Conservation laws and their boundaries

A conservation claim needs an account

When a photon redshifts, a comoving observer later measures less photon energy than a corresponding earlier comoving observer. Where did that energy go? Before inventing a destination, identify the quantity being compared, the observers and the region whose account is being kept.

Use the same homogeneous relativistic fluid description as the preceding lesson. Let q = a/a_ref and follow a region moving with the background fluid. Its proper volume is V = V_ref q³. Write its fluid energy as E = εV, with energy density ε in J/m³.

For a separately conserved ideal fluid, the continuity equation gives dE = −P dV, where P is pressure. This relates changes of energy and proper volume along the chosen expansion history. It does not require the region to be a literal container with material walls.

The equation is a form of the local fluid balance. Its use here assumes the stated homogeneous fluid, a following volume and no exchange with another component. Adding interactions or following a different boundary requires the corresponding terms and definitions.

The component’s pressure changes the result

Compare three separate hypothetical component accounts. In each case choose V_ref = 1 m³ and ε_ref = 8 J/m³ at q = 1. Each therefore starts with E_ref = 8 J. They are alternatives with the same reference numbers, not three descriptions of the same substance.

For conserved pressureless matter, ε = 8q⁻³ J/m³ and P is approximately zero. At q = 2, the volume is 8 m³ and ε = 1 J/m³, so E = 8 J. Dilution lowers the density while the rest-energy total in this following region stays fixed.

For separately conserved free radiation, ε = 8q⁻⁴ J/m³ and P = ε/3. At q = 2, ε = 1/2 J/m³ and E = 4 J. The photon energy total decreases as the volume grows; the pressure is positive and also changes during expansion.

For an ideal cosmological-constant component, ε = 8 J/m³ remains constant and P = −ε = −8 Pa. At q = 2, E = 64 J. The following volume contains more of this component’s energy because its energy density stays fixed while the volume increases. The negative pressure gives the corresponding sign in dE = −P dV.

Endpoint pressure is not the whole history

For the radiation account, pressure falls during expansion. Multiplying its final pressure, 1/6 Pa, by the volume increase, 7 m³, would give 7/6 J. That is not the correct integrated pressure term, because the earlier pressure was larger.

The finite change is ΔE = −∫P dV along the history. Radiation gives ΔE = 4 J − 8 J = −4 J, so the integral of P dV is 4 J. A definite integral adds the contributions with the pressure appropriate to each intermediate volume.

An optional mathematical check writes E(V) = 8 J × (V/V_ref)⁻¹ᐟ³. Differentiation gives −dE/dV = [8 J/(3V_ref)] × (V/V_ref)⁻⁴ᐟ³, which is ε/3 and has pressure units. Integrating between V_ref and 8V_ref reproduces the 4 J decrease.

The cosmological-constant example is different because its pressure is constant. There ΔE = −PΔV = −(−8 Pa)(7 m³) = 56 J. The shortcut works because the pressure does not change over this particular interval, not because endpoint multiplication is a universal rule.

Local conservation remains valid

General relativity expresses local stress-energy conservation through a covariant divergence equation, often written ∇_μ T^{μν} = 0. The tensor notation is optional here: it states a local balance of energy and momentum using the spacetime geometry. The fluid continuity equation used above is its homogeneous energy-balance form.

A single total energy for an entire curved, evolving spacetime is an additional question. Constructing a conserved global energy requires appropriate structure, such as a suitable time-translation symmetry or specified boundary conditions and a valid definition of the associated charge.

A generic expanding cosmological background does not supply the same preferred time-translation symmetry as a stationary situation. One therefore cannot assume that adding locally measured photon energies on different cosmic-time slices produces a universal conserved scalar for the whole Universe.

This does not mean that energy conservation is meaningless or that energy is never conserved in general relativity. Local conservation applies, and global conserved quantities can exist when the required conditions hold. The lesson is to state those conditions before extending a local or familiar mechanical account to all of cosmology.

Do not create an unsupported destination

The radiation calculation establishes a particular result: the photon energy in a specified following region decreases as its volume expands, consistently with the local pressure–volume relation. It does not identify a separate reservoir receiving an equal amount of energy.

The cosmological-constant calculation is a separate ideal account. Combining its increasing energy with the radiation example does not show that redshift energy turns into dark energy. No such conversion law was assumed or derived.

Likewise, the chosen region’s boundary is a bookkeeping surface moving with the smooth fluid, not a physical piston. The pressure term has a precise place in the relativistic fluid balance; a picture of an external container can add mechanisms that are absent from the model.

A useful first-reading procedure is to ask: Which observers measure energy? Which region is followed? What are ε and P? Are there exchanges? Does a proposed global conservation statement have the extra structure it needs? These questions preserve the valid conservation law while exposing an unjustified extension.

Worked reasoning

Try the reasoning

1. Why is final radiation pressure times the total volume change insufficient in the stated example?

   Show the answer and reasoning

   Radiation pressure changes during the expansion.

   The finite pressure term must be integrated along the history; its value is 4 J, not 7/6 J.

2. When the pressureless matter density falls from 8 to 1 J/m³, has the following-region rest energy fallen by a factor of eight?

   Show the answer and reasoning

   No. It remains 8 J in this approximation.

   The following volume increases from 1 to 8 m³, compensating the density decrease.

3. Does local stress-energy conservation by itself establish one universally conserved total energy for every cosmological spacetime?

   Show the answer and reasoning

   No.

   A global conserved quantity requires a suitable definition and additional structure, such as the relevant symmetry or boundary conditions.

What to carry forward

• For a separately conserved homogeneous fluid in a following volume, dE = −P dV.
• Changing pressure requires an integral; density and following-region total energy are different quantities.
• Local conservation remains valid while global conservation claims need their own definitions and conditions.