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WannabeNewton

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## Main Question or Discussion Point

Hi all. It is well known that in Schwarzschild space-time, a torque-free gyroscope in circular orbit at

Coming back to the photon radius in Schwarzschild space-time, it is clear from the above prescription that Fermi-transport of the gyroscope implies the gyroscope always points tangent to the circle because a photon emitted from my photon gun will travel along this circle, and so will the reflected photon, irrespective of my angular velocity. I have two questions with regards to the above:

(1) I don't quite understand what constraints there are on the worldline of the mirror in the above operational procedure, which is basically Pirani's "bouncing photon" method. Certainly it cannot follow an arbitrary worldline in my infinitesimal neighborhood, so what kind of restrictions are there on said mirror's worldline? Consider for example an observer and a mirror at rest on a rigidly rotating disk in flat space-time that are separated by an infinitesimal radial amount. Then there exist both future and past directed radial null geodesics given by ##\frac{dr}{dt} = \pm (1 - \omega^2 r^2)^{1/2}## from my worldline to that of the mirror's so a photon emitted by myself towards the mirror will come back to me in the same radial direction. Will not then the gyroscope remain fixed in the radial direction? But this does not constitute Fermi-Walker transport as the polar axes of the local frame fixed to the disk rotate relative to the momentarily comoving local inertial frame due to Thomas precession. So where in the above is my incorrect application of Pirani's "bouncing photon" method?

(2) Let ##(M,g_{\mu\nu})## be a static axisymmetric space-time having a time-like and hypersurface orthogonal Killing field ##\xi^{\mu}## and an axial Killing field ##\psi^{\nu}##; it is easy to see that ##M## being static is equivalent to ##\psi_{\mu}\xi^{\mu} = 0##. Now consider a congruence of circular orbits in the rest space of ##\xi^{\mu}## given covariantly by the time-like Killing field ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}##. Then, given the above conditions, it can be shown that ##\eta_{[\alpha}\nabla_{\beta}\eta_{\gamma]} = 0## for

However intuitively I do not understand why the space-time must be static as opposed to just stationary. Consider a freely falling circular photon orbit in some stationary (but not static) axisymmetric space-time. If an observer were to be in circular orbit at this photon radius then why couldn't he just use Pirani's "bouncing photon" procedure from above to conclude that Fermi-Walker transport of a comoving gyroscope implies that its axis remains tangent to the circle? Intuitively speaking, under what circumstances would this fail in a stationary but non-static space-time, and why?

*any*permissible angular velocity at the photon radius (also known as the photon sphere i.e. ##r = 3M##) will, if initially tangent to the circle, remain tangent to the circle everywhere along its world-line; see e.g. http://arxiv.org/pdf/0708.2490.pdf. This is usually explained as follows. Say I'm described by an arbitrary worldline in an arbitrary space-time and I orient a gyroscope and a photon gun along some direction at an initial event; there is a swarm of mirrors surrounding me in my infinitesimal neighborhood and at each instant of my proper time, I shoot out a photon in the direction of the gyroscope axis at that instant and reorient the gyroscope axis in the direction of the reflected photon when it arrives back to me, with the direction being relative to my worldline of course. In doing this, the gyroscope axis gets Fermi-Walker transported along my worldline; see section 2 of https://www.zarm.uni-bremen.de/uploads/tx_sibibtex/2001LaemmerzahlNeugebauer.pdf and problem 5 on p.161 of Geroch's general relativity notes http://home.uchicago.edu/~geroch/Links_to_Notes.html [Broken] and p.164 of said notes for the solution.Coming back to the photon radius in Schwarzschild space-time, it is clear from the above prescription that Fermi-transport of the gyroscope implies the gyroscope always points tangent to the circle because a photon emitted from my photon gun will travel along this circle, and so will the reflected photon, irrespective of my angular velocity. I have two questions with regards to the above:

(1) I don't quite understand what constraints there are on the worldline of the mirror in the above operational procedure, which is basically Pirani's "bouncing photon" method. Certainly it cannot follow an arbitrary worldline in my infinitesimal neighborhood, so what kind of restrictions are there on said mirror's worldline? Consider for example an observer and a mirror at rest on a rigidly rotating disk in flat space-time that are separated by an infinitesimal radial amount. Then there exist both future and past directed radial null geodesics given by ##\frac{dr}{dt} = \pm (1 - \omega^2 r^2)^{1/2}## from my worldline to that of the mirror's so a photon emitted by myself towards the mirror will come back to me in the same radial direction. Will not then the gyroscope remain fixed in the radial direction? But this does not constitute Fermi-Walker transport as the polar axes of the local frame fixed to the disk rotate relative to the momentarily comoving local inertial frame due to Thomas precession. So where in the above is my incorrect application of Pirani's "bouncing photon" method?

(2) Let ##(M,g_{\mu\nu})## be a static axisymmetric space-time having a time-like and hypersurface orthogonal Killing field ##\xi^{\mu}## and an axial Killing field ##\psi^{\nu}##; it is easy to see that ##M## being static is equivalent to ##\psi_{\mu}\xi^{\mu} = 0##. Now consider a congruence of circular orbits in the rest space of ##\xi^{\mu}## given covariantly by the time-like Killing field ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}##. Then, given the above conditions, it can be shown that ##\eta_{[\alpha}\nabla_{\beta}\eta_{\gamma]} = 0## for

*all*permissible values of ##\omega## on a time-like integral 2-manifold of ##\eta^{\mu}## if and only if the null vector field ##k^{\mu} = \xi^{\mu} + \Omega \psi^{\mu}## is a geodesic on this 2-manifold, where ##\Omega^2 = -\xi_{\mu}\xi^{\mu}/\psi_{\nu}\psi^{\nu}##. This is a generalization of the situation at the photon radius in Schwarzschild space-time due to ##\eta_{[\alpha}\nabla_{\beta}\eta_{\gamma]} = 0## being equivalent to Fermi-Walker transport.However intuitively I do not understand why the space-time must be static as opposed to just stationary. Consider a freely falling circular photon orbit in some stationary (but not static) axisymmetric space-time. If an observer were to be in circular orbit at this photon radius then why couldn't he just use Pirani's "bouncing photon" procedure from above to conclude that Fermi-Walker transport of a comoving gyroscope implies that its axis remains tangent to the circle? Intuitively speaking, under what circumstances would this fail in a stationary but non-static space-time, and why?

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