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An Investigation into Temperature and Accelerations (Gravity/Curvature)


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Summary of Investigations into Curves and Accelerations

In our case, we have already noticed before that these ''curves'' are equivalent to accelerations at infinitessimal regions. We linked those curvature dynamics to the temperature in the following way:

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

For the special case of [math]m = 1[/math]. We'll also set the Boltzmann constant to natural units for simplicity in the future. 

It was noted, that the term [math]\frac{ds^{\mu}}{d\tau} \cdot \frac{ds^{\mu}}{d\tau}[/math] is nothing but the square of the metric curve. This too, was a squared metric definition [math]<\dot{\psi}|\dot{\psi}>[/math]. 

The latter expression was encountered in our investigation of possible curvature in the Hilbert space:

[math]\sqrt{<\dot{\psi}|\dot{\psi}>} = \int \int\ |W(q,p)^2| \sqrt{<\psi|\Gamma^2|\psi>}\ dqdp \geq \frac{1}{ \pi \hbar}\sqrt{<\psi|H^2|\psi>}[/math]

Where we make use of the Wigner function. What's interesting is that it is often an unfounded assumption by many that the Hilbert space needs to be flat, or that there is any credible argument against a single system exerting their own curvatures on spacetime (since it is not illegal that particles can couple to their own gravitational fields, see the Newton-Schrodinger equation as an example), or that even infinitessimal regions need to be flat. A good example are the minimum distance candidates, like the Bure Metric, or the Fubini-Study metric. 

The geodesic length was

[math]|\psi_0 - \psi_1| = \sqrt{2 - 2\cos \theta} = 2 sin\frac{\alpha}{2}[/math]

In which you can calculate the angle

[math]\cos \alpha = \frac{<\psi_0|\psi_1> + <\psi_1|\psi_0>}{2}[/math]

In which an inequality holds 

[math]\cos \alpha \leq |<\psi_0|\psi_1>|[/math]

and so that the length of the curve on some unit sphere was

[math]\arccos |<\psi_0|\psi_1>|[/math]

and once again

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

We also constructed the Mandelstam-Tamm inequalities, in a strong inequality between the functions:

[math]|<\psi(0)|\psi(t)>|^2 \geq  \cos^2\ \int \int (|W(q,p)|<\Gamma^2> - <\psi|\Gamma^2|\psi> )\Delta t\ dqdp \geq \cos^2(\frac{[<H> - <\psi|H|\psi> ]\Delta t}{\pi \hbar}) = \cos^2(\frac{\Delta H \Delta t}{\hbar})[/math]

for [math]0 < t < \frac{\pi \hbar}{2 \Delta H}[/math]. And is similar to a collapse time equation for the survival probabilities of the geometry of the system.  Note also, we ended up constructing a theory of the acceleration which looked a lot like a gravitational version of the Schrodinger equation and was akin to talking to covariant derivative on some curve on a metric which took the form

[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]


Do Particles Experience Curvature at Infinitesimal Regions?

Yes, I believe they can experience small curvatures - in fact, I want to bring our attention to Sakharov and his metric elasticity [math]\kappa = 8 \pi G[/math] and is probably related to the idea of the string tension [math]\frac{8 \pi G}{c^2}[/math] in which we have the elasticity related the gravitational permeability [math]\mu_0[/math]  weighted by a factor of the speed of light squared [math]\frac{\mu_0}{c^2}[/math] - it implies that all interactions causing gravity by their contributions to the energy-momentum tensor curve space with the same elasticity coefficient, which K. Arun has suggested may imply a microscopic origin for [math]G[/math]. Certainly, [math]P^{-4}[/math] propagators can be entirely described by their space time curvatures (see references), and Nobel Prize Winner A. Salam has suggested his own version of ''strong gravity'' which involves a cosmological and particle scale value for [math]G[/math].

What threw me off in my initial investigations was a continued assumption I was reading on the internet suggesting infinitesimally small spaces are only flat. This seems though to be an assumption at best.  

It is even suggested by some authors that even a fluctuation should exert a curvature in the vacuum! So long as the stress energy tensor is not zero, then all volumentric forms of energy exert deformations on the surrounding gemetry. 

The energy associated to the heat [math]Q[/math] is often represented by the enthalpy

[math]\Delta Q = \Delta H = H_0 - H[/math]

And the heat capacity is featured in the following equation in which measures how much energy is required to raise the temperature of a system

[math]\Delta H = c_p m \Delta T = c_p m(T_0 - T)[/math]

and becomes the volumetric measure of the change of enthalpy if 

[math]\Delta \mathbf{H} = c_p \rho \Delta T = c_p \rho(T_0 - T)[/math]
Which is a form more encountered on the internet. The rate of flow of heat (as from Fourier's law) per unit area through some surface is proportional to the negative teperature gradient

[math]\frac{\Delta Q}{\Delta t} = -\kappa S \frac{\Delta T}{\mathbf{x}} = -\kappa S \nabla T[/math]

In which [math]\mathbf{x}[/math] is a measure of thickness of the medium in which [math]\frac{\Delta T}{\mathbf{x}}[/math] is the temperature gradient [math]\nabla T[/math] and is always negative since heat flows one way (flows from the higher to lower temperatures always). [math]S[/math] is the shape operator.

Let's quickly compare these equations to one suggested by Arun et el. The metric elasticity can be understood in terms of the conductivity which is further related to the temperature in the following way:

[math]\Delta T = \kappa(r^2_0 - r^2)[/math]

This is suggested as a direct solution to the heat gradient equation

[math]\frac{\partial R}{\partial t} = \kappa \nabla^2 R[/math]

They suggested that curvature flows, in the sense of a Ricci flow and [math]\kappa[/math] will play the role of the conductivity. I am still attempting to decode these equations as they did not define their variables very well - my initial opinion is that their form of equation is very similar to the separation of variables solution for the heat equation. If the second equation is the standard gradient equation for heat flow then we can compare it with the differential form of the heat equation,

[math]\frac{\partial T(x,t)}{\partial t} = \kappa \nabla^2 T(x,t)[/math]

This would suggest Arun and Sivaram have used a notation [math]R[/math] for the temperature which is... pretty irregular. Moreover, when they come to their proposed gravitational analogue of the equation in terms of the metric elasticity, they denote a small [math]r[/math] and so is confusing whether they are supposed to denote the same variable. You can get a similar equation through a process of the separation of variables, but is still unclear to me and since they have not properly denoted their variables, it will remain unclear until the lightbulb above my head sparks up. 


Either way, this is a very good place to start an investigation on temperature and accelerations. 





Salam, Abdus; Sivaram, C. (1 January 1993). "Strong Gravity Approach to QCD and Confinement".


Edited by Dubbelosix
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The Ricci flow is

[math]\frac{\partial p}{\partial t} = \Delta p[/math]

which is analogous to the heat equation

[math]\frac{\partial u}{\partial t} = \Delta u[/math]

The reason why this approach may be fruitful becomes clear from a passage in wiki


''The reader may object that the heat equation is of course a linear partial differential equation—where is the promised nonlinearity in the p.d.e. defining the Ricci flow?

The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking p(x,y)=0. So if p is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation.''


Note then, it has the possibility to describe deviations in the flat geometry of Euclidean flat space.

It seems then their notation of [math]R[/math] may just be another notation to denote a ''Ricci Flow.''

A more interesting form of the Ricci flow does exist using the Ricci curvature

[math]\frac{d}{dt}g_{ij} = -2R_{ij}[/math]



The actual definition they have chosen is confusing because technically speaking the diffusion equation should have a factor of [math]\rho c[/math] attached to the LHS, which would allow you to construct the thermal diffusion [math]\alpha[/math]

[math]\rho c_p \frac{\partial T}{\partial t} = \kappa \nabla^2 T[/math]

Such as

[math] \frac{\partial T}{\partial t} = \alpha \nabla^2 T[/math]

[math]\alpha = \frac{\kappa}{\rho c_p}[/math]


This last equation is equivalent to the volumetric heat [math]\dot{Q}_V[/math].

Edited by Dubbelosix
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Still trying to decode the equation suggested by Arun and Sivaram. It turns out there is some history of the combining of the heat equation with Ricci flow. 

[math]\frac{\partial T}{\partial t} = -\nabla^2 T + R T[/math]

Where [math]R[/math] is the Ricci scalar. So as we can see the curvature acts like the Laplacian. 





Edited by Dubbelosix
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