Linear GR Frame Dragging ?

[ro...@tessien.com]

I'm wondering what the effect would be for linear frame dragging.
Virtually all discussion of frame dragging applies to rotating objects.
Suppose a linear interaction, e.g. in the center of a relativistic jet,
how would spacetime be curved?

If I have a test mass A, and a pair of other masses B flying like two
jet aircraft flying in formation side by side such that they fly past A,
with A at the mid position between the two B "jet particles". Is there
a way to calculate the degree of spacetime curvature imposed for
simplistic examples?

What would the overall effect be? e.g., would the two jet B particles
impart a small amount of energy to A such that it's motion is altered?
Or would A move in the direction of B during passage, then return to
stationary non motion as it had initially?

I'm trying to sort out what would happen if A was immersed in a
continuous "machine gun" like flux of B particles............ignoring
direct hits and only considering how spacetime ought to curve based on
the energy and fluence of B particles.

thanks,
rt

[Roland Franzius]

A frame dragging, in local coordinates in the vacuum regions of space time

ds^2 = ( a(y) dt + b(y) dx)^2 - (c(y) dt + f(y) dx)^2 -dy^2...

can of course be transformed away locally by a linear transformation
into Minkowski form

ds^2 = dt^2 - dx^2 - dy^2 ...

exept if on closed y-submanifolds
((a,b),(c,d))(y)
is a non trivial SO(1,3) bundle.

This is certainly the case if
1) the y=const manifolds are spacelike rotational surfaces x infinite
time or deformations
and
2) a complete turn for y=0->2pi along a these directions does not result
in an identity transformation for the local frames.

In this case one needs an atlas of more than one overlapping
differentiable coordinate patches to cover the manifold.

The linear case seems to be similar to linear rotational flow in
Newtonian fluid dynamics.

Probably it plays no role because in vacuum it will reduce to a trivial
bundle as solution to Einsteins vacuum equations.

The alternative of a infinite space, uniformly filled with matter,
rotating locally in one t-x-plane, would probably violate all
cosmological symmetry principles.

If such a solution is possible mathematically with a reasonable Ricci
tensor as momentum tensor of matter is decidable with twentysomething
lines of curvature calculation.

--

Roland Franzius

[Anon E. Mouse]

> What would the overall effect be? e.g., would the two jet B particles
> impart a small amount of energy to A such that it's motion is altered?
> Or would A move in the direction of B during passage, then return to
> stationary non motion as it had initially?
>
> I'm trying to sort out what would happen if A was immersed in a
> continuous "machine gun" like flux of B particles............ignoring

This problem would be computationally similar to the calculation of a
Bernoulli effect, with c as the limit rather than speed of sound.
Perhaps researching Bernoulli calculation methods will help you find a
solution.

Plasma jets of the type you describe are a rare but much studied topic
in cosmology perhaps researching this literature will turn up more
qualified opinion than my own. As you mention, this is an uncommon
application for GRT.

This comment by Roland;

"The linear case seems to be similar to linear rotational flow in
Newtonian fluid dynamics."

Seems correct to me also, this observation forms the basis of
Maxwell's magnetic force derivations, although his geometry predates
Newton and was subtly different than the common interpretation of
Newtonian mechanics. Perhaps the curl function or a Lorentz transform
of this will function for you in GRT.

For these plasma's like charge repulsion is presumed and is greater
than gravitational confinement and the plasma jets diverge over time
or distance.

Sincerely,

AAG

[Gerry Quinn]

In article <7a008ed3-b483-496e-8d27-421d7c848184
@m13g2000yqc.googlegroups.com>, agal...@gmail.com says...

> > What would the overall effect be? e.g., would the two jet B particles
> > impart a small amount of energy to A such that it's motion is altered?
> > Or would A move in the direction of B during passage, then return to
> > stationary non motion as it had initially?
> >
> > I'm trying to sort out what would happen if A was immersed in a
> > continuous "machine gun" like flux of B particles............ignoring
>
> This problem would be computationally similar to the calculation of a
> Bernoulli effect, with c as the limit rather than speed of sound.
> Perhaps researching Bernoulli calculation methods will help you find a
> solution.

It should also be computationally similar to a particle equidistant
between two counter-rotating massive objects (except that such a
particle would be dynamically unstable to small perturbations in
position).

If so, then if you add the frame dragging effects of two such counter-
rating objects, the result should be qualitatively similar to your two
jets, or particle flux.

- Gerry Quinn