Equation 42 · Every Crystal Has Its Own Speed of Light
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subscript
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
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The third register fails, and it fails by proof rather than by degree. A crystal has a ground state and a lattice, and both pick out a rest frame: no continuous symmetry carries the crystal’s description in one inertial frame into its description in another, because the lattice Hamiltonian is manifestly not invariant under a boost at any speed, including . Relativity of simultaneity fails along with it outside the linear window, and so does the idea of one universal energy-momentum relation: a crystal has many bands, each with its own dispersion and its own emergent speed, so a magnon, a phonon, and a Dirac electron sharing one lattice do not share a light cone. “The” speed…
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The third register fails, and it fails by proof rather than by degree. A crystal has a ground state and a lattice, and both pick out a rest frame: no continuous symmetry carries the crystal’s description in one inertial frame into its description in another, because the lattice Hamiltonian is manifestly not invariant under a boost at any speed, including . Relativity of simultaneity fails along with it outside the linear window, and so does the idea of one universal energy-momentum relation: a crystal has many bands, each with its own dispersion and its own emergent speed, so a magnon, a phonon, and a Dirac electron sharing one lattice do not share a light cone. “The” speed of light of a material is, on this reading, a category error — there are as many as there are bands, and the census’s own table already shows it. What the crystal loses is not a technicality. It is the principle of relativity itself, and losing it is what makes the crystal an ether.
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