Pratyusava Baral
Compact orbiting binaries like the black hole binary system observed in GW150914 carry large amount of orbital angular momentum. The post-ringdown compact object formed after merger of such a binary configuration has only spin angular momentum, and this results in a large orbital angular momentum excess. One significant possibility is that the gravitational waves generated by the system carry away this excess orbital angular momentum. An estimate of this excess is made. Arguing that plane gravitational waves cannot possibly carry any orbital angular momentum, a case is made in this paper for gravitational wave beams carrying orbital angular momentum, akin to optical beams. Restricting to certain specific beam-configurations, we predict that such beams may produce a new type of strain, in addition to the longitudinal strains measured at aLIGO. Current constraints on post-ringdown spins, derived within the plane-wave approximation of gravitational waves, therefore stand to improve.
For detection of these gravitational events, the plane wave approximation is widely used. Unfortunately, this approximation throws away any information regarding angular momentum. In plane waves, the direction of momentum transfer (direction of Poynting vector) is parallel to the direction of propagation and thus carries no angular momentum. More specifically the total orbital angular momentum is proportional to ʃ(x^i k^j – x^jk^i) d^3x which is zero as d^3x is a rotationally invariant measure while the others are three vector components. Thus in other words, the plane wave approximation throws away information about angular momentum.
This to solve this problem, we need to look for solutions of the linearized vacuum Einstein’s equations which are not plane. To carry angular momentum, a radiation has to have a position dependent polarisation tensor which is equivalent to having a spacetime dependent phase term. Before looking for such solutions one needs to remember that GWs propagate with the speed of light (null) on 4D flat (Minkowskian) spacetime and oscillations are perpendicular to the direction of propagation (transverse). 4D Minkowskian spacetime can be written as a product of a 2D light cone and a 2D flat (Euclidean) surface. The polarization which lies on the Euclidean Surface has to have non-trivial spatial dependence for such a transverse wave to carry angular momentum. Its simple to write down a general solution after this decomposition. We refer to this solution as GW beams. It can be shown that these solutions are analogous to electromagnetic beams used in the field of laser optics.
GWs stretch and contract spacetime thus creating a strain which can be detected by a Michelson-like interferometer. For plane waves strain induced due to a gravitational wave is constant. If we consider a GW beam, strain is not constant or is a function of coordinates. What this means is a circular ring of particles which changes to an ellipse upon the passage of plane waves gets distorted. This distortion from an ellipse shall give a measure of OAM carried by the wave. This in turn shall be helpful in constraining the angular momentum of the source from a new handle thus improving present constraints.