Preservation of bifurcate horizons and surface gravity

Fix $q\in [0,1]$ and use the common pole and primitive in F7. Since $f(p)=0$, expansion of the modulus in the denominator definition gives $D_q=((1+q)-(1-q)f{)}^2+(1-q{)}^2{\psi }^2$ and $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2>0$. Thus $d_q-D_q=(1-q{)}^2(d_0-D_0)+4q(1-q)f$. Evaluating the supplied family on $T$, with $T^{\mathrm{♭}}(T)=-f$, proves $W_q(T)=d_q-D_q$. By F8 and $d{T}^{\mathrm{♭}}=2F$, its exterior derivative is $d{W}_q=4(1-q{)}^2((f-1)F+\psi {\star }_{g}F)-8q(1-q)F=4(1-q)(((1-q)f-(1+q))F+(1-q)\psi {\star }_{g}F)$, exactly the required derivative equation in F3.

The local flow of $T$ preserves its generator: differentiate its flow law in the orbit parameter, as in F5. It preserves $g$ because $T$ is Killing; explicitly, the flow law and Lie-derivative definition give $\frac{d}{dt}{\mathrm{\Psi }}_t^{\ast }g={\mathrm{\Psi }}_t^{\ast }({\mathcal{L}}_{T}g)=0$, where ${\mathrm{\Psi }}_t$ denotes this flow. Hence it preserves $T^{\mathrm{♭}}$. Since $W_0$ is stationary, the family in F7 is stationary as well. It is a smooth real one-form on the common neighborhood in F2. Together with step 1.1, this verifies all three defining Harrison one-form conditions on $D_q>0$, for the same normalized potential and pole as in F3. The stationary Lorentzian data, $T_p=0$, the range $q\in [0,1]$ and positivity in F2 therefore match every hypothesis of the Lorentzian-algebra supplier. It gives invertibility of $A_q$, the pullback identity and smooth Lorentzian signature for the metric in F3.

By the flow law and the tensor Lie-derivative definition in F5, $\frac{d}{dt}{\mathrm{\Phi }}_t^{\ast }Q={\mathrm{\Phi }}_t^{\ast }({\mathcal{L}}_{K}Q)$ on each local flow domain: differentiate ${\mathrm{\Phi }}_{t+h}^{\ast }Q={\mathrm{\Phi }}_t^{\ast }({\mathrm{\Phi }}_h^{\ast }Q)$ at $h=0$. Thus ${\mathcal{L}}_{K}Q=0$ makes ${\mathrm{\Phi }}_t^{\ast }Q=Q$ on the interval containing zero. In particular, $[K,T]=0$ implies ${\mathrm{\Phi }}_t^{\ast }T=T$. Since $K$ commutes with $T$ and preserves $g$, it preserves $T^{\mathrm{♭}}$ and $f$. It also preserves $D_q$ and $W_q$, so the displayed formula gives ${\mathcal{L}}_K{g}_q=0$. At $S$, $T$ is tangent to $S$. Hence $A_q$ maps $TS$ into itself; its invertibility makes its restriction to the two-dimensional $TS$ invertible. Consequently $g_q{\mathrm{\mid }}_{TS}=(D_q\mathrm{/}4)(A_q{\mathrm{\mid }}_{TS}{)}^{\ast }(g{\mathrm{\mid }}_{TS})$ is positive definite.

The flow carries its generator $K$ to itself, by differentiating the flow law in the orbit parameter. Since ${\mathrm{\Phi }}_t^{\ast }{W}_q=W_q$ by F2 and step 3.1, $W_q(K)\circ {\mathrm{\Phi }}_t=W_q(K)$. This scalar is therefore constant along the local $K$-flow and vanishes on $S$. Use the affine profiles in [F1]. Along a fixed generator on ${\mathcal{H}}^{+}$, $K=\kappa \underset{‾}{u}{\mathrm{\partial }}_{\underset{‾}{u}}$, so its orbit has $\underset{‾}{u}(\tau )=\underset{‾}{u}(0)e^{\kappa \tau }$ and tends to that generator's point on $S$ as $\tau \to -\mathrm{\infty }$. On ${\mathcal{H}}^{-}$ the profile $K=-\kappa u{\mathrm{\partial }}_u$ gives $u(\tau )=u(0)e^{-\kappa \tau }$ and the limit as $\tau \to +\mathrm{\infty }$. Restricting to short affine generator segments about $S$ keeps each approaching half-orbit in the common neighborhood. Smoothness therefore gives $W_q(K)=0$ on both branches. Since $T$ is tangent to each branch and $K$ is its $g$-normal, $g(K,T)=0$ there.

Let $\mathrm{\Phi }={\mathrm{\Phi }}_t$ be a local $K$-flow map. It is a $g_q$-isometry by step 3.1. Write $({\mathrm{\Phi }}_{\ast }X{)}_{\mathrm{\Phi }(p)}=d{\mathrm{\Phi }}_p{X}_p$ and define a connection on its source by ${\widehat{\mathrm{\nabla }}}_{X}Y={\mathrm{\Phi }}_{\ast }^{-1}({\mathrm{\nabla }}_{{\mathrm{\Phi }}_{\ast }X}^q({\mathrm{\Phi }}_{\ast }Y))$. The chain rule gives $({\mathrm{\Phi }}_{\ast }X)(h)\circ \mathrm{\Phi }=X(h\circ \mathrm{\Phi })$ for every smooth target function $h$. Applying it twice gives $[{\mathrm{\Phi }}_{\ast }X,{\mathrm{\Phi }}_{\ast }Y](h)\circ \mathrm{\Phi }=[X,Y](h\circ \mathrm{\Phi })$, hence $[{\mathrm{\Phi }}_{\ast }X,{\mathrm{\Phi }}_{\ast }Y]={\mathrm{\Phi }}_{\ast }[X,Y]$. The same chain rule transfers the affine-connection rules to $\widehat{\mathrm{\nabla }}$. Torsion freedom of ${\mathrm{\nabla }}^q$ and the bracket identity give ${\widehat{\mathrm{\nabla }}}_{X}Y-{\widehat{\mathrm{\nabla }}}_{Y}X=[X,Y]$. Since ${\mathrm{\Phi }}^{\ast }{g}_q=g_q$, metric compatibility of ${\mathrm{\nabla }}^q$ pulls back to $X{g}_q(Y,Z)=g_q({\widehat{\mathrm{\nabla }}}_{X}Y,Z)+g_q(Y,{\widehat{\mathrm{\nabla }}}_{X}Z)$. Uniqueness in F6, applied on this open four-dimensional source, gives $\widehat{\mathrm{\nabla }}={\mathrm{\nabla }}^q$, or $d\mathrm{\Phi }({\mathrm{\nabla }}_X^{q}Y)={\mathrm{\nabla }}_{{\mathrm{\Phi }}_{\ast }X}^q({\mathrm{\Phi }}_{\ast }Y)$.

Substitution in the metric formula yields $g_q(K,\cdot )=(D_q\mathrm{/}4)g(K,\cdot )$ on either branch. Thus $K$ is tangent and normal there for $g_q$, and the nonzero-generator part is unchanged. The Lorentzian metric and nonzero normal show that each branch is null; together with step 3.1 this makes each a $g_q$-Killing horizon. Differentiating the scalar metric formula, every term involving $g(K,K)$, $g(K,T)$ or $W_q(K)$ as an undifferentiated factor vanishes on the branch. Therefore $d(g_q(K,K))=(D_q\mathrm{/}4)d(g(K,K))$ there. The original signed equations give $-2\kappa g_q(K,\cdot )$ on ${\mathcal{H}}^{+}$ and $+2\kappa g_q(K,\cdot )$ on ${\mathcal{H}}^{-}$. At $S$ these equations are zero equations; the signed values are understood by their branch extensions.

By F1 and F4, $\mathrm{\Phi }$ fixes $S$ pointwise. Its isometry differential therefore preserves $TS$, its $g_q$-orthogonal complement and the corresponding normal projection. For tangent fields $X,Y$ along $S$, $B(X,Y)=({\mathrm{\nabla }}_X^{q}Y{)}^{\perp }$ is independent of their extensions: a field vanishing on $S$ has zero covariant derivative in tangent directions there. The affine-connection rules make $B$ tensorial, since the term $X(a)Y$ is tangent and disappears under normal projection. Projecting the connection identity of step 4.2 thus gives $d\mathrm{\Phi }B(X,Y)=B(d\mathrm{\Phi }X,d\mathrm{\Phi }Y)$. Its action on $TS$ is the identity. Its action on $TM{\mathrm{\mid }}_S\mathrm{/}TS$ has boost eigenvalues $e^{\pm \kappa t}$ by the original normal boost; the quotient identifies equivariantly with the $g_q$-normal plane because $S$ is $g_q$-spacelike. No nonzero normal vector is fixed by every such boost. Invariance therefore forces $B=0$, proving total geodesicity. Metric compatibility then gives $g_q({\mathrm{\nabla }}_X^{q}L,Y)=-g_q(L,{\mathrm{\nabla }}_X^{q}Y)=0$ for each null normal $L$ and tangent $X,Y$, so both null second fundamental forms vanish.