Single-potential construction of the Harrison one-form family

PDF p. 9: (2.9) and definition of Θ0

Let $(M,g,N,T)$ be the stationary Lorentzian data of Stationary Lorentzian data and tensor conventions, and let $U$ carry the supplied normalized Ernst potential $\sigma =f+i\psi$ of Killing two-form and normalized Ernst notation, with $f=-g(T,T)$ and real $\psi$. Thus $F_{ab}={\mathrm{\nabla }}_a{T}_b$ is the Killing two-form and ${\star }_{g}F$ is its metric Hodge dual with the specified orientation. Write $T^{\mathrm{♭}}=g(T,\cdot )$.

For $q\in [0,1]$, define the real functions $D_q=\mathrm{\mid }q(1+\sigma )+(1-\sigma ){\mathrm{\mid }}^2$ on $U$. Let a supplied point $p\in U$, called a pole for this construction, satisfy $T_p=0$, and put $d_q=D_q(p)$. These are pointwise definitions from the supplied potential and point. Define the real two-form ${\mathrm{\Theta }}_0=4((f-1)F+\psi {\star }_{g}F)$.

A normalized primitive for this construction is a supplied smooth real one-form $W_0$ on $U$ satisfying both $d{W}_0={\mathrm{\Theta }}_0$ and $W_0(T)=d_0-D_0$. Here $d$ is exterior differentiation, and $W_0(T)$ is evaluation of the one-form on the vector field. For such a choice define $W_q=(1-q{)}^2{W}_0-4q(1-q)T^{\mathrm{♭}}$. In stationary coordinates with $T={\mathrm{\partial }}_t$, the time component is $(W_0{)}_0=d_0-D_0$. Neither a primitive nor one satisfying the normalization is asserted to exist by this definition.

**Given:**The data, pole and normalized primitive stipulated in the Definition.

Goal: Justify the displayed forms and the time-component convention, and record the resulting contraction of the family.

1.1 The Killing form and its Hodge dual are smooth real two-forms, and $f$ and $\psi$ are smooth real functions. Their displayed linear combination therefore defines a smooth real two-form ${\mathrm{\Theta }}_0$. The two conditions on $W_0$ are requirements on supplied data; they do not follow from the definition of ${\mathrm{\Theta }}_0$. The displayed linear combination of $W_0$ and $T^{\mathrm{♭}}$ gives a unique smooth real one-form $W_q$ for each specified $q$. [given, F1, F2, F3]

2.1 In a coordinate chart with $T={\mathrm{\partial }}_t$, evaluation of a one-form on $T$ is its time component, so the stipulated normalization gives $(W_0{)}_0=d_0-D_0$. At the pole $f(p)=0$. Expanding the modulus gives $D_q=((1+q)-(1-q)f{)}^2+(1-q{)}^2{\psi }^2$ and $d_q=(1+q{)}^2+(1-q{)}^2\psi (p{)}^2$. Hence $d_q-D_q=(1-q{)}^2(d_0-D_0)+4q(1-q)f$. Since $T^{\mathrm{♭}}(T)=-f$, the definition of $W_q$ yields $W_q(T)=d_q-D_q$. [given, F1, F4, F6, step 1.1, algebra] ∎

The cited article attributes the construction. The calculation above uses the stated data and definitions; it does not prove existence or the regularity and decay properties of a normalized primitive.

The cited article attributes the construction. The calculation above uses the stated data and definitions; it does not prove existence or the regularity and decay properties of a normalized primitive.