Can we repulse metter by specially crafted gravitational waves? Assuming we could generate arbitrary gravitational wave patterns (somewhat like we can do with electromagnetic radiation now), what effects would we be able to achieve? In particular, would it be possible to repulse matter away from the beam source, instead of attracting it? Or to create the famous tractor beam that holds a remote object in a fixed relative position wrt the source?
 A: A graviton is a particle like any other particle. A gravitational wave is quantized into the graviton, though all we approximately understand is an asymptotic weak field limit. The Weyl tensor $C_{abcd}$ for conformal spacetime without matter defines a gravi-electric tensor $E_{bc} ~=~ C_{abcd}U^aU^d$ where this field tensor obeys 
$$
\frac{dV^b}{ds} ~=~ {E^b}_cV^c,
$$
which is a form of the geodesic deviation equation. The Hodge star operation on the Weyl tensor defines a gravi-magnetic tensor with $B_{bc} ~=~ ^*C_{abcd}U^aU^d$. These fields are traceless tensors corresponding to the two different directions of polarization in the quadrupole moment of a gravity wave. The traceless condition tells us the two polarizations can't be in the same direction.
This is classical, and we really only have glimpses of quantum gravity by various means.  Even just weak field naive quantization gives some ideas, and BMS theory is being worked around weak gravitons of this sort. String theory has some ideas. There is the $AdS_n\sim CFT_{n-1}$ correspondence that says gravitation in the anti-de Sitter spacetime (the bulk)  has the structure of a conformal gauge field at the boundary. There are also some curious developments that the more local a field is in the bulk the more nonlocal it is on the boundary, and visa versa. So there are hints of quantization here. But if anyone tells you they know what quantum gravity is they are either most likely lying or there is a chance they are the next ultra-great physicist along side Einstein, Bohr and the rest.
So what I am going to write might be taken with a grain of salt, maybe a whole tablespoon of it. The graviton will impart momentum from one particle to another. As spin-1 or $s=\hbar$ can flip its spin $m = 1$ projected along its momentum to $m=-1$ in the absorption or emission of a graviton. If the graviton has some momentum that is absorbed by the particle. If one could devise a sort of graviton beam then one could move particles and matter around with this, at least presumably. There are all sorts of questions concerning Cauchy data and the rest that on a classical level need to be addresse, not to mention how is is even possible to generate gravitons this way.
