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.
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