Dubbelosix Posted December 5, 2017 Report Share Posted December 5, 2017 (edited) The length of a curve in a Hilbert space is independent of the parameter choice, [math]\dot{\psi} = \frac{d}{dt} \psi_t \in \mathcal{H}[/math] And so the length can be known [math]length\ of\ the\ curve = \int \sqrt{(\dot{\psi}_t,\dot{\psi}_t)}\ dt[/math] The vector [math]\dot{\psi}_t[/math] is the contravariant tangent in which its length is the velocity with which [math]\psi_t[/math] travels in the Hilbert space [math]\frac{ds}{dt} = \sqrt{(\dot{\psi}_t,\dot{\psi}_t)}\ dt[/math] Introducing an equation by Anandan now, in which the energy of the system is related to the curvature [math]E = \frac{k}{G} (\Delta \Gamma)^2[/math] This equation uses a constant of proportionality [math]k[/math] which we will fix as it will yield the upper limit of the gravitational force. His equation can also be written then as [math]E = \frac{c^4}{G} \int (\nabla \Gamma)^2\ dV = \frac{c^4}{G} \int \frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})\ dV[/math] It was possible to combine the physics of Anandan's equation into the Hilbert by space by recognizing the geometry can be related to the Hamiltonian in the following way, by constructing a relationship with the curve and a Schrodinger like equation: [math]\frac{ds}{dt} \equiv \sqrt{<\dot{\psi}|\dot{\psi}>} = \frac{c^4}{8 \pi G} \int \int\ |W(q,p)^2| \sqrt{<\psi|\Gamma^2|\psi>}\ dqdp \geq \frac{1}{\pi \hbar}\sqrt{<\psi|H^2|\psi>}[/math] In which we have made use of the Wigner function [math]\int \int |W(q,p)^2|\ dqdp \geq \frac{1}{\pi \hbar}[/math] It was natural to construct the curve equation as shown so that it could provide solutions to the Schrodinger equation ~ [math]\frac{1}{ i \hbar}H|\psi>\ = |\dot{\psi}>[/math] It was noticed that if you squared the curve equation you could decompose the equation to form two solutions [math]<\dot{\psi}|[/math] and [math]|\dot{\psi}>[/math]. Then both the vectors correspond to the same curve in the Hilbert space. It becomes interesting when you take one of the solutions and hit the Covariant derivative on the curve you end up with an analogous geodesic acceleration equation, [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}{ \pi \hbar}(\frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i})dV|\psi>[/math] Notice, the wave cannot be linear any more and the Covariant derivative acts on the curvature and stress energy tensors in the following way; [math]\nabla_n\Gamma^{ij} = \frac{d}{dx^n}\Gamma^{ij} + \Gamma^{i}_{n\rho} \Gamma^{\rho j} + \Gamma^{j}_{n \rho}\Gamma^{\rho}_{i}[/math] [math]\nabla_nT^{ij} = \frac{d}{dx^n}T^{ij} + \Gamma^{i}_{n\rho} T^{\rho j} + \Gamma^{j}_{n \rho}T^{\rho}_{i}[/math] It is like the geodesic equation because the covariant derivative acting on a curve are formally identical, [math]\frac{d^2x^{\mu}}{d\tau^2} = - \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau}[/math] Or as [math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math] The covariant derivative is made of the acceleration term plus a correction term, [math]\frac{d}{d\tau} \frac{dx^{\mu}}{d\tau} + (correcting\ term)[/math] From the curve equation it is also possible to construct the following relationship for the curve, [math]\nabla_n \dot{\gamma}(t) = \nabla_n\frac{dx^{\mu}}{d\tau} \equiv\ min\ \sqrt{<\dot{\psi}|[\nabla_j,\nabla_j]|\dot{\psi}>}[/math] Because we are dealing with Covariant connection with derivatives in time, it is easy to argue that there is a classical commutation going on [math][\nabla_j,\nabla_j] = 0[/math] which may be itself an indication that this theory of gravity is classical in structure(?) - nevertheless, we have spoke about non-commutation in the past and how it is important to describe the quantum phase space. Finally, there has also been some investigation into possible incorporation of temperature into gravity: [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] Here, the mass is assumed constant [math]m=1[/math] and the temperature is proportional to the square of the curve in the metric! The energy associated to heat is called 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 temperature gradient [math]\frac{\Delta Q}{\Delta t} = -\kappa S \frac{\Delta T}{\mathbf{x}} = -\kappa S \nabla T[/math] In which [math]x[/math] is a measure of the distance which can increase and therefore temperature decreases, [math]\nabla T[/math] is the temperature gradient ∇T and is always negative since heat flows one way (flows from the higher to lower temperatures always, except for the strange case of entangled particles). [math]S[/math] is the shape operator. Arun and Sivaram have suggested that curvature flows, in the sense of the Ricci flow. They suggest an equation of the form [math]\frac{\partial R}{\partial t} = \kappa \nabla^2 R[/math] I have struggled interpreting the equation since other treatments I have read, [math]R[/math] tends to be a Ricci scalar curvature which tends to replace the definition of the Laplacian in the following way [math]\frac{\partial T}{\partial t} = -\nabla^2 T + R T[/math] I propose it is true that the following flow holds for the temperature: [math]\frac{\partial T}{\partial t} g_{ij} = -2 \nabla^2 T_{ij}[/math] Because of the Ricci flow definition of the form [math]\frac{\partial}{\partial t} g_{ij} = -2R_{ij}[/math] Also, it must be noted that some treaments of the equations ''appear'' to be simplified since the full heat equation involves a density and a heat capacity coefficient [math]\rho c_p \frac{\partial T}{\partial t} = \kappa \nabla^2 T[/math] This allows you to construct [math]\frac{\partial T}{\partial t} = \alpha \nabla^2 T[/math] in which [math]\alpha = \frac{\kappa}{\rho c_p}[/math] is the diffusivity. By Fouriers law, the rate of flow of heat through a surface proportional to the negative temperature gradient is [math]Q = -k\nabla T = -k \frac{\partial T}{\partial x}[/math] The units of this equation are [math]\frac{J}{m^2s} = -\frac{J}{msK} \frac{K}{m}[/math] Again, the previous equation could be related to a curvature in the following way [math]Q = -k \Gamma T[/math] In this simple form, we can see the entropy may be related to curvature in the following way [math]S = \frac{Q}{T} = -k\Gamma[/math] If the temperature at any point changed, the local gradient heat flow is [math]\frac{\partial T}{\partial t} = -\frac{1}{\rho C_p} \frac{\partial Q}{\partial x}[/math] We will use this equation in a moment, we can see if we use the equation we have just had: [math]Q = -k \Gamma T[/math] and create a gradient such that [math]\nabla Q = \frac{\partial Q}{\partial x} = -k R T[/math] Then we can retrieve the definition of the heat flow equation in terms of the Ricci curvature again [math]R[/math], keep in mind, [math]k[/math] is the thermal conductivity. If the temperature at any point changed, the local gradient heat flow is [math]\frac{\partial T}{\partial t} = -\frac{1}{\rho C_p} \frac{\partial Q}{\partial x}[/math] We will use this equation in a moment, we can see if we use the equation we have just had: [math]\nabla Q = \frac{\partial Q}{\partial x} = -k R T[/math] Multiply through by [math]-\frac{1}{\rho C_p}[/math] [math]-\frac{1}{\rho C_p}\nabla Q = -\frac{1}{\rho C_p}\frac{\partial Q}{\partial x} = \frac{k}{\rho C_p} R T = \alpha R T = \frac{\partial T}{\partial t}[/math] Though I was never sure what was truly intended by Arun and Sivaram, I have noticed their form of the equation holds up to the analogous comparing of the diffusion equations, the Ricci flow and the heat equation, which wiki even makes a mention: [math]\frac{\partial p}{\partial t} = \Delta p[/math] [math]\frac{\partial T}{\partial t} = \Delta T[/math] (here [math]\Delta = \nabla \cdot \nabla = \nabla^2[/math]) In our case, our ''analogous equations'' would be [math]\partial_t R g_{ij} = \frac{\partial R}{\partial t} g_{ij} = -2 \nabla^2 R_{ij}[/math] [math]\partial_t T g_{ij} = \frac{\partial T}{\partial t} g_{ij} = -2 \nabla^2 T_{ij}[/math] These are the simplified versions that were provided by wiki. https://en.wikipedia.org/wiki/Ricci_flow here [math]p[/math] is related to the curvature tensor, in which case, the equation would be formally identical to the one suggested by Arun and Sivaram. It seems in the equations, there is a Ricci flow [math]\frac{\partial}{\partial t} g_{ij}[/math] associated to the temperature. [math]T[/math]. In the last reference provided, the flow equation has been noted as [math]\frac{\partial R}{\partial t} = \nabla^2 + 2R^2[/math] which is the renormalized form. http://intlpress.com/site/pub/files/_fulltext/journals/sdg/1993/0002/0001/SDG-1993-0002-0001-a002.pdf. Instead of saying the two equations are analogous, you can simply argue, the Ricci flow is really the heat equation for a Riemannian manifold. (See last reference) References http://mathworld.wolfram.com/RicciFlow.html http://www.sciencedirect.com/science/article/pii/S0001870811002763 https://arxiv.org/ftp/arxiv/papers/1205/1205.4624.pdf https://www.uni-muenster.de/Physik.TP/archive/fileadmin/lehre/NumMethoden/WS0910/ScriptPDE/Heat.pdf http://www.physik.uni-leipzig.de/~uhlmann/PDF/UC07.pdf http://g9toengineering.com/resources/heattransfer.htm http://intlpress.com/site/pub/files/_fulltext/journals/sdg/1993/0002/0001/SDG-1993-0002-0001-a002.pdf Edited December 7, 2017 by Dubbelosix Quote Link to comment Share on other sites More sharing options...
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