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Appendix A. ADM Evolution and σ-Reparameterization (Derivations)
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This appendix collects standard ADM equations and shows how CTGP's growth parameter σ can be used to
reparameterize evolution. The presentation follows standard canonical GR notation and then adds
CTGP's σ-change of variables.
A.1 ADM Variables and Constraints
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
g_{μν} → (h_{ij}, N, N^i); K_{ij} = (1/2N)(∂_t h_{ij} − ∇_i N_j − ∇_j N_i)
Hamiltonian constraint:
H = R^{(3)} + K² − K_{ij}K^{ij} − 16πG ρ = 0
Momentum constraint:
H_i = ∇_j (K^j_i − δ^j_i K) − 8πG j_i = 0
A.2 Evolution Equations
~~~~~~~~~~~~~~~~~~~~~~~
∂_t h_{ij} = 2N K_{ij} + ∇_i N_j + ∇_j N_i
∂_t K_{ij} = −∇_i ∇_j N + N( R^{(3)}_{ij} + K K_{ij} − 2K_{ik}K^k_j ) + (matter terms)
A.3 Cosmological Time and Its Relabelings
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The discussion in §6.7 establishes that the present hypersurfaces are Cauchy surfaces whose induced
data satisfy the ADM constraints and evolve consistently under the Einstein equations. This appendix
records the explicit form of the generation parameter in cosmological settings.
In FLRW with a big-bang singularity, cosmological time is cosmic time: τ = t. A relabeling σ = F(t)
with F′ > 0 converts time derivatives by ∂_σ = (1/F′) ∂_t; for the canonical choice F = identity, σ
= t.
A.3a Well-Definedness, Dimensions, and Reparameterization of σ
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Three properties of the generation parameter are relied upon elsewhere and are stated explicitly
here. The first is well-definedness. Cosmological time is defined as a supremum over all past-
directed causal curves, not by integration along a chosen congruence, so no path-independence
condition is required: τ(p) is well defined whenever it is finite. Its level sets are Cauchy
surfaces when τ is regular (Theorem 17.4). A rotating cosmology such as Gödel’s dust solution admits
no regular cosmological time, consistent with the global hyperbolicity requirement of §6.1.
The second is dimension. For the canonical choice σ = τ, σ carries dimensions of time; for other
relabelings, its dimensions are those of F. The two are related monotonically, induce the same
foliation, and differ only in normalization and units. Statements that σ “corresponds to”
cosmological proper time are to be read in this sense.
The third is reparameterization. Any strictly increasing F yields an equally admissible parameter,
so σ is fixed only up to monotonic relabeling. This is harmless for every claim the framework makes,
because those claims depend on the level sets of σ and the direction of ∇_μσ, both of which are
relabeling-invariant. The magnitude |∇_μσ| = F′ is conventional and carries no physical content.
For reference, the chain from geometry to foliation is as follows: g_{μν} → causal structure → τ,
the maximal elapsed proper time, defined without choice → Σ_τ, Cauchy surfaces when τ is regular → σ
= F(τ), a labeling convention. The matter congruence enters only as a consequence: in cosmology, the
maximizing geodesics are the comoving worldlines.
A.4 σ-Evolution of ADM Equations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substituting ∂_t = F′ ∂_σ into ADM evolution, with the lapse normalized to cosmic time:
∂_σ h_{ij} = (2N/F′) K_{ij} + (1/F′)(∇_i N_j + ∇_j N_i)
∂_σ K_{ij} = (1/F′)(standard ADM right-hand side)
A.5 Expansion Scalar and Raychaudhuri Calibration
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let u^μ be the timelike unit vector field aligned with the matter congruence. The expansion scalar θ
= ∇_μu^μ satisfies θ = d(ln √h)/dτ, measuring the fractional rate of local volume change of the
foliation along the congruence. Its evolution is governed by the Raychaudhuri equation:
dθ/dτ = -(1/3)θ² - σ_{μν}σ^{μν} + ω_{μν}ω^{μν} - R_{μν}u^μu^ν + ∇_μa^μ
In the irrotational, geodesic cosmological limit (ω_{μν} = 0, a^μ = 0) this reduces to:
dθ/dτ = -(1/3)θ² - σ_{μν}σ^{μν} - R_{μν}u^μu^ν
CTGP uses this result to interpret geometric growth in terms of foliation expansion rather than as a
derivative dM/dσ of the generated domain treated as a scalar quantity. The expansion scalar θ
provides the GR-native local measure of how the present hypersurface is extending relative to
neighboring hypersurfaces: θ > 0 in expanding cosmologies, θ < 0 in collapsing regimes. Via the
Einstein equations, the R_{μν}u^μu^ν term links the evolution of θ directly to the matter content
through the Ricci curvature, grounding CTGP’s growth picture in standard relativistic dynamics.
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Appendix B. Causal Structure Theorems and Their Application to CTGP
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B.1 Global Hyperbolicity and Causal Order
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A spacetime (M, g) is globally hyperbolic if it admits a Cauchy hypersurface — a spacelike surface
intersected exactly once by every inextendible causal curve. This property is widely believed to
hold for our universe on cosmological scales (Hawking and Ellis 1973).
Theorem B.1 (Hawking–King–McCarthy, 1976). In suitably well-behaved relativistic spacetimes, the
chronological relation I+(p) — the set of events reachable from p by future-directed timelike curves
— determines the conformal structure of the spacetime metric. The topology and light-cone structure
can be reconstructed uniquely up to an overall conformal factor from knowledge of the chronological
order alone.
Theorem B.2 (Malament, 1977). The causal ordering relation alone suffices to determine the conformal
metric structure of relativistic spacetime under mild conditions. Temporal precedence is not a
coordinate convention but a geometric fact encoded in causal relations.
Theorem B.3 (Geroch, 1970). In a globally hyperbolic spacetime there exists a continuous time
function — a function that strictly increases along every future-directed causal curve. Moreover, M
≅ ℝ × Σ, where each slice is a Cauchy hypersurface. This splitting is not imposed but follows from
the causal structure.
Theorem B.4 (Bernal and Sánchez, 2003, 2005). In any globally hyperbolic spacetime, there exists a
smooth temporal function whose gradient is everywhere timelike and whose level sets are smooth
spacelike Cauchy hypersurfaces. Any Geroch splitting can be refined to a smooth one. Moreover, any
two smooth Cauchy foliations can be smoothly deformed into each other.
B.2 Application to CTGP
~~~~~~~~~~~~~~~~~~~~~~~
These theorems guarantee that the mathematical structure CTGP requires — a global foliation by
present hypersurfaces Σ_σ with σ increasing causally — exists in any globally hyperbolic spacetime.
This is not an additional CTGP postulate but a consequence of GR’s causal structure under physically
reasonable assumptions. Standard interpretations treat the entire product manifold ℝ × Σ as
ontologically complete. CTGP instead interprets the parameter labeling the slices as indexing the
progressive generation of spacetime structure: at stage σ, only the present slice exists; the
hypersurfaces in its past belong to the formal history M(σ), and the future hypersurfaces Σ_{σ′>σ},
while mathematically definable within the full spacetime manifold, are not part of the generated
domain.
The freedom to choose among smooth temporal foliations corresponds to the freedom to pick different
smooth temporal functions. CTGP resolves this ambiguity with cosmological time: among all time
functions whose level sets are Cauchy surfaces, it is singled out by maximal elapsed proper time,
and in cosmology it coincides with the matter rest frame. The logical priority is clear: causal
geometry (D.1–D.4) precedes cosmological time (Theorem 17.4), which precedes the labeling convention
σ = F(τ). σ is not a novel object introduced by CTGP; it is a distinguished Cauchy time function
that general relativity already provides.
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