Relativistic Poisson Equation

[Dubbelosix]

I wondered whether anyone had considered an equation of the form:

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi[/math]

 

Which is analogous to the Ricci flow but instead focuses on the potential - I soon found out it had indeed been suggested (see ref). The idea would invoke a relativistic Newton-Poisson equation of the form

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi = - \alpha (\frac{\partial^2 \phi}{\partial \tau^2} - \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}) = \alpha ( \frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2\phi)[/math]

 

We should note now, what the Newton-Poisson equation is in a standard three dimensional form

 

[math]\nabla^2 \phi = 4 \pi G \rho[/math]

 

It means we can write it as

 

[math]\alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + 4\pi G \rho)[/math]

 

This would be equivalent to a relativistic Poisson equation.

 

You can consider a mass parameter as well and this would look like:

 

[math]\alpha (\eta^{\mu \nu}\partial_{\mu} \partial_{\nu} \phi + |\mathbf{k}|^2 + \frac{\omega^2}{c^2}) = \alpha( \Box^2 \phi + \frac{m^2c^2}{\hbar}\phi )  = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2 \phi + \frac{m^2c^2}{\hbar}\phi)[/math]

 

In which

 

[math]- |\mathbf{k}|^2 + \frac{\omega^2}{c^2} =  \frac{m^2c^2}{\hbar^2}[/math]

 

Is the dispersion relation.

 

In the context of curved space in general relativity, this turns into

 

First without mass parameter. 

 

[math]\partial_t \phi \equiv \alpha g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \phi  = \alpha g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \phi)  = \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi) = \alpha(\frac{-1}{\sqrt{-g}}\partial_{\mu}(g^{\mu \nu} \sqrt{-g} \partial_{\nu} \phi)) [/math]

 

And now with the mass parameter. 

 

[math] \alpha(g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \phi + \frac{p^{\mu}p_{\mu}}{\hbar^2}\phi) = \alpha (g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \phi) + \frac{p^{\mu}p_{\mu}}{\hbar^2}\phi) = \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi + \frac{p^{\mu}p_{\mu}}{\hbar^2}\phi)[/math]

 

[math]= \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi + \frac{m^2}{\hbar^2} \frac{dx}{d \tau^{\mu}} \frac{d x}{d \tau^{\nu}}g^{\mu \nu}\phi)[/math]

 

which is the curved diffusion equation with mass parameter. I have never seen the diffusion Ricci flow written like this. 

 

 

 

Ref: http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter2.pdf

 

https://cmouhot.files.wordpress.com/1900/10/harmonicmaps.pdf

Edited January 8, 2018 by Dubbelosix

[Vmedvil]

 

I wondered whether anyone had considered an equation of the form:

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi[/math]

 

Which is analogous to the Ricci flow but instead focuses on the potential - I soon found out it had indeed been suggested (see ref). The idea would invoke a relativistic Newton-Poisson equation of the form

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi = - \alpha (\frac{\partial^2 \phi}{\partial \tau^2} - \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}) = \alpha ( \frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2\phi)[/math]

 

We should note now, what the Newton-Poisson equation is in a standard three dimensional form

 

[math]\nabla^2 \phi = 4 \pi G \rho[/math]

 

It means we can write it as

 

[math]\alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + 4\pi G \rho)[/math]

 

This would be equivalent to a relativistic Poisson equation.

 

You can consider a mass parameter as well and this would look like:

 

[math]\alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2 \phi + \frac{m^2c^2}{\hbar}\phi)[/math]

 

In the context of curved space in general relativity, this turns into

 

[math]\alpha \partial^{\mu}\partial_{\mu} \phi = \alpha (g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \phi) + \frac{m^2c^2}{\hbar^2}\phi) = \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi + \frac{m^2c^2}{\hbar^2}\phi)[/math]

 

 

 

 

Ref: http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter2.pdf

 

 

Ya, See I get what dubblesoix  is saying here, basically Probability over Space then over time as torque.

Edited December 31, 2017 by Vmedvil

[Vmedvil]

[math]\phi[/math] is not a wave function, it is a gravitational potential. There is no torque in these equations either. It's a wave equation ... a four dimensional Poisson equation with diffusion coefficient. And the last equation describes it in terms of the curvature of spacetime. 

 

then why did you use the symbol phi that is probability and tau or τ  which is torque?

 

In any case, what is that like a 3rd or 4th or 5th integral?

 

 

 

About phi in probability theory

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

Edited December 31, 2017 by Vmedvil

[Shustaire]

Look for the 4 gradient.

 

Anyways here is the 4d D'Alambertian operator

 

[math]\square=\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}[/math]

[Vmedvil]

[math]\tau[/math] is spacelike time. It is composed of [math]c t[/math]. You can read about that on wiki's page on relativistic heat equation which even tells you about the difference between [math]\Box[/math] and [math]\Box^2[/math]

 

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

 

Though their construction is using the temperature denoted as [math]\Theta[/math], in this work, in the OP, we are considered the diffusion of the potential field [math]\phi[/math]. 

Yes, so the time component is phi/tau = Probability per torque @ a unit of time which is the grad part.

Edited December 31, 2017 by Vmedvil

[Vmedvil]

Read the link I provided. It has it all there. 

No, that is torque it is a function of theta.

 

Edited December 31, 2017 by Vmedvil

[Shustaire]

Yes you can rewrite this as

 

[math]\square = \frac{\partial^2}{\partial \tau^2}-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}-\frac{\partial^2}{\partial z^2}[/math]

 

As you can see the time derivative has absorbed the inverse speed of light squared and is the definition of the spacelike time variable. 

 

yes that's correct

 

https://arxiv.org/pdf/1509.01543.pdf

Edited December 31, 2017 by Shustaire

[Shustaire]

proper time

[Vmedvil]

Not in this work, in this work [math]\phi[/math] is the gravitational potential. And [math]\tau[/math] is a space derivative, it not a torque. 

The obviously you don't know what that was derived from. 

 

 

 

Edited December 31, 2017 by Vmedvil

[Shustaire]

Not in this case Vmedvil

 

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

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

 

anyways probably last you all will see of me have fun

Edited December 31, 2017 by Shustaire

[Vmedvil]

Not in this case Vmedvil

Did you even read the paper you sent me, either of you?

Edited December 31, 2017 by Vmedvil

[Vmedvil]

Those equations are not related Vmedvil with my work. The gravitational potential has dimensions of [math]-\frac{Gm}{R}[/math]. They come from Newtons laws. Has nothing to do with Torque 

 

are you so sure?

 

[Vmedvil]

I constructed the equations, I should know what they are made of Vmedvil. 

 

It is missing a mass, or it would be torque.

[Vmedvil]

And when I say dimensions, I don't mean the three space dimensions, I mean dimensional analysis. 

 

That is false, F = M a , Torque =  F R 

 

C² = V²   , a  R = V²

[Vmedvil]

Oh, Dimensional analysis is important but it can if you convert the units.

[Super Polymath]

 

I wondered whether anyone had considered an equation of the form:

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi[/math]

 

Which is analogous to the Ricci flow but instead focuses on the potential - I soon found out it had indeed been suggested (see ref). The idea would invoke a relativistic Newton-Poisson equation of the form

 

[math]\frac{\partial \phi}{\partial t} = \alpha \Box^2 \phi = - \alpha (\frac{\partial^2 \phi}{\partial \tau^2} - \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}) = \alpha ( \frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2\phi)[/math]

 

We should note now, what the Newton-Poisson equation is in a standard three dimensional form

 

[math]\nabla^2 \phi = 4 \pi G \rho[/math]

 

It means we can write it as

 

[math]\alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + 4\pi G \rho)[/math]

 

This would be equivalent to a relativistic Poisson equation.

 

You can consider a mass parameter as well and this would look like:

 

[math]\alpha \Box^2 \phi = \alpha (\frac{\partial^2 \phi}{\partial \tau^2} + \nabla^2 \phi + \frac{m^2c^2}{\hbar}\phi)[/math]

 

In the context of curved space in general relativity, this turns into

 

[math]\partial_t \phi \equiv \alpha \partial^{\mu}\partial_{\mu} \phi = \alpha (g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \phi) + \frac{m^2c^2}{\hbar^2}\phi) = \alpha(g^{\mu \nu} \partial_{\mu}\partial_{\nu} \phi + g^{\mu \nu} \Gamma^{\sigma}_{\mu \nu}\partial_{\sigma}\phi + \frac{m^2c^2}{\hbar^2}\phi)[/math]

 

which is the curved diffusion equation with mass parameter. I have never seen the diffusion written like this. 

 

 

 

Ref: http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter2.pdf

 

What about an ADS/CFT duality treatment for a 5 dimensional (one positive & one negative ~2.5 [2+/-1 as opposed to 3+1 dimensions [not dimensional analysis] to allow for causal sets, sub-planck micro causality for gw waves, for divisions of 1 planck length per 1 planck time aka C regimes for a non-discrete/lorentz invariant continuous spacetime with non-instantaneous quantum jumps albeit local realism] fractal dimension connected via wormhole metric) relativistic poisson equation?

 

Could you build that for my theory in the event that I end up taking some calculus & can use it in the future? 

[Shustaire]

Ads/cft would not describe your personal hypothesis.