  Recently, pulsar timing array (PTA) collaborations announced evidence for an
isotropic stochastic gravitational wave (GW) background. The origin of the PTA
signal can be astrophysical or cosmological. In the latter case, the so-called
secondary scalar-induced GW scenario is one of the viable explanations, but it
has a potentially serious issue of the overproduction of primordial black holes
(PBHs) due to the enhanced curvature perturbation. In this letter, we present a
new interpretation of the PTA signal. Namely, it is originated from an extra
spectator tensor field that exists on top of the metric tensor perturbation. As
the energy density of the extra tensor field is always subdominant, it cannot
lead to the formation of PBHs. Thus our primordial-tensor-induced scenario is
free from the PBH overproduction issue.
