Manuscript IX-Free-2@1 · IX-Free-2.tex · section · abb2397a

Nonzero charge forces a timelike boundary singularity

Gate S: boundary-charge separation › Nonzero charge forces a timelike boundary singularity · lines 962–980

IX-Free-2@1#eq:Bsurface

For the component convention in (eq:C), integration by parts gives

(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .

Its surface coefficient in (eq:boundary) is

B (b)^{00} = - \frac{1}{2} b_{0}, B (b)^{0 i} = \frac{1}{2} b_{i}, B (b)^{ij} = - \frac{1}{2} δ^{ij} b_{0} .

In particular, $B$ is injective on the boundary covector $b$ .

Displayed formulas in this passage

IX-Free-2@1#eq:Bsurface · line 972
$B (b)^{00} = - \frac{1}{2} b_{0}, B (b)^{0 i} = \frac{1}{2} b_{i}, B (b)^{ij} = - \frac{1}{2} δ^{ij} b_{0} .$
unlabeled · line 966
$(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .$