  Non-Hermitian systems, with symmetric or antisymmetric Hamiltonians under the
parity-time ($\mathcal{PT}$) operations, can have entirely real eigenvalues.
This fact has led to surprising discoveries such as loss-induced lasing and
topological energy transfer. A merit of anti-$\mathcal{PT}$ systems is free of
gain, but in recent efforts on making anti-$\mathcal{PT}$ devices, nonlinearity
is still required. Here, counterintuitively, we show how to achieve
anti-$\mathcal{PT}$ symmetry and its spontaneous breaking in a linear device by
spinning a lossy resonator. Compared with a Hermitian spinning device,
significantly enhanced optical isolation and ultrasensitive nanoparticle
sensing are achievable in the anti-$\mathcal{PT}$-broken phase. In a broader
view, our work provides a new tool to study anti-$\mathcal{PT}$ physics, with
such a wide range of applications as anti-$\mathcal{PT}$ lasers,
anti-$\mathcal{PT}$ gyroscopes, and anti-$\mathcal{PT}$ topological photonics
or optomechanics.
