Is it possible to produce circularly polarized gravitational wave and how to determine if it is CW or CCW? I saw the GIFs showing both "plus" and "cross" linearly polarized gravitational wave using ring of dots, how about circularly polarized gravitational wave, if so how are they created and how can we determine if it is clockwise or counter clockwise circularly polarized or left/right handed if you will? If possible a GIF speaks a thousand words and put label on each components and the difference in phase too ;D
 A: Gravitational waves (GWs) can be "circularly polarised". In general, a GW can be expressed as a mixture of two waves with the $h_+$ and $h_\times$ tensor polarisation states. If one of these is zero or the two waves are in phase with each other then this corresponds to "linear polarisation". A ring of test masses will oscillate in a cruciform pattern, with its axes at a rotation angle determined by the relative amplitude of the two components.
A more general state for GWs is "elliptical polarisation", where the two components have arbitrary amplitudes and an arbitrary phase difference. For the special case that the two linear polarisation components are of equal amplitude but are $\pi/2$ out of phase then you get "circular polarisation". The ring of test masses oscillates in a cruciform pattern but the axes of the oscillation rotate as the wave passes through.
I found an illustrative gif produced by www.einstein-online.info. The animated image is too large to be uploaded but can be found here for example.

A circularly polarised gravitational wave would be observed coming from a binary system (e.g. a pair of black holes), where the system geometry has the orbital plane in the plane of the sky. i.e. a "face-on" orbit. An edge-on binary would produce linearly polarised GWs.
Since gravitational waves are sensitive to the difference in arm lengths between two perpendicular arms, they do have sensitivity to polarisation, though it is to some extent degenerate with respect to the position of the source with respect to the detector. A network of detectors can in principle determine both the direction and orbital plane inclination of the binary system - which in turn determines the degree of linear vs circular polarisation. However I'm not sure that the handedness (determined by whether the components are $\pm \pi/2$ out of phase) has been determined.
