How Does Linear Framedragging Affect Parallel Moving Masses in Flat Spacetime?

[BoraxZ]

TL;DR Summary

Will a linear framedragging effect influence the motion of two non-colinearly relativistically moving masses? Will it induce rotation in both masses?

Imagine two equal masses, m, moving through flat spacetime with opposite 3-momenta, as seen from an inertial frame in the COM.

In the massless case of two parallel, non-colinearly infinitely long moving bundles of light, Bonnor Beams, the beams are curved if the momenta are opposite, and stay straight when the momenta point in the same direction. The situation is not exactly the massless equivalent of the two masses, because the masses have no infinite extension.

If two identical massive particles move parallel with equal momenta the only effect is, obviously, that they move towards each other because of their mass. We can choose a reference frame in which both particles have zero momentum.

Let's look at the stress-energy tensor (from Wikipedia):

The only component in the stress-energy tensor in the case is the tt-component (00).

If the two masses travel colinearly and with opposite momenta, as seen in the COM frame (so they move directly at us or away from us), and assume the motion to be in the x-direction, then besides the tt-component (00), a tx- and xt-component (01 and 10), and an xx (11). The last one is the pressure in the x direction.

In the case I'm interested in we can choose the inertial COM frame so the velocities are directed parallel to the x-axis. They travel on parallel lines and have a closest approach d. We choose the origin of our frame to be in the middle of d.

The stress-energy tensor will contain tt, tx, ct, xx, ty, yt, xy, yx, and yy-components, so only the z-components are left out.

The problem is how to derive from this the metric as seen from the COM frame. Can we say something a priori? For example, will, as in the case of a rotating mass, a rotation be induced in the masses after they have passed each other? Which won't happen in classical mechanics, obviously.

 

[PeterDonis]

TL;DR Summary: Will a linear framedragging effect influence the motion of two non-colinearly relativistically moving masses? Will it induce rotation in both masses?

Imagine two equal masses, m, moving through flat spacetime

There is no frame dragging of any kind in flat spacetime.

If you want to analyze this kind of scenario, you need to solve the Einstein Field Equation with the given stress-energy tensor to find the curved spacetime that your ansatz about the stress-energy tensor results in. Then you need to see what, if any, frame dragging effects there are in that curved spacetime.

If you have the necessary background for an "A" level thread in the relativity forum, you should be able to do this.

 

[BoraxZ]

There is no frame dragging of any kind in flat spacetime.

If you want to analyze this kind of scenario, you need to solve the Einstein Field Equation with the given stress-energy tensor to find the curved spacetime that your ansatz about the stress-energy tensor results in. Then you need to see what, if any, frame dragging effects there are in that curved spacetime.

If you have the necessary background for an "A" level thread in the relativity forum, you should be able to do this.

Yes, indeed. Can we say a priori that the metric should contain off-diagonal components?

I was thinking about what you wrote about flat spacetime. I guess what I meant was that they are moving in an asymptotically flat spacetime with no other mass-energy currents

 

[PeterDonis]

Can we say a priori that the metric should contain off-diagonal components?

I'm not sure you can say anything a priori without doing some math.

I guess what I meant was that they are moving in an asymptotically flat spacetime with no other mass-energy currents

That's fine, but it doesn't change what you need to do as I described in my previous post.

 

[BoraxZ]

I'm not sure you can say anything a priori without doing some math.

I don't know. I think for most spacetime manifolds the metric is diagonizable. For the static Schwarzschild metric this is the case. It's static. But for the stationary Kerr-metric this seems impossible. Are there criteria for telling this in advance? But that's another question.

 

[PeterDonis]

I think for most spacetime manifolds the metric is diagonizable.

No, for "most" of them it is not. Most manifolds don't have the high degree of symmetry of the particular examples you appear to be familiar with.

Are there criteria for telling this in advance?

Not in general, no.

 

[BoraxZ]

No, for "most" of them it is not. Most manifolds don't have the high degree of symmetry of the particular examples you appear to be familiar with.Not in general, no.

I see what you mean. I couldn't find any examples in the net of linear framedragging. It's My guess that two parallel moving massive 2D plates (while holding them somehow at a constant distance) will come to rest asymptotically. Like two anti-parallel Bonnor beams get curved (though that happens because they are 3D, and a 2D beams would probably get frequency shifted only).

Well, I'll see. Just applying the EFE should do the trick. We can ignore, in case of relativistic motion and small masses, the direct gravity contribution, coming from the masses only..

 

[PeterDonis]

It's My guess that two parallel moving massive 2D plates (while holding them somehow at a constant distance) will come to rest asymptotically.

You can't just magically hold them at a constant distance. Either that motion is consistent with the spacetime geometry you get from the stress-energy tensor, or it isn't. In GR, unlike in, for example, electromagnetism, you can't arbitrarily specify the motion of matter; the entire solution, including matter motions, has to be self-consistent.

We can ignore, in case of relativistic motion and small masses, the direct gravity contribution, coming from the masses only..

No, we can't do any such thing, since what you are calling the "direct gravity contribution" will be a larger effect on the spacetime geometry than any possible frame dragging effects you are interested in.

 

[BoraxZ]

No, we can't do any such thing, since what you are calling the "direct gravity contribution" will be a larger effect on the spacetime geometry than any possible frame dragging effects you are interested in.

I'm not sure about this. If two, say, bricks move close to the speed of light past each other, it seems that the framedragging effects beats the direct gravity. The direct gravity depends on their masses only. Of course there could form a black hole if the relative velocity are hyper-relativistic. But that's not because of their mass.

 

[BoraxZ]

You can't just magically hold them at a constant distance

Can't you put a construction between them over which both planes can slide frictionlessly?
Anyhow, it seems the relative velocity of plates wil reduce, producing gravitational waves.

I think two bricks will induce rotation in one another.

 

[PeterDonis]

If two, say, bricks move close to the speed of light past each other, it seems that the framedragging effects beats the direct gravity.

Don't go by what "seems". Do the math.

The direct gravity depends on their masses only.

No, it doesn't. Gravity is not a force in GR. The overall spacetime curvature depends on the stress-energy tensor. With mass in motion, at a minimum you either need to include its momentum, or you need to model it as a continuous fluid and include its pressure, in the "source" of what you are thinking of as "direct gravity".

Of course there could form a black hole if the relative velocity are hyper-relativistic.

No, they can't. We have had umpteen previous threads on this.

Can't you put a construction between them over which both planes can slide frictionlessly?

If you include the stress-energy of this "construction" in the overall stress-energy tensor, sure. But then the "construction" will also be a source of gravity, as it should be.

Anyhow, it seems the relative velocity of plates wil reduce, producing gravitational waves.

"Seems" is pointless. Do the math.

I think two bricks will induce rotation in one another.

Don't "think". Do the math.

 

[PeterDonis]

@BoraxZ I am closing this thread as you have done nothing but wave your hands and speculate. If and when you have some actual math to present, PM me and I can reopen the thread so you can post it.