ℋ-Space Metric

D $ℋ$ -Space Metric

Update

In the following the derivation of the $ℋ$ -space metric, is given.

We begin with the cut function, $-- -- uB = Z (ξa(τ),ζ,ζ) = X (τ,ζ,ζ)$ that satisfies the good cut eqation $-- ∂2Z = σ(Z, ζ,ζ)$ . The ( $-- ζ,ζ)$ are (for the time being) completely independent of each other though $-- ζ$ is to be treated as being “close” the complex conjugate of $ζ$ .

(Later we will introduce $∗ ζ$ instead of $ζ$ via

for the purpose of simplifying an integration.)

Taking the gradient of $a -- Z (z ,ζ,ζ)$ , multiplied by an arbitrary four vector $a v$ , (i.e., $a V = v Z,a$ ), we see that it satisfies the linear Good Cut equation,

Let $V0$ be a particular solution, and assume for the moment that the general solution can be written as

with the four components of $l∗a$ to be determined. Substituting Equation (381

) into the linearized GCE equation we have

2 ∗ ∗ ∂ (V0la) = σ,Z V la, ∂(l∗a∂ (V0) + V0 ∂l∗a) = σ,Z V l∗a, ∗ 2 ∗ 2∗ ∗ la∂ (V0) + 2∂V0 ∂la + V0 ∂ la = σ,Z V0la, 2∂V0 ∂l∗a + V0 ∂2l∗a = 0, ∗ 2 2∗ 2V0 ∂V0∂la + V0 ∂ la = 0, ∂V 20 ∂l∗a + V02∂2l∗a = 0, ∂(V 2∂l∗) = 0, 0 a

which integrates immediately to

The $∗ m a$ are three independent $l = 1$ , $s = 1$ functions. By taking linear combinations they can be written as

∗ b b m a = Tamˆb = Ta ∂ˆlb

where $ˆla$ is our usual $√- ( - --- -) ˆla = 22- 1,− ζ1++ζζζ,− i(ζ1+−ζζζ), 11−+ζζζζ$ . The coefficients $Tab$ are functions only of the coordinates, $za$ .

Assuming that the monople term in $2 V$ is sufficiently large so that it has no zeros and then by rescaling $V$ we can write $V −2$ as a monopole plus higher harmonics in the form