  We construct a holographic dual of the Schwinger-Keldysh effective action for
the dissipative low-energy dynamics of relativistic charged matter at strong
coupling in a fixed thermal background. To do so, we use a mixed signature bulk
spacetime whereby an eternal asymptotically anti-de Sitter black hole is glued
to its Euclidean counterpart along an initial time slice in a way to match the
desired double-time contour of the dual field theory. Our results are
consistent with existing literature and can be regarded as a fully-ab initio
derivation of a Schwinger-Keldysh effective action. In addition, we provide a
simple infrared effective action for the near horizon region that drives all
the dissipation and can be viewed as an alternative to the membrane paradigm
approximation.
