Stability of relativistic stars with scalar hairs. (arXiv:2007.09864v1 [gr-qc])
<a href="http://arxiv.org/find/gr-qc/1/au:+Kase_R/0/1/0/all/0/1">Ryotaro Kase</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Kimura_R/0/1/0/all/0/1">Rampei Kimura</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Sato_S/0/1/0/all/0/1">Seiga Sato</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Tsujikawa_S/0/1/0/all/0/1">Shinji Tsujikawa</a>
We study the stability of relativistic stars in scalar-tensor theories with a
nonminimal coupling of the form $F(phi)R$, where $F$ depends on a scalar field
$phi$ and $R$ is the Ricci scalar. On a spherically symmetric and static
background, we incorporate a perfect fluid minimally coupled to gravity as a
form of the Schutz-Sorkin action. The odd-parity perturbation for the
multipoles $l geq 2$ is ghost-free under the condition $F(phi)>0$, with the
speed of gravity equivalent to that of light. For even-parity perturbations
with $l geq 2$, there are three propagating degrees of freedom arising from
the perfect-fluid, scalar-field, and gravity sectors. For $l=0, 1$, the
dynamical degrees of freedom reduce to two modes. We derive no-ghost conditions
and the propagation speeds of these perturbations and apply them to concrete
theories of hairy relativistic stars with $F(phi)>0$. As long as the perfect
fluid satisfies a weak energy condition with a positive propagation speed
squared $c_m^2$, there are neither ghost nor Laplacian instabilities for
theories of spontaneous scalarization and Brans-Dicke (BD) theories with a BD
parameter $omega_{rm BD}>-3/2$ (including $f(R)$ gravity). In these theories,
provided $0<c_m^2 le 1$, we show that all the propagation speeds of
even-parity perturbations are sub-luminal inside the star, while the speeds of
gravity outside the star are equivalent to that of light.
We study the stability of relativistic stars in scalar-tensor theories with a
nonminimal coupling of the form $F(phi)R$, where $F$ depends on a scalar field
$phi$ and $R$ is the Ricci scalar. On a spherically symmetric and static
background, we incorporate a perfect fluid minimally coupled to gravity as a
form of the Schutz-Sorkin action. The odd-parity perturbation for the
multipoles $l geq 2$ is ghost-free under the condition $F(phi)>0$, with the
speed of gravity equivalent to that of light. For even-parity perturbations
with $l geq 2$, there are three propagating degrees of freedom arising from
the perfect-fluid, scalar-field, and gravity sectors. For $l=0, 1$, the
dynamical degrees of freedom reduce to two modes. We derive no-ghost conditions
and the propagation speeds of these perturbations and apply them to concrete
theories of hairy relativistic stars with $F(phi)>0$. As long as the perfect
fluid satisfies a weak energy condition with a positive propagation speed
squared $c_m^2$, there are neither ghost nor Laplacian instabilities for
theories of spontaneous scalarization and Brans-Dicke (BD) theories with a BD
parameter $omega_{rm BD}>-3/2$ (including $f(R)$ gravity). In these theories,
provided $0<c_m^2 le 1$, we show that all the propagation speeds of
even-parity perturbations are sub-luminal inside the star, while the speeds of
gravity outside the star are equivalent to that of light.
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