U}_{\mu})[/math]

 
 
 
Edited by Dubbelosix
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Of course, you can create a contravariant form of the Einstein field equations from this by:

 

[math]G^{\mu \nu} = \frac{8 \pi G}{c^4}T^{\mu \nu} = \frac{8 \pi G}{c^4}\ <\dot{\psi}|\rho^{\mu \nu}|\dot{\psi}>\ = \frac{8 \pi G}{c^4}\  <\psi|\mathbf{H}^{\mu \nu}|\psi>[/math]

 

As an ordinary solution to the Schrodinger equation [would imply] complexification on the right hand side of 

 

[math]\frac{1}{\hbar} \mathbf{H}|\psi>\ = i|\dot{\psi}>[/math]

 

But as always, I cannot find vindication for it appearing in a wave theory of gravity. We will use the identity operator:

 

[math]|\psi><\psi| = \mathbf{I}[/math]

 

we get

 

[math]G^{\mu \nu}|\psi>\ =\ \frac{8 \pi G}{c^4}T^{\mu \nu}|\psi>\ =\  \frac{8 \pi G}{c^4}\mathbf{H}^{\mu \nu}|\psi>[/math]

 

Using a Hamiltonian density. rearranging the equation

 

[math]\frac{c^4}{8 \pi G}\mathbf{G}^{\mu \nu} |\psi>\ =\  T^{\mu \nu}|\psi>\ =\   \mathbf{H}^{\mu \nu}|\psi>[/math]

Edited by Dubbelosix
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Even physicists working for their pay grade, do things like this so they can explore their own theory with more depth. While there is no conclusions, it should be pretty obvious that we have derived a form of the stress energy from our theory and we have also explored wave solutions. 

Edited by Dubbelosix
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So let's define the curve - we have defined it in another approach which served as the original toy model, which in the Hilbert space is defined by the wave function:

 

[math]\frac{ds}{dt} = \sqrt{<\dot{\psi}|\dot{\psi}>}[/math]

 

We then retrieve the definition of the square of the velocity from the work cited above:

 

[math](\frac{S(r,t)}{m})^2[\Gamma_{\mu}, \Gamma^{\mu}] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

This means it is the same thing as saying:

 

[math](\frac{ds}{dt})^2 = \frac{ds}{dt} \cdot \frac{ds}{dt} =\ <\dot{\psi}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} [/math]

 

Distribution of the mass tensor could suffice from here and it would yield a similar result from a previous result approach concerning the mass tensor:

 

[math]\hat{H}^{\mu \nu} =\ <\dot{\psi}|M^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} M^{\mu \nu}[/math]

 

This can be considered as an approach to a quantum theory of gravity (which could easily remain classical) - the last equation has dimensions of energy. The stress energy tensor is related to Einstein's tensor as

 

[math]G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}[/math]

 

And the mass tensor is related to the stress energy as

 

[math]\int\ T^{\mu \nu}\ dV = M^{\mu \nu}c^2[/math]

 

So it's also true that we can define a mass density tensor through the stress energy relationship:

 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} \rho^{\mu \nu}[/math]

 

We can determine a wave equation for the propagation of the mass depends on the gravitational field:

 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu} \Box \phi^{\mu \nu}[/math]

 

In which [math]\partial^{\mu}\partial_{\mu} = \Box[/math] is the d'Alembertian wave operator (which involves time derivatives as well as the usual space derivatives). The density tensor is

 

[math]\rho^{\mu \nu} = \Box \phi^{\mu \nu}[/math]

 

dimensions of

 

[math] \Box \phi = \frac{G \times mass}{volume} = G \rho[/math]

 

So we can see there is in fact an additional value of the gravitational constant which is attached to the mass density term. But for conventional terms, we can set [math]G = 1[/math] and happily satisfy the approach we are using. 

 

The partial derivatives can in normal form be replaced by the gravitational connections of the field: 

 

[math]\Box = g^{\mu \nu}\nabla_\mu\nabla_\nu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu)[/math]

 

The rules and crucial identities for the operator in curved space is

 
[math]\Box \phi = g^{\mu \nu}\nabla_{\mu}\nabla_{\nu} \phi = g^{\mu \nu}\nabla_{\mu}(\partial_{\nu}\phi)[/math]
 
[math]= g^{\mu \nu}(\partial_{\mu}\partial_{\nu} \phi - \Gamma^{\sigma}_{\mu \nu} \partial_{\sigma}\phi) = (\partial_{\mu}\partial^{\mu} - g^{\mu \nu}\Gamma^{\sigma}_{\mu \nu}\partial_{\sigma})\phi[/math]
 
This then also allows us to write:
 

[math]T^{\mu \nu} =\ <\dot{\psi}|\mathbf{M}^{\mu \nu}|\dot{\psi}>\ =  \frac{\phi^{\mu \nu}}{\sqrt{-g}}\partial_\mu \mathbf{U}^{\mu}(\sqrt{-g}\partial^{\mu}\mathbf{U}_{\mu})[/math]

 
 
 

 

 

 

[math]T^{\mu \nu} =\ <\dot{\psi}|\rho^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu}[g^{\mu \nu} \nabla_{\nu}(\partial_{\nu}\phi^{\mu \nu})] = \mathbf{U}_{\mu}\mathbf{U}^{\mu}(g^{\mu \nu} [\nabla_{\nu},\nabla_{\nu}]\phi^{\mu \nu})[/math]

 
[math] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} (\partial_{\mu}\partial^{\mu} - g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu} \partial_{\sigma})\phi^{\mu \nu} = g^{\mu \nu}\mathbf{U}_{\mu}\mathbf{U}^{\mu} \nabla_{\mu}(\partial_{\nu} \phi ) = \mathbf{U}_{\mu}\mathbf{U}^{\mu}[\partial_{\mu}\partial^{\mu}\phi - g^{\mu \nu}\Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}]\phi^{\mu \nu}[/math]
 
 
We can also identify the curvature tensor as
 
[math][\nabla_{\mu}, \nabla_{\nu}] = \mathbf{R}_{\mu \nu}[/math]
 
 
This as I explained in my first essay to the gravitational research foundation, is that covariant derivatives, just from pure definition, do not generally commutate. The case above, since they have different subscripts, they will follow a non-commutative algebra. This is the first case I have came across naturally that imposed a possible ''smearing'' of gravity into the quantum phase space. I thought about it for a long time and this is why I decided if the wave function was a gravitational wave, then this would mean there are some deviations of how we understand gravity into the obscurity of quantum mechanics - such as gravitational pilot waves predict they may be responsible for the quantum leap of an electron due to a perturbation, or may even explain why we cannot see the quantum wave function, only measure its effects and speculations within the framework of quantum mechanics. Under a pilot wave model that is gravitational in nature, the most surprising thing I came to realize was that the quantum gravitational wave must be capable of being ''in phase';' with matter, to explain how a gravitational wave could even guide a particle, since they are expected in classical physics to move at light speed. I almost gave up on the idea of gravitational waves, but found some ways out of complicated brick walls. 
Edited by Dubbelosix
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Let's now talk a bit about that curvature tensor. But first, what do I mean when I avoid a quantization of an aether field? First of all, there can be no motion associated to a true relativistic aether - ie. it cannot be made from something more fundamental. Some have went as far to say it even violates relativity from first principles to think there can be any particle associated to an aether theory. The one we chose was a gravitational aether, where even the absence of a graviton still does not mean ''nothing.'' To try and prove this concept which while not new, I independently came to the conclusion it [may] be possible to unify gravitational waves as being manifestly the wave function itself. This tied in with the experiment known as the double slit experiment in which deBroglie showed that even when a particle is measured. there is still an interference band suggesting there is a so-called ''empty wave,'' that travels through the other slit. When not being watched, the ''gravitational wave function'' appears to have more space to spread out and this is because no measurement has been made on the localization of the particle. So long as the particle is allowed to smear out in space, and assuming the empty wave is in entanglement with the particle itself, then the wave should also do the same. The entanglement ensures that both the wave guiding the particle and the empty wave guiding itself through the other slit are simultaneously affected when the localization of the particle is determined. Again, these perturbations are expected to be far to small to detect with current technology as a true gravitational wave.

 

If the gravitational wave additionally has no quantization associated to it, then it would be the closest thing to the ''empty wave'' I can imagine; but remember, empty does not mean it is technically ''nothing'' but instead, implies it has no test particle associated to it. My personal opinion on this matter was formed very early on from only the first principles of relativity, that is, gravity is not even a force by the true quantum definition - which maybe surprisingly makes gravity on a different league to the rest of the so-called, fundamental forces of nature. Because of this, I became sceptical of gravity being a quantized gauge Boson of spin-2 - after all, gravity is a pseudo force, it does not technically require a graviton mediator.

The Christoffel symbol can be 'loosely' though of as being analogous to a force in Newtons equations (where mass has been set to 1 to denote that it is a constant in this formulation):

[math]\Gamma = \frac{1}{2} \frac{\partial g_{00}}{\partial x}[/math]

Newtonian formulation of this acceleration is

[math]F = -\frac{\partial \phi}{\partial x}[/math]

However as mentioned, the gravitational force is not actually a true definition of a force as we come to expect say, in the proposed fundamental fields of nature, which are inherently complex (when quantum gravity is not) and that require quantization of field particles acting as mediators of the force (something which gravity is expected to use to form the unification theory in the opinions of many scientists). It's actually a crucial component of many theories, most notably string theory.

Gravity is a pseudo force and can be understood in the following (neat) and (concise and short) way:

[math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math]

where

[math]\Gamma^{\mu}_{\nu \lambda} = \frac{\partial x^{\mu}}{\partial \eta^{a}}\frac{\partial^2 \eta^a}{\partial x^{\nu}\partial x^{\lambda}}[/math]

or more compactly

[math]\Gamma^{\mu}_{\nu \lambda} = J^{\mu}_{a} \partial_{\nu} J^{a}_{\lambda} = J^{\mu}_{a} \partial_{\lambda} J^{a}_{\nu} \equiv J^{\nu}_{a} J^{a}_{\nu \lambda}[/math]

which represents a pseudo force for gravity which makes it in the same league as the Coriolis and the Centrifugal forces in which it is not customary to quantize such fields. Again, from first principles this was never something required within a realistic model of particle physics.

This is why, in this model, I investigated gravity in the context of the phase space using Von Neumann-like operators. The commutation properties smears the classical vacuum into the quantum - attempts to measure gravity at the atomic and quantum scales are on-going. There have been interesting investigations into trying to attempt to find the influence of gravity on superpositioned rubidium atoms - this experiment shows that gravity (as accelerations in the superpositioned states of two interacting clouds) pulled on the clouds at the same rate as other clouds at different energy levels. As you can see, attempting to measure the actual effects of gravity on the quantum scale, is difficult because it appears to be so weak, at least on the scale we can probe spacetime. I say this, because some theories of gravity hold that it may only become quantum mechanically-significant at Planck scales. Later Wheeler created a concept of this in his quantum vacuum foam hypothesis, which was actually an early theory about the existence of quantum fluctuations.

In this work, I will lay out the foundation to this toy model - we will start off by making clear that this model uses (but does not need to depend) on a possible non-trivial spacetime uncertainty principle that is predicted by both string theory and quantum loop gravity. I link this dynamically to gravity in the following way by using the antisymmetric curvature tensor

[math]R_{\mu \nu} = [\nabla_x\nabla_0 - \nabla_0 \nabla_x] \geq (g^{\mu \nu})^{-1} \frac{1}{\ell^2}[/math]

 

Contracted version

 

[math]\mathbf{R} = g^{\mu \nu} R_{\mu \nu} \geq \frac{1}{\ell^2}[/math]

And there may be some indication (see http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf) that there are eigenvalues attached to the definition of the Planck length in the phase space,

[math]R_{\mu \nu} = [\nabla_{\mu}\nabla_{\nu} - \nabla_{\nu} \nabla_\mu] \geq (g_{\mu \nu})^{-1} \frac{(\ell^{2})^{-1}}{\sum_i \sqrt{n_i(n_i + 1)}}[/math]

 

This form above is simply calculated in the normal way from the Christoffel symbols which form two connections of the gravitational field, still following of course, the commutation laws,

[math][\nabla_i, \nabla_j] = (\partial_i + \Gamma_i)(\partial_j + \Gamma_j) - (\partial_j + \Gamma_j)(\partial_i + \Gamma_i)[/math]

[math]= (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i\Gamma_j) - (\partial_j \partial_i + \partial_j\Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)[/math]

[math]= -\partial_i, \Gamma_j + \partial_j, \Gamma_i + [\Gamma_i, \Gamma_j][/math]

 

In which the last term (the two gamma matrices) are made up from the derivatives of space and so they too obey non-commutation laws.

Edited by Dubbelosix
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The whole point of this, exists on whether in principle the physics holds up - for there to be a true spacetime anti-commutation requires also that not only space as we know it is an observable, but also time itself! In fact, its ironic that while time is treated exactly like space unified into a single continuum or metric, when three of those components are generally considered observables when time is not.

I want people to consider geometry as an observable (which it is under the treatment of general relativity) and for a full transition into quantum theory would require that geometry be described by Hermitian matrices.

There appears to be slight change in notation when considering the Hermitian Ricci Curvature and you can follow that in the first reference. You don't need to do anything fancy, we just impose there exists a Hermitian manifold - which is the complex definition of a Riemannian manifold and so you can also have the complex definition of the Ricci curvature. This means at least in principle, the space time non-commutivity can still remain since it is known that two Hermitian operators may not commute. It also means geometry can in principle be described as an observable which I feel is important for the unification theories that involve ''measurable lengths.''

Moving on, we now have a possible application of the spacetime uncertainty principle, in a new kind of form. We showed at the very start of these investigations, how you might interpret it as two connections of the gravitational field, one with spatial derivatives and another with time.  It also exists that there can be non-commuting Hermitian operators (since space and time are treated as observables [x, ct] ≠ 0) - also keep in mind, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle): In gravitationally-warped spacetime the motion through time manifests as motion through space.

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[math]T^{\mu \nu} =\ <\dot{\psi}|\rho^{\mu \nu}|\dot{\psi}>\ = \mathbf{U}_{\mu}\mathbf{U}^{\mu}[g^{\mu \nu} \nabla_{\nu}(\partial_{\nu}\phi^{\mu \nu})] = \mathbf{U}_{\mu}\mathbf{U}^{\mu}(g^{\mu \nu} [\nabla_{\nu},\nabla_{\nu}]\phi^{\mu \nu})[/math]

 
[math] = \mathbf{U}_{\mu}\mathbf{U}^{\mu} (\partial_{\mu}\partial^{\mu} - g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu} \partial_{\sigma})\phi^{\mu \nu} = g^{\mu \nu}\mathbf{U}_{\mu}\mathbf{U}^{\mu} \nabla_{\mu}(\partial_{\nu} \phi ) = \mathbf{U}_{\mu}\mathbf{U}^{\mu}[\partial_{\mu}\partial^{\mu}\phi - g^{\mu \nu}\Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}]\phi^{\mu \nu}[/math]
 
 
We can also identify the curvature tensor as
 
[math][\nabla_{\mu}, \nabla_{\nu}] = \mathbf{R}_{\mu \nu}[/math]
 
 
This as I explained in my first essay to the gravitational research foundation, is that covariant derivatives, just from pure definition, do not generally commutate. The case above, since they have different subscripts, they will follow a non-commutative algebra. This is the first case I have came across naturally that imposed a possible ''smearing'' of gravity into the quantum phase space. I thought about it for a long time and this is why I decided if the wave function was a gravitational wave, then this would mean there are some deviations of how we understand gravity into the obscurity of quantum mechanics - such as gravitational pilot waves predict they may be responsible for the quantum leap of an electron due to a perturbation, or may even explain why we cannot see the quantum wave function, only measure its effects and speculations within the framework of quantum mechanics. Under a pilot wave model that is gravitational in nature, the most surprising thing I came to realize was that the quantum gravitational wave must be capable of being ''in phase';' with matter, to explain how a gravitational wave could even guide a particle, since they are expected in classical physics to move at light speed. I almost gave up on the idea of gravitational waves, but found some ways out of complicated brick walls. 

 

 

In the L2 Cauchy Schwarz space inequalities, it was possible to form bounds on the connections. It showed that the quantum version of gravity has twice the upper bound of that predicted from classical physics. We showed how the curvature tensor is related to the anti-commutation in the phase space:

 

[math]\mathbf{R}_{\mu \nu} = [\nabla_{\mu}\nabla_{\nu} - \nabla_{\nu} \nabla_{\mu}] \geq (g^{\mu \nu})^{-1} \frac{1}{\ell^2}[/math]

 

Contracted version

 

[math]\mathbf{R} = g^{\mu \nu} \mathbf{R}_{\mu \nu} \geq \frac{1}{\ell^2}[/math]

 

It also means for the first equation in the quote, that it can also be expressed as an inequality:

 

[math]T^{\mu \nu} = \mathbf{U}_{\mu}\mathbf{U}^{\mu}(g^{\mu \nu} [\nabla_{\nu},\nabla_{\nu}]\phi^{\mu \nu}) = \mathbf{U}_{\mu}\mathbf{U}^{\mu}(g^{\mu \nu}\mathbf{R}_{\mu \nu}\phi^{\mu \nu}) \geq (g_{\mu \nu})^{-1} \frac{1}{\ell^2}[/math]

 

It might also interest you to know, that not only has the spacetime uncertainty shown up within the models of both string theory and quantum loop gravity, is a suggestion that the prediction is possibly non-trivial and equally has been supported from scattering experiments.

Edited by Dubbelosix
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The contracted form of the last equation here is simply

 

[math]\mathbf{T} = g_{\mu \nu}\ T^{\mu \nu} = \mathbf{U}_{\mu}\mathbf{U}^{\mu}( [\nabla_{\nu},\nabla_{\nu}]\phi^{\mu \nu}) = \mathbf{U}_{\mu}\mathbf{U}^{\mu}(\mathbf{R}_{\mu \nu}\phi^{\mu \nu}) \geq (\frac{1}{\ell^2}) \mathbf{U}_{\mu}\mathbf{U}^{\mu}\phi[/math]

 

The contracted form is boring to some people, but in the special case of dimensions with [math]D = 4[/math] gives the contracted Einstein tensor becomes the negative of the Ricci tensor:

[math]\mathbf{G} = \frac{2 - D}{2}\ \mathbf{R} = \kappa \mathbf{T}[/math]

 

It does show for the equation we derived at the beginning of the post has some equivalent forms we can derive:

 

[math]\mathbf{G} = \kappa\ \mathbf{T} = \kappa\ g_{\mu \nu}\ T^{\mu \nu} \equiv \frac{2 - D}{2}\ \mathbf{R} \geq \frac{8 \pi G}{c^4}\ (\frac{1}{\ell^2_{planck}}) \mathbf{U}_{\mu}\mathbf{U}^{\mu}\ \phi[/math]

Edited by Dubbelosix
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There's loads of different approaches I could take to describe how the particle becomes entangled with the empty wave, but sometimes a logical argument alone can suffice. Below, I use some notation some people may not be so acquainted with so I give it some explanation. It does involve the concepts of entanglement.

 

[math]H(\Psi) = H(\psi_{empty}|\psi_{occup}) + H(\psi_{empty} : \psi_{occup})[/math]

Here, [math]H(\psi_{empty}|\psi_{occup})[/math] is the entropy in [math]\psi_{empty}[/math] (after) having measured the system that became correlated in the occupied wave function [math]\psi_{occup}[/math].

Further, [math]H (\psi_{empty} : \psi_{occup})[/math] is the information gained about [math]\psi_{empty}[/math] by measuring the occuped state [math]\psi_{occup}[/math]. This reveals a conservation in the second law not so dismilar to the Shannon entropy in terms of information theory. Likewise from information theory, we can talk about the empty wave and occupied wave in terms of the upper bound of correlation which is given as:

[math]H(\Psi) = H(\rho_{\psi_{empty}}) + H(\rho_{\psi_{occup}}) - H(\rho_{\psi_{empty,\ occup}})[/math]

Which is a quantum discord attempt to unify an empty wave with an occupied wave.

Edited by Dubbelosix
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