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 FLRW universe has none; there is no global “total energy of the universe” whose books must balance, so no account the photon’s loss must be debited from. (In a static spacetime the ledger exists: Schwarzschild redshift really is bookkept, via the Killing field, between photon and gravitating source.)

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. 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. Nothing is lost to anywhere; there was never one number 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