Distinctness of boundary-eliminated solution branches for θ-dependent modes in 4D Einstein–Gauss–Bonnet slow rotation

Determine whether, in four-dimensional Einstein–Gauss–Bonnet gravity with the static black hole metric f(r)=h(r)=1+(r^2/(2α))(1−√(1+8αM/r^3) and scalar profile ψ′(r)=(√f−1)/(r√f), the l≥2 Legendre modes of a potentially θ-dependent frame-dragging function ω(r,θ) admit any regular, differentiable solution on the domain r∈[r_H,∞) that satisfies asymptotic flatness and horizon regularity. Equivalently, ascertain whether the solution branch eliminated by the boundary condition at spatial infinity is distinct from the solution branch eliminated by the boundary condition at the horizon, or whether they coincide—thereby leaving (or not) a regular branch for l≥2 and determining if ω can have θ-dependence in this theory.

Background

In Section 4, the authors analyze the possibility that the frame-dragging function ω may depend on the polar angle θ and decompose ω(r,θ) into Legendre modes ω_l(r). For l≥2, imposing regularity at infinity and at the horizon typically eliminates one of the two independent solution branches of the radial equation for ω_l, leaving the question of whether any regular, differentiable solution remains.

Specializing to four-dimensional Einstein–Gauss–Bonnet gravity, they give the static black hole solution f(r)=h(r)=1+(r^2/(2α))(1−√(1+8αM/r^3), the scalar profile ψ′(r)=(√f−1)/(r√f), and the functions Q(r)=r^4√(1+8αM/r^3) and ĤQ(r)= (r^2/f) [1 + (2α/r^2)(f−1−rf′)] that enter the l≥2 radial equation y_l'' + [(Q'^2 − 2QQ'')/(4Q^2) + (2 − l(l+1))(ĤQ/Q)] y_l = 0 (after the change of variables y_l=√Q ω_l). One branch is eliminated by asymptotic flatness at infinity; another is eliminated by differentiability at the horizon due to a logarithmic term.

The authors note they do not have a proof that these eliminated branches are distinct in the EGB case; if they coincide, a regular branch could remain, implying possible θ-dependence of ω. Establishing whether these branches are distinct (as in GR) or coincide would settle the existence of any regular θ-dependent modes for l≥2 in this theory.