A Geometric Interpretation Of The Gravitational Field Strength In 3D

[Dubbelosix]

 

The equation as topic of interpretation and investigation, comes from the current topic I have been reading vast literature on, gravitomagnetism - in the following equation, we specifically have a triple scalar product;

 

[math]\nabla \cdot (\mathbf{H} \times (\frac{\phi}{c^2}))  =  (\frac{\phi}{c^2}) \cdot (\nabla \times \mathbf{H})  = \frac{m}{r^3} = \frac{1}{2c}\frac{\mathbf{J} - 3(\mathbf{J} \cdot \frac{\mathbf{r}}{r}) \frac{\mathbf{r}}{r}}{r^4} \approx \nabla \cdot \mathbf{E} =  4 \pi \rho[/math]

 

A cross product involved in the scalar product of course, produces another scalar, but it does possess a geometric meaning as well. The volume of space of a parallelepipe is:

 

[math]V = |a \times b|\ |c|\ |\cos\ \theta| = (a \times b )\cdot c[/math]

 

Looking at it in geometric terms like this, should not be so strange if you consider geometric approaches have existed in theoretical physics, one particular one on mind is the geometrization of Einstein'g relativistic equation:

 

[math]E^2 = p^2c^2 + m^2c^4[/math]

 

Which theoretically can be seen equivalent to a geometric interpretation under the normal Pythagorean relationship:

 

[math]a^2 = b^2 + c^2[/math]

 

What's interesting is that a parallelepipe is a three dimensional figure formed by six parallelograms - named the rhomboid. How do you visualize this in three dimensional space? It is actually pretty difficult. 

 

I did come across a very helpful site to visualize this; you can direct the graph (see last reference), when you make the length of the vectors zero (the arrows), you find that [math](a \times b ) \cdot c = 0[/math] at the point of origin. Geometrically, especially within relativity, I predict certain solutions from the triple scalar product equation could be at least in theory, describable under Parallel Transport. I in fact worked on a topic related to this for my first paper to the gravitational research foundation, in which I derived an equation satisfying the Schrodinger equation for Parallel Transport in a curved spacetime interval (see also third reference):

 

[math]\nabla_n|\dot{\psi}>\ = \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2|\ (\frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i})|\psi>\ dqdp \geq \int\ \frac{1}{\hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math]

 

Which had been derived under fundamental assumptions concerning the uncertainty of the system, due to the presence of the Wigner function ~ giving theoretically, a quantum solution to non-commutative geometries. 

 

When [math]|W(q,p)^2|[/math] appears, we intend the quantum uncertainty in [math]~\hbar[/math]

 

This uncertainty is known as the Wigner function, found here to have implication with ''quantum gravity'' using non-commutation rules. 

 

it isn't hard to see why these are solutions similar to

 the ordinary Schrodinger equation of the form:

 

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

 

Showing also that it was bound by the commutation rules applied (surprisingly and remarkably concisely) as:

 

[math]\nabla_n \dot{\gamma}(t) \equiv\ min\ g^{ij}\sqrt{<\dot{\psi}|[\nabla_i,\nabla_j]|\dot{\psi}>}[/math]

 

 

REFERENCES:

 

https://en.wikipedia.org/wiki/Parallel_transport

 

https://hilbertspace.quora.com/

 

https://mathinsight.org/scalar_triple_product

Edited October 22, 2018 by Dubbelosix

[Dubbelosix]

A bit more on this gravity stuff? I feel like getting back into the non-linear Hilbert space I had been investigating for the first essay. 

 

[math]\mathbf{D}V^n = dV^n + \Gamma^n_{mr}V^rdx^m[/math]

 

For a smooth manifold, the tangent bundle of [math]M[/math] is the affine connection, itself distinguishes a class of curves called affine geodesics, (Kobayashi and Nomizu). The curve is given as:

 

[math]\Gamma(\gamma)\dot{\gamma}(s) = \dot{\gamma}(t)[/math]

 

and the derivative yields the ordinary notation, which featured in my previous work on a non-linear Schrodinger equation

 

[math]\nabla_{\gamma(t)} \dot{\gamma} = 0[/math]

 

Where [math]\Gamma[/math] is the usual gravitational field (connection) and [math]\nabla[/math] is the Covariant derivative. 

 

[math]\nabla_{\gamma(t)} \dot{\gamma}(t) \equiv\ min\ g^{ij}\sqrt{<\dot{\psi}|[\nabla_i,\nabla_j]|\dot{\psi}>}[/math]

 

The previous equation is a curve-distance equation, defining the minimum of the geodesic. The product of commutators [math][\nabla_i,\nabla_j][/math] not only has intrinsic uncertainty attached to the spacetime, but as is well-known, they also form the Riemann tensor [math]R_{ij}[/math]. It simply takes form as

 

[math]R_{ij} = [\nabla_i,\nabla_j] = [\nabla_i \nabla_j - \nabla_j \nabla_i] \geq g_{ij}\ (\frac{1}{\ell^2})[/math]

 

With [[math]\ell[/math]] a notation for the Planck length. The commutation relationships are calculated the following (usual) way, equivalent to the Riemann curvature tensor:

 

[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]

 

Also with, I found a non-trivial inequality bound identical in form to the quantum bound:

 

[math]<\psi|[\nabla_i, \nabla_j] |\psi>\ =\ <\psi| R_{ij}| \psi>\ \leq 2 \sqrt{|<\nabla^2_i><\nabla^2_j>|}[/math]

 

Concerning the curve equation [math]\nabla_{\gamma(t)}\dot{\gamma} = 0[/math] the product of the wave functions which have lengths of velocity in the Hilbert space is given by:

 

[math]K_BT = \frac{1}{2}(\frac{dx^{\mu}}{d\tau} \cdot \frac{dx^{\mu}}{d\tau}) \equiv\ \frac{1}{2}<\dot{\psi}|\dot{\psi}>[/math]

 

... But (maybe) more fundamentally, from the same geometrized Hilbert space, we have found a definition of ''how temperature arises'' within the theory. Certainly motion is included for those wave functions who have a velocity and time derivative in the length of the Hilbert space... and motion of atomic and molecular systems is the reigning explanation to how objects may heat up. 

Edited October 25, 2018 by Dubbelosix