EOE-007 Find c using electrons

We now know that lightspeed c is a property not just of light but of spacetime itself, as literally the number of meters in a second as one trades off motion through traveler-time (cδτ/δt) with motion through space (δx/δt) from the vantage point of a flat spacetime observer. This tradeoff follows directly from the flat-spacetime version of Pythagoras' theorem (i.e. the metric equation) written as a square of distance per unit-change in map-time δt to give c2 = (cδτ/δt)2 + (δx/δt)2.  

When coordinate-speed δx/δt is much less than c, the free-particle dispersion relation (e.g. frequency versus wavenumber) plotted as kinetic energy K = (dt/dτ-1)mc2 versus momentum p = mδx/δτ for a given mass, would (as Newton expected) be a straight line on a log-log plot with slope α = 2 of the form K = p2/(2m). For any given value of p, all values of K would have been possible.

At high speeds, however, this line curves over to a mass-independent "photon line", of the form K = pc. This has a slope of 1 instead of 2 on same log-log plot (see below). That photon-line represents the bottom of the "lightspeed curtain", which requires instead that no objects may be found with K greater than the upper limit of pc. 

Practical values for electron energy/momenta in transmission electron microscopes (found in many universities and hospitals) lie on the curved part of this plot, which suggests that one may use their electrons to determine the value of spacetime constant c, which after all is a property of the spacetime for all objects (literally the number of meters in a second), rather than merely a property (like the speed) of light itself. 

Aside: We may soon give folks on-line a chance to do this experiment using our electron optics simulator, simply by allowing the user to change the electron accelerating voltage (now fixed at 300 keV). 

The poster for our presentation at the national 2017 Microscopy and Microanalysis conference follows:

References

• • Tavish L. E. Hill and P. Fraundorf (2015) "A Spacetime-Constant Experiment Using Electrons", American Association of Physics Teachers 2015 Summer Meeting Program Book, page 57 pdf.

  • David Osborn, Tianna McBroom, and P. Fraundorf (2017) "Sensitivity of TEM data on lightspeed to camera-length's voltage variation", Microscopy and Microanalysis 23:S1, 2308-2309 pdf discussion.