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 spacetime 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 enhancement saturate at a maximum value max(Delta J) ~ (M/R) J. 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 lengthscale, xi^{1/4} <~ 10^8 km.
Gravity, one of the four fundamental forces of nature, is elegantly described by Einstein's theory of general relativity (GR). Nearly a century after its discovery, GR has successfully passed the more and more subtle and precise tests that it has been submitted to (for a review, see for example Ref. [1]). Nevertheless, Einstein's theory is probably not the final word on gravity. We expect that a more fundamental theory unifying all forces should be able to describe not only gravity but also quantum phenomena that may take place, for example, at the center of black holes. Since gravity does seem to describe nature quite faithfully at the energy and length scales that are accessible to us, we expect that it may be the low-energy limit of such a fundamental theory. If this is the case, gravity should be described by an effective theory, for which the action contains higher-order curvature terms than standard GR, the effect of which can become apparent in strong gravity situations.
One such theory is modified Chern-Simons (CS) gravity (for a review on the subject, see Ref. [2]). In this theory, the metric is coupled to a scalar field through the Pontryagin density R R (to be defined below). The dynamical and nondynamical versions of the theory (which in fact are two separate classes of theories), then differ in the prescription for the scalar field. In nondynamical CS gravity, the scalar field is assumed to be externally prescribed. It is often taken to be a linear function of coordinate time (the so-called “canonical choice”), which selects a particular direction for the flow of time [3], and induces * yacine@ias.edu parity violation in the theory. Nondynamical CS gravity then depends on a single free parameter, which has been constrained with measurement of frame-dragging on bodies orbiting the Earth [4], and with the doublebinary-pulsar [5,6]. Nondynamical CS theory is quite contrived as a valid solution for the spacetime must satisfy the Pontryagin constraint R R = 0; it should therefore rather be taken as a toy model used to gain some insight in parity-violating gravitational theories. Dynamical CS gravity, which is the subject of the present paper, is a more natural theory where the scalar field itself is given dynamics (even though arbitrariness remains in the choice of the potential for the scalar field). The scalar field evolution equation is sourced by the Pontryagin density, which is non-vanishing only for spacetimes which are not reflection-invariant, as is the case in the vicinity of rotating bodies. Contrary to the nondynamical theory, though, dynamical CS gravity is not parity breaking, but simply has different solutions than GR for spacetimes which are not reflection-invariant (see discussion in Sec. 2.4 of Ref. [2]). Dynamical CS gravity has only recently received some attention, as it is more complex than the nondynamical version. Refs. [7,8] computed the CS correction to the Kerr metric in the slowrotation approximation. Ref. [9] studied the effect on CS gravity on the waveforms of extreme-and intermediatemass ratio inspirals. Recently, Ref. [10], proposed a solution for the spacetime inside slowly-rotating neutron stars, assuming that the solution outside the star was identical to that of a black hole of the same mass and angular momentum. All the aforementioned studies were done in the small-coupling limit, i.e. considering the CS modification as a perturbation around standard GR.
The two main points of the present work are as fol-arXiv:1110.5329v1 [astro-ph.HE] 24 Oct 2011 lows. First, we show that, in contrast with GR, the spacetime around a slowly-rotating relativistic star is different from that of a slowly-spinning black hole in CS gravity, owing to different boundary conditions (regularity conditions at the horizon for a black hole versus continuity and smoothness conditions at the surface of a star). Second, we solve for the CS modification to the metric and the scalar field simultaneously, in the fully-coupled case (nonperturbative with respect to the CS coupling strength), for a slowly-rotating star or black hole. Our motivation in doing so is that frame-dragging effects are difficult to measure and are not highly constrained; it is therefore still possible that they differ significantly from the GR prescription. The solutions obtained would be exact (modulo the slow-rotation approximation) if CS gravity were taken as an exact theory. If one asumes the CS action is only the truncated series expansion of an exact theory, then our solutions are only meaningful at the linear order in the CS coupling strength.
As shown in previous works, the CS correction only affects the gravitomagnetic sector of the metric at leading order in the slow-rotation limit. We find that for both stars and black holes, the CS correction leads to a suppression of frame-dragging at a distance of a few stellar radii, or a few times the black hole horizon radius. This suppression is perturbative in the small-
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