# I Direction of rotational frame dragging

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1. Nov 15, 2017

### Sonderval

I have the following question considering frame dragging:
A test mass starting at rest near a rotating mass or with an initial velocity pointing towards the center of the rotating mass will be deflected in such a way that it begins to move around the mass in the rotational direction. This is normal frame dragging.
Now consider the opposite situation: A test mass starts near the rotating mass but is moving outwards with a velocity in radial direction. In which direction would this mass be deflected?

According to standard formulations of gravitoelectromagnetism like here
http://www.arxiv.org/abs/gr-qc/0311030
the acceleration is proportional to v×B (with B being the analogue of the magnetic field); so if I reverse the velocity, the acceleration should be reversed as well, i.e., if a moving mass is deflected to the right relative to its velocity on falling in, a mass moving out should also be deflected to the right relative to its velocity.

But when I try to see this effect (for example using the program GRorbits
http://stuleja.org/grorbits/), the outwards moving mass is deflected in the co-rotating direction.

So I'm confused which answer is correct and where the mistake in my argument concerning the cross-product lies, if there is one.

I suspect that part of the problem may be that the "outward radial direction" depends on whether I look at the situation from the point of view of a distant Schwarzschild observer or a local frame, but I am not exactly sure how to see this more clearly or make this idea exact.

2. Nov 16, 2017

### Paul Colby

If I read (I'm using the word read rather loosely here, more like a momentary glance) the paper linked to correctly, $B$ should be along $J$, the angular momentum of the mass. In this case the rotation of $v$ is about $J$ which means the change in $v$ would have the opposite sign if the direction is reversed.

3. Nov 17, 2017

### Sonderval

@Paul Arveson
Thanks. Yes, and this means that if a particle flying in one direction is accelerated to the right, a particle flying in the opposite direction is also accelerated to the right.
Actually, I think I found the problem: I looked only at the deflection of the orbit in GRorbits - but I should look at how the deflection changes over time. This shows that the direction of the acceleration is actually opposite to the rotation of the mass as the formula suggests.
The initial deflection of the orbit is due to the fact that the "outward direction" is not the same for an observer close to the mass as for a distant Schwarzschild observer.