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GRDoer
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Homework Statement
Consider the Godel Metric in spherical coordinates as on page 6 here;
[tex]ds^2=4a^2\left[-dt^2+dr^2+dz^2-(\sinh^{4}(r)-\sinh^{2}(r))d\phi^2+2\sqrt{2}\sinh^{2}(r)dt d\phi)\right][/tex]
This is a solution to Einstein's Equations if we have ##a=\frac{1}{2\sqrt{2\pi\rho}}## and ##\Lambda =-\frac{1}{2a^2}##, where ##\Lambda## is the cosmological constant and ##\rho## is the density of the uniform dust with which we fill the universe.
It is often stated that this represents a rotating universe, but to me that is not immediately obvious, and so I want to prove it. To do so I'm considering an observer stationary with respect to the Godel coordinates at some point ##(r,\phi,z)## and equipped with a gyroscope. If the universe is rotating, the gyroscope should obviously precess. I want to find the rate of precession.
Homework Equations
Einstein's Equations and those mentioned throughout the post.
The Attempt at a Solution
The observer is stationary, so we have ##-1=\vec{u}\cdot\vec{u}=g_{\alpha\beta}u^{\alpha}u^{\beta}=g_{tt}(u^{t})^{2}=-4a^{2}(u^{t})^{2}##, so ##u^{t}=\frac{1}{\sqrt{2}a}## and so ##\vec{u}=(\frac{1}{\sqrt{2}a},0,0,0)##.
##a## is constant so clearly ##\vec{a}=0##, so the gyroscope is in free fall. This means its spin, ##\tilde{S}## undergoes parallel transport; ##\nabla_{\vec{u}}\tilde{S}=0##. Spin is a 1-form so we have:
[tex]\frac{S_{\alpha}}{dt}=\Gamma^{\gamma}_{\alpha\beta}S_{\gamma}u^{\beta}u[/tex]
This means that ##s_{z}## and ##s_{t}## are constant, and as we must have ##\tilde{S}\cdot\vec{u}=0##, ##s_{t}=0##.
The resulting ODES are;
[tex]\frac{dS_{r}}{dt}=\frac{\sqrt{2}}{a\sinh(2r)}S_{\phi}[/tex]
[tex]\frac{dS_{\phi}}{dt}=-\frac{1}{2\sqrt{2}a}\sinh(2r)S_{r}[/tex]
They have general solution:
[tex]S_{r}(t)=-2i(A\exp({\frac{it}{a\sqrt{2}}})-B\exp({-\frac{it}{a\sqrt{2}}}))[/tex]
[tex]S_{\phi}(t)=\sinh(2r)(A\exp({\frac{it}{a\sqrt{2}}})+B\exp({-\frac{it}{a\sqrt{2}}}))[/tex]
I'm not sure how to go from here to the angular velocity/momentum associated with the spin.
The spin 1-form is given component wise by ##s_{\alpha}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}J^{\beta\gamma}u^{\delta}##, which let's us use the above result to find the components of ##J^{\alpha\beta}##, and I believe I should find something along the lines of ##\frac{dS}{dt}=\Omega\times S## for some angular velocity ##\Omega##, but classical mechanics never was my strong suit so I'm not sure of the specifics. Honestly the whole concept of spin and precession in GR is doing my head in.
I think the result should be ##2\sqrt{\pi\rho}##, which I have seen while researching online, but yeah, I have no idea how to prove it.
I guess the crux of my question is how do we relate ##S_{\alpha}## to the precession velocity of the gyroscope of which it is the spin?
Any pointers, general or specific would really help, thanks.
##a## is constant so clearly ##\vec{a}=0##, so the gyroscope is in free fall. This means its spin, ##\tilde{S}## undergoes parallel transport; ##\nabla_{\vec{u}}\tilde{S}=0##. Spin is a 1-form so we have:
[tex]\frac{S_{\alpha}}{dt}=\Gamma^{\gamma}_{\alpha\beta}S_{\gamma}u^{\beta}u[/tex]
This means that ##s_{z}## and ##s_{t}## are constant, and as we must have ##\tilde{S}\cdot\vec{u}=0##, ##s_{t}=0##.
The resulting ODES are;
[tex]\frac{dS_{r}}{dt}=\frac{\sqrt{2}}{a\sinh(2r)}S_{\phi}[/tex]
[tex]\frac{dS_{\phi}}{dt}=-\frac{1}{2\sqrt{2}a}\sinh(2r)S_{r}[/tex]
They have general solution:
[tex]S_{r}(t)=-2i(A\exp({\frac{it}{a\sqrt{2}}})-B\exp({-\frac{it}{a\sqrt{2}}}))[/tex]
[tex]S_{\phi}(t)=\sinh(2r)(A\exp({\frac{it}{a\sqrt{2}}})+B\exp({-\frac{it}{a\sqrt{2}}}))[/tex]
I'm not sure how to go from here to the angular velocity/momentum associated with the spin.
The spin 1-form is given component wise by ##s_{\alpha}=\frac{1}{2}\epsilon_{\alpha\beta\gamma\delta}J^{\beta\gamma}u^{\delta}##, which let's us use the above result to find the components of ##J^{\alpha\beta}##, and I believe I should find something along the lines of ##\frac{dS}{dt}=\Omega\times S## for some angular velocity ##\Omega##, but classical mechanics never was my strong suit so I'm not sure of the specifics. Honestly the whole concept of spin and precession in GR is doing my head in.
I think the result should be ##2\sqrt{\pi\rho}##, which I have seen while researching online, but yeah, I have no idea how to prove it.
I guess the crux of my question is how do we relate ##S_{\alpha}## to the precession velocity of the gyroscope of which it is the spin?
Any pointers, general or specific would really help, thanks.