don't read unless you expert in Einstein-Hilbert action
The order parameter for SO(4N) to SO(4) spontaneous symmetry lowering in my previous paper is the symmetric tensor $h_{\mu\nu}$. Once one of its diagonal elements acquire the observation value the SO(4N) get lowered to SO(4N-1).
Just got two insights.
1st. Both metrics tensor $g_{\mu\nu}$ and graviton field $h_{\mu\nu}$ are good candidates to break the 4N-Lorentz symmetry and low the dimension. Just need to find solutions to the free Einstein equation with $g$ variation from subspace to subspace.
2nd. Above idea is wrong. The metrics of the space with the weak gravity is $g_00 = 1 + (2/c^2)U(r)$, $g_11=g_22=g_33=-1$, but the Lorentz invariance isn't broken. In other words we can always rotate the space even if it is curved.
3rd. I was surprised to find papers with gravitons breaking the Lorentz invariance (eg http://arxiv.org/abs/0905.0955v1)
Just got two insights.
1st. Both metrics tensor $g_{\mu\nu}$ and graviton field $h_{\mu\nu}$ are good candidates to break the 4N-Lorentz symmetry and low the dimension. Just need to find solutions to the free Einstein equation with $g$ variation from subspace to subspace.
2nd. Above idea is wrong. The metrics of the space with the weak gravity is $g_00 = 1 + (2/c^2)U(r)$, $g_11=g_22=g_33=-1$, but the Lorentz invariance isn't broken. In other words we can always rotate the space even if it is curved.
3rd. I was surprised to find papers with gravitons breaking the Lorentz invariance (eg http://arxiv.org/abs/0905.0955v1)