# Generalized PDFs Imply a Gravitomagnetic Moment!?!?!?

Today I’m writing conference proceedings, which are boring me to write, so they will probably be inhumanely boring to read, and lethally boring to publish. I may try to write them so the first letters spell out a hidden message, just to stay focused.

Part of the way I’m constructively procrastinating is skimming a review paper on Generalized Parton Distributions. They’re a pretty cool idea. So QCD can’t be perturbatively calculated at arbitrarily soft scales, so nobody knows how to directly calculate from first principles whats happening inside of hadrons. The lattice folk are making progress here, but that technique takes a lot of power, so those calculations can’t easily get incorporated into general calculations.   You can parametrize what’s happening in a hadron, measure it, and the factorization theorem tells you your resulting functions are universal, modulo an evolution of the factorization scale. So we can measure parton distributions functions here at HERA, and then you can roll them up to the LHC/TeVatron scales or down to a fixed target, and everyone agrees on what these functions are and do. If you are operating at first order, the vanilla type of PDFs naively tell you what the probability of finding a quark or gluon of a certain type and certain longitudinal momentum is. At higher orders the interpretation isn’t so clear, but they still return a real scalar. There’s no interference, no helicity, no transverse momentum. You can tack on spin or other stuff, but its always a bit of a blunt object.  How the proton gets its spin out of the quark-gluon shimmy is still a big mystery, so theorists have been experimenting with difference ways to combine PDFs and form factors, to include interference terms, and understand all components of nulceon spin. The situation is stabilizing a bit, and this paper seems to imply that the parameterization they describe is widely used.

I got a bit shocked by the following couple of lines:

…according to an extension of the equivalence principle of general relativity to describe the interaction of the nucleon with the external gravitational ﬁeld one arrives to the interpretation of B(0) as an anomalous gravitomagnetic moment being the analog of the anomalous magnetic moment [47]. There is also evidence supporting the conjecture that the equivalence principle is valid separately for quarks and gluons resulting in exact equipartition of momenta and angular momenta in the nucleon. The most precise numerical support comes from lattice calculations [48].

AH!  What!!?!?!  Who said anything about gravity!?!?!  But it’s not really what it looked like at first glance.  B(0) is zero, btw, so whatever you want to call it is moot, but the cool thing is that [47] paper, where the author sees a relationship similar to the equivalence principle, and this cancels out that B(0) thing at all orders.  I can’t do GR, so I can’t comment on the validity of the approach, but its a cool idea…..

### 7 responses to “Generalized PDFs Imply a Gravitomagnetic Moment!?!?!?”

1. anon

hi,
this has been bothering me for a while and
its sort of related to this.
similar to a current of charge,
a moving ‘current’ of
mass would lorentz contract and cause larger
than expected gravitational attraction.
is that maybe related to dark matter observations?
or is that already included in general relativity
by definition?
or way too small on galactic rotation scales
to affect anything?
thanks

2. Super Toll

@anon

You may be referring to the Schwarzschild solution of GR which is spherical symmetric. The soultion indicates that the “gravitational attraction” as you put it depends both on the mass and the radius of the object, which would wrongfully imply that a contraction of the radius would change the gravitaional attraction. This is wrong, because the solution is totally lorentz invariant. When boosting the system, whoever, the spherical symmetric coordination system we had chosen for the solution is no longer a good choice… Remember not to copnfuse the coordinates with the actual geometry, the usual coordinates have, e.g. a singularity in the event horizon which is not present in the geometry…

3. anon

gravity changes if the object is rotating.

i don’t know GR so using simpler
(possibly wrong) arguments:
if a ‘current’ of mass is flowing by you,
it will lorentz contract and the mass denstiy
you see will be larger than the mass density in
the ‘currents’ rest frame. and you
would feel more attraction than straight
newtonian gravity would give you.
this is just like how magnetism can be introduced
in E&M.
my thought was that similarly a rotating
mass would also have larger than expected
gravitational attraction. its gravitomagnetism,
but every search i did for it always came back
to some kind of esoteric superconductor experiment. i was just wondering why you
never see it mentioned in galactic rotation
curve mass calculations.

thanks

4. Hi Anon,

Short answer: cool question, I don’t know.

This really isn’t my specialty, so I’ll hedge my bets with incompleteness, rather than being full stop wrong. I’ll think out loud and you can sift the chaff.

Rotational binary stars systems are predicted to produce gravitational waves, which are a purely non-classical effect.* This effect has only been indirectly verified, but I don’t think its contentious. This leads me to believe that any spatial rotation is a part of GR and should be expected to produce plausible predictions. I don’t know what the predicted GR-effect of a single rotating body _is_, though.

I have also haven’t done galactic rotation calc myself. If you don’t see the rotation treated GR-stylie, maybe 1) it’s _is_ being treated in GR, but is done implicitly. 2)Somewhere they do a classical approximation, and somewhere there’s an decent argument why it produces the same result. I think it’s 1), because galactic rotation observations produce unexpected results, so I’d be pretty surprised if people didn’t do a full GR treatment with all the dressing before they decided to introduce dark matter.

Anywho, I recommend you do a googly-search for “Physics Forum” and see if you can find an active GR forum. Here’s a conglomerate-forum: http://www.physicsforums.com/
And, I’d like to hear back if you get a reasonable answer.

Good Luck!

*Thanks to Axel for a cool conversation.

5. Super Toll

@Homer

Just a short comment on your answer: There’s *no* way that anyone made a full GR treatment on the rotation of galaxies. GR is simply way too non-linear for that. For dark matter the data has been compared to GR in the Newtonian limit, which should suffice, and dark matter was found… or at least postulated.

@Anon. You do indeed ask an interesting question, but you should remember: A Lorentz contraction is a change of coordinates, *not* a change of the geometry. How the geometry gets affected by a rotating system is beyond me tho…

Hmmm….

6. By “full” I meant full numerical.

7. Anon,

Rotating bodies are expected to exhibit a frame-dragging effect, which causes objects in free-fall to revolve around the body in question. I have heard that there is experimental evidence for this, but I do not have a citation. However, Gravity Probe B is currently in orbit doing one such test. Here is a Wikipedia article on the geometry of spacetime around rotating bodies.

The effect of the Kerr geometry decays with distance, so it shouldn’t have a large effect on galactic rotation curves (disclaimer: I’m not a GR specialist). At any rate, the evidence for existence of dark matter is reasonably solid, because of phenomena like the Bullet cluster.