- #1

WannabeNewton

Science Advisor

- 5,800

- 537

As you may already know, given a time-like congruence describing some extended body with world-tube ##\mu## embedded in space-time, there are various different characterizations of what it means for this extended body to be non-rotating. Of particular interest for me is the setting of non-rotation criteria as discussed in the following paper: http://scitation.aip.org/content/aip/journal/jmp/16/2/10.1063/1.522521 Now I don't know how many of you will be able to access the article as it is pay-walled and I have university access but I will try to summarize the parts of the paper relevant to my question, as well as spell out the details of the calculations that the paper completely left out.

Consider a stationary axisymmetric asymptotically flat space-time with time-like and axial Killing fields ##\xi^{\mu},\psi^{\mu}## and let ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}## be another time-like Killing field. Let ##\mu## be a 2-dimensional time-like integral manifold of ##\eta^{\mu}## i.e. each point of the 2-manifold ##\mu## follows an integral curve of ##\eta^{\mu}##; then ##\mu## represents the world-tube of what the authors call a "Sagnac tube" and we can think of ##\omega## as the angular velocity of the Sagnac tube relative to spatial infinity (the Sagnac tube can be imagined as a 1-dimensional axisymmetric ring with perfectly reflecting internal walls, surrounding some isolated body). Finally, let ##\lambda = -\eta_{\mu}\eta^{\mu}##.

Consider now a half-silvered mirror placed at some point on the Sagnac tube, with world-line ##\gamma##, and two null curves ##C_+, C_-## in ##\mu## representing corotating and counterrotating light beams emerging from the mirror; let ##k^{\mu},k'^{\mu}## represent null vector fields on ##\mu## chosen so as to be tangent to ##C_+, C_-##. We can always choose ##k^{\mu},k'^{\mu}## such that ##k^{\mu}\eta_{\mu} = k'^{\mu}\eta_{\mu} = -1## by an appropriate normalization. Because ##\mu## is 2-dimensional, all 2-forms on ##\mu## are proportional; letting ##\tilde{\nabla}_{\mu}## be the derivative operator on ##\mu## we then have that ##\tilde{\nabla}_{[\mu}k_{\nu]} = \varphi k_{[\mu}\eta_{\nu]}## hence ##-\frac{1}{2}\varphi = k^{\mu}\eta^{\nu}\tilde{\nabla}_{[\mu}k_{\nu]} = -\frac{1}{2}k^{\mu}k^{\nu}\tilde{\nabla}_{\mu}\eta_{\nu} =0 ##, where the final equality comes from ##\eta^{\mu}## being a Killing field. So ##\tilde{\nabla}_{[\mu}k_{\nu]} = 0## meaning ##\oint _{C} k_a dS^a## is independent of the closed curve ##C## in ##\mu##, and similarly for ##k'^{\mu}##. Ashtekar and Magnon then show that ##\Delta \tau = 2(\lambda|_{\gamma})^{1/2}\oint_C \lambda^{-1}\eta_{\mu}dS^{\mu}## where ##\Delta \tau## is the Sagnac shift between the corotating and counterrotating light beams upon arrival back at the mirror. From Stokes' theorem we thus have that ##\Delta \tau = 2(\lambda|_{\gamma})^{1/2}\int_{\Sigma} \nabla_{[\mu}(\lambda^{-1}\eta)_{\nu]}dS^{\mu\nu}## where ##\Sigma## is the interior of ##C##. Finally, let ##\epsilon_{\mu\nu\alpha} = \lambda^{-1/2}\epsilon_{\mu\nu\alpha\beta}\eta^{\beta}## and ##\omega^{\mu} = \epsilon^{\mu\nu\alpha\beta}\eta_{\nu}\nabla_{\alpha}\eta_{\beta}##.

We then have

##(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-3/2}\omega^{\mu}\epsilon_{\mu\nu\alpha}dS^{\nu\alpha}= (\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-2}\epsilon^{\mu \gamma \delta \sigma}\epsilon_{\mu \nu\alpha\beta}\eta^{\beta} \eta_{\gamma}\nabla_{\delta}\eta_{\sigma}dS^{\nu\alpha}= -6(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-2}\eta^{\beta} \eta_{[\nu}\nabla_{\alpha}\eta_{\beta]}dS^{\nu\alpha}##

and furthermore ##3\eta^{\beta} \eta_{[\nu}\nabla_{\alpha}\eta_{\beta]} = \eta_{\nu} \eta^{\beta} \nabla_{\alpha} \eta_{\beta} - \eta_{\alpha} \eta^{\beta}\nabla_{\nu} \eta_{\beta} -\lambda \nabla_{\nu} \eta_{\alpha} = \eta_{[\nu}\nabla_{\alpha]}\lambda - \lambda \nabla_{[\nu}\eta_{\alpha]}## since ##\eta^{\mu}## is a Killing field

hence ##(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-3/2}\omega^{\mu}\epsilon_{\mu\nu\alpha}dS^{\nu\alpha}= 2(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\nabla_{[\mu}(\lambda^{-1}\eta_{\nu]})dS^{\mu\nu} = \Delta \tau ##

Now ##\omega^{\mu} = 0## means that gyroscopes do not precess in the rest frame of the Sagnac tube (since ##\eta^{\mu}## is a Killing field, it makes sense to talk about the rest frame of the entire Sagnac tube). From the above, ##\omega^{\mu} = 0 \Rightarrow \Delta \tau = 0## so it would seem that when dealing with time-like Killing fields, non-rotation relative to local gyroscopes implies non-rotation in the sense of vanishing Sagnac shift. But according to pp.231-232 of the notes http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf, in Godel space-time there exists a Sagnac tube which is non-rotating relative to local gyroscopes ("CIR" or "compass of inertia on the ring" criterion for non-rotation) but rotating according to the Sagnac effect ("ZAM" or "zero angular momentum" criterion for non-rotation); the ##k## in the notes is the ##\omega## in ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}##. But this example is clearly at odds with the result above that ##\omega^{\mu} = 0 \Rightarrow \Delta \tau = 0## so what's going on? Why is there this apparent contradiction?

Consider a stationary axisymmetric asymptotically flat space-time with time-like and axial Killing fields ##\xi^{\mu},\psi^{\mu}## and let ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}## be another time-like Killing field. Let ##\mu## be a 2-dimensional time-like integral manifold of ##\eta^{\mu}## i.e. each point of the 2-manifold ##\mu## follows an integral curve of ##\eta^{\mu}##; then ##\mu## represents the world-tube of what the authors call a "Sagnac tube" and we can think of ##\omega## as the angular velocity of the Sagnac tube relative to spatial infinity (the Sagnac tube can be imagined as a 1-dimensional axisymmetric ring with perfectly reflecting internal walls, surrounding some isolated body). Finally, let ##\lambda = -\eta_{\mu}\eta^{\mu}##.

Consider now a half-silvered mirror placed at some point on the Sagnac tube, with world-line ##\gamma##, and two null curves ##C_+, C_-## in ##\mu## representing corotating and counterrotating light beams emerging from the mirror; let ##k^{\mu},k'^{\mu}## represent null vector fields on ##\mu## chosen so as to be tangent to ##C_+, C_-##. We can always choose ##k^{\mu},k'^{\mu}## such that ##k^{\mu}\eta_{\mu} = k'^{\mu}\eta_{\mu} = -1## by an appropriate normalization. Because ##\mu## is 2-dimensional, all 2-forms on ##\mu## are proportional; letting ##\tilde{\nabla}_{\mu}## be the derivative operator on ##\mu## we then have that ##\tilde{\nabla}_{[\mu}k_{\nu]} = \varphi k_{[\mu}\eta_{\nu]}## hence ##-\frac{1}{2}\varphi = k^{\mu}\eta^{\nu}\tilde{\nabla}_{[\mu}k_{\nu]} = -\frac{1}{2}k^{\mu}k^{\nu}\tilde{\nabla}_{\mu}\eta_{\nu} =0 ##, where the final equality comes from ##\eta^{\mu}## being a Killing field. So ##\tilde{\nabla}_{[\mu}k_{\nu]} = 0## meaning ##\oint _{C} k_a dS^a## is independent of the closed curve ##C## in ##\mu##, and similarly for ##k'^{\mu}##. Ashtekar and Magnon then show that ##\Delta \tau = 2(\lambda|_{\gamma})^{1/2}\oint_C \lambda^{-1}\eta_{\mu}dS^{\mu}## where ##\Delta \tau## is the Sagnac shift between the corotating and counterrotating light beams upon arrival back at the mirror. From Stokes' theorem we thus have that ##\Delta \tau = 2(\lambda|_{\gamma})^{1/2}\int_{\Sigma} \nabla_{[\mu}(\lambda^{-1}\eta)_{\nu]}dS^{\mu\nu}## where ##\Sigma## is the interior of ##C##. Finally, let ##\epsilon_{\mu\nu\alpha} = \lambda^{-1/2}\epsilon_{\mu\nu\alpha\beta}\eta^{\beta}## and ##\omega^{\mu} = \epsilon^{\mu\nu\alpha\beta}\eta_{\nu}\nabla_{\alpha}\eta_{\beta}##.

We then have

##(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-3/2}\omega^{\mu}\epsilon_{\mu\nu\alpha}dS^{\nu\alpha}= (\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-2}\epsilon^{\mu \gamma \delta \sigma}\epsilon_{\mu \nu\alpha\beta}\eta^{\beta} \eta_{\gamma}\nabla_{\delta}\eta_{\sigma}dS^{\nu\alpha}= -6(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-2}\eta^{\beta} \eta_{[\nu}\nabla_{\alpha}\eta_{\beta]}dS^{\nu\alpha}##

and furthermore ##3\eta^{\beta} \eta_{[\nu}\nabla_{\alpha}\eta_{\beta]} = \eta_{\nu} \eta^{\beta} \nabla_{\alpha} \eta_{\beta} - \eta_{\alpha} \eta^{\beta}\nabla_{\nu} \eta_{\beta} -\lambda \nabla_{\nu} \eta_{\alpha} = \eta_{[\nu}\nabla_{\alpha]}\lambda - \lambda \nabla_{[\nu}\eta_{\alpha]}## since ##\eta^{\mu}## is a Killing field

hence ##(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\lambda^{-3/2}\omega^{\mu}\epsilon_{\mu\nu\alpha}dS^{\nu\alpha}= 2(\lambda|_{\gamma})^{1/2}\int _{\Sigma}\nabla_{[\mu}(\lambda^{-1}\eta_{\nu]})dS^{\mu\nu} = \Delta \tau ##

Now ##\omega^{\mu} = 0## means that gyroscopes do not precess in the rest frame of the Sagnac tube (since ##\eta^{\mu}## is a Killing field, it makes sense to talk about the rest frame of the entire Sagnac tube). From the above, ##\omega^{\mu} = 0 \Rightarrow \Delta \tau = 0## so it would seem that when dealing with time-like Killing fields, non-rotation relative to local gyroscopes implies non-rotation in the sense of vanishing Sagnac shift. But according to pp.231-232 of the notes http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf, in Godel space-time there exists a Sagnac tube which is non-rotating relative to local gyroscopes ("CIR" or "compass of inertia on the ring" criterion for non-rotation) but rotating according to the Sagnac effect ("ZAM" or "zero angular momentum" criterion for non-rotation); the ##k## in the notes is the ##\omega## in ##\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}##. But this example is clearly at odds with the result above that ##\omega^{\mu} = 0 \Rightarrow \Delta \tau = 0## so what's going on? Why is there this apparent contradiction?

Last edited by a moderator: