8. ADM Decomposition and Hypersurface Evolution
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In ADM form, the spacetime metric is decomposed into lapse N, shift N^i, and the induced 3-metric
h_{ij} on Σ_t. The extrinsic curvature K_{ij} encodes how Σ_t is embedded in the ambient spacetime:
g_{μν} → (h_{ij}, N, N^i)
K_{ij} = (1/2N)(∂_t h_{ij} − ∇_i N_j − ∇_j N_i)
The evolution equations and constraints provide a natural slice-to-slice evolution picture. CTGP
maps this to an ontological claim: Σ_σ is literally produced from Σ_{σ−Δσ} by lawful dynamics,
without reference to a pre-existing future slice.
The ADM Hamiltonian and momentum constraints ensure that the generated data on each new Σ_{σ+Δσ} are
consistent with the field equations. These constraints are not boundary conditions from the future
but internal consistency requirements on the present slice, further supporting the generative
interpretation.
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9. Action Principle on Generated Domains and Cosmological Foliation
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9.1 Action Principle
~~~~~~~~~~~~~~~~~~~~
The action principle for CTGP restricts integration to the generated domain M(σ). Within this action
principle, the kinematic equation governing σ is obtained as an Euler–Lagrange condition rather than
stipulated. The CTGP framework is formulated by requiring the physical generative history to satisfy
the stationary-action condition δS = 0, where the action is taken to be:
S(σ) = ∫_{M(σ)} √(−g) [ L_{GR} + λ(∇_μσ ∇^μσ + f(σ)) ] d⁴x
where L_{GR} = (R − 2Λ)/16πG + L_{matter} is the Einstein–Hilbert Lagrangian density with matter, λ
is a Lagrange multiplier field, f(σ) > 0 is the generation law (f = 1 for the canonical choice σ =
τ), and √(−g) d⁴x is the invariant volume element. The integration is restricted to M(σ), ensuring
no future teleology. The term λ(∇_μσ ∇^μσ + f(σ)) enforces the norm condition on σ. Variation with
respect to the fields yields the constraint and propagation relations below.
This action and all subsequent variational results are valid within the emergent geometric regime (ρ
≪ ρ_P), where the smooth Lorentzian manifold (M, g_{μν}) is a valid continuum approximation. The
pre-classical regime is addressed in §9.2 and §10.2; the extended action S_{ext}(σ) that models the
crossover via the suppression function α(ρ) is developed in §9.2.
Varying S(σ) with respect to λ yields the Euler–Lagrange condition:
δS/δλ = 0 ⟹ ∇_μσ ∇^μσ + f(σ) = 0
That is, ∇_μσ ∇^μσ = −f(σ), the norm condition of §6.3, arising here as a necessary condition for
stationarity of S(σ). Varying with respect to σ yields the propagation equation ∇_μ(λ∇^μσ) =
½λf′(σ), which transports λ along the generation flow. Varying with respect to g^{μν} recovers
G_{μν} = 8πG T_{μν} together with the λ-dependent term of Proposition 17.7, which vanishes
identically for the configuration λ = 0 adopted in §6.9.
The variational grounding answers the question of why σ should obey any particular equation at all:
σ obeys ∇_μσ ∇^μσ = −f(σ) for the same reason that geodesics obey the geodesic equation — because it
is what the variational principle demands. The formal statement is given in Proposition 17.7 (§17).
The restriction of the integration domain to the generated manifold M(σ) is the central ontological
commitment of the framework — not merely a heuristic convenience. In the standard block universe
interpretation, the action is implicitly integrated over the entire four-dimensional manifold,
treating all events as equally real elements of the variational principle. By explicitly limiting
the action to M(σ), CTGP asserts that future spacetime regions do not contribute to the dynamical
definition of the theory at stage σ. The future is not simply unknown; it is not part of the domain
of the physical laws that generate the present. This transforms the interpretation from a passive
description of a pre-existing structure to an active generation of new structure from existing
structure. The Euler-Lagrange equations derived from this action are therefore not constraints on a
static block, but rules for constructing the next increment of reality from the existing one. The
domain of the action is itself a growing entity, and the field equations are defined only over that
domain. Because Euler–Lagrange equations are local, this restriction does not alter the field
equations at any point of M(σ); the distinction it draws concerns the domain over which the laws are
defined, not their local form. A completed block would serve as the domain for an action defined
over the whole of M; the CTGP action has no such domain. This distinction is formalized in Theorem
17.9 (§17).
9.2 Domain of Validity and the Extended Action
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The effective action S(σ) of §9.1 is valid within a precisely specified domain: the emergent
geometric regime, where ρ ≪ ρ_P (the Planck density), and the smooth Lorentzian manifold (M, g_{μν})
is a valid continuum approximation. Outside this domain — specifically in the pre-geometric regime
where ρ ∼ ρ_P — the continuum integral, the metric volume element √(−g) d⁴x, and the Ricci scalar R
are themselves emergent structures that cannot be taken as primitive inputs to the variational
principle. The two regimes and their respective formal machinery are distinguished as follows:
• Pre-geometric regime (ρ ∼ ρ_P): σ is primitive; no metric assumed; no continuum action defined;
causal ordering is the operative structure.
• Emergent geometric regime (ρ ≪ ρ_P): σ represented by cosmological time; metric available; S(σ)
applies.
The transition between regimes is not a discontinuity but a crossover parameterized by the
dimensionless ratio ρ/ρ_P. This crossover can be represented in the effective action by introducing
a density-dependent suppression factor α(ρ) on the GR Lagrangian:
S_{ext}(σ) = ∫_{M(σ)} √(−g) [ α(ρ) L_{GR} + λ(∇_μσ ∇^μσ + f(σ)) ] d⁴x
α(ρ) = 1 / (1 + (ρ/ρ_P)^n), n ≥ 1
The suppression function α(ρ) satisfies: α → 1 as ρ/ρ_P → 0 (low-energy limit, GR fully restored),
and α → 0 as ρ/ρ_P → ∞ (Planck regime, GR suppressed). The σ-constraint sector λ(∇_μσ ∇^μσ + f(σ))
is retained at all densities, reflecting the fact that the generative ordering structure of σ is
active across both regimes — it is never suppressed, because it is the structure from which the
geometric regime itself emerges. The exponent n controls the sharpness of the transition; n = 2
gives a smooth crossover centered at ρ = ρ_P.
An important ontological clarification is required by S_{ext}(σ): the metric g_{μν} still appears in
the extended action through √(−g) and the σ-constraint sector. This creates a choice between two
interpretations of the metric’s status in the pre-classical regime. Under the emergent-metric
reading (consistent with §10.3), g_{μν} in S_{ext}(σ) is to be understood as a continuum
approximation that becomes progressively less valid near ρ_P; the action itself is correspondingly
an effective field theory expression whose UV completion lies in the pre-geometric structure. Under
the constrained-metric reading, g_{μν} remains a fundamental but dynamically unconstrained quantity
whose constraints emerge alongside GR as α(ρ) → 1. CTGP adopts the emergent-metric reading as
primary (Emergent Metric Principle, §10.3), treating S_{ext}(σ) as valid only where the continuum
approximation holds. The extended action therefore serves a specific formal function: it models the
crossover continuously rather than imposing a sharp regime boundary, and it identifies α(ρ) as the
natural emergence function for GR within the CTGP framework. S_{ext}(σ) is the candidate effective
action for the crossover regime; its pre-geometric limit is not fully specified by the continuum
formalism alone, and remains subject to the Scope of the Research Program principle of §10.4.
9.3 Cosmological Foliation and the CMB Frame
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In homogeneous, isotropic cosmology, the FLRW metric provides a preferred foliation by cosmic time
t:
ds² = −dt² + a(t)² dΩ_k²
This does not violate local Lorentz invariance; rather, it reflects a physical symmetry of our
universe's matter distribution. The CMB frame singles out a preferred rest frame not as a kinematic
artifact but as a physical fact about the distribution of matter and energy. CTGP's global present
Σ_σ is defined by cosmological time, and on large scales it coincides with this frame, providing a
non-absolute but physically grounded global 'now.' Different observers need not synchronize their
local frames; they share the cosmological foliation without needing operational access to it.
[FIGURE: Figure 5]
Figure 5. FLRW foliation: surfaces of constant cosmic time t, with comoving worldlines diverging
as the scale factor a(t) grows. The comoving worldlines are the maximizing geodesics that define
cosmological time and provide the physical basis for CTGP’s global present.
9.3.1 Emergent Cosmological Foliation
A clarification bears on one of the most persistent objections: the charge that CTGP introduces a
preferred frame. The correct response is that CTGP does not introduce a preferred frame — it
identifies the cosmological foliation as an emergent physical structure generated by the large-scale
matter distribution.
The logical chain that establishes this is as follows:
Einstein equations are foliation-neutral → the actual matter distribution fixes the metric → the metric fixes cosmological time τ → on large scales, the level sets of τ coincide with the CMB rest frame
CTGP does not use the cosmological foliation as an external input. Rather, CTGP identifies the
cosmological foliation as an emergent physical structure generated by the large-scale matter
distribution. This framing aligns CTGP with spontaneous symmetry breaking, a mechanism physicists
already accept throughout physics: a rotationally symmetric Lagrangian can produce a ground state
that breaks rotational symmetry, with the symmetry-breaking direction selected by the physical
configuration rather than by the Lagrangian itself. The CMB rest frame is precisely analogous: the
Einstein equations are foliation-neutral, but the physical solution — a universe filled with matter
in a particular configuration — fixes a metric whose cosmological time is, on large scales, the CMB
rest frame.