A constant Lorentz-invariant vacuum stress has pressure minus its energy density and is equivalent to Lambda in Einstein gravity. Predicting its value requires an absolute metric-sensitive source law, not only a normalized completion response.
Keep Lambda, energy density, and mass density distinct
The cosmological constant Λ has units of inverse length squared. Its equivalent vacuum energy density εΛ has units of energy per volume, while the corresponding mass density ρΛ is εΛ/c². In the Einstein-equation convention used here, their relationship is:
These factors matter when translating an inner-time energy scale into curvature. A formula for mass density cannot be inserted into an equation expecting energy density without the conversion. The source should also specify whether a vacuum term is already included in the matter stress or written separately as Λ.
Derive the pressure of a constant vacuum contribution
Suppose a homogeneous component has constant energy density εv and is separately conserved. In a comoving volume V, its energy is E=εvV. The local perfect-fluid balance dE=−p dV then gives pv=−εv. The same equation follows from ε̇+3H(ε+p)=0 when H is nonzero.
This negative pressure is a response to changing physical volume. It does not mean negative energy density: εv may be positive. A Lorentz-invariant vacuum stress has the corresponding metric-proportional form Tμν=−εvgμν in compatible relativistic conventions. An arbitrary nonstationary source state need not have this form.
Worked example: the pure positive-vacuum solution
Take a spatially flat Einstein-gravity model with only a positive constant Λ. Its expansion equation is H²=Λc²/3. The expanding branch has H=c√(Λ/3), so a(t)=a*exp[H(t−t*)]. The proper Hubble and future event radii are c/H=√(3/Λ) for this indefinite exponential future.
Multiplying Λ by four doubles H and halves this radius. The example translates a vacuum coefficient into a definite geometric effect without selecting its numerical value. It also assumes a vacuum-only regime; adding matter, radiation, or evolving fields changes the full history and its horizon relations.
Prove why normalized responses miss a constant
For a finite model, shift H to H+C I. At inverse temperature β, every Boltzmann weight gains the same factor exp(−βC), which cancels from normalized probabilities. Unitary evolution gains one global phase exp(−iCt/ħ), leaving ordinary expectation values unchanged. Transition energies remain differences and are also unchanged.
This establishes an information obstruction for those observables. The physical gravitational question is different: if the added term multiplies a geometry-dependent volume in the action, varying the metric can reveal it. The full geometric functional must decide which normalization is physically relevant.
Temporal smoothing does not lower the mean
Let completed vacuum samples be εn=ε̄+ηn, with independent zero-mean fluctuations of variance σ². Averaging M samples keeps expectation ε̄ and reduces variance to σ²/M. For M=100, the RMS noise falls by ten, but the mean does not fall at all.
Adding a common offset C shifts every average by C, regardless of how many completions are included. Correlations can change the noise-suppression law without removing this fact. DTT renewal can therefore motivate a mechanism for smooth records, while a small gravitating mean requires an additional cancellation, selection, or protected relation in the metric-sensitive dynamics.
Test a proposed inverse-fourth-power scale
Consider the dimensionally allowed expression ε=Aħc/L⁴ with dimensionless A. Doubling L reduces the density by sixteen, so the physical definition of L is decisive. If L=c/H, the expression becomes AħH⁴/c³. Compare it with the critical energy density εcrit=3c²H²/(8πG):
This ratio is a calculable dimensionless combination, not automatically unity. Choosing L from a desired density is an inverse fit. A source derivation must select L and A independently and explain why the resulting contribution has the pressure and conservation law used in the cosmological equations.
Match curvature and vacuum coefficients together
The constants guide writes a dimensionless effective action with a curvature coefficient κ and constant coefficient r. With action scale a* and length ℓ*, its stated Einstein-action matching gives G=c³ℓ*²/(16πa*κ) and Λ=r/(2κℓ*²). Their product isolates a source-level ratio:
The identity shows which combination one source must determine. It does not calculate r from a relative phase stiffness. A successful theory should also show how this ratio responds when matter sectors change, rather than tune the vacuum normalization independently at each stage.
A focused source-level vacuum calculation
The next construction should retain the absolute geometric action throughout inner generation and completion, specify its regulator and state, and calculate both volume and curvature responses. Conservation must hold for the total retained system, including any compensating sector proposed to reduce the vacuum contribution.
The six nested inner roles could constrain which terms are generated together; outer completion could select a stable state. Their explanatory value lies in a derived relationship among coefficients, not simply a large number of repeated cycles. A protected ratio or a controlled response to a matter threshold would be a concrete advance toward a cosmological constant calculation.
Continue with the articles
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- Completion cost and the trace
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