Analytic HTEE in Moving Plasmas and Its Transition

We investigate the holographic timelike entanglement entropy of a boosted $(1+1)$-dimensional plasma within the AdS$_3$/CFT$_2$ correspondence, considering a Lorentz-boosted BTZ black hole background. We solve the coupled extremal-surface equations analytically and obtain a closed-form expression for the HTEE. We show that below a critical boost velocity, the turning points of the extremal surface are purely imaginary, leading to the universal imaginary contribution to the entropy, while the real part acquires a nontrivial dependence on the boost velocity. Moreover, we identify a critical boost velocity at which the complex extremal-surface branch terminates. Beyond this critical velocity, the physically admissible extremal surfaces become purely real, signalling a transition from the holographic timelike entanglement entropy to a spacelike-like real saddle that is geometrically similar to the standard holographic entanglement entropy. The known static BTZ result is recovered in the zero-velocity limit, as expected.