Visual of Green's function

27 Sep 2009

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Sun, 09/27/2009 - 16:35

Lowell

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Joined: 04/25/2009

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Just thinking that it would be cool to create a visualization of the Green's function.

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Lowell

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Lowell

Sat, 10/03/2009 - 23:29

doug

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Joined: 03/13/2009

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Re: Visual of Green's function

Hello Lowell:

That does sound like an interesting task. Given a particular differential equation, there might be a Green's function. The Maxwell equations in the Lorenz gauge look like this:

$\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial y^2} - \frac{\partial^2 \phi}{\partial z^2} = \rho$

$\frac{\partial^2 Ax}{\partial t^2} - \frac{\partial^2 Ax}{\partial x^2} - \frac{\partial^2 Ax}{\partial y^2} - \frac{\partial^2 Ax}{\partial z^2} = Jx$

$\frac{\partial^2 Ay}{\partial t^2} - \frac{\partial^2 Ay}{\partial x^2} - \frac{\partial^2 Ay}{\partial y^2} - \frac{\partial^2 Ay}{\partial z^2} = Jy$

$\frac{\partial^2 Az}{\partial t^2} - \frac{\partial^2 Az}{\partial x^2} - \frac{\partial^2 Az}{\partial y^2} - \frac{\partial^2 Az}{\partial z^2} = Jz \quad eq. ~ 1-4$

Feynman called this equation beautiful because it is. I have been doing work on a unified field theory paper, and my equation for gravity in this same gauge looks like this:

$-\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} - \frac{\partial^2 \phi}{\partial y^2} - \frac{\partial^2 \phi}{\partial z^2} = \rho$

$\frac{\partial^2 Ax}{\partial t^2} + \frac{\partial^2 Ax}{\partial x^2} - \frac{\partial^2 Ax}{\partial y^2} - \frac{\partial^2 Ax}{\partial z^2} = Jx$

$\frac{\partial^2 Ay}{\partial t^2} - \frac{\partial^2 Ay}{\partial x^2} + \frac{\partial^2 Ay}{\partial y^2} - \frac{\partial^2 Ay}{\partial z^2} = Jy$

$\frac{\partial^2 Az}{\partial t^2} - \frac{\partial^2 Az}{\partial x^2} - \frac{\partial^2 Az}{\partial y^2} + \frac{\partial^2 Az}{\partial z^2} = Jz \quad eq. ~ 5-8$

A one-term variation on the 4D wave equation, but it applies to gravity, cool! It should be clear that the $1/R^2$ solution will work for both equation 1 and equation 5. Nice, but we knew that already. That is a Green's function of the differential equation. What I did not know were solutions for 2-4 and 6-8. Here is the set of Green's functions for the 8 differential equations:

$\phi = 1 + \frac{k q}{\sqrt{x^2 + y^2 + z^2}}$

$Ax = 1 + \frac{k q}{\sqrt{t^2 - y^2 - z^2}}$

$Ay = 1 + \frac{k q}{\sqrt{t^2 - x^2 - z^2}}$

$Az = 1 + \frac{k q}{\sqrt{t^2 - x^2 - y^2}} \quad eq.~ 9-12$

This observation made my month of September! The 1+ is needed so that energy is positive, whether the k is positive and like charges repel as happens in EM, or k is negative and like charges attract for gravity. Maxwell knew about this issue according to this nice historical account. I really should through some numbers into this think and see what flies out. I have NO idea, so that make me an honest experimenter.

Doug