Minkowskara

Minkowskara, a twin-faced glass golem of the Paradox Parlour, carries light through a glass slab, and her two faces give two answers for its momentum.

The Physics

How much momentum does light carry inside a medium of refractive index n? Hermann Minkowski (1908) gave a momentum that grows with n; Max Abraham (1909–1910) gave one that shrinks as 1/n. Both follow from consistent electrodynamics, and experiments have seemed to support each in different setups.

p_Minkowski = nħω/c, p_Abraham = ħω/(nc)

• p: momentum of one photon in the medium
• n: refractive index of the medium
• ħ: reduced Planck constant (exact in the 2019 SI)
• ω: angular frequency of the light
• c: speed of light in vacuum (exact)

The widely accepted resolution (Barnett, Phys. Rev. Lett. 104, 070401, 2010) is that both are right: the Abraham value is the kinetic momentum of the light alone, and the Minkowski value is the canonical momentum, which includes the momentum the medium takes up. Which one an experiment measures depends on what it moves.

Honest caveat: Some authors still disagree about how the total momentum should be split between field and matter, and the split is not unique. What is not in doubt is that the total momentum of light plus medium is conserved.

On the sticker

Minkowskara holds the glass slab, with a cyan beam of light crossing inside it. Printed text: MINKOWSKARA with the line n vs 1/n. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.