Slowly rotating stars and black holes in dynamical Chern-Simons gravity

Chern-Simons (CS) modified gravity is an extension to general relativity (GR) in which the metric is coupled to a scalar field, resulting in modified Einstein field equations. In the dynamical theory, the scalar field is itself sourced by the Pontryagin density of the space-time. In this paper, the coupled system of equations for the metric and the scalar field is solved numerically for slowly rotating neutron stars described with realistic equations of state and for slowly rotating black holes. An analytic solution for a constant-density nonrelativistic object is also presented. It is shown that the black hole solution cannot be used to describe the exterior space-time of a star as was previously assumed. In addition, whereas previous analysis were limited to the small-coupling regime, this paper considers arbitrarily large coupling strengths. It is found that the CS modification leads to two effects on the gravitomagnetic sector of the metric: (i) Near the surface of a star or the horizon of a black hole, the magnitude of the gravitomagnetic potential is decreased and frame-dragging effects are reduced in comparison to GR. (ii) In the case of a star, the angular momentum $J$, as measured by distant observers, is enhanced in CS gravity as compared to standard GR. For a large coupling strength, the near-zone frame-dragging effects become significantly screened, whereas the far-zone enhancements saturate at a maximum value $\Delta J_{\max }\sim (M/R)J_{\mathrm{GR}}$. Using measurements of frame-dragging effects around the Earth by Gravity Probe B and the LAGEOS satellites, a weak but robust constraint is set to the characteristic CS length scale, ${\xi }^{1/4}\lesssim {10}^8\mathrm{km}$.

Figure 1

CS scalar field ${\vartheta }_1$ as a function of radius, for a constant-density nonrelativistic object. The curves are parametrized by the dimensionless coupling constant $\zeta$ defined in Eq. (31). For the ease of visualization, we have used dotted lines for $\zeta \le 1$ (the linear or quasilinear regime, where ${\vartheta }_1$ increases uniformly with $\zeta$), and solid lines for $\zeta >1$ where nonlinearity results into a damping of the scalar field in the near-zone and a plateauing of its asymptotic behavior at large radii.

Figure 2

Function $\omega (r)$ in CS gravity around a constant-density nonrelativistic object, for various values of the dimensionless coupling strength $\zeta$ defined in Eq. (31), marked as labels. The dotted curve corresponds to standard GR ($\zeta =0$). Because of the sharp boundary of a constant-density object, this function is continuous but not smooth, and the jump in its derivative at the surface is given by Eq. (28). All curves asymptote to $\omega \approx 2J/r^3$ for $r\gg R{\zeta }^{1/3}$ but differ significantly close to the surface. The vertical dashed lines show the locations probed by the LAGEOS [23] and GPB [22] satellites, which orbit the Earth at an altitude of 6000 km and 640 km, respectively, and are used to set constraints to the CS length scale in Sec. 7.

Figure 3

Neutron star mass-radius relation for the two equations of state used in this work: Lattimer-Swesty (LS220) [18] and Shen et al [19, 20].

Figure 4

Moment of inertia as a function of neutron star mass in standard GR, for the two equations of state used in this work: Lattimer-Swesty (LS220) [18] and Shen et al [19, 20].

Figure 5

Radial dependence of the CS scalar field ${\vartheta }_1(r)$ for a 1 $M_{\odot }$ neutron star described by the Shen et al. EOS (the LS220 EOS shows a very similar behavior and we have not plotted it here for more clarity). The curves are labeled by the dimensionless coupling strength $\zeta$ defined in Eq. (31). For $\zeta \lesssim 1$, the scalar field increases uniformly over the whole range $r>0$. For $\zeta \gtrsim 1$, the scalar field gets suppressed in the vicinity of the star; in the far-field, $\underset{r\to \infty }{lim}{r}^2{\vartheta }_1(r)$ asymptotes to a constant value for large values of $\zeta$. We have shown the two regimes $\zeta \le 1$ and $\zeta >1$ with different line styles for more visual clarity.

Figure 6

Radial dependence of the normalized metric coefficient $\omega (r)/\Omega$, for a 1 $M_{\odot }$ neutron star described by the Shen et al. EOS (the LS220 EOS shows a very similar behavior and the corresponding result was not plotted here). The curves are labeled by the dimensionless coupling strength $\zeta$ defined in Eq. (31). For $\zeta \gtrsim 1$ frame dragging becomes strongly suppressed in the vicinity of the star. At large radii $\omega \approx 2J/r^3$, where $J=J_{\mathrm{GR}}+\Delta J_{\mathrm{CS}}$ is enhanced with respect to the value in GR.

Figure 7

Change in the moment of inertia $\Delta I_{\mathrm{CS}}/I_{\mathrm{GR}}$ induced by the CS modification as a function of the CS coupling strength $\zeta$, for the two EOSs considered. We plot $\Delta I_{\mathrm{CS}}/I_{\mathrm{GR}}$ for the values $M/R=0.1$ and 0.15. The dashed lines show the analytic result derived in Sec. 4c for a constant-density nonrelativistic star (also for $M/R=0.1$ and 0.15 from bottom to top).

Figure 8

Relative change in the moment of inertia $\Delta I_{\mathrm{CS}}$ induced by the CS modification in the small-coupling limit, as a function of neutron star compactness $M/R$, for the two EOSs considered. We show the dimensionless quantity $\underset{\zeta \to 0}{lim}{R}^4/{\ell }_{\mathrm{cs}}^4\Delta I_{\mathrm{CS}}/I_{\mathrm{GR}}$. The dashed line shows the analytic result derived in Sec. 4c for a constant-density nonrelativistic star. The amplitude is off by approximately an order of magnitude but the overall behavior is relatively well reproduced.

Figure 9

Asymptotic value of the CS change in moment of inertia, $\max (\Delta I_{\mathrm{CS}}/I_{\mathrm{GR}})$, as a function of neutron star compactness. Also shown is our analytic approximation for nonrelativistic constant-density objects.

Figure 10

CS scalar field outside the horizon of a slowly rotating black hole, for various values of the dimensionless coupling parameter $\zeta \equiv \sqrt{2\pi }{\ell }_{\mathrm{cs}}^2/M^2$. For better clarity, we have shown the small-coupling regime $\zeta \le 1$ with dotted lines and the nonlinear regime $\zeta >1$ with solid lines.

Figure 11

Metric coefficient $\omega (r)$ outside the horizon a slowly rotating black hole, for various values of the dimensionless CS coupling parameter $\zeta \equiv \sqrt{2\pi }{\ell }_{\mathrm{cs}}^2/M^2$.