Regularity of the η term at the bifurcation surface for Λ ≠ 0

Establish whether the (n−1,1)-form η that appears in the on‑shell decomposition of the total symplectic potential in Einstein–Æther theory with nonzero cosmological constant is regular on the bifurcation surface H of a stationary black hole. Specifically, in the Λ ≠ 0 case where the total symplectic potential satisfies Θ_tot ≈ dA_Æ + η with dη = d[(Λ/(8πG_N)) ε_M], prove the regularity of η on H so that the horizon symplectic flux reduces to ι_ξ A_Æ as in the Λ = 0 case.

Background

In the symplectic analysis of Einstein–Æther theory, the bulk Lagrangian is on‑shell exact when the cosmological constant vanishes, allowing the total symplectic potential to be written as Θtot ≈ dAÆ. This makes the horizon symplectic flux integrable and expressible in terms of A_Æ.

When Λ ≠ 0, the authors find Θtot ≈ dAÆ + η with dη = d[(Λ/(8πG_N)) ε_M]. Their derivation of the first law and the integration of the symplectic flux at the bifurcation surface H relies on the assumption that η is regular there. They verified in an example (BTZ black hole) that neglecting η causes no inconsistency, but they lack a general proof of η’s regularity.

A proof of regularity (or a precise characterization of conditions ensuring it) would remove an assumption in the derivation of the first law and Smarr relations for Λ ≠ 0 backgrounds and would clarify the role of η in the covariant phase space framework for Einstein–Æther gravity.