From the Monge transportation problem to Einstein's gravitation through Euler's Hydrodynamics

The quadratic Monge optimal transportation problem can be revisited in Euler's language of Hydrodynamics as was explained by Jean-David Benamou and the speaker about 20 years ago. It turns out that Einstein's theory of gravitation, at least in vacuum, can be treated in a very similar way as a kind of quadratic matrix-valued optimal transportation problem.