Dynamical Chern-Simons Modified Gravity I: Spinning Black Holes in the Slow-Rotation Approximation (0902.4669v3)

Abstract: The low-energy limit of string theory contains an anomaly-canceling correction to the Einstein-Hilbert action, which defines an effective theory: Chern-Simons (CS) modified gravity. The CS correction consists of the product of a scalar field with the Pontryagin density, where the former can be treated as a background field (non-dynamical formulation) or as an evolving field (dynamical formulation). Many solutions of general relativity persist in the modified theory; a notable exception is the Kerr metric, which has sparked a search for rotating black hole solutions. Here, for the first time, we find a solution describing a rotating black hole within the dynamical framework, and in the small-coupling/slow-rotation limit. The solution is axisymmetric and stationary, constituting a deformation of the Kerr metric with dipole scalar "hair," whose effect on geodesic motion is to weaken the frame-dragging effect and shift the location of the inner-most stable circular orbit outwards (inwards) relative to Kerr for co-rotating (counter-rotating) geodesics. We further show that the correction to the metric scales inversely with the fourth power of the radial distance to the black hole, suggesting it will escape any meaningful bounds from weak-field experiments. For example, using binary pulsar data we can only place an initial bound on the magnitude of the dynamical coupling constant of $\xi^{1/4} \lesssim 10^{4} {\textrm{km}}$. More stringent bounds will require observations of inherently strong-field phenomena.

• The paper presents a stationary, axisymmetric solution for spinning black holes in dynamical Chern-Simons gravity that introduces scalar hair, deviating from the classical Kerr metric.
• Using slow-rotation and small-coupling approximations, the study quantifies corrections that weaken frame-dragging and shift ISCO positions for co-rotating and counter-rotating orbits.
• The findings have significant observational implications, informing constraints from pulsar data and guiding future gravitational wave template development for strong-field tests.

Analysis of Dynamical Chern-Simons Modified Gravity

The paper "Dynamical Chern-Simons Modified Gravity I: Spinning Black Holes in the Slow-Rotation Approximation" explores the implications of Dynamical Chern-Simons (CS) modified gravity, particularly in the context of rotating black holes. This work is situated within the framework of string theory, where CS modifications arise as necessary anomaly-canceling corrections. While many solutions from General Relativity (GR) persist in CS-modified frameworks, the Kerr metric does not, prompting investigation into alternative rotating black hole solutions.

The authors present a stationary, axisymmetric solution to the field equations in dynamical CS gravity, achieved through slow-rotation and small-coupling approximations. They find that the CS correction leads to a deformation of the Kerr metric characterized by scalar "hair," which affects geodesic motion. Specifically, this correction weakens the frame-dragging effect and shifts the inner-most stable circular orbit (ISCO) outwards for co-rotating geodesics and inwards for counter-rotating ones relative to the Kerr solution. Importantly, the correction scales as the inverse fourth power of radial distance, implying limited observational constraints from weak-field experiments. The paper uses pulsar data to place initial bounds on the dynamical coupling constant, suggesting that stronger constraints will necessitate observations of strong-field phenomena.

Theoretical and Practical Implications

The introduction of a dynamical scalar field in the CS framework distinguishes it from non-dynamical formulations, allowing the field to evolve rather than being statically prescribed. This dynamic nature alleviates concerns over the well-posedness of initial value problems, which are significant in non-dynamical versions. The authors' approach provides a more rigorous and self-consistent treatment of CS modifications, enabling the exploration of generalized relativistic corrections like those affecting ISCO locations.

The implications are profound for astrophysical systems, particularly those involving strong gravitational fields such as accretion disks around black holes. The modified ISCO dynamics might alter accretion processes, impacting observational signatures like X-ray emissions. Thus, predictions in this modified framework can potentially guide the interpretation of observational data from current and future astrophysical surveys.

Future Directions

Further research is needed to explore solutions without the slow-rotation approximation, which could provide comprehensive insights into the nature of fast-rotating black holes in dynamical CS gravity. Numerical techniques will likely be essential for addressing such scenarios. Another important avenue is the development of gravitational wave (GW) templates that incorporate CS corrections, which could significantly affect GW searches and parameter estimation efforts in observing campaigns by instruments like LIGO and Virgo.

Additionally, establishing concrete constraints on CS coupling constants from GW events, such as binary black hole mergers, is a crucial next step. Such constraints would not only enhance our understanding of CS gravity but also confirm or refute the viability of alternative gravitational theories.

Conclusion

The paper presents an important advancement in the paper of modified gravitational theories. By providing a first solution of a spinning black hole in dynamical CS modified gravity, the authors lay the groundwork for further theoretical and observational investigations. This research highlights the subtle yet significant modifications introduced by CS corrections, underscoring the potential for these theories to reveal new aspects of gravitational physics in strong fields. As such, it invites both theoretical scrutiny and empirical evaluation, potentially offering new insights into the fabric of spacetime as explored through high-energy astrophysical phenomena.