= D-brane supergravity solution
{c}
{title2=$ds_s^2=H^{-1/2}dx_\parallel^2+H^{1/2}dx_\perp^2$}
For the asymptotically flat extremal D$p$ solution with $p<7$, $H=1+Q/r^{7-p}$ is harmonic away from the source and $e^\Phi=g_sH^{(3-p)/4}$. The electric potential can be chosen as $C_{0\cdots p}=g_s^{-1}(H^{-1}-1)$; magnetic descriptions use the dual flux, and a <D3-brane> has self-dual five-form flux. An equally oriented probe has DBI density $-\mu_p g_s^{-1}H^{-1}$ and Wess-Zumino density $+\mu_p g_s^{-1}(H^{-1}-1)$, whose sum is position independent. This proves the corresponding <parallel D-brane no-force identity> in the classical background. The supergravity approximation requires controlled curvature and coupling; it does not resolve every brane core.
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