$$\gamma = (1 - v^2/c_{\mathrm{eff}}^2)^{-1/2}$$

Equation 34 · Crossed Fields

Every Crystal Has Its Own Speed of Light

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A fourth fact transfers, but only with a correction the paper computes rather than sets aside. A quasiparticle wavepacket moving at speed v inside the linear window carries an internal clock — a spinor phase, a zitterbewegung oscillation — that runs slow by the familiar factor $\gamma$ = (1 - $v^2$/$c_{\mathrm{eff}}^2$)^{-1/2} : 1.15 at v = 0.5\,$c_{\mathrm{eff}}$ , 2.29 at 0.9\,$c_{\mathrm{eff}}$ . That dilation is real and, in principle, measurable, but only inside a window whose edge the band’s own curvature sets. Writing the correction as a term proportional to (k/$\Lambda_{\mathrm{curv}}$)^2 , the fractional error in $\gamma$ stays under one percent as long as the wavepacket’s momentum sits below…

Meanings in this article

• $c_{\mathrm{eff}}^2$: the square of $c_e$ff; the including.

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