  Weyl particles exhibit chiral transport property under external curved
space-time geometry. This effect is called chiral gravitational effect, which
plays an important role in quantum field theory. However, the absence of real
Weyl particles in nature hinders the observation of such interesting phenomena.
In this paper, we show that chiral gravitational effect can be manifested in
Weyl metamaterials with spatially controlled nonlocality. This inhomogeneous
modulation results in a spatially dependent group velocity in the Weyl cone
dispersion, which is equivalent to introducing a curved background space-time
(or gravitational field) for Weyl pseudo-spinors. The synthetic gravitational
field leads to the quantization of energy levels, including chiral zeroth order
energy modes (or simply chiral zero modes) that determine the chiral transport
property of pseudo-spinors. The inhomogeneous Weyl metamaterial provides an
experimentally realizable platform for investigating the interaction between
Weyl particles and gravitational field, allowing for observation of chiral
gravitational effect in table-top experiments.
