From S2 to RP2: The non-orientable Schwarzschild spacetime

Abstract

Einstein's field equations are local, so they retain their standard form on non-orientable manifolds, but the global structure of the solution need not be the same. We show that the orientability assumption in Hawking's horizon topology theorem is not redundant: dropping it produces a spacetime that satisfies the vacuum equations yet differs globally from the orientable case. Specifically, we construct the non-orientable Schwarzschild spacetime MnoSchw=M~Kr/⟨ιD⟩MnoSchw=M~Kr/⟨ιD⟩ by quotienting the Kruskal extension under the orientation-reversing isometry ιD:(V,U,θ,ϕ)↦(−V,−U,π−θ,ϕ+π)ιD:(V,U,θ,ϕ)↦(−V,−U,π−θ,ϕ+π) of Jacobian −1−1. A Z2Z2 quotient with an involution of Jacobian −1−1 is non-orientable; all previously studied involutions of the Kruskal extension---including the elliptic interpretation of Rindler and Gibbons, the RP3RP3 geon, and Socolovsky's antipodal identification---have Jacobian +1+1 and produce orientable quotients. Non-orientable black-hole instantons in the Euclidean path integral were studied by Chamblin and Gibbons, but these are Euclidean (Riemannian) manifolds, not Lorentzian spacetimes. To the best of our knowledge, the present construction provides the first genuinely non-orientable Lorentzian spacetime quotient of the Schwarzschild/Kruskal extension with a free orientation-reversing isometric involution in 3+1 dimensions, yielding an RP2RP2 Killing horizon cross-section. For MnoSchwMnoSchw, four categories of global features differ from the orientable case: topologically, the horizon is RP2RP2 rather than S2S2 and its area is halved (A=8πμ2A=8πμ2); causally, the spacetime is not time-orientable (w1(L)≠0w1(L)=0), which obstructs the standard global positive-frequency decomposition underlying the conventional Hawking-state derivation; in the mass sector, the T=0T=0 Cauchy slice is orientable despite the spacetime being non-orientable, so the standard ADM formula applies directly (the density Stokes' theorem provides the general framework for non-orientable slices, as we prove in Section 3.3); and structurally, the orientation-reversing surface ΣΣ does not exist (ιDιD acts freely).