  We investigate three proposals of distance on the moduli space of metrics:
(1) a distance derived from the symplectic form of phase space, (2) a distance
obtained by moving BPS objects at small velocity, (3a) a distance proposed by
DeWitt and (3b) the distance used in the context of the generalised Swampland
distance conjecture. In particular, we calculate these distances on a space of
geometries that have the same asymptotics as the supersymmetric black hole in
five dimensions. These moduli spaces contain a locus where there exists an
infinite tower of massless particles, which emerges at finite distance
according to proposals (1) and (3a), and at infinite distance from proposal (2)
and (3b): distances (1) and (3) agree, and they disagree with distance (3b).
