Open-string endpoint momentum flux ($J_m=e{T}^2{X}_m^{\prime }+u{P}_m$)

For the Nambu-Goto phase-space action, ${\dot{P}}_m={\partial }_{\sigma }{J}_m$ with $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$. Thus ${\dot{\mathcal{P}}}_m=J_m(\pi )-J_m(0)$. Free endpoint variations force $J_m=0$ individually. A Dirichlet boundary condition allows an external support or D-brane to absorb momentum in the fixed direction.

Free endpoint variations set the open string flux $e{T}^2{X}_m^{\prime }+u{P}_m$ to zero. In a boundary-adapted temporal gauge, freedom to vary time forces $u=0$ at the end. Nonzero lapse then gives $X^{\prime m}=0$, a Neumann boundary condition.

In a boundary-adapted temporal gauge, a free open string endpoint has $X^{\prime m}=0$. The Nambu–Goto phase-space constraints then imply $P^2=0$, while $\dot{X}=eP$ implies ${\dot{X}}^2=0$. For $X^0=t$, its spatial speed is one. This null boundary is compatible with a timelike interior worldsheet.