4.1.2. the energy vs. momentum half-plane

Plots of kinetic energy versus momentum for a free particle can provide students with their first example of a dispersion relation. By covering particles ranging in size from radio-wave photons to the visible universe, such plots can also illustrate why lasers will have much less recoil than projectile weapons of comparable energy, how the ``jump to lightspeed" even for a human without their spaceship might require the energy of 1000 atomic bombs, and how deBroglie's relation between momentum p and wavelength λ = h/p results in macroscopic wavelengths for only the smallest particles.

Such plots can also illustrate something else that Galileo and Newton likely did not imagine. Vector proper-velocity w like vector momentum p = mw = mγv has no size upper-limit. This is also true of scalar Lorentz-factor γ, which like kinetic energy K = (γ − 1)mc2 has no intrinsic upper-limit. Hence Newton likely figured that by adding kinetic energy to an object of appropriate mass m, objects may be found with any desired mix of kinetic energy K and translational momentum p.

As shown in the figure, however, Minkowski’s metric equation (cδτ)2 = (cδt)2 − δr·δr in effect lowers a curtain on kinetic-energy/momentum space by making only the lower right half of it accessible to moving objects. The log-log plot, which also has lines of constant mass and constant coordinate-velocity, thus provides students with an integrative view of kinetic-energy and momentum space (not to mention deBroglie wavelengths) for a wide range of objects in and beyond everyday experience.

Thus for example if one points these relationships out as early as possible in an intro-physics course (perhaps as early as the kinematics section on relative velocities if one takes the time to distinguish traveler-time τ from map-time t in defining Lorentz factor γ ≡ dt/dτ as in the previous section), then one may find opportunities again-and-again to refer back to it as new phenomena come up in the course. 

Example Problem 4.1.2a: How does the recoil of a U = 1.8[MJ] laser pulse (grey dot high up on the curtain) with the momentum of a speeding automobile of comparable energy.

A 1000 pound car at 70[mph] ~ 31[m/s] has a kinetic energy of about 0.2[MJ] but a momentum of about 14194[kg m/s], while the 1.8[MJ] laser pulse has a momentum of about U/c ~ 0.006 [kg m/s]. Hence the laser has the momentum of a 200[mg] object traveling at 70[mph]. That's an object with the mass of a pencil eraser without the pencil.

If you could reflect the laser pulse without absorbing it's kinetic energy, the 70 mph pencil-eraser impulse would be a half-million times lower than the impulse of a 70mph car with 1/9th the laser pulse's kinetic energy. Reflection therefore is a much better strategy for laser beams than it is for massive projectiles. 

Example Problem 4.1.2b: Can you estimate from this plot the energy it would take to accelerate you up to the relativistic proper-speed of one lightyear per traveler year?

In this case, the green diagonal line on the plot for an 80 kg human might help. 

You might also ask what electron energies e.g. in a TV tube would start to show effects of that relativistic curtain. For instance we use 300 keV electrons in a transmission electron microscope almost everyday. How does the speed of these compare to the speed-of-light in water?