![v = 1√-[0,1,0,− 1], v = √1-[1,0,− 1,0],
+ 2 × 2](unification-of-gr-qm29x.svg)
The Weyl tensor of a gravitational wave has exactly two independent polarisation states (+ and ×). These emerge from the eigenstructure of ρtidal.
For the wave metric pair g(0) = [1.2,0.8,1.2,0.8], g(1) = [0.8,1.2,0.8,1.2], the eigenvalues of ρtidal are [−0.250,−0.250,−0.211,+0.712]. The two degenerate eigenvalues λ = −1∕4 correspond to the two graviton polarisation modes, with eigenvectors
the discrete analogues of alternating compression and expansion along perpendicular axes.
Corollary 4 (Two polarisations without spin-2 postulate). The two graviton polarisations emerge from the degeneracy of the symmetric off-diagonal density matrix. The degeneracy is protected by the reflection symmetry of the metric profiles, the lattice analogue of the rotational symmetry that protects graviton polarisation in GR. No spin-2 field is postulated.