Adding spacetime curvatures

[geordief]

I understand  that spacetime curvature is mathematically  modelled as a local effect.

But the  effect  of gravity  is felt at great distances

 

Is it necessary  for all the local spacetime curvatures to be summed geometrically  for this large gravitational  field to be modelled?

 

If so,is there some  technique for doing this ?

 

Suppose we are looking at the curvatures produced by the moon  and the earth and we pick a point midway between the line that joins their respective  centres,how would one calculate the curvature due to the earth and the curvature due to the moon at that point?

 

And how do we add them?

 

[MigL]

It would be much simpler to model/sum gravitational potentials.

[Markus Hanke]

13 hours ago, geordief said:

And how do we add them?

You cannot add them. GR is a nonlinear model, which means that, in general, the sum of two valid solutions to the field equations isn’t itself a valid solution.

What you’d have to do is solve the equations using a distribution of multiple sources as boundary condition. This is quite difficult, and can, in general, only be done numerically.

[studiot]

5 hours ago, Markus Hanke said:

You cannot add them. GR is a nonlinear model, which means that, in general, the sum of two valid solutions to the field equations isn’t itself a valid solution.

What you’d have to do is solve the equations using a distribution of multiple sources as boundary condition. This is quite difficult, and can, in general, only be done numerically.

+1

[geordief]

5 hours ago, Markus Hanke said:

You cannot add them. GR is a nonlinear model, which means that, in general, the sum of two valid solutions to the field equations isn’t itself a valid solution.

What you’d have to do is solve the equations using a distribution of multiple sources as boundary condition. This is quite difficult, and can, in general, only be done numerically.

Do the multiple sources all  have to be calculated individually ?

 

How does the boundary  allow one to get to the curvature of the point in space we are interested in?

 

Is there a spacetime distance between that point and the boundary?

 

Do directions come into play ?(is/are  there a spatial or tempero-spatial relationship between the sources and the point  of which we wish to calculate the curvature?)

[Markus Hanke]

20 hours ago, geordief said:

Do the multiple sources all  have to be calculated individually ?

I do not actually know precisely how one would go about doing this in a numerical algorithm. 

If I was to be tasked with figuring this out, my approach would be to use a linear approximation. I would linearise the field equations, and solve for each source in isolation initially taking into account only lower-order correction terms to keep things simple. Since this is now a linear model, you can simply add up the solutions. I would then redo this in iterations, taking into account more and more high-order correction terms with each iteration. With each step this will become increasingly more complicated - so I’d terminate once the calculation takes too long, or I reach the required accuracy.

Another idea would be to “pixelate” my spacetime, ie do a lower-resolution approximation rather than use continuous functions.

This is just brainstorming.

[geordief]

1 hour ago, Markus Hanke said:

I do not actually know precisely how one would go about doing this in a numerical algorithm. 

If I was to be tasked with figuring this out, my approach would be to use a linear approximation. I would linearise the field equations, and solve for each source in isolation initially taking into account only lower-order correction terms to keep things simple. Since this is now a linear model, you can simply add up the solutions. I would then redo this in iterations, taking into account more and more high-order correction terms with each iteration. With each step this will become increasingly more complicated - so I’d terminate once the calculation takes too long, or I reach the required accuracy.

Another idea would be to “pixelate” my spacetime, ie do a lower-resolution approximation rather than use continuous functions.

This is just brainstorming.

So ,in practice one would just revert to the Newtonian model?

There is just too much  gravitational self interaction for the process to be fruitful?

Does that change at all if the system reduces to fewer and fewer sources of mass and energy-momentum?

Might  the gravitational  wave simulation that so successfully predicted the wave pattern  when those two black holes collided have been done the way you suggested?(they didn't fall back on Newtonian  mechanics for that ,surely did they?)

 

I realize that modeling gravitational  waves is not the same thing as adding spacetime  curvatures. 

[studiot]

1 hour ago, geordief said:

So ,in practice one would just revert to the Newtonian model?

There is just too much  gravitational self interaction for the process to be fruitful?

Does that change at all if the system reduces to fewer and fewer sources of mass and energy-momentum?

Might  the gravitational  wave simulation that so successfully predicted the wave pattern  when those two black holes collided have been done the way you suggested?(they didn't fall back on Newtonian  mechanics for that ,surely did they?)

 

I realize that modeling gravitational  waves is not the same thing as adding spacetime  curvatures. 

The point of Newtonian (gravity) and Maxwellian (electromagnetic) treatments is that they are couched in terms of mechanical forces.

Mechanical forces are vectors that can be added in vector fashion to produce a vector resultant to work with.

So when applied to field theory you have a vector field of forces for each source contributing to the overall result.

[Markus Hanke]

21 hours ago, geordief said:

So ,in practice one would just revert to the Newtonian model?

There are different numerical algorithms depending on what you are trying to achieve, and the required level of accuracy - you pick the one that’s appropriate for the task at hand.

Linearised GR is not the same as Newtonian gravity - it works with spacetime geometries instead of forces, but treats these as small deviations from flat Minkowski spacetime, so it works only for weak fields. Its advantage is that the dynamics are linear, so the maths are easier.

21 hours ago, geordief said:

Does that change at all if the system reduces to fewer and fewer sources of mass and energy-momentum?

Well, if you have only a single source, or some special case of two or three sources, then often you can solve the GR equations directly. So no need for numerics in these cases.

21 hours ago, geordief said:

Might  the gravitational  wave simulation that so successfully predicted the wave pattern  when those two black holes collided have been done the way you suggested?

Almost certainly not. They probably did a full numerical solution of the full Einstein equations for the BH merger, using whatever algorithm works for this - hence the need for powerful computers. My guess though is that they probably used “lattice GR”, ie they treated spacetime not as continuous, but as a finite lattice made up of small volumes. That kind of approximation reduces the computational load considerably. But that’s just a guess on my part, I don’t know for sure - I’ve never really studied numerical GR.