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*for all possible values of*##\omega## if the orbit lies on the photon radius ##r = 3M##. Then the paper directed me to the following short paper, http://www.dmf.unisalento.it/~giordano/allow_listing/AJP000936.pdf, which explains intuitively how for a circular orbit with the above ##u^{\mu}## at the photon radius, the "centrifugal force" on a spacecraft in the orbit vanishes

*for all values of*##\omega##.

The reason I put centrifugal force in scare quotes is the paper never actually defines what it means by "the centrifugal force on the spacecraft " in a circular orbit of any allowed angular velocity ##\omega## at ##r = 3M##! The centrifugal force, both in GR and in Newtonian mechanics, only arises in rotating frames wherein the rotation is relative to local gyroscopes. Therefore what rotating frame is the paper referring to when it talks about "the centrifugal force on the spacecraft "? If we go to the rest frame of a spacecraft in circular orbit at the photon radius then from the first paper we know that ##u_{[\gamma}\nabla_{\nu}u_{\nu]} = 0## for all possible values of ##\omega## and since ##u^{\mu}## follows an orbit of the Killing field ##\xi^{\mu} + \omega \eta^{\mu}## (taking ##\omega## to be constant throughout space-time) this implies that the rest frame of the spacecraft is non-rotating for all values of ##\omega##. Hence there would be no centrifugal forces in the rest frame of the spacecraft for all allowed angular velocities at the photon orbit. Is this what the paper means when it claims that "the centrifugal force on the spacecraft " vanishes for all angular velocities at the photon orbit?

I would like to think so but then I came upon the following paper, http://arxiv.org/pdf/gr-qc/9808036v1.pdf, which managed to confuse me beyond repair. The relevant part of the paper, as far as this thread is concerned, is section 3. The paper attempts to give covariant formulations of the familiar inertial forces from Newtonian mechanics, particularly the centrifugal and Coriolis forces. Immediately however I don't get the basis of the covariant formulation because it's done relative to an irrotational congruence. If the congruence is irrotational then why would there be centrifugal or Coriolis forces arising relative to it? As is well known, these inertial forces only arise in rotating reference frames.

For example, let's apply the formula (40) for the centrifugal force relative to the irrotational time-like Killing field of a static space-time, in particular flat space-time. More precisely, let the static space-time be flat space-time and consider a rigidly rotating disk relative to a global inertial frame with origin at the center of the disk. Then the irrotational time-like Killing field ##\xi^{\mu}## corresponds to orbits of inertial observers at rest in this global inertial frame. The 4-velocity field of observers at rest on the disk is given as usual by ##u^{\mu} = \gamma (\xi^{\mu} + \omega \eta^{\mu})##. Then ##e^{\psi} = \gamma## and ##e^{\alpha} = r## hence applying (40) we get ##Z = -\frac{1}{2}\gamma^2 \omega^2 r \partial_r ## but this is not the centrifugal force on the observers at rest on the disk. Rather it's the centripetal force acting on said observers. Obviously it would make no sense for there to be a centrifugal force because we've done our calculation relative to an irrotational congruence and in particular in a non-rotating global inertial frame in flat space-time.

In Schwarzschild space-time the irrotational time-like Killing field ##\xi^{\mu}## corresponds to the integral curves of the observers at rest in the gravitational field. The rest frames of these observers are non-rotating. Therefore there should be no centrifugal forces arising in these rest frames. If we applied (40) relative to ##\xi^{\mu}## in Schwarzschild space-time for a 4-velocity ##u^{\mu} = \gamma(\xi^{\mu} + \omega \eta^{\mu})## correspond to a circular orbit then presumably ##Z## would again resemble the centripetal force on this orbit. So what gives? Why is this paper calling this the centrifugal force? And on that note why would the above example in flat space-time yield a centrifugal force when it clearly gives the centripetal force instead?

What's really confusing to me is in section 5.1 of the paper (bottom of p.12), it is proven that ##Z## vanishes for any and all circular orbits at the photon radius in Schwarzschild space-time but as discussed directly above since ##Z## is defined relative to the non-rotating observers at rest in the gravitational field, it shouldn't correspond to a centrifugal force since the centrifugal force only shows up in rotating frames. If like in the flat space-time case ##Z## corresponds rather to a centripetal force then I'm having trouble seeing intuitively why ##Z## would vanish for circular orbits at the photon radius since these orbits most certainly have a centripetal acceleration!

Thanks in advance!