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Two-center exact exterior with same-sign fluxes

Example

Let $d, m_{1}, m_{2} > 0$ have length units, set $a_{1} = (0, 0, 0)$ and $a_{2} = (d, 0, 0)$ , and take $σ = + 1$ . On $D = R^{3} ∖ {a_{1}, a_{2}}$ , put

$H (x) = 1 + \frac{m_{1}}{∣ x - a_{1} ∣} + \frac{m_{2}}{∣ x - a_{2} ∣} .$

The resulting datum is an exact source-free electrovac exterior. For either center $a_{A}$ , every coordinate sphere $S_{A} (r) = {x : ∣ x - a_{A} ∣ = r}$ with $0 < r < d$ , oriented by its outward Euclidean normal, has electric flux $m_{A}$ and magnetic flux zero. Precisely, fix any $s \in R_{x^{0}}$ and use the oriented embedded sphere $i_{s, S_{A} (r)} (S_{A} (r))$ in the spacetime, where $i_{s, S_{A} (r)} (x) = (s, x)$ as in Finite positive multicenter MP exterior and fluxes; the fluxes are the pullback integrals under that inclusion and are independent of $s$ . The centers are excluded from the manifold; this example asserts no center regularity, horizon extension, equilibrium stability, or observation.

Facts & Assumptions

Given: The distinct centers, positive lengths, sign, and the exact harmonic MP family.

[F1]

Positive harmonic conformastatic electrovac data defines $H$ , the metric, and the global potential on a positive harmonic exterior.

[F2]

Finite positive multicenter MP exterior and fluxes proves smoothness, harmonicity, source-free electrovac equations, and the flux $σ m_{A}$ for each isolated center.

[A1]

$d, m_{1}, m_{2} > 0$ and the two centers are distinct.

Verification

1.1givenF1F2A1construct

The Euclidean distances $∣ x - a_{1} ∣$ and $∣ x - a_{2} ∣$ are positive and smooth on $D$ . The formula is the $N = 2$ instance of the function in [F2], with positive coefficients and $σ = + 1$ , so it is positive and harmonic on $D$ and defines the metric-field data of [F1].

2.1givenF2A1step 1.1∎

Apply [F2] with $N = 2$ , $a_{1} \neq a_{2}$ , $m_{1}, m_{2} > 0$ , and $σ = + 1$ . For every displayed $0 < r < d$ , the closed radius- $r$ ball about either center excludes the other center. Thus [F2] applies to the time-slice embeddings specified in the Example, giving the geometric electrovac equations on the full punctured domain and $q_{e} = m_{A}$ , $q_{m} = 0$ on every such sphere; these values are independent of $s$ . The result is an exact mathematical example; none of the excluded centers is included as a regular point.

Two-center exact exterior with same-sign fluxes

Example

Facts & Assumptions

Verification

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