Hayato Motohashi, Teruaki Suyama
We study linear perturbations around the static and spherically symmetric
spacetime for the gravitational theories whose Lagrangian depends on Ricci
scalar and the parity violating Chern-Simons term. By an explicit construction,
we show that Hamiltonian for the perturbation variables is not bounded from
below in general, suggesting that such a background spacetime is unstable
against perturbations. This gives a strong limit on a phenomenological
gravitational model which violates parity. We also provide a necessary and
sufficient condition for the theory to belong to a special class in which no
such instability occurs. For such theories, the number of propagating modes for
$\ell \ge 2$ is three, one from the odd and the other two from the even. Unlike
in the case of $f(R)$ theories, those modes are coupled each other, which can
be used as a distinctive feature to test the parity violating theories from
observations. All the modes propagate at the speed of light. No-ghost condition
and no-tachyon condition are the same as those in $f(R)$ theories. For the
dipole perturbations, the odd and the even modes completely decouple. The odd
mode gives a slowly-rotating BH solution whose metric is linearized in its
angular momentum. We provide an integral expression of such a solution. On the
other hand, the even mode propagates at the speed of light. For the monopole
perturbation, in addition to a mode which just shifts the mass of the
background BH, there is also one even mode that propagates at the speed of
light.
View original:
http://arxiv.org/abs/1107.3705
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