Cosmos

Local vs global energy

Disconnect

0.9446 eV

E_e − E_now = 0 eV accounted + 0.9446 eV unsunk

Local

closes

∇_μ T^{μν} = 0

continuity: ∇ · T = 0
residual: 0.0e+0
a ω (conformal): 0.9446 eV
U / U_e (box): 0.5000
−∫ p dV: -0.5000

Global

open

no ADM Hamiltonian

timelike Killing: none
k₀ (comoving): ∝ 1/a
photon ΔE sink: none

E emitted

1.8892 eV

E now

0.9446 eV

ΔE photon

0.9446 eV

Local covariant conservation holds: radiation in a comoving volume loses exactly −∫p dV, residual 0. There is still no global Hamiltonian. The missing photon energy is not in the galaxies and is not stored as gravitational field energy. That number is the disconnect.

z on arrival

1.0000

redshift

photon stretch so far a/a_e − 1: 1.0000

The photon follows a radial null geodesic. Energy measured by comoving observers falls as 1/a. The two books next to the observer are the model: local continuity closes, the global photon budget does not.

emitted

656.3 nm

observed

1.31 μm · infrared

Observed light is outside the visible window (infrared).

The two books

local: ΔU = −∫ p dV · residual 0.0e+0

E_e − E_o = 0 eV + 0.9446 eV

a ω is conserved because the conformal metric is Minkowski. ω itself is not. Distances: d_c = 3401 Mpc · d_L = 6802 Mpc · d_A = 1701 Mpc · lookback 7.95 Gyr · age 13.79 Gyr.

Disconnect

0.9446 eV

Local residual 0.0e+0. Global book has no sink — that is the exact disconnect.