Periodic isometries and invariant normal averaging

Work in ZFC. Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$: $N$ is a smooth timelike field, and a timelike vector $v$ is future-directed when $g(v,N)<0$. A normalized circle action by isometries is a smooth global flow $\varphi :\mathbb{R}\times M\to M$ of a smooth vector field $Z$, with ${\varphi }_{s+2\pi }={\varphi }_s$ for all $s$, such that $2\pi$ is the least positive common period and ${\varphi }_s^{\ast }g=g$ for every $s$. The generator is $Z_p={\frac{d}{ds}\mid }_{s=0}{\varphi }_s(p)$. A fixed point is a point $p$ for which ${\varphi }_s(p)=p$ for every $s$; consequently $Z_p=0$. The least-period condition concerns the whole action, so individual orbits may have shorter periods or be fixed.

Let $S\subseteq M$ be an embedded submanifold invariant under the action, meaning ${\varphi }_s(S)=S$ for all $s$. A smooth field along $S$ is an assignment $p\mapsto n_p\in T_{p}M$ such that, in every ambient slice chart $(U,x)$ with $S\cap U$ given by $x^{k+1}=\cdots =x^4=0$, the coefficients in $n_p={\sum }_{a=1}^4{n}^a(p){\mathrm{\partial }}_{x^a}{\mathrm{\mid }}_p$ are smooth functions of the slice variables $(x^1,\dots ,x^k)$. Such a field is normal when $g_p(n_p,v)=0$ for every $p\in S$ and $v\in T_{p}S$. Suppose $n$ is future timelike at every point. Define its circle average and normalized invariant normal by

${\bar{n}}_p=\frac{1}{2\pi }{\int }_0^{2\pi }d{\varphi }_s{\mathrm{\mid }}_{{\varphi }_{-s}(p)}\left(n_{{\varphi }_{-s}(p)}\right)ds,{\widehat{n}}_p=\frac{{\bar{n}}_p}{\sqrt{-g_p({\bar{n}}_p,{\bar{n}}_p)}}\mathrm{.}$

Here $d{\varphi }_s{\mathrm{\mid }}_q:T_{q}M\to T_{{\varphi }_s(q)}M$ is the differential. The isometry condition uses tensor pullback: $({\varphi }_s^{\ast }g{)}_q(u,v)=g_{{\varphi }_s(q)}(d{\varphi }_s{\mathrm{\mid }}_{q}u,d{\varphi }_s{\mathrm{\mid }}_{q}v)$. The integral is taken in the fixed finite-dimensional vector space $T_{p}M$: for a continuous curve $v(s)={\sum }_{a=1}^4{v}^a(s)e_a$ in any fixed basis, set ${\int }_0^{2\pi }v(s)ds={\sum }_{a=1}^4({\int }_0^{2\pi }{v}^a(s)ds)e_a$, using scalar Riemann integrals. An invariant field along $S$ is one satisfying $d{\varphi }_t{\mathrm{\mid }}_p(v_p)=v_{{\varphi }_t(p)}$ for all $p\in S$ and $t\in \mathbb{R}$. The construction assumes the supplied field $n$; it asserts no existence of such a field.

The Axiom of Choice

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Local and global flows generated by a vector field

Embedded submanifolds and slice charts

The differential of a smooth map

The pullback of a covariant tensor field

A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion

Every continuous scalar component on a closed finite interval is bounded and Riemann integrable. The tagged-sum characterization identifies the coordinatewise vector integral with the limit of the vector sums ${\sum }_{i}v({\xi }_i){\mathrm{\Delta }}_i$ as the mesh tends to zero. A fixed linear map commutes with this limit: its matrix acts by finite sums of constant multiples of convergent scalar components. Applying this to a change-of-basis matrix proves that the integral is independent of the chosen basis; applying it to any linear functional or differential proves commutation with that map. Scalar order inequalities pass to the limit from the positive-weight sums. Likewise, the finite-sum triangle inequality gives $\mathrm{\mid }\int w(s)ds\mathrm{\mid }\le \int \mathrm{\mid }w(s)\mathrm{\mid }ds$ for a continuous Euclidean-vector-valued $w$. All these scalar integrands are continuous, so their integrability is supplied. These sum arguments also give $\mathrm{\mid }\int h\mathrm{\mid }\le (b-a)\sup \mathrm{\mid }h\mathrm{\mid }$ on $[a,b]$. Translating tagged partitions proves invariance under translation of the integration variable, and joining tagged partitions at a split point proves additivity across that point; the componentwise construction carries both properties to vector integrals. A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion The Darboux and Riemann definitions agree: a bounded f on [a,b] is Darboux integrable with integral I if and only if for every real ε > 0 there is a real δ > 0 such that |S(f,P,ξ) - I| < ε for every tagged partition of mesh below δ The inner-product norm is definite, homogeneous, and satisfies the triangle inequality

The coordinate formula for the differential shows that the slice-coordinate definition of smoothness is unchanged on overlaps: the new components are the old components multiplied by the smooth Jacobian matrix of the chart transition. It also shows that $v(s,p):=d{\varphi }_s{\mathrm{\mid }}_{{\varphi }_{-s}(p)}{n}_{{\varphi }_{-s}(p)}$ has smooth components jointly in $(s,p)$. This is local near each $(s,p)$; the moving source point may use different charts for different $s$, while the output is expressed in one chart around $p$. The flow law and differential chain rule make each ${\varphi }_s$ a diffeomorphism with inverse ${\varphi }_{-s}$, and its restriction to $S$ maps tangent spaces onto tangent spaces. For $q={\varphi }_{-s}(p)$ and $X\in T_{p}S$, the pullback formula gives $g_p(v(s,p),X)=g_q(n_q,d{\varphi }_{-s}{\mathrm{\mid }}_{p}X)=0$. The integrand at $s=0$ is $n_p$. As $s$ varies continuously, $v(s,p)$ is timelike by isometry and remains in the same timelike component as $n_p$, hence in the future cone. Commutation with the linear functional $g_p(\cdot ,X)$ makes ${\bar{n}}_p$ normal. Coordinate formula for the differential The differential of a smooth map The chain rule for differentials of smooth maps The pullback of a covariant tensor field Local and global flows generated by a vector field Embedded submanifolds and slice charts Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

In a future-oriented orthonormal basis at $p$, write $v(s,p)=(a(s),b(s))$ with $b(s)\in {\mathbb{R}}^3$. The future-cone description gives $a(s)>\mathrm{\mid }b(s)\mathrm{\mid }$. The continuous positive function $a(s)-\mathrm{\mid }b(s)\mathrm{\mid }$ attains a positive minimum ${\delta }_p$ on $[0,2\pi ]$. The integral order and norm inequalities just established give $(2\pi {)}^{-1}\int a\ge (2\pi {)}^{-1}\int \mathrm{\mid }b\mathrm{\mid }+{\delta }_p\ge \mathrm{\mid }(2\pi {)}^{-1}\int b\mathrm{\mid }+{\delta }_p$. Hence ${\bar{n}}_p$ is future timelike, the positive square root in the normalization exists, and $g(\widehat{n},\widehat{n})=-1$. No uniform lower bound in $p$ is required. The timelike set has exactly two convex components, described in orthonormal coordinates Continuous functions on a closed interval attain their extrema The Darboux and Riemann definitions agree: a bounded f on [a,b] is Darboux integrable with integral I if and only if for every real ε > 0 there is a real δ > 0 such that |S(f,P,ξ) - I| < ε for every tagged partition of mesh below δ The inner-product norm is definite, homogeneous, and satisfies the triangle inequality

On a compact coordinate rectangle in $S$, repeated differentiation under the integral sign, one slice variable at a time, shows that every partial derivative of the components of $\bar{n}$ is the integral of the corresponding derivative of $v(s,p)$. The derivatives of the integrand are continuous, hence uniformly continuous, on the rectangle times $[0,2\pi ]$, and their integrals are continuous: a uniform change of at most $\epsilon$ in a component changes its integral by at most $2\pi \epsilon$, by the sum bound above. Thus $\bar{n}$ is smooth. Since $-g(\bar{n},\bar{n})$ is a smooth positive function, its positive square root and the normalized field are smooth as well. Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous Coordinate formula for the differential Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

For $p^{\mathrm{\prime }}={\varphi }_t(p)$, commutation of a fixed differential with integration and the flow chain rule give $d{\varphi }_t{\mathrm{\mid }}_p{\bar{n}}_p=(2\pi {)}^{-1}{\int }_0^{2\pi }v(s+t,p^{\mathrm{\prime }})ds$. To see that a full-period integral is independent of its starting point, reduce $t$ modulo $2\pi$ to $r\in [0,2\pi )$, split the interval $[r,r+2\pi ]$ at $2\pi$, and translate its second piece by $-2\pi$; periodicity identifies that piece with $[0,r]$. The integral therefore equals ${\bar{n}}_{p^{\mathrm{\prime }}}$. Isometry preserves the normalization factor, so $\widehat{n}$ is invariant too. The chain rule for differentials of smooth maps Local and global flows generated by a vector field The Darboux and Riemann definitions agree: a bounded f on [a,b] is Darboux integrable with integral I if and only if for every real ε > 0 there is a real δ > 0 such that |S(f,P,ξ) - I| < ε for every tagged partition of mesh below δ The pullback of a covariant tensor field

The ZFC convention includes the choice assumptions inherited from the stated manifold interface. The Axiom of Choice