# Frame dragging — is there a “non-tiny” example?

Now. As I understand it, in fact, the earth (10^25 kg) creates a very small, very tiny, frame dragging effect. Indeed, we have measured this using satellite experiments.

So, the Earth (10^25 kg) creates a very tiny, minuscule, frame dragging effect.

I ask -- is there anything big enough, that it creates a let's say "non-tiny" frame dragging effect?

So for example, you astronomers who work with (say) super-clusters of galaxies, do you have to, as a matter of course, figure "frame dragging" when doing calculations regarding the size/shape/etc of these mega-objects?

Finally, I've tried to read what I can about the frame dragging effect of black holes - but I find it confusing and sort of not applicable to what I'm asking here.

Again to recap what I'm asking:

So, I know the Earth has a (tiny, minuscule) frame-dragging effect.

What about a star like the sun? A galaxy? A super-cluster?

How big does an object have to be to have "non-tiny" frame dragging effects?

Are frame-dragging effects an every day calculation for those who deal with (say) galaxies? clusters?

Thanks! I hope this is clear.

Note! here are some actual answers, thank to the amazing J.R.:

The sun and Jupiter - both still have extremely tiny frame dragging effects (about 100 times bigger than the Earth's extremely tiny frame dragging effect).

Galaxies - in fact - and I couldn't find this answer anywhere on the internet - galaxies in fact have trivial / basically zero frame dragging effect. (Since they are so thin.) Astounding!

Note! I previously included the following prelude to this question: If you spin a bucket of water, of course, the water "forms a concave shape" - Newton's Bucket thought experiment. As I understand it, physicists now believe that if you spin an astronomically large bucket of water then in fact very surprisingly ... it does not make the concave shape ... due to "frame dragging" cancelling the inertial force. However, it now appears I was totally confused on this issue, so I have deletd the prelude to avoid confusion. Sorry. (It's very much worth noting that all the discussion you can google up about "newton's bucket + frame dragging" -- seems very confused, so research with care on this topic!)

The spacetime outside a spinning mass is described by the Kerr metric. To explain how the Kerr metric produces frame dragging is hard, because it's not something for which there's an easy intuitive model. Frame dragging arises because the spacetime geometry links the angle measured around the spinning object to time, and this means the angle changes with time. Points initially at some fixed angle get dragged in the direction of rotation.

The magnitude of the frame dragging effect is calculated from the Kerr metric, but it's not simply a case of how massive the object is. All rotating black holes contain a region called the ergosphere within which the frame dragging effect is so strong that nothing can resist it. The more massive a black hole is the bigger will be its ergosphere, but even tiny black holes still have an ergosphere.

The frame dragging angular velocity in the equatorial plane at a distance $r$ is given by (this equation is in the Wikipedia article I linked):

$$\Omega = \frac{r_s\alpha c}{r^3 + \alpha^2 (r + r_s)}$$

where $\alpha$ is related to the angular momentum:

$$\alpha = \frac{J}{Mc}$$

and $r_s$ is the Schwarzschild radius:

$$r_s = \frac{2GM}{c^2}$$

The frame dragging gets larger as the distance $r$ gets smaller, but there's obviously a minumum value of $r$ that corresponds to the radius of the object. For the Earth you can't have $r < 6378$km because that's what the radius of the Earth is. You ask how the frame dragging changes with the size of the object: if you're thinking about an astronomical object like a star then as the mass increases it's size also increases, so the calculation isn't a trivial one. Plus big objects like stars are a lot less dense than small objects like the Earth so the relationship between mass and radius is different.

So how the frame dragging at the surface of a star compares to the frame dragging at the surface of the Earth depends on various different factors. However frame dragging is normally a very small effect for everything except superdense objects like neutron stars and of course black holes. It's the density that is the biggest factor.

Lastly, I've never come across the idea that water in a large enough bucket won't form a concave surface. Can you provide a link to whatever article it was you read that in?

• Thank you very much for that information -- thanks - to save time I will try to follow up crisply: (1) so, with Earth, roughly near the surface, we know the "frame dragging effect" is tiny. what about with the sun, roughly near the surface? mot importantly, what about a galaxy, roughly near the surface? {are frame-dragging effects enormous, for galaxies?? or, totally trivial??} (2) so (only if it's a trivial calculation!) for a planet or sun (say density about 1 gcm3), how big would it have to be to have non-trivial F.D.E. near surface? supercluster sized? solarsystem sized? thanks! – Fattie Jul 21 '14 at 16:57
• Regarding massive buckets of water: you know, I'm afraid now I'm confused because every thing I'm checking on Pfister/Braun etc. (even an ingenue like me can google "Newton's bucket" "Mach!" and so on! :) ) seems to be muddled up and has varying descriptions. For example .. www-history.mcs.st-and.ac.uk/HistTopics/Newton_bucket.html (that one seems to have a plain typo?? - they're saying there would be a paraboloid shape to the water even if it was NOT spinning?!) – Fattie Jul 21 '14 at 17:09
• @JoeBlow: as it happens I did a quick calculation of the frame dragging at the surface of the Sun and also at the surface of Jupiter. Both are about 100 times greater than the Earth, though note this is still only 100 times a ridiculously small number. The average density of a galaxy is so low that any frame dragging effect is effectively zero. To get a significant frame dragging you want a high density i.e. a combination of high mass and small radius. – John Rennie Jul 21 '14 at 17:10
• @JoeBlow: re the bucket, if you put a stationary bucket inside a rotating spherical shell then the frame dragging will start to rotate the water in the bucket and make the surface curve. But your question says: if you have a bucket of water big enough, and spin it, it WILL NOT form the concave shape and that's what I don't understand. – John Rennie Jul 21 '14 at 17:12
• Re: megabuckets. Clearly (a) I know nothing, and (b) all the references I can find just now, are, frankly, a mess. So I'll quickly remove it from the question to avoid woes for future googlers. BUT. In your example: forget a "stationary bucket", imagine the whole spherical shell is the bucket. You explain, the frame dragging "magic" will make it curve - BUT - it would curve "normally" due to being spun; so they cancel at some point?! Note: en.wikipedia.org/wiki/Frame-dragging .. note the passage which concludes ....... – Fattie Jul 21 '14 at 17:26

Just to add to John Rennie's answer, the objects where we expect to see the largest frame dragging effects are spinning black holes. There, there is actually a surface called the ergosphere (outside of the event horizon), where it is impossible for observers to stay stationary with respect to observers far from the black hole. In a sense, their reference frame is being dragged faster than the speed of light.

This is interesting because there is a technique called the Penrose process where it is possible to rob angular momentum from the black hole while inside of the erogsphere, and convert that to energy that can escape the system. In principle, this could be used to get more energy than would even be possible with nuclear reactions. Of course, you need a spinnning black hole first...

• Thanks again, these answers and comments mean the world to me. This is a tricky site - because, you experts have to deal with THREE CLASSES of questions. (1) actual expert physics questions. (2) worthy popular-science -level questions from jokers like me. (3) full-on ufo-leaning, my-theory-on-negative-energy speculative / conspiracy theory questions! Also because you are all real scientists, you have to be graceful even when dealing with category three :) {On, like, an iphone game engineering site or whatever, if someone asks a nutty question - I just swear at them :) } THANKS AGAIN! – Fattie Jul 22 '14 at 8:53