A smooth radial primitive can be returned to a weak end gauge

The mass-one weak metric bounds satisfy the end-normalization hypotheses; its auxiliary chart is for the same stationary field. The differentiated convention requires each Cartesian derivative of order $k$ to gain a factor $r^{-k}$ (or $R^{-k}$). Work in ZFC, carrying the normalization supplier's choice assumption; the local transition calculation below uses no additional choice principle. Stationary end normalization from quotient compactification Differentiated asymptotic bounds and parameter uniformity The Axiom of Choice

The supplied normalized Ernst potential and Killing two-form use the stated complex-linear Hodge convention. The vacuum Ernst identity gives $\mathrm{Re}\sigma =-g(T,T)$ up to a real constant on a connected end; normalization fixes that constant to zero. Real and imaginary parts and modulus have their ordinary complex meanings. Ernst sublevels and the Mars–Simon tensor interface Vacuum Killing and Ernst differential identities Real and imaginary parts, complex conjugation, and modulus Complex self-dual forms and metric-dependent contraction

For two charts of the same timelike stationary end whose quotient metrics are uniformly Euclidean-comparable and whose stationary connection covectors are $O(r^{-2})$ and $O(R^{-2})$, the unconditional time-gauge conclusion gives bounded $Dq,D{q}^{-1}$, comparable radii and $d{F}_{\ast }=\phi -q^{\ast }{\phi }^{\mathrm{\prime }}$. No interior product or Cauchy hypothesis is required for that conclusion. A decaying stationary time gauge gives bounded section displacement

A smooth Riemannian metric has a unique Levi–Civita connection with ${\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}{h}^{k\mathrm{\ell }}({\mathrm{\partial }}_i{h}_{j\mathrm{\ell }}+{\mathrm{\partial }}_j{h}_{i\mathrm{\ell }}-{\mathrm{\partial }}_{\mathrm{\ell }}{h}_{ij})$. The Christoffel transformation law applies to this same connection in overlapping smooth charts. Fundamental theorem of riemannian geometry Christoffel formula for the levi civita connection Christoffel symbol transformation law Here a Riemannian metric is a smooth positive-definite symmetric covariant two-tensor; its pullback is $(q^{\ast }{h}^{\mathrm{\prime }}{)}_x(v,w)=h_{q(x)}^{\mathrm{\prime }}(D{q}_{x}v,D{q}_{x}w)$. Riemannian metric and riemannian manifold Pullback of a riemannian metric as a tensor

The total chain rule and the one-variable sum, product and quotient derivative rules apply componentwise and along coordinate lines to smooth functions and finite matrices. Reciprocal factors are used only where the denominator is bounded away from zero. The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0

On a stationary product with ${\iota }_T\mathrm{\Theta }=dD$ and stationary $D$, the form $\mathrm{\Theta }-d{t}^{\mathrm{\prime }}\wedge dD$ descends to a smooth closed spatial two-form $b^{\mathrm{\prime }}$. A smooth spatial primitive $a^{\mathrm{\prime }}$ of $b^{\mathrm{\prime }}$ lifts to $W^{\mathrm{\prime }}=(c_0-D)d{t}^{\mathrm{\prime }}+a^{\mathrm{\prime }}$ with $d{W}^{\mathrm{\prime }}=\mathrm{\Theta }$, stationarity and $W^{\mathrm{\prime }}(T)=c_0-D$. Stationary lifting of a spatial primitive

A closed spatial two-form with coefficients $O_5(R^{-4})$ has the radial primitive with coefficients $O_5(R^{-3})$. If its differentiated bounds hold at every finite order, the same radial integral is smooth. Radial primitive and five differentiated decay bounds

Pullback of a one-form has coefficients $(q^{\ast }{a}^{\mathrm{\prime }}{)}_i=a_a^{\mathrm{\prime }}(q(x)){\mathrm{\partial }}_i{q}^a$. Exterior differentiation commutes with smooth pullback and satisfies $d^2=0$, so subtracting an exact smooth one-form preserves exterior derivative. The pullback of a differential form The exterior derivative commutes with pullback The exterior derivative squares to zero

The smooth Lorentzian metric and Killing field have their geometric meanings; the orientation is a supplied smooth choice of tangent-space orientation. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model Killing fields of a Lorentzian metric Oriented smooth manifolds and oriented charts The connection is the torsion-free metric-compatible connection of $g$, and Ricci-flat means its Ricci tensor vanishes. Metric compatibility, torsion freedom and the metric-selected connection class Lower $T$ by $T^{\mathrm{♭}}=g(T,\cdot )$ and take its covariant derivative using the induced tensor derivative. The inverse metric, musical isomorphisms and index raising Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields Stationarity means vanishing Lie derivative along $T$; in the supplied stationary charts its flow translates time, so this is independence of the tensor coefficients from that time coordinate. The Lie derivative of a tensor field If the original section is supplied as maximal, this means its mean curvature vanishes; retaining that same section retains the supplied property. Spacelike sections, maximality and lapse–shift data

Smooth $k$-forms are smooth sections of ${\bigwedge }^k{T}^{\ast }M$; a form is closed when $d\omega =0$ and exact when $\omega =d\eta$. A smooth differential k-form Closed and exact differential forms Exterior differentiation is the invariant vector-field derivative on forms, and ${\iota }_T\omega$ inserts $T$ in the first slot; for complex forms use the complex-linear extension fixed in [F2]. The exterior derivative by the invariant vector-field formula Interior product of a form by a vector field The wedge in $d{t}^{\mathrm{\prime }}\wedge dD$ is the pointwise wedge of alternating covectors. The wedge product of differential forms