10. The missing energy of an old photon

A CMB photon has been traveling since recombination, and the expansion has stretched its wavelength about 1,100-fold: it has lost 99.9% of the energy it started with. Where did that energy go?

Solution

Nowhere — the question assumes a ledger the universe does not keep.

Conservation laws are bought with symmetries: energy conservation, in particular, with time-translation invariance — in geometric language, a timelike Killing field. An expanding universe filled with matter and radiation has none (the exceptions — empty Milne, de Sitter inside a static patch — are precisely the maximally symmetric cases, and ours is neither); there is no global “total energy of the universe” whose books must balance, so no account the photon’s loss must be debited from. Nor can one say “the gravitational field took it”: gravitational energy has no local density in GR — the equivalence principle kills any covariant energy density for the field, leaving only coordinate-dependent pseudotensors. There is no bank the energy could have been deposited in. (In a stationary spacetime the ledger exists: the Killing field fixes a conserved $E = -p_\mu \xi^\mu$, constant along the photon’s whole flight, so emitter and absorber sit on one balance sheet — the Schwarzschild redshift is just the difference between two static observers’ local readings of that one number.)

What survives everywhere is the local law $\nabla_\mu T^{\mu\nu} = 0$ — a continuity equation, not a global conservation law. For the cosmic fluid it reads $\dot\rho = -3H(\rho + p)$: dilution for dust ($\rho \propto a^{-3}$), dilution plus one extra factor for radiation ($\rho \propto a^{-4}$) — and that extra $a^{-1}$ is exactly the redshift. Do not read it as a first law with the photon gas doing $p\,dV$ work “against the expansion”: there is no piston and no exterior to receive the work, only more universe — and dust, with $p = 0$, loses its ledger just the same. Kinematically, each comoving observer the photon passes is receding from the last one who measured it: energy is observer-dependent, $E = -p_\mu u^\mu$, and a chain of mutually receding observers simply records ever-smaller values.

What the photon does carry unchanged is not an energy. FLRW is conformally flat, and $\partial_\eta$ in $ds^2 = a^2(\eta)\left(-d\eta^2 + d\mathbf{x}^2\right)$ is a conformal Killing field — for null geodesics that is enough to conserve $aE$ exactly. The photon has held one number constant since recombination: its comoving momentum. The redshift is simply what that conservation law looks like to observers who insist on calling the thing they measure an energy. Nothing is lost to anywhere; there was never an energy to conserve.

Deeper in the notebook: 02. Redshift · 02. Noether’s Theorem in GR · 07. Killing Vectors and Isometries · 01. The Friedmann Equations and Cosmic Dynamics