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From the work of Phong and Sturm in 2007, for a polarised projective manifoldand an ample test configuration, one can associate the geodesic ray ofplurisubharmonic metrics on the polarising line bundle using the solution ofthe Monge-Amp\`ere equation on an equivariant resolution of singularities ofthe test configuration. We prove that the Mabuchi chordal distance between thegeodesic rays associated with two ample test configurations coincides with thespectral distance between the associated filtrations on the section ring.  This gives an algebraic description of the boundary at the infinity of thespace of positive metrics, viewed - as it is usually done for spaces ofnegative curvature - through geodesic rays.

Geometry at the infinity of the space of positive metrics: test configurations, geodesic rays and chordal distances