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 $\varepsilon$ each, flying in opposite directions, form a system with $\mathbf{p}_{\mathrm{tot}} = 0$ and $E_{\mathrm{tot}} = 2\varepsilon$, hence invariant mass $2\varepsilon/c^2$ — massless constituents, massive whole. The box of light is a composite at rest whose rest energy is up by exactly the $E$ you paid to fill it (ideal mirrors: none of it leaks into the walls). Invariant mass is inertial mass; that a scale reads it too is the equivalence principle — and the mechanism by which the scale finds out is worth spelling out.

The scale learns it through radiation pressure. A photon that has fallen the box’s height arrives at the floor blueshifted by $gh/c^2$; one that has climbed arrives at the ceiling redshifted. The floor is pushed harder than the ceiling — but integrate the imbalance over the isotropic gas and it comes out at $\tfrac{4}{3}(E/c^2)\,g$, an overshoot: pressure gravitates too, and for a photon gas $P = u/3$. The surplus third is returned by the walls, which the gas holds in tension, and tension weighs negatively. Laue’s theorem — $\int T^{ij}\,d^3x = 0$ for any static bounded system — guarantees the stresses cancel in the total, and the books close at precisely $(E/c^2)\,g$. That such accounts cannot help but close is the point of the stress-energy tensor: energy density, momentum flux, pressure, and stress gravitate together, and only together.

Papers: Laue, Zur Dynamik der Relativitätstheorie (1911) is the theorem’s original home — the complete static system as the carrier of the four-momentum; Kolbenstvedt, The mass of a gas of massless photons (1995) weighs exactly this box, walls and all.

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