Ordered horizon rods and twist-jump labels

An ordered finite vacuum rod datum consists of an integer $N\ge 1$, real numbers $z_i^{-}\le z_i^{+}$ for $1\le i\le N$ and $z_i^{+}<z_{i+1}^{-}$ for $1\le i<N$, and intervals ${\mathcal{H}}_i=[z_i^{-},z_i^{+}]$. A strict interval is called a nondegenerate horizon rod and an interval with equal endpoints a degenerate puncture $p_i=(0,0,z_i^{-})$. In this abstract rod datum these names label the intervals; their identification with geometric event horizons is a separate theorem.

The complementary axis intervals are ${\mathrm{\Gamma }}_1=(-\mathrm{\infty },z_1^{-})$, ${\mathrm{\Gamma }}_{j+1}=(z_j^{+},z_{j+1}^{-})$ for $1\le j<N$, and ${\mathrm{\Gamma }}_{N+1}=(z_N^{+},\mathrm{\infty })$. Supply real constants $c_1,\dots ,c_{N+1}$. The labels ${\mathcal{J}}_i=(c_{i+1}-c_i)\mathrm{/}4$ are the angular-momentum parameters in the HKWX convention. A shift of every $c_j$ by the same constant is a twist gauge change and leaves these labels unchanged. Their identification with Komar integrals of a geometric spacetime is not asserted by this definition.

Given: The integer, ordered endpoints and constants.

Goal: Verify the interval decomposition and gauge-invariant labels.

1.1 By the supplied interval definitions, every intervening interval is nonempty: its endpoints satisfy $z_j^{+}<z_{j+1}^{-}$ and their midpoint lies strictly between them. The two exterior intervals are semi-infinite. All complementary axis intervals together with the horizon intervals exhaust the real axis: $\mathbb{R}=({\bigcup }_{i=1}^N{\mathcal{H}}_i)\cup ({\bigcup }_{j=1}^{N+1}{\mathrm{\Gamma }}_j)$. Indeed, a real number below $z_1^{-}$ or above $z_N^{+}$ lies in an exterior interval; otherwise, if it belongs to no horizon interval, the ordered finite endpoints place it strictly between two consecutive horizon intervals. Equal endpoints give singletons and do not collapse either neighboring open interval because the adjacent strict inequalities remain. [given, F1, F2, algebra]

2.1 Each difference $c_{i+1}-c_i$ is a definite real number, and a common additive shift cancels in the difference. Division by four defines the stated labels without a choice of primitive. Their geometric interpretation requires reconstruction data in addition to these finite labels. [given, step 1.1, algebra] ∎