Redshifting of forces in stationary space - times

[WannabeNewton]

Agreed. I just still find it so weird that taking the gradient of the energy of the particle, as measured at infinity, actually gave the right answer but even with the conservation of energy argument via radiation I still cannot justify why the equality would be true on physical grounds for an arbitrary asymptotically flat stationary space-time and for a general force (general in the sense that it need not be purely radial when holding and equivalently lowering the stationary particle).

 

[PeterDonis]

for an arbitrary asymptotically flat stationary space-time and for a general force (general in the sense that it need not be purely radial when holding and equivalently lowering the stationary particle).

One other piece of food for thought: does an asymptotically flat stationary spacetime have to be at least axisymmetric?

 

[WannabeNewton]

I can't find anything in Carroll or Wald about it Peter. At first glance I cannot see any connection between the two a priori but intuition is useless for these matters. However even if it was true we would still have a bit of an issue. I say this because of the following:

IF our space-time was also axisymmetric then we can always find a coordinate system $(t,\phi,x^{2},x^{3})$ adapted to both the time-like and axial killing vector fields so that in these coordinates $\xi ^{a} = (\frac{\partial }{\partial t})^{a},\psi ^{a} = (\frac{\partial }{\partial \phi})^{a}$ and $\partial _{t}g_{ab} = \partial _{\phi}g_{ab} = 0$. The same calculation from above holds showing that $F_{t} = 0$ in these coordinates but we also now have that $F_{\mu}\psi ^{\mu} = F_{\phi} = -\frac{1}{2}mV^{-2}\psi ^{\mu}\nabla_{\mu}g_{tt} = -\frac{1}{2}mV^{-2}\partial _{\phi}g_{tt} = 0$ in these coordinates. Similarly, $\nabla_{\phi}E = \nabla_{t}E = 0$ in these coordinates. However we still don't know anything about the $x^{2},x^{3}$ coordinates so we can't really conclude anything about the other components of the force or gradient of energy nor can we really conclude anything further about the components of $(\delta s)^{\mu}$ because we don't have, in this general case, a notion of a "radial" direction / "radial" coordinate function since there is no guarantee of spherical symmetry. I am really starting to doubt on physical grounds why $(F_{\infty})_{a} = -\nabla_{a}E$ would hold true under such a general case of any asymptotically flat stationary space-time. It is just eating away at me why it gave us the right answer for this problem.

 

[PeterDonis]

However even if it was true we would still have a bit of an issue. I say this because of the following:

I need to look at these derivations some more. For the stationary axisymmetric (but not static) case I don't see physically how F_{\phi} can be zero for an object following an orbit of the timelike KVF. There should be a tangential force because of frame dragging.

 

[WannabeNewton]

I don't think the stationary particle hanging at the other end of the string, following an orbit of the time-like KVF, undergoes frame dragging in the coordinate system adapted to the time-like and axial KVFs. \frac{d\phi}{dt} = \frac{d\phi/d\tau}{dt/d\tau} = \frac{u^{\phi}}{u^{t}} = \frac{\xi ^{\phi}}{\xi ^{t}} = \frac{\delta ^{\phi}_{t}}{\delta ^{t}_{t}} = 0.

On the other hand for the locally non-rotating observers in the adapted coordinates, that is the ones following orbits of $\nabla^{a}t$, we have \frac{d\phi}{dt} = \frac{d\phi/d\tau}{dt/d\tau} = \frac{u^{\phi}}{u^{t}} = \frac{\nabla^{\phi}t}{\nabla^{t}t} = \frac{g^{\mu\phi}\nabla_{\mu}t}{g^{\nu t}\nabla_{\nu}t} = \frac{g^{\mu\phi}\delta ^{t}_{\mu}}{g^{\nu t}\delta^{t}_{\nu}}= \frac{g^{t\phi}}{g^{tt}} = -\frac{g_{t\phi}}{g_{\phi\phi}}

 

[PeterDonis]

I don't think the stationary particle hanging at the other end of the string, following an orbit of the time-like KVF, undergoes frame dragging in the coordinate system adapted to the time-like and axial KVFs.

No, it should, because such a particle is not a ZAMO. Only a ZAMO should experience no tangential force.

\frac{d\phi}{dt} = \frac{d\phi/d\tau}{dt/d\tau} = \frac{u^{\phi}}{u^{t}} = \frac{\xi ^{\phi}}{\xi ^{t}} = \frac{\delta ^{\phi}_{t}}{\delta ^{t}_{t}} = 0.

This just shows that the orbits of the timelike KVF have constant angular coordinate, which we knew anyway because we've chosen a chart specially so that orbits of the timelike KVF have all spatial coordinates constant. It doesn't show that the component of proper acceleration in the tangential direction is zero. (Remember that proper acceleration will have terms in the connection coefficients as well as terms in the partial derivatives of the coordinates. For example, even in Schwarzschild spacetime the orbits of the timelike KVF, in Schwarzschild coordinates, have constant spatial coordinates, but the proper acceleration in the r direction is nonzero.) The tangential proper acceleration shouldn't be zero, because the "flow of frame dragging" in the particle's vicinity has a nonzero tangential component.

Conversely, a ZAMO (what you are calling a "locally non-rotating observer") has nonzero angular velocity (its angular coordinate changes), but its proper acceleration should have zero angular component, because it is "moving along with the flow" of frame dragging in its vicinity.

We really need to compute the proper acceleration, maybe starting with a special case like Kerr spacetime.

 

[PeterDonis]

I can't find anything in Carroll or Wald about it Peter.

Wald Ch. 12 discusses the fact that a stationary *vacuum* spacetime must be axisymmetric; this is part of the proof of the "no-hair" theorems for black holes. However, I don't see anything about stationary non-vacuum spacetimes.

 

[WannabeNewton]

It doesn't show that the component of proper acceleration in the tangential direction is zero. (Remember that proper acceleration will have terms in the connection coefficients as well as terms in the partial derivatives of the coordinates. For example, even in Schwarzschild spacetime the orbits of the timelike KVF, in Schwarzschild coordinates, have constant spatial coordinates, but the proper acceleration in the r direction is nonzero.)

I don't think I am particularly interpreting you correctly. Are you basically saying that the only reason $d\phi/dt = 0$ for the stationary particle is that the observer at infinity is exerting some non-zero tangential force at infinity in order to keep the particle stationary hence in these coordinates at constant $\phi$? This should then imply the particle felt some non-zero local force $m\frac{\mathrm{d} U^{\phi}}{\mathrm{d} \tau } = F^{\phi} = mg^{\mu\phi}\nabla_{\mu}\ln V$ in these coordinates? Hence isn't the observer at infinity exerting a force so that the particle won't be frame dragged (so as not to have a changing tangential position in this coordinate system)?

I'm either making a silly error or something because I keep getting that $F^{\phi} = 0$ in these coordinates for the stationary particle hanging off the string. Consider again the local force felt by the particle expressed in these coordinates, F^{\phi} = mg^{\mu\phi}\nabla_{\mu}\ln V = \frac{1}{2}mV^{-2}g^{\mu\phi}\nabla_{\mu}V^{2} = -\frac{1}{2}mV^{-2}g^{\mu\phi}\nabla_{\mu}(g_{\alpha\beta}\xi ^{\alpha}\xi ^{\beta}) = -\frac{1}{2}mV^{-2}g^{\mu\phi}\partial_{\mu}(g_{tt}) Here we can freely replace $\triangledown _{\mu}\rightarrow \partial _{\mu}$ because $V^{2} = -g_{\alpha\beta}\xi ^{\alpha}\xi ^{\beta}$ is a scalar. Note that because we are in the coordinates adapted to both the time-like and axial KVFs, $\partial _{t}g_{ab} = \partial _{\phi}g_{ab} = 0$ and $g_{ti} = g_{\phi i} = 0$ where $i = 2,3$ because of the orthogonality of $(\frac{\partial }{\partial x^{2}})^{a},(\frac{\partial }{\partial x^{3}})^{a}$ with $(\frac{\partial }{\partial t})^{a},(\frac{\partial }{\partial \phi})^{a}$. This implies $g^{ti} = g^{\phi i} = 0$ as well (same $i$'s) and I checked on wolfram to make sure (I'm using the matrix on page 165 of Wald); here was the wolfram check: http://www.wolframalpha.com/input/?i=inverse+{{-V,W,0,0},+{W,X,0,0},+{0,0,a,b},+{0,0,b,c}}. Then, using these facts, F^{\phi} = -\frac{1}{2}mV^{-2}g^{\mu\phi}\partial_{\mu}g_{tt} = -\frac{1}{2}mV^{-2}(g^{t\phi}\partial _{t}g_{tt} + g^{\phi\phi}\partial _{\phi}g_{tt} + g^{2\phi}\partial _{2}g_{tt} + g^{3\phi}\partial _{3}g_{tt}) = 0 so I'm at a loss as to what's going wrong here in the computation, in these coordinates.

 

[Ben Niehoff]

Wald Ch. 12 discusses the fact that a stationary *vacuum* spacetime must be axisymmetric; this is part of the proof of the "no-hair" theorems for black holes. However, I don't see anything about stationary non-vacuum spacetimes.

There are known counterexamples with electromagnetic fields. Israel-Wilson metrics, for example. There's another example, too, but its name escapes me at the moment.

Basically, you can take a bunch of extremal Reissner-Nordstrom black holes, and their electrostatic repulsion will balance out their gravitational attraction such that they feel no force. You can put them anywhere you like.

Edit: Majumdar-Papapetrou was the name I was looking for.

 

[PeterDonis]

This should then imply the particle felt some non-zero local force ... in these coordinates? Hence isn't the observer at infinity exerting a force so that the particle won't be frame dragged (so as not to have a changing tangential position in this coordinate system)?

That's my intuitive guess, but I haven't tried to actually compute the proper acceleration. I'm hoping to have time this evening to plug the Kerr metric into maxima to compute the connection coefficients, which are the key things I don't have a handy reference for.

I'm at a loss as to what's going wrong here in the computation, in these coordinates.

I'm having trouble seeing it too, which is why I want to crank through the computation another way, analogous to the way I'm familiar with doing it in the static case (in Schwarzschild coordinates), where I know there is no tangential force on a static object. I'm hoping that will make it clearer to me either why there is a difference if the spacetime is not static, or why my intuition about why there ought to be a difference is wrong.

 

[WannabeNewton]

That's my intuitive guess, but I haven't tried to actually compute the proper acceleration. I'm hoping to have time this evening to plug the Kerr metric into maxima to compute the connection coefficients, which are the key things I don't have a handy reference for.

It makes intuitive sense to me too. If you try and "picture" it, the particle will start twirling around about the rope and the observer at infinity should have to exert a tangential force to keep it remaining stationary. However I am not well versed on the actual "physical-ness" of frame dragging vs. it being purely a coordinate effect so I don't know if that might be a reason why the computation screams negative. Thanks for the effort Peter, if you have time then that would indeed be very helpful.

I'm having trouble seeing it too, which is why I want to crank through the computation another way, analogous to the way I'm familiar with doing it in the static case (in Schwarzschild coordinates), where I know there is no tangential force on a static object. I'm hoping that will make it clearer to me either why there is a difference if the spacetime is not static, or why my intuition about why there ought to be a difference is wrong.

Sounds like a plan

 

[PeterDonis]

the actual "physical-ness" of frame dragging vs. it being purely a coordinate effect

It certainly isn't just a coordinate effect; it's been measured experimentally by Gravity Probe B in the vicinity of Earth, though with a pretty large error bar (19%) due to the smallness of the effect vs. the various sources of noise in the measurement:

http://einstein.stanford.edu/

However, it is a rather un-intuitive effect, which is why I want to crank through the computation. I'll post an update when I've done so.

 

[WannabeNewton]

It would be nice if there was a way to interpret it physically in a manner similar to the fictitious forces in rotating reference frames one sees in Newtonian theory e.g. Coriolis forces.

 

[PeterDonis]

It would be nice if there was a way to interpret it physically in a manner similar to the fictitious forces in rotating reference frames one sees in Newtonian theory e.g. Coriolis forces.

After further computation, I think there is a workable interpretation using centrifugal force. I plan to give the details in a post on my PF blog, following on to my previous post on "centrifugal force reversal" near a Schwarzschild black hole:

https://www.physicsforums.com/blog.php?b=4327

I'm still working through computations, but I can give a quick update on the key question, which is, what are the components of proper acceleration for an object following an orbit of the timelike KVF in a stationary but non-static spacetime? I have done the computation for Kerr spacetime in the "equatorial plane" (where the math is easier), and the answer is, there is still only one component of the proper acceleration, the radial component. So I was wrong to think there was a tangential component. I'll briefly summarize the computation and then give what I think is going to come out of my further computations as an interpretation of what frame dragging does.

First, the general form of the proper acceleration for an object following an orbit of the timelike KVF in any stationary axisymmetric spacetime, in the natural chart adapted to the two KVFs, is:

a^a = u^b \nabla_b u^a = u^b ( \partial_b u^a + \Gamma^a{}_{bc} u^c ) = u^t \Gamma^a{}_{tt} u^t

The only connection coefficients that fit the pattern above are \Gamma^r{}_{tt} and \Gamma^{\theta}{}_{tt}. In the equatorial plane, however, \Gamma^{\theta}{}_{tt} goes to zero, so the r component is the only one left.

The line element for the Kerr metric in the equatorial plane, where \theta = \pi / 2 and d \theta = 0, is

ds^2 = - V^2 dt^2 - \frac{4 M a}{r} dt d\phi + \frac{dr^2}{W^2} + r^2 H^2 d\phi^2

where for convenience I have defined

V^2 = 1 - \frac{2M}{r}

W^2 = 1 - \frac{2M}{r} + \frac{a^2}{r^2} = V^2 + \frac{a^2}{r^2}

H^2 = 1 + \frac{a^2}{r^2} \left( 1 + \frac{2M}{r} \right)

(These definitions are a bit different than the standard convenience functions that are usually defined for the Kerr metric, but they will work better for this particular problem.)

We then have the connection coefficient

\Gamma^r_{tt} = \frac{M}{r^2} W^2

and the 4-velocity for the static observer

u^t = \frac{1}{V}

which gives for the proper acceleration

a = \sqrt{g_{rr}}a^r = \sqrt{g_{rr}} u^t \Gamma^r{}_{tt} u^t = \frac{1}{W} \frac{M}{r^2} W^2 \frac{1}{V^2} = \frac{M}{r^2 V} \frac{W}{V}

Note that for the case of a static spacetime, i.e., a Schwarzschild black hole, a = 0 and hence W = V, and we recover the familiar result for the Schwarzschild case. Note also that, in the general case of a rotating black hole, we have W > V and hence the proper acceleration for a static observer is *larger* than for the Schwarzschild case, for a given value of the r coordinate. (This is not really an "apples to apples" comparison, because the r coordinate does not have the same direct physical meaning in Kerr spacetime that it does in Schwarzschild spacetime. But it's still suggestive--at least, I think it is, subject to further computations which I'll briefly discuss now.)

What I'm trying to do now is to extend the above analysis to the more general case of an observer circling the hole with a constant tangential velocity, to correspond with the analysis I did for the Schwarzschild case in the blog post I linked to above. Here's what I'm guessing will come out of that: in the Schwarzschild case, outside of r = 3M, increasing one's tangential velocity decreases the proper acceleration required to hold altitude, i.e., the maximum proper acceleration is experienced by a static observer. (Interesting things happen at and inside r = 3M, but I'll save those for the follow-up blog post.) In the Kerr case, at least in the equatorial plane, I'm guessing that, outside of some limiting radial coordinate (which probably won't be r = 3M), the maximum proper acceleration required to hold altitude will be experienced, not by static observers, but by ZAMOs.

In other words, if I'm right, the effect of frame dragging is to change how "centrifugal force" works; it changes the *magnitude* of proper acceleration required for a given equatorial trajectory, not its direction. What actually gets "dragged" tangentially is the "zero point" of centrifugal force: the closer to the rotating hole you are, the more the "zero point" of centrifugal force is biased in the direction of the hole's rotation. We'll see how it pans out; I plan to put the details in a PF blog post, and I'll post a link and a quick summary here when I do.

 

[Passionflower]

I have done the computation for Kerr spacetime in the "equatorial plane" (where the math is easier), and the answer is, there is still only one component of the proper acceleration, the radial component. So I was wrong to think there was a tangential component.

I thought a test mass in the Kerr spacetime undergoes no proper acceleration except for a rotational acceleration around its own axis.

Am I wrong?

 

[pervect]

In other words, if I'm right, the effect of frame dragging is to change how "centrifugal force" works; it changes the *magnitude* of proper acceleration required for a given equatorial trajectory, not its direction. What actually gets "dragged" tangentially is the "zero point" of centrifugal force: the closer to the rotating hole you are, the more the "zero point" of centrifugal force is biased in the direction of the hole's rotation. We'll see how it pans out; I plan to put the details in a PF blog post, and I'll post a link and a quick summary here when I do.

I think this makes sense from the electromagnetic analogy which should work in the weak-field case as gravitomagnetism.

The frame dragging field should act like the B field of a rotating charge, i.e. it should point along the spin axis. The force on a moving, orbiting charge should be in the direction of velocity x B, which is a radial direction.

I'm not sure if I entirely trust this analogy yet, but in the 5 minutes I've thought about it, I haven's seen any obvious holes.

 

[PeterDonis]

I thought a test mass in the Kerr spacetime undergoes no proper acceleration except for a rotational acceleration around its own axis.

It depends on the trajectory the test mass is following, doesn't it? In this particular case, we are talking about objects that are "hovering" at a constant altitude, i.e., obviously not in free fall. So they will have nonzero proper acceleration.

 

[PeterDonis]

The frame dragging field should act like the B field of a rotating charge, i.e. it should point along the spin axis. The force on a moving, orbiting charge should be in the direction of velocity x B, which is a radial direction.

Yes, this is a good way of putting it, I think. (The term "gravitomagnetism" is used for just this reason.)

But there is one important thing to note: the analysis I did above was for an object in the equatorial plane. An object not in the equatorial plane will have a \theta component to its proper acceleration as well as an r component. (This corresponds to a charge orbiting in a plane not perpendicular to the spin axis of the B field source.)

 

[Passionflower]

It depends on the trajectory the test mass is following, doesn't it? In this particular case, we are talking about objects that are "hovering" at a constant altitude, i.e., obviously not in free fall. So they will have nonzero proper acceleration.

Oh I see you want the test mass hovering, yes then they must accelerate.

But it seems to me that the rotational acceleration does not go away with this.

 

[PeterDonis]

But it seems to me that the rotational acceleration does not go away with this.

If by "rotational acceleration" you mean a tangential \phi component to the proper acceleration, yes, I thought so too until I did the computation. See my previous posts.

 

[Passionflower]

If by "rotational acceleration" you mean a tangential \phi component to the proper acceleration, yes, I thought so too until I did the computation. See my previous posts.

I mean the rotation around the test mass' axis.

 

[PeterDonis]

I mean the rotation around the test mass' axis.

Meaning rotation relative to Fermi-Walker transport along the test mass' worldline? Yes, that's still present, but it's a whole other can of worms that I'm not trying to open at this time. I'm only looking at the proper acceleration of the trajectory, i.e., only at the 4-velocity vector, not the entire orthonormal tetrad for each observer.

 

[Passionflower]

Meaning rotation relative to Fermi-Walker transport along the test mass' worldline? Yes, that's still present, but it's a whole other can of worms that I'm not trying to open at this time. I'm only looking at the proper acceleration of the trajectory, i.e., only at the 4-velocity vector, not the entire orthonormal tetrad for each observer.

Seems to me that if it rotates fast it could become a factor in calculating the Doppler shift.

 

[WannabeNewton]

Very nice, thanks for the computations Peter! I'll take a look at the blog post regarding the centrifugal forces. I'm still trying to understand what you said regarding the centrifugal force at the very end of your post #44, the last paragraph that is, and trying to connect it to Pervect's analogy with the electromagnetic field.

Until then however, regarding the issue of why $F^{b} = -\nabla^{b} E$ worked in giving us $F_{\infty} = VF$: even for an arbitrary stationary axisymmetric space-time it is eluding me why this holds true (because so far all we have worked out is the time-like and tangential components vanish). Maybe it was just a coincidence?

 

[WannabeNewton]

I mean the rotation around the test mass' axis.

By this, do you mean precession of the spin axis due to frame dragging? This is non zero even for stationary observers. Here is a thread where I had to find the precession for an observer at rest, with respect to the background minkowski metric, at the center of a rotating shell in the linearized approximation: https://www.physicsforums.com/showthread.php?t=675475

If you'll notice, the precession $\frac{\mathrm{d} \vec{S}}{\mathrm{d} t} = \frac{4}{3}\frac{M}{R}\vec\omega \times \vec{S}$ is very similar in form to the expression relating the rate of change of a vector as seen in an inertial frame to its rate of change in a frame rotating with constant angular velocity $\Omega$ in the case where the vector is at rest in the rotating frame: $\frac{\mathrm{d} \vec{Q}}{\mathrm{d} t} = \vec\Omega \times \vec{Q}$

 

[PeterDonis]

Maybe it was just a coincidence?

I don't think so, but I am holding off on commenting further until I've done some more computations.

 

[PeterDonis]

if I'm right, the effect of frame dragging is to change how "centrifugal force" works; it changes the *magnitude* of proper acceleration required for a given equatorial trajectory, not its direction. What actually gets "dragged" tangentially is the "zero point" of centrifugal force: the closer to the rotating hole you are, the more the "zero point" of centrifugal force is biased in the direction of the hole's rotation.

The blog post is still not complete--in fact, it has now grown into two blog posts because the first one went over the 10000 character PF limit on blog posts! However, I can at least report enough results to address the topic of this thread. (The blog posts are also going to address some more general questions about the behavior of proper acceleration in Kerr spacetime, which I find interesting in their own right since there are some key differences from the non-rotating case that I didn't see coming. But more on some of that below.)

Regarding the intuitive guesses I made, quoted above, I was half right. The effect of frame dragging, at least in the equatorial plane of Kerr spacetime, does not affect the direction of proper acceleration, only its magnitude. However, the effect on the "zero point" of centrifugal force, i.e., the point of maximum proper acceleration required to maintain altitude, is the opposite of what I guessed: the maximum point moves "backwards" relative to the hole's rotation, i.e., the more the hole rotates, the more the maximum point moves in the retrograde direction--the direction of tangential motion opposite to the hole's rotation.

In retrospect, of course, this is obvious. What frame dragging does is to make it easier (i.e., to require less force) to maintain altitude if you are circling the hole along with the hole's rotation, and to make it harder if you are circling against the hole's rotation. This can be interpreted as "adding some centrifugal force" due to the hole's rotation, so of course a ZAMO, who is rotating with the hole, will find it easier to maintain altitude than a static observer. (The part that isn't quite obvious is that even a static observer gets some "help" from the hole's rotation, since the maximum is always at some nonzero retrograde angular velocity.)

This means that, in the equatorial plane, the force required to hold an object static is indeed purely radial (as I posted before), and so is the gradient of the energy at infinity (since in the equatorial plane we have E = \sqrt{g_{tt}} = \sqrt{1 - 2M/r}). It's then straightforward to show (as I think we've already done in this thread) that the force at infinity is equal to minus the gradient of energy at infinity.

However, for objects held static at a point that is not in the equatorial plane, both the force and the gradient of energy at infinity will have a non-radial component (a \theta component in the Boyer-Lindquist chart for Kerr spacetime). This part will not be in my forthcoming blog post, but it looks like a computation of the components will show that, although the individual components are not identical, the ratio of the \theta component to the r component is the same for both (so both vectors point in the same direction--or opposite directions if we take the minus sign into account), and the magnitude of the gradient of the energy at infinity is indeed the "redshift factor" times the magnitude of the local force vector (rest mass times proper acceleration). In other words, the equation (force at infinity = - gradient of energy at infinity) does not hold at the component level, but it *does* hold as an invariant equation summed over all nonzero components. The computations are rather messy so I won't post them unless someone asks, but this at least suggests that the same should be true for a generic stationary spacetime.

 

[Mentz114]

...
...

In other words, if I'm right, the effect of frame dragging is to change how "centrifugal force" works; it changes the *magnitude* of proper acceleration required for a given equatorial trajectory, not its direction. What actually gets "dragged" tangentially is the "zero point" of centrifugal force: the closer to the rotating hole you are, the more the "zero point" of centrifugal force is biased in the direction of the hole's rotation. We'll see how it pans out; I plan to put the details in a PF blog post, and I'll post a link and a quick summary here when I do.

I read this post with interest ( although I cut it here for brevity).

Working with the Kerr metric in Boyer-Lindquist coordinates, I got the acceleration of a hovering observer
in the local static frame basis to be $\frac{m\,r-{a}^{2}}{{r}^{2}\,\sqrt{{r}^{2}-2\,m\,r+{a}^{2}}}\ \partial_r$.

For a test body in orbit an constant r with angular velocity $\dot{\phi}$, the expression for the radial acceleration is a bit long, but the Taylor expansion in $a$ around zero is

$-\frac{\left( {r}^{3}-3\,m\,{r}^{2}\right) \,{\dot{\phi}}^{2}-m}{{r}^{2}-2\,m\,r} + \frac{2\,a\,m\,\dot{\phi}\,\sqrt{{r}^{2}\,{\dot{\phi}}^{2}+1}}{\sqrt{r}\,{\left( r-2\,m\right) }^{\frac{3}{2}}}+\frac{2\,{a}^{2}\,m\,{\dot{\phi}}^{2}}{{\left( r-2\,m\right) }^{2}} + ...$
which shows the contribution of $a$ to the acceleration clearly. This expression depends on the signs of $a$ and $\dot{\phi}$, so there are different forces required in the spin and anti-spin directions.

In the equatorial plane only the r-component of proper acceleration is non-zero.

 

[WannabeNewton]

So what you are saying is that based on your calculations for kerr space-time in the usual coordinates, $\nabla_{\mu}E\nabla^{\mu}E = (F_{\infty})_{\mu}(F_{\infty})^{\mu}$ held true? While it isn't immediately physically obvious to me, I will try to show that $\nabla_{a}E\nabla^{a}E = (F_{\infty})_{a}(F_{\infty})^{a}$ holds for all stationary (and if needed asymptotically flat) space-times or at least come up with some physical argument for it.

 

[PeterDonis]

So what you are saying is that based on your calculations for kerr space-time in the usual coordinates, $\nabla_{\mu}E\nabla^{\mu}E = (F_{\infty})_{\mu}(F_{\infty})^{\mu}$ held true?

With a minus sign on the LHS, yes.