Looking for linear frame dragging formula

[Jonathan Scott]

Linear frame dragging is the effect predicted by General Relativity that if a gravitational source is accelerated, this produces a small gravitational induction effect on other objects which tends to accelerate them in the direction of the source acceleration. The induced acceleration is a multiple of the gravitational potential due to the source times the acceleration of the source. It is analogous to the $\partial A/\partial t$ term in the electric field expression for electromagnetism.

At present, it is too small an effect to be confirmed experimentally, being normally swamped by the usual gravitational field, so it seems to be largely ignored. However, for a clean theoretical situation one could for example consider a test mass inside a spherical shell, where there should be no static field, and then accelerate the shell.

The induced acceleration is presumably approximately of the following form:
$$n \frac{Gm}{rc^2} \frac{dv}{dt}$$

I would like to know what $n$ is in this context. Can anyone provide a definitive reference, please? I've looked in various text books (for example MTW, Wald, Ciufolini & Wheeler "Gravitation and Inertia") and Googled without finding anything useful. I also have an old paper on frame dragging by Nordtvedt, but I don't understand it enough to extract the information I need.

If gravity was the same as electromagnetism, we would have $n = 1$. However, the gravitational equivalent of the four-potential is a tensor quantity, so it does not transform in the same way as the electromagnetic potential. My guess is $n = 2$, as I think the approximate equivalent of the four-potential of a moving gravitational source involves the square of the classical velocity, $\phi(1+\tfrac{2v}{c}+\tfrac{v^2}{c^2})$, which of course does not transform as a four-vector.

 

[renormalize]

Here's a link to a 2005 paper with a comprehensive treatment of the subject: A model for linear dragging that contains references to earlier work. Be warned that the derivations and results in the paper are rather involved. As the authors state:

 

[Jonathan Scott]

Thanks very much - that's much better than anything I found so far. But it's a lot to digest to get the value of a simple constant! Their model for the method of acceleration seems very contrived, being a charge distribution inside the shell. I know that constant linear acceleration in free fall needs energy, but I would have hoped they could have assumed something like a distant mechanism and light strings for approximation purposes (which would still of course contribute source terms involving tension and pressure).

From a very quick look, it appears to conclude that for some specific model n=4/3, the same factor as for the Lense-Thirring rotation, but it could take me a while to understand whether that model matches my question.

 

[PeterDonis]

if a gravitational source is accelerated

Note that doing this means you need to include whatever is producing the acceleration in the gravitational source (the stress-energy tensor). For example, in the scenario you pose later on, with a spherical shell with a test mass inside, and the shell getting accelerated, whatever accelerates the shell also has stress-energy and has to be included. And since accelerating the shell will involve accelerating something else in the opposite direction (by conservation of momentum), whatever linear dragging occurs due to the shell ought to, at least to first order, be cancelled by the dragging due to the something else.

A common mistake in posing scenarios involving gravity in GR is to forget that a self-consistent solution for an object with nonzero proper acceleration has to also include whatever is producing the acceleration.