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Slowly-Rotating Black Holes in Einstein-Dilaton-Gauss-Bonnet Gravity: Quadratic Order in Spin Solutions

Dimitry Ayzenberg, Nicolas Yunes

TL;DR

This work derives a new slowly rotating black hole solution in Einstein-Dilaton-Gauss-Bonnet gravity, accurate to quadratic order in spin and to leading order in the coupling, using a perturbative framework that builds on known nonspinning and linear-in-spin EDGB solutions. The authors decompose the metric and scalar field into tensor-harmonic components, solve the resulting Schwarzschild perturbation equations, and verify the solution against the EDGB field equations. They compute horizon, ergosphere, and quadrupole modifications, establish the absence of outside-horizon closed timelike curves, and analyze test-particle orbits, including energy, angular momentum, Kepler’s law, and ISCO shifts. A notable result is that the full solution is Petrov type I, unlike Kerr or lower-order EDGB cases, with implications for geodesic integrability and potential observational tests via accretion disk signatures and gravitational waves.

Abstract

We derive a stationary and axisymmetric black hole solution in Einstein-Dilaton-Gauss-Bonnet gravity to quadratic order in the ratio of the spin angular momentum to the black hole mass squared. This solution introduces new corrections to previously found nonspinning and linear-in-spin solutions. The location of the event horizon and the ergosphere are modified, as well as the quadrupole moment. The new solution is of Petrov type I, although lower order in spin solutions are of Petrov type D. There are no closed timelike curves or spacetime regions that violate causality outside of the event horizon in the new solution. We calculate the modifications to the binding energy, Kepler's third law, and properties of the innermost stable circular orbit. These modifications are important for determining how the electromagnetic properties of accretion disks around supermassive black holes are changed from those expected in general relativity.

Slowly-Rotating Black Holes in Einstein-Dilaton-Gauss-Bonnet Gravity: Quadratic Order in Spin Solutions

TL;DR

This work derives a new slowly rotating black hole solution in Einstein-Dilaton-Gauss-Bonnet gravity, accurate to quadratic order in spin and to leading order in the coupling, using a perturbative framework that builds on known nonspinning and linear-in-spin EDGB solutions. The authors decompose the metric and scalar field into tensor-harmonic components, solve the resulting Schwarzschild perturbation equations, and verify the solution against the EDGB field equations. They compute horizon, ergosphere, and quadrupole modifications, establish the absence of outside-horizon closed timelike curves, and analyze test-particle orbits, including energy, angular momentum, Kepler’s law, and ISCO shifts. A notable result is that the full solution is Petrov type I, unlike Kerr or lower-order EDGB cases, with implications for geodesic integrability and potential observational tests via accretion disk signatures and gravitational waves.

Abstract

We derive a stationary and axisymmetric black hole solution in Einstein-Dilaton-Gauss-Bonnet gravity to quadratic order in the ratio of the spin angular momentum to the black hole mass squared. This solution introduces new corrections to previously found nonspinning and linear-in-spin solutions. The location of the event horizon and the ergosphere are modified, as well as the quadrupole moment. The new solution is of Petrov type I, although lower order in spin solutions are of Petrov type D. There are no closed timelike curves or spacetime regions that violate causality outside of the event horizon in the new solution. We calculate the modifications to the binding energy, Kepler's third law, and properties of the innermost stable circular orbit. These modifications are important for determining how the electromagnetic properties of accretion disks around supermassive black holes are changed from those expected in general relativity.

Paper Structure

This paper contains 23 sections, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. EDGB Gravity
  3. Rotating Black Hole Solutions
  4. Approximation schemes
  5. BH solutions to $\mathcal{O}(\alpha'^2\chi^0)$ and $\mathcal{O}(\alpha'^2\chi')$
  6. BH solutions at $\mathcal{O}(\alpha'^2\chi'^2)$
  7. Scalar field
  8. Metric tensor
  9. Accuracy of the approximate solution
  10. Properties of the Solution
  11. Singularity, horizon, and ergosphere
  12. Lorentz signature
  13. Closed timelike curves
  14. Multipolar structure
  15. Petrov type
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Curves of zero velocity ($V_{{\hbox{\tiny eff}}}=0$) for the Kerr (top) and EDGB (bottom) metric with $\zeta = 0.1$. The red shaded regions show the allowed bound-orbit regions ($V_{{\hbox{\tiny eff}}}\geq0$) with $E=0.95$, $L_z=3m$, and $\chi=0.3$. The left panels corresponds to the region outside the horizon, while the right ones show the region inside the horizon. The thick black lines at $r/m=1.955$ (top) and $r/m=1.832$ (bottom) correspond to the location of the horizon for this example.