addition

The passage around this formula

The tilt is where the geometry stops being a figure of speech. Once the tilt varies smoothly from place to place across a crystal, an electron propagating through it is, at leading order in the standard low-energy expansion, equivalent to a particle moving through a curved effective spacetime of the same mathematical form — a Painlevé–Gullstrand metric — that describes an object falling into a black hole, with the tilt itself playing the role of the metric’s off-diagonal, “flow” components. The paper writes the whole statement as a single Hamiltonian, fixing the Fermi velocity $v_F$ , the Pauli matrices $\boldsymbol{\sigma}$ acting on the electron’s internal degree of freedom, the momentum-space…

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