$$c_{\mathrm{LR}} t$$

Equation 11 · Crossed Fields

Every Crystal Has Its Own Speed of Light

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where C and $\xi$ are constants fixed by the microscopic couplings and d(x,y) is the distance between the two sites on the lattice [ 1 ] . Outside the cone, where d(x,y) exceeds $c_{\mathrm{LR}}$ t , the right-hand side collapses toward zero; inside it, the inequality simply has nothing to say. The velocity $c_{\mathrm{LR}}$ plays exactly the causal role the vacuum speed c plays in relativity, with one difference that matters for everything that follows: it is a property of the material, not of empty space, and it changes from one lattice to the next.

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Equation 30 · Crossed Fields

Every Crystal Has Its Own Speed of Light

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Two facts transfer exactly, with no correction term anywhere, because they follow from locality alone rather than from any emergent Lorentz symmetry. Front-speed causality is simply the Lieb-Robinson bound restated: $c_{\mathrm{LR}}$ is a hard limit, and the statement that nothing propagates outside the cone is a proof about commutators, not an analogy borrowed from relativity. Effective causal order follows immediately — two operations separated by more than $c_{\mathrm{LR}}$ t commute up to an exponentially small tail, so a well-defined partial order of what can affect what exists inside the crystal for the same structural reason it exists in the vacuum. Massless-cone kinematics is the third…

Meanings in this article

• $c_{\mathrm{LR}}$: a hard limit, and the statement that nothing propagates outside the cone is a proof about commutators, not an analogy borrowed from relativity.

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