Gravitational Simulation help with inhomogeneous wave equation?

[The victorious truther]

Dear scienceforums.net, 

So i've tried indulging in doing simple python simulations of gravitational phenomenon and have been thinking about simulating a form of gravitation given by an inhomogeneous wave equation version of gauss law of gravity. This is for a personal interest of mine and not because I think such a representation of gravitation would actually be more successful in matching observations. Though i'm fairly curious as to what you would expect in terms of galactic rotation curves with such a simple time retarded potential theory of gravitation. The equation is given below, 

\[  \nabla^{2} \phi - \frac{1}{ c^{2} } \frac{\partial^{2}  \phi }{ \partial t^{2} } = 4 \pi G \rho. \]

Basically its a wave equation with a source and thusly I would expect that if I were to simulate this with a finite difference approximation, given no restrictions on the borders, \( f(x) \) initially is that of a single point of density at the origin, you would get something like this (click play on t). However this isn't what I get from simulating a single point at the origin with some finite value for all t, 

\[ \frac{\partial^{2}  \phi }{ \partial x^{2} } - \frac{1}{ c^{2} } \frac{\partial^{2}  \phi }{ \partial t^{2} } = f(x,t) . \]

Instead it just blows up so I must be thinking about this wrong. 

The equation for the discretization of the 1 dimensional wave equation with a discretized source is, assuming \( \frac{c \Delta t}{\Delta x} = 1 \), 

\[ \phi^{t+1}_{i} = \phi^{t}_{i+1} + \phi^{t}_{i-1} - \phi^{t-1}_{i} - g^{t}_{i} \]

Sincerely, freshman going on sophomore year

Edited August 10, 2020 by The victorious truther

[Duda Jarek]

Looks like GEM approximation of general relativity (confirmed e.g. by Gravity Probe B): https://en.wikipedia.org/wiki/Gravitoelectromagnetism

You can use standard methodology/tools from electromagnetism for that, like avoiding PDEs with Green's function e.g. in static https://en.wikipedia.org/wiki/Biot–Savart_law or general https://en.wikipedia.org/wiki/Jefimenko's_equations

ps. Kepler problem simulator using this approximation: https://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/

Edited August 10, 2020 by Duda Jarek

[joigus]

Duda is right AFAIK. You cannot picture solutions to inhomogeneous eq. by propagating the profile of the static source, which seems what you're naively doing in your link. Inhomogeneous eq. behaves differently. You need the Green function. Then you have retarded solutions in terms of the density. Or you could just plot Lienard-Wiechert-like potentials by relating constants and eliminating indices.

[The victorious truther]

9 hours ago, Duda Jarek said:

Looks like GEM approximation of general relativity (confirmed e.g. by Gravity Probe B): https://en.wikipedia.org/wiki/Gravitoelectromagnetism

You can use standard methodology/tools from electromagnetism for that, like avoiding PDEs with Green's function e.g. in static https://en.wikipedia.org/wiki/Biot–Savart_law or general https://en.wikipedia.org/wiki/Jefimenko's_equations

ps. Kepler problem simulator using this approximation: https://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/

Damn you are perceptive, yes i've heard of the GEM approximations and have thought about different approaches to defining the gravitational field. Given gauss law for gravitation, with the analogy to therefore electromagnetic waves, it makes sense that as an approximation or for personal investigation that I would merely use an inhomogeneous wave equation. Guess i'll just jump into it with the Jefimenko's Equations you linked to assuming they are synonymous with a gravito-electromagnetic field. The problem would be then discretizing it and then integrating. If you know how to do that with these equations or have resources that would be appreciated but i'll see what I can do on my own first then get back to you. 

6 minutes ago, joigus said:

Duda is right AFAIK. You cannot picture solutions to inhomogeneous eq. by propagating the profile of the static source, which seems what you're naively doing in your link. Inhomogeneous eq. behaves differently. You need the Green function. Then you have retarded solutions in terms of the density. Or you could just plot Lienard-Wiechert-like potentials by relating constants and eliminating indices.

Eliminating indices? If you are talking about tensors you are going to lose me rather quickly. I've sort'a looked into tenors but not much. 

Edited August 10, 2020 by The victorious truther

[joigus]

1 hour ago, The victorious truther said:

 

Eliminating indices? If you are talking about tensors you are going to lose me rather quickly. I've sort'a looked into tenors but not much. 

No. Not tensors. Don't worry. Just take the electrostatic potential and forget about the vector potential. That's what I meant. The mathematical problem is the same.

For example, eq. (4) in:

http://users.wfu.edu/natalie/s13phy712/lecturenote/lecture27/lecture27latexslides.pdf

Like Duda said, electrostatic scalar potencial. That's what I meant by dropping indices. Take the scalar part and leave the rest alone.

Where \delta means "evaluate at argument equals zero and cancel the integral sign". That's called Dirac's delta function.

Edited August 10, 2020 by joigus

[Duda Jarek]

12 hours ago, The victorious truther said:

Guess i'll just jump into it with the Jefimenko's Equations you linked to assuming they are synonymous with a gravito-electromagnetic field.

It would be interesting to derive (and confirm) general relativity corrections to Jefimenko equations ...

Many GR confirmations like Gravity Probe B are in fact of GEM - what confirmations of higher order terms are there?

 

GEM was introduced by Oliver Heaviside in 1893 and is Lorentz invariant ... interesting thought experiment: how physics would look like without Einstein?

I believe they would build on GEM - adding corrections to Lagrangian, starting with EM-GEM coupling to bend photon trajectories by Sun.

GEM has no renormalization problem, trivially unifies with the rest of physics as it is just another F_munu F^munu in Lagrangian - many approaches to solve this problem like string theory might never appear (?)

Where the problems would start?

[Markus Hanke]

22 hours ago, Duda Jarek said:

GEM was introduced by Oliver Heaviside in 1893 and is Lorentz invariant [...]

GEM has no renormalization problem, trivially unifies with the rest of physics as it is just another F_munu F^munu in Lagrangian - many approaches to solve this problem like string theory might never appear (?)

No, this isn't correct. Please refer to my comments on the other thread about this.

[Strange]

On 8/11/2020 at 8:04 AM, Duda Jarek said:

It would be interesting to derive (and confirm) general relativity corrections to Jefimenko equations ...

Many GR confirmations like Gravity Probe B are in fact of GEM - what confirmations of higher order terms are there?

 

GEM was introduced by Oliver Heaviside in 1893 and is Lorentz invariant ... interesting thought experiment: how physics would look like without Einstein?

I believe they would build on GEM - adding corrections to Lagrangian, starting with EM-GEM coupling to bend photon trajectories by Sun.

GEM has no renormalization problem, trivially unifies with the rest of physics as it is just another F_munu F^munu in Lagrangian - many approaches to solve this problem like string theory might never appear (?)

Where the problems would start?

!

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You have you own thread for this topic. Do not hijack other people's threads with your own speculative ideas.