05. A box full of light

A perfectly mirrored box sits on a scale. You fill it with photons of total energy $E$ — particles with zero mass. Does the scale read more?

Solution

Yes: heavier by exactly $E/c^2$.

Mass is not the sum of the masses of the parts; it is the norm of the total four-momentum. Already two photons of energy $E$ flying in opposite directions form a system with $\mathbf{p}_{\mathrm{tot}} = 0$ and $E_{\mathrm{tot}} = 2E$, hence invariant mass $2E/c^2$ — massless constituents, massive whole. The box of light is a composite at rest with rest energy increased by $E$, and it gravitates, and weighs, accordingly.

The scale learns this through radiation pressure. Photons bouncing off the floor have fallen through the box’s height and arrive slightly blueshifted; photons hitting the ceiling arrive slightly redshifted. The floor is pushed harder than the ceiling, and integrating the imbalance over the photon gas gives precisely $(E/c^2)\,g$. The bookkeeping device that makes all such accounts come out consistent is the stress-energy tensor: energy density, momentum flux, and pressure all gravitate together.

Deeper in the notebook: 06. Relativistic Mechanics and the Stress-Energy Tensor · 01. Gravitational Redshift