Mentz114 said:
What is ##\hat{e}_z## ? I'm not used to seeing a z-coordinate in the Schwarzschild spacetime.
Technically, it's a unit vector in cylindrical coordinates, but in the equatorial plane (##\theta = \pi / 2##), these are basically the same as the standard spherical coordinates; ##\hat{e}_z## is then just the "vertical" unit vector, the one pointing in the ##\theta## direction. I should have clarified that, sorry.
(Also, the vorticity is really an antisymmetric 2nd-rank tensor in the 3-space of constant coordinate time; but in 3-D space of course you can always convert an antisymmetric 2nd-rank tensor to a vector. A vector in the "vertical", ##z## or ##\theta## direction, is equivalent to a 2nd-rank antisymmetric tensor in the ##r - \phi## plane.)
Mentz114 said:
From BillK's blog
This agrees with the ##\omega## I found in the local frame basis.
That ##\omega## is the angular velocity of rotation, not the vorticity. The vorticity is what Bill_K calls ##\Omega## in his blog post (actually his ##\Omega## is minus the vorticity because he is working in a rotating frame).
Mentz114 said:
I still don't understand if this number is related to the orbital period or the rotation of the axes of the transported frame, or both.
I more or less summarized what the different angular velocities mean in my post #117, but I'll briefly recap some of that here, this time referring to Schwarzschild spacetime (where de Sitter precession as well as Thomas precession is present):
(1a) As seen from infinity, an observer orbiting the hole geodesically at radius ##r## orbits with angular velocity ##\omega = \sqrt{M / r^3}##.
(1b) As seen by the observer orbiting the hole, if he keeps his line of vision pointed radially outward, the "fixed stars" at infinity are rotating about him with angular velocity ##- \gamma \omega## (i.e., in the opposite sense to the rotation seen at infinity in #1a), where ##\gamma = 1 / \sqrt{1 - 2M / r - \omega^2 r^2}##. The factor of ##\gamma## is due to time dilation; the ##2M / r## term is usually called gravitational time dilation, and the ##\omega^2 r^2## term is the usual time dilation due to relative motion.
(2a) If such an observer is part of a congruence of observers, all of whom are circling the hole with the same angular velocity ##\omega## (note that this means members of the congruence at smaller and larger radial coordinates will *not* be moving on geodesics--think of them as all being on a rotating disk similar to the flat spacetime case, though here the disk has to have a hole in the center where the central mass is), then the observer can define a set of spatial vectors by sticking out connecting rods to neighboring members of the same congruence. These rods will be fixed in place on the rotating disk, and so this set of spatial vectors will also rotate, as seen from infinity, with angular velocity ##\omega##. Call these spatial vectors the "congruence vectors".
(2b) As seen by the observer orbiting the hole at radius ##r##, once again, the "fixed stars" at infinity will rotate, relative to the congruence vectors, with angular velocity ##- \gamma \omega##.
(3b) If the orbiting observer also carries a set of gyroscopes, and uses them to define a second set of spatial vectors, then these vectors, if Newtonian physics were exactly correct, would always point in the same direction relative to infinity; i.e., they would not rotate at all relative to infinity. This would mean that, relative to the orbiting observer, the second set of vectors--call them the "gyro vectors"--would rotate relative to the congruence vectors with angular velocity ##- \gamma \omega##. The vorticity of a congruence is standardly defined as the angular velocity of rotation of the congruence vectors relative to the gyro vectors, as seen by the orbiting observer, so it would be ##\Omega_{Newton} = \gamma \omega##.
However, there are two relativistic effects that change this: Thomas precession and de Sitter precession. Thomas precession adds a retrograde component to the rotation of the gyro vectors, and de Sitter precession adds a prograde component; the net result is that the orbiting observer sees the gyro vectors rotate, relative to the congruence vectors, with angular velocity ##- \Omega = - \gamma^2 \omega \left( 1 - 3M / r \right)##. The vorticity of the congruence is minus this, so it is ##\Omega = \gamma^2 \omega \left( 1 - 3M / r \right)##.
This also means that, as seen by the orbiting observer, the "fixed stars" at infinity will rotate, relative to the gyro vectors, with angular velocity ##\Omega - \gamma \omega = \omega \gamma \left[ \gamma \left( 1 - 3M / r \right) - 1 \right] = \omega \gamma \left( \gamma - 1 - 3 \gamma M \ r \right)##.
(3a) As seen from infinity, the congruence vectors rotate, relative to the gyro vectors, with angular velocity ##\Omega / \gamma = \gamma \omega \left( 1 - 3M / r \right)##. That means the gyro vectors rotate, relative to infinity, with angular velocity ##\omega - \Omega / \gamma = \omega \left[ 1 - \gamma \left( 1 - 3M / r \right) \right] = - \left( \gamma - 1 \right) \omega + 3 \gamma \omega M / r##. (If we flip the sign and add a factor of ##\gamma## for time dilation, we obtain the angular velocity given at the end of #3b, as we should.) The first term is the Thomas precession and the second is the de Sitter precession.
) I still think this is a matter of terminology, and different people do have different preferences for that.
