Gravitomagnetism equations-wikipedia wrong?

[tim_lou]

gravitomagnetism equations--wikipedia wrong?

Hi, I read the wikipedia article from:
http://en.wikipedia.org/wiki/Gravitoelectromagnetism

(scroll down to the "Maxwell's eq" for gravity)

where it says
$$\nabla \times \mathbf{B}=-4\pi G\mathbf{J}$$
(neglect the other terms)

however, when I compare that to Sean Carroll's book of GR (spacetime and geometry),
on page 282, (7.31) it reads something like
$$\nabla^2 \mathbf{\omega}=-16 \pi G T_{0j}=-16 \pi G \mathbf{J}$$
Carroll defines,
$$\mathbf{B}=\nabla \times \mathbf{\omega}$$

so in effect, we have something like
$$\nabla \times \mathbf{B}=-16 \pi G \mathbf{J}$$

while a factor of two is reasonable since the the force law in Carroll's book is without the 2 in front of B, the additional factor of 4 is just weird... I could not get around that at all. Is wikipedia wrong or am I missing something?

 

[Jonathan Scott]

Hi, I read the wikipedia article from:
http://en.wikipedia.org/wiki/Gravitoelectromagnetism

(scroll down to the "Maxwell's eq" for gravity)

where it says
$$\nabla \times \mathbf{B}=-4\pi G\mathbf{J}$$
(neglect the other terms)

however, whenever comparing to Sean Carroll's book of GR,
on page 282, (7.31) it reads something like
$$\nabla^2 \mathbf{\omega}=-16 \pi G T_{0j}=-16 \pi G \mathbf{J}$$
Carroll defines,
$$\mathbf{B}=\nabla \times \mathbf{\omega}$$

so in effect, we have something like
$$\nabla \times \mathbf{B}=-16 \pi G \mathbf{J}$$

while a factor of two is reasonable since the the force law in Carroll's book is without the 2 in front of B, the additional factor of 4 is just weird... I could not get around that at all. Is wikipedia wrong or am I missing something?

I also noticed the same oddity recently. Even with their factor of 2 on B_g, something seems to be wrong there. When I tried myself to make gravity look like Maxwell's equations long ago, just assuming an overall factor of 2 was not enough to make it work, and I'm sure there were additional factors of 2 or 4 which appeared in the equations.

My suspicion is that the equations quoted in the Wikipedia case have ignored the curvature of space and are therefore already missing some factors of 2 compared with a more accurate version of those equations. I'll see if I can find time to check it out more carefully later.

 

[Jonathan Scott]

OK, I found a useful refresher in arXiv:gr-qc/0311030, "Gravitoelectromagnetism: A Brief Review" by Bahram Mashhoon, referenced in the Wikipedia article, and I've also compared it with chapter 6 of Ciufolini & Wheeler "Gravitation and Inertia". (I don't have the Carroll book here). There definitely seems to be factor of 2 anomaly in one part of the Wikipedia entry compared with the other two. There are also some sign inconsistencies, but I think this is simply because the convention for the direction of the ordinary gravitational field varies between the sources.

The gravitomagnetic fields is called B in Mashhoon's paper, B_g in the Wikipedia entry and H in Ciufolini & Wheeler. It appears that the definitions of these versions of the field are all different, giving H = -2B = 4B_g.

The "Maxwell's equations" equivalent match between Mashhoon and the Wikipedia entry seems to be correct at least as far as factors of two are concerned (although the sign convention for E is reversed).

The amount of gravitomagnetic field due to a rotating body also matches, in that the factor applied to J/r³ is 2 for H, -1 for B and 1/2 for B_g.

However, the Wikipedia entry expression in terms of B_g for the "Lorentz force law" seems to be out by a factor of 2 compared with the other two sources. The cross-product factor in Ciufolini & Wheeler is H and in Mashhoon it is -2B, which means that in the Wikipedia entry it should be 4B_g.

If the quantity which Carroll is using for the field is equivalent to H in Ciufolini and Wheeler, the equation would be consistent with the Wikipedia entry, as this would be equal to 4B_g. The Wikipedia entry is therefore slightly in error in a second way, in that some of the literature uses a field which is four times rather than twice B_g.

 

[tim_lou]

Thanks for the reply. that clarifies a lot! so the mistaken part is the wikipedia force law... someone seriously should change it and perhaps mention the different "B" fields occurring in the literature. I am not that good at writing explanations so maybe someone else can take on this task? (plus.. I'm feeling lazy right now)

 

[Jonathan Scott]

Thanks for the reply. that clarifies a lot! so the mistaken part is the wikipedia force law... someone seriously should change it and perhaps mention the different "B" fields occurring in the literature. I am not that good at writing explanations so maybe someone else can take on this task? (plus.. I'm feeling lazy right now)

I already added a note to the talk page about the apparent error in the force law, to make sure it is at least recorded in some way.

 

[cristo]

There's a lesson here: don't trust wikipedia for advanced topic like this. Instead, why not use a review paper, like the one Jonathon links to?