Equation 5 · Part 6 · A Strained Crystal Is a Designer Spacetime
=
=
What this part means
The expressions on both sides represent the same quantity under the stated assumptions.
Its job in the formula
The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.
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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 , the Pauli matrices acting on the electron’s internal degree of freedom, the momentum-space…
Learn the underlying idea
An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.
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Sources cited in the surrounding passage
These citations provide research context; check each source for the exact claim it supports.