# Analytic Closure of General Relativity and the Limits of Spacetime Geometry

Joakim Munkhammar

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# Analytic Closure of General Relativity and the Limits of Spacetime Geometry

Joakim Munkhammar

Private communication address

joakim.munkhammar@gmail.com

(Dated: February 5, 2026)

We investigate the global structure of General Relativity as an analytically completed theory. Admissible gravitational configurations are taken to be closed under local analytic continuation in field space, subject to diffeomorphism covariance. Physical spacetime geometry is defined by a strict covariant projection to a real Lorentzian metric, which alone determines proper time, causal structure, and observable curvature. Within this analytic-gravity framework, the Einstein equations admit a natural decomposition into real and imaginary sectors after projection. In homogeneous cosmology, the resulting analytic phase sector yields a generic stiff branch with $\rho \propto a^{-6}$, while global admissibility constrains late-time behavior and leaves a residual metric-proportional integration constant, interpreted as an effective cosmological constant. Classical curvature singularities are reclassified as projection boundaries: loci where spacetime observability and proper time cease to be defined even though analytic continuation of the underlying configuration may persist. The analysis is entirely classical and preserves the local content of General Relativity. It clarifies the limits of spacetime geometry under analytic completion and establishes a foundation for subsequent extensions involving structures not representable within spacetime geometry alone.

## I. INTRODUCTION

### A. Background, scope, and analytic completeness

General Relativity (GR) is formulated as a principle theory: a small set of structural principles determines the geometric objects, invariances, and local dynamics of gravitation. In particular, gravitation is described by a real Lorentzian metric, the laws are generally covariant, local dynamics are governed by the Einstein field equations, and physical observables are defined through invariant geometric quantities such as proper time and causal structure [1, 2]. From these principles follow both the form of the field equations and the class of configurations that may be regarded as physically admissible.

While the local dynamical content of GR is fully fixed by these principles, its global structure is less tightly constrained. The standard formulation leaves open questions concerning the admissibility of analytically extended configurations, the interpretation of spacetime singularities, and the origin and status of global integration constants such as the cosmological constant [3, 4]. In practice, analytic continuation is routinely employed to construct maximal extensions, distinguish coordinate artifacts from genuine curvature singularities, and analyze global structure, as in the Schwarzschild and FLRW spacetimes [5, 6]. These constructions rely essentially on analyticity of the metric and its continuation beyond particular coordinate charts.

At the same time, analyticity is not a generic property of all smooth solutions of the Einstein equations and is usually imposed as a technical assumption rather than elevated to a physical principle [7]. The standard framework therefore does not state explicitly whether the space of admissible geometries is to be regarded as closed under the analytic extensions that the equations themselves naturally generate.

Choosing to forbid analytic continuation constitutes an additional *restricting principle*. Such a restriction is not implied by general covariance, locality, or the geometric nature of gravitation, and requires independent justification. In the absence of an invariant criterion for exclusion, the alternative pursued here is to adopt *analytic completeness* as a closure principle: within a chosen real-analytic category, the admissible configuration space is taken to be closed under local analytic continuation in field space, subject to diffeomorphism covariance.

This stance follows Einstein's methodological distinction between principle theories and constructive theories: one first formulates a minimal set of structural principles, and then derives their maximal consequences before introducing additional structure [8, 9]. The present work adheres to this discipline by preserving the principles of General Relativity while removing an implicit limitation on their maximal consequences through analytic closure.

Analytic completeness introduces no new fundamental geometric structures and no new dynamical postulates. Instead, it demands that the existing principles of GR be applied maximally. The result is an enlargement of the admissible configuration space and, consequently, an enlargement of the dynamical consequences that follow from the Einstein equations within this analytic hull.

The present work is classical in scope. It concerns the analytic completion and global structure of General Relativity at the level of spacetime geometry alone and does not introduce quantum postulates, matter representations, or modifications of local dynamics. Analytic closure is adopted here as a methodological consistency requirement on the space of admissible configurations, not as a claim of physical necessity or empirical completeness.
## B. Overview and outline

Imposing analytic completeness at the level of configuration space leads naturally to a complexified analytic envelope for the metric. Physical spacetime observables are defined by a strict projection to a real Lorentzian metric, which alone determines causal structure and proper time. Within this analytic-gravity hull, the projected field equations acquire a scalar phase sector and a corresponding geometric phase contribution in the projected real equations. In homogeneous cosmology this structure admits a stiff analytic branch, while global admissibility (cosmological selection) constrains late-time behavior and leaves a residual integration constant interpreted as an effective cosmological constant. Classical singularities are reinterpreted as projection boundaries: points at which the real Lorentzian spacetime description ceases to exist even though analytic continuation of the envelope may persist.

The paper is organized as follows. Section II defines the analytic envelope and the projection principle, and clarifies covariance, branch structure, and the special-relativistic limit. Section III derives the projected real and imaginary field equations and identifies the scalar phase sector. Sections IV and V develop homogeneous cosmology, the stiff analytic branch, cosmological selection, and the emergence of an effective cosmological constant. Section VI treats the Big Bang as a global projection boundary and clarifies the emergence and breakdown of proper time, including a cosmological arrow of time in the sense of global admissibility. Section VII applies the same logic to black hole spacetimes. Section VIII concludes with discussion and outlook to further extensions.

# II. ANALYTIC COMPLETION AND PROJECTION

## A. Analytic envelope and complexified field equations

We work within a real-analytic category of metric configurations and impose analytic completeness as a closure condition on the gravitational configuration space. Within this setting, admissible configurations are required to be closed under local analytic continuation in field space, subject to diffeomorphism covariance. Analytic completeness is not elevated to a universal axiom of physics; rather, it defines a conservative analytic hull within which questions of global structure and continuation can be addressed unambiguously.

Analytic continuation in field space naturally admits complexified metric configurations. Accordingly, we introduce a (possibly complex) metric tensor $g_{\mu\nu}$ as an analytic envelope around real Lorentzian spacetime geometry. Throughout this work, the complex metric is not interpreted as a physical spacetime; it serves solely as an analytic carrier required by the closure condition.

As a minimal and covariant analytic extension of the gravitational dynamics, we assume that the analytic envelope satisfies the Einstein field equations in their analytically continued form,

$$G_{\mu\nu}[g] = \frac{8\pi G}{c^4} T_{\mu\nu}, \qquad (1)$$

where $G_{\mu\nu}[g]$ is the Einstein tensor constructed from $g_{\mu\nu}$, $G$ denotes Newton's gravitational constant, and $T_{\mu\nu}$ is the analytically continued stress-energy tensor. Equation (1) is not proposed as a new physical theory; it is adopted as the simplest covariant analytic envelope consistent with the principles of General Relativity.

## B. Projection to real Lorentzian observables

Physical spacetime observables are defined by a strict projection

$$P: g_{\mu\nu} \mapsto h_{\mu\nu}, \qquad (2)$$

where $h_{\mu\nu}$ is required to be real, Lorentzian, and non-degenerate. Proper time, causal structure, and measurable curvature are defined exclusively in terms of $h_{\mu\nu}$. The projection is adopted as a constitutive principle of the framework rather than as a derived dynamical condition.

Locally in field space, the complex metric may be written in polar form as

$$g_{\mu\nu} = h_{\mu\rho} (e^{i\Theta})^{\rho}_{\nu}, \qquad (3)$$

where $\Theta^{\mu}_{\nu}$ is a real mixed-index tensor encoding analytic phase information. The exponential denotes a matrix exponential. This decomposition is diffeomorphism covariant and separates observable Lorentzian geometry from analytic phase structure in a multiplicative manner.

The projected metric $h_{\mu\nu}$ defines the physical spacetime. While the analytic envelope $g_{\mu\nu}$ may carry phase information, no complex or modified version of Minkowski spacetime is introduced: in the absence of curvature and phase gradients, the projection reduces identically to standard special relativity.

Throughout this framework, spacetime itself is taken to be real: the underlying manifold and its coordinates are real, and physical spacetime geometry is defined exclusively by the projected real Lorentzian metric $h_{\mu\nu}$. No complexification of spacetime coordinates or of the physical line element is introduced.

## C. Branch structure and analytic continuation

The polar decomposition (3) is generically multi-valued due to the logarithmic nature of the matrix exponential. Analytic continuation in field space may therefore generate multiple branches of the analytic envelope related by shifts of $\Theta^{\mu}_{\nu}$ by integer multiples of $2\pi$.
along suitable directions in configuration space. These branches correspond to distinct analytic sheets of the completed configuration space.

Branch multiplicity is a structural consequence of analytic completeness and does not imply the existence of multiple physical spacetimes. Physical admissibility is determined only after projection: distinct analytic branches may or may not admit a real Lorentzian metric under P. Branches for which the projection fails to yield a real, non-degenerate $h_{\mu\nu}$ are excluded from physical interpretation.

### D. Analyticity versus holomorphicity

The analyticity assumed throughout this work refers to analyticity in field space rather than holomorphicity in spacetime coordinates. No complex structure is imposed on spacetime itself, and the metric is not required to be holomorphic as a function of spacetime coordinates. Consequently, no Cauchy-Riemann conditions are imposed. Compatibility between real and imaginary sectors is instead enforced by the Einstein field equations themselves, viewed as analytic relations on the space of metric configurations.

This distinction ensures that the framework preserves the Lorentzian character, causal structure, and local covariance of General Relativity while allowing analytic continuation at the level of configuration space.

Throughout this work, spacetime coordinates are taken to be real, while the metric field $g_{\mu\nu}$ and the associated Einstein tensor $G_{\mu\nu}[g]$ are allowed to be complex-valued tensor fields on a real spacetime manifold. Analyticity refers to the dependence of the field equations on the metric as a configuration variable, not to holomorphic structure in spacetime coordinates.

## III. PROJECTED FIELD EQUATIONS AND THE PHASE SECTOR

### A. Expansion and projection of the Einstein equations

Substituting the polar decomposition (3) into the analytic Einstein equations (1) and expanding in derivatives of the phase tensor $\Theta^{\mu}_{\nu}$ yields coupled real and imaginary equations after projection. We retain terms up to second order in phase gradients and denote higher-order contributions collectively by $O(\nabla^3\Theta)$. This truncation is appropriate for the long-wavelength and homogeneous settings considered below and preserves the leading nontrivial effects of analytic phase structure.

The projected real part takes the form

$$G_{\mu\nu}[h] = \frac{8\pi G}{c^4} T^{(m)}_{\mu\nu} + Q^{(\Theta)}_{\mu\nu} + O(\nabla^3\Theta), \quad (4)$$

where $T^{(m)}_{\mu\nu}$ is the ordinary matter stress-energy tensor constructed from $h_{\mu\nu}$ and $Q^{(\Theta)}_{\mu\nu}$ denotes an effective geometric contribution arising from the analytic phase sector. The latter has the dimensions of curvature and should not be interpreted as a physical stress-energy tensor.

The projected imaginary part yields a tensorial constraint,

$$\mathcal{I}_{\mu\nu}[\Theta, h] = 0 + O(\nabla^3\Theta), \quad (5)$$

which enforces compatibility between the analytic envelope and the projected real geometry. Equations (4)-(5) together encode the leading effects of analytic phase structure on spacetime geometry without modifying the local dynamical content of General Relativity.

### B. Geometric phase contribution

At leading nontrivial order, the geometric phase contribution in the real sector is given by

$$Q^{(\Theta)}_{\mu\nu} = \frac{3}{32\pi G} \left( \nabla_{\mu}\theta \nabla_{\nu}\theta - \frac{1}{2}h_{\mu\nu} \nabla_{\rho}\theta \nabla^{\rho}\theta \right), \quad (6)$$

where $\nabla_{\mu}$ denotes the Levi-Civita connection of $h_{\mu\nu}$ and

$$\theta \equiv \frac{1}{4} \Theta^{\mu}_{\mu} \quad (7)$$

is the scalar trace of the full tensorial phase. Expression (6) has the same algebraic form as the stress-energy tensor of a massless scalar field, but its interpretation here is purely geometric: it represents the leading correction to the projected Einstein tensor induced by analytic phase structure and introduces no new matter degrees of freedom.

### C. Imaginary sector and scalar reduction

The imaginary constraint (5) simplifies considerably under symmetry reduction. In homogeneous and isotropic settings, vector and tensor components of $\Theta^{\mu}_{\nu}$ would source anisotropic stresses and are therefore excluded at leading order. The scalar trace $\theta$ is the only phase component compatible with homogeneity and isotropy.

Contracting the imaginary equation yields the scalar wave equation

$$\Box_h \theta = 0, \quad (8)$$

where $\Box_h = h^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ is the d'Alembertian associated with the projected metric. Equation (8) is time-reversal invariant and admits solutions labeled by integration data that characterize distinct analytic branches of the completed configuration space.
The scalar phase $\theta$ is not introduced as an independent dynamical field. It arises as the symmetry-reduced remnant of the tensorial analytic phase $\Theta^\mu_\nu$ and encodes global analytic information associated with branch structure. Its influence on spacetime geometry is entirely mediated through the geometric contribution $Q_{\mu\nu}^{(\Theta)}$ in the projected real equations.

### D. Covariance and conservation

Diffeomorphism covariance of the projected equations follows directly from the covariance of the analytic envelope and the projection principle. The contracted Bianchi identity $\nabla^\mu G_{\mu\nu}[h] = 0$ implies

$$ \nabla^\mu Q_{\mu\nu}^{(\Theta)} = -\frac{8\pi G}{c^4} \nabla^\mu T_{\mu\nu}^{(m)} + O(\nabla^3 \Theta), \quad (9) $$

ensuring consistency of the projected equations at the order retained. In the absence of matter sources, the geometric phase contribution is covariantly conserved to leading order.

## IV. HOMOGENEOUS COSMOLOGY AND THE STIFF ANALYTIC BRANCH

### A. FLRW reduction

We consider homogeneous and isotropic cosmology described by the spatially flat FLRW metric

$$ ds^2 = -c^2 dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2), \quad (10) $$

where $a(t)$ is the scale factor. Ordinary matter and radiation are described by the usual perfect-fluid stress-energy tensor with densities $\rho_m$ and $\rho_r$.

Under these symmetry assumptions, the scalar phase $\theta$ depends only on cosmic time. The wave equation (8) reduces to

$$ \ddot{\theta} + 3H\dot{\theta} = 0, \quad (11) $$

where $H = \dot{a}/a$ is the Hubble parameter. The general solution is

$$ \dot{\theta} = \frac{C}{a^3}, \quad (12) $$

with integration constant $C$ labeling distinct homogeneous analytic branches.

### B. Geometric phase contribution and stiff scaling

Substituting (12) into the geometric phase contribution (6) yields an effective homogeneous contribution with energy density and pressure

$$ \rho_{\theta} = p_{\theta} = \frac{3c^2}{32\pi G} \dot{\theta}^2 = \frac{3c^2 C^2}{32\pi G} a^{-6}. \quad (13) $$

The phase sector therefore behaves as a stiff component with equation of state $p_\theta = \rho_\theta$ and scaling $\rho_\theta \propto a^{-6}$.

Components with this scaling are commonly referred to as stiff fluids and have been studied extensively in early-universe cosmology, most often in the context of kination scenarios where the kinetic energy of a scalar field dominates the expansion [10–12]. The present framework differs in origin: the stiff scaling arises here as a generic homogeneous analytic branch of the analytic-gravity hull rather than as an independently postulated matter sector.

### C. Friedmann equation with phase contribution

Including ordinary matter, radiation, and the geometric phase contribution, the Friedmann equation takes the form

$$ H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2} (\rho_m + \rho_r + \rho_\theta) .. \quad (14) $$

Here and throughout this section, $\rho$ and $p$ denote energy densities and pressures (in SI units), so that $H^2 = (8\pi G/3c^2)\rho$. Standard cosmology gives

$$ \rho_m(a) = \rho_{m0} a^{-3}, \quad \rho_r(a) = \rho_{r0} a^{-4}, \quad (15) $$

where $\rho_{m0}$ and $\rho_{r0}$ denote present-day matter and radiation densities (assuming $a_0 = 1$) [13, 14].

As $a \to 0$, the stiff analytic branch dominates the expansion and fixes the local early-time asymptotic behavior,

$$ a(t) \sim (3Ct)^{1/3}. \quad (16) $$

This asymptotic form controls the analytic structure near the Big Bang projection boundary and underlies the branch behavior discussed in subsequent sections. It should be emphasized that the existence of this branch does not imply its observational realization.

### D. Special case $C = 0$

The special branch $C = 0$ corresponds to a constant phase, $\dot{\theta} = 0$, and hence a vanishing geometric phase contribution, $\rho_\theta = p_\theta = 0$. In this case, the cosmological evolution reduces exactly to standard GR with the same matter content. This branch is admissible within the analytic-gravity hull but is nongeneric from the standpoint of analytic completion, since generic analytic branches carry nonzero integration data.

Importantly, the interpretation of the Big Bang as a projection boundary does not rely on $C \neq 0$. It persists in the $C = 0$ case, although the local analytic structure near $a = 0$ is then governed by the dominant matter component rather than by the stiff phase. We therefore treat the stiff analytic branch primarily as a diagnostic
of the enlarged solution space implied by analytic completeness rather than as a phenomenological prediction. No dedicated observational search for an $a^{-6}$ component is advocated here.

# V. GLOBAL ADMISSIBILITY AND COSMOLOGICAL SELECTION

## A. Global admissibility of analytic branches

Analytic completeness enlarges the space of locally admissible solutions of the Einstein equations by admitting analytic continuation in field space. When combined with the projection principle defining physical observables, this enlargement generically produces multiple analytic branches that differ in their global behavior. Not all such branches admit a consistent real Lorentzian spacetime description over extended intervals of cosmological time.

This criterion is not a new dynamical law and does not modify the local Einstein equations. In this respect, cosmological selection plays the same logical role as global boundary conditions in classical General Relativity, such as asymptotic flatness or global hyperbolicity—restricting the solution space without altering the field equations. Rather, it acts on the space of analytically admissible solutions, distinguishing those branches that correspond to physically observable spacetimes from those that do not.

## B. Cosmological selection

In homogeneous cosmology, the relevant admissibility requirement is necessarily global in nature, as it concerns the existence of a consistent spacetime description over cosmological time scales. We therefore introduce cosmological selection as the minimal admissibility condition appropriate to the analytic-gravity hull: only those analytic branches are regarded as physically admissible for which the projected real Lorentzian metric exists at late times and defines a spacetime with finite energy densities and well-defined proper time.

Cosmological selection does not require that the integration constant $C$ vanish identically at all times, nor does it impose a preferred initial condition. It requires only that the realized cosmological history admit a late-time regime in which the projected metric is well approximated by a standard FLRW spacetime. Analytic branches for which the stiff phase dominates indefinitely at late times fail to admit such a regime and are therefore excluded from physical admissibility.

## C. Emergence of an effective cosmological constant

Once cosmological selection has acted, the projected real Einstein equations (4) admit a residual metric-proportional integration constant. This constant arises because the projected equations determine the traceless part of the Einstein tensor, while the trace remains unconstrained by the phase sector once its late-time influence is suppressed. The contracted Bianchi identity then permits the appearance of a covariantly constant term proportional to $h_{\mu\nu}$.

Accordingly, the late-time projected equations take the form

$$G_{\mu\nu}[h] + \Lambda_{\text{eff}} h_{\mu\nu} = \frac{8\pi G}{c^4} T^{(m)}_{\mu\nu}, \quad (17)$$

where $\Lambda_{\text{eff}}$ is an effective cosmological constant. The present analysis does not determine the numerical value of $\Lambda_{\text{eff}}$ nor address its quantum stability. Rather, it identifies its structural origin as a residual integration mode that becomes admissible once analytic branch selection has removed the stiff phase contribution from the late-time cosmology.

This mechanism is closely related to, but conceptually distinct from, unimodular gravity: in both cases the cosmological constant appears as an integration constant rather than as a fundamental parameter of the action. Here, however, the integration freedom arises from analytic completion and projection rather than from a constraint on the metric determinant.

# VI. GLOBAL PROJECTION BOUNDARIES AND THE COSMOLOGICAL ARROW OF TIME

## A. Analytic continuation versus spacetime observability

Within the analytic-gravity hull it is essential to distinguish between analytic continuation of the underlying configuration and the existence of physical spacetime. Analytic completeness requires that the complexified configuration $g_{\mu\nu}$ be admitted whenever it arises as a local analytic continuation in field space. This requirement places no a priori restriction on continuation across loci where the projected real metric becomes degenerate or ill-defined.

Spacetime observability, however, is defined solely by the projected real Lorentzian metric $h_{\mu\nu} = P(g_{\mu\nu})$. Proper time, causal structure, and the ordering of events are meaningful only on domains where $h_{\mu\nu}$ is real, Lorentzian, and non-degenerate. Analytic continuation of $g_{\mu\nu}$ does not, by itself, guarantee the existence of such a spacetime.
## B. The Big Bang as a global projection boundary

In homogeneous cosmology, the Big Bang corresponds to the limit $a(t) \to 0$ of the projected FLRW metric (10). In this limit, curvature invariants constructed from $h_{\mu\nu}$ diverge and the Lorentzian spacetime description breaks down.

Within the analytic-gravity hull, this breakdown is reinterpreted as a projection boundary. Even though the analytic configuration $g_{\mu\nu}$ may admit continuation across the locus $a = 0$, no real Lorentzian metric can be extracted there by the projection P. The Big Bang therefore marks the boundary of spacetime observability rather than a failure of analytic geometry or of the underlying field equations.

The stiff analytic asymptotic behavior (16) illustrates this structure explicitly. Near the Big Bang,

$$a(t)^3 = 3Ct, \qquad (18)$$

so that the scale factor behaves as $a(t) \sim t^{1/3}$. While $a^3(t)$ is analytic in $t$, the scale factor $a(t)$ itself exhibits a branch point at $t = 0$. Analytic continuation of the solution around this point leads to distinct analytic branches related by monodromy. Only those branches for which the projection yields a real Lorentzian metric correspond to physically admissible spacetimes.

## C. Emergence, breakdown, and cosmological arrow of time

Time in the present framework is treated strictly as an operational notion. As in standard General Relativity, coordinate time serves only as a parameterization, while physical time is defined by proper time along timelike curves. Proper time is determined by the projected real Lorentzian metric and is the only notion of time with direct physical meaning.

Proper time along a timelike curve $\gamma$ is given by

$$d\tau = \frac{1}{c} \sqrt{-h_{\mu\nu} dx^\mu dx^\nu}. \qquad (19)$$

This definition is meaningful only where $h_{\mu\nu}$ possesses Lorentzian signature. As the Big Bang is approached and $a(t) \to 0$, the projected metric becomes degenerate and (19) ceases to define a real quantity. In this precise sense, proper time breaks down at the Big Bang: the ordering of events and the notion of temporal evolution are not defined beyond the projection boundary. In this operational sense, proper time emerges at the Big Bang as soon as the projected metric becomes real and Lorentzian.

Although the local field equations are time-reversal invariant, the framework admits a natural cosmological arrow of time. This arrow does not arise from a dynamical violation of time-reversal symmetry, but from global admissibility: cosmological selection favors branches in

which the stiff analytic contribution becomes subdominant at late times and the projected metric admits an extended regime of well-defined proper time. In this limited sense, the arrow of time is associated with the direction in which spacetime observability improves rather than degrades.

Questions concerning the origin of initial conditions, possible pre-Big Bang phases, or the interpretation of analytic structure beyond the projection boundary require additional degrees of freedom and are deferred to subsequent extensions of the theory. No claim is made that this geometric arrow coincides with thermodynamic, statistical, or quantum-mechanical arrows of time, which depend on additional structures beyond the analytic-gravity hull considered here.

# VII. BLACK HOLE SPACETIMES AND LOCALIZED PROJECTION BOUNDARIES

## A. Analytic continuation and event horizons

In classical General Relativity, event horizons are not curvature singularities but arise from the global causal structure of spacetime. In appropriate coordinate systems, such as Kruskal-Szekeres coordinates for the Schwarzschild solution, the metric is smooth across the horizon and admits analytic extension [5].

Within the analytic-gravity hull, this distinction remains unchanged. The analytic envelope $g_{\mu\nu}$ admits continuation across event horizons, and the projected real metric $h_{\mu\nu}$ remains real, Lorentzian, and non-degenerate. Event horizons therefore do not constitute projection boundaries. Proper time along timelike worldlines remains well defined, and the classical description of infalling observers is preserved exactly.

Thus, within the analytic-gravity hull considered here, restricted to spacetime geometry and its analytic completion, the framework does not alter the local or global behavior of event horizons in General Relativity and introduces no modification to black hole exterior or near-horizon physics.

## B. Curvature singularities as localized projection boundaries

The situation differs qualitatively at curvature singularities, such as $r = 0$ in the Schwarzschild solution. At such loci, curvature invariants constructed from the projected metric $h_{\mu\nu}$ diverge and the Lorentzian spacetime description breaks down.

In the analytic-gravity hull, this breakdown is again reinterpreted as a localized projection boundary. While the analytic envelope $g_{\mu\nu}$ may admit continuation across the singular locus in field space, no real Lorentzian metric can be extracted there by the projection P. The failure is
therefore not attributed to analytic geometry itself, but
to the operational definition of spacetime observables.

It should be emphasized here that this perspective dif-
fers sharply from approaches that seek to regularize or
eliminate curvature singularities by modifying the dy-
namics or introducing new degrees of freedom. In the
present framework, singularities are neither resolved nor
smoothed; they are reclassified as boundaries of space-
time observability inherent to the geometric description.

C. Observer worldlines and termination

Timelike worldlines of classical observers are defined
by the projected metric $h_{\mu\nu}$ and terminate when they
encounter a projection boundary. For an observer falling
into a black hole, proper time remains well defined across
the event horizon but ceases to be defined upon reaching
the curvature singularity.

Beyond this point, analytic continuation of the envelope $g_{\mu\nu}$ may persist, but the framework does not assign physical meaning to such continuation in the absence of a real Lorentzian metric. No continuation of the observer's history is defined within the analytic-gravity hull. This reflects a limitation of spacetime geometry itself rather than an incompleteness of the analytic description.

VIII. DISCUSSION AND OUTLOOK

A. Summary of results and scope of the analytic–gravity hull

The aim of this work has been to make explicit a struc-
tural choice that is already implicit in much of classical
General Relativity: that admissible gravitational config-
urations are taken to be closed under analytic contin-
uation wherever such continuation exists. When com-
bined with a strict operational definition of observables
via projection to a real Lorentzian metric, this require-
ment yields a coherent analytic-gravity hull whose local
dynamics coincide exactly with those of standard GR,
whose global structure is more tightly constrained.

Within this framework, the complexified Einstein
equations decompose naturally into coupled real and
imaginary sectors after projection. The imaginary sec-
tor reduces, under symmetry assumptions appropriate to
homogeneous cosmology, to a scalar phase mode obeying
a covariant wave equation. Its associated geometric con-
tribution behaves as a stiff analytic branch with $\rho \propto a^{-6}$
at early times. This behavior is not introduced as an
independent matter component, but arises as a generic
homogeneous branch implied by analytic completion and
projection.

Analytic completeness also sharpens the interpretation
of classical singularities. Both the Big Bang and black
hole curvature singularities are reclassified as projection
boundaries: loci at which the real Lorentzian spacetime

description ceases to exist, even though analytic continu-
ation of the underlying configuration may persist. Event
horizons, by contrast, remain nonsingular and admit an-
alytic continuation and projection exactly as in classical
GR. Proper time emerges operationally wherever the pro-
jected metric is real and Lorentzian, and breaks down at
projection boundaries. In this sense, time is an emergent
notion tied to spacetime observability rather than a fun-
damental parameter of the analytic configuration space.

Finally, the framework clarifies the status of the cosmo-
logical constant. After cosmological selection has acted,
the projected real Einstein equations admit a residual
metric-proportional integration constant, interpreted as
an effective cosmological constant. The present analy-
sis does not predict its numerical value nor address its
quantum stability; rather, it identifies its structural ori-
gin within a maximally extended classical geometric de-
scription.

B. Status and closure of the analytic–gravity hull

The analytic–gravity hull constructed here is closed in
a precise but limited sense. Once analytic completeness
and the projection principle are adopted, no further in-
dependent degrees of freedom are available at the level of
spacetime geometry alone. All admissible configurations
and their global properties must arise from the same ge-
ometric objects and the Einstein field equations within
the chosen analytic category.

This statement of closure should not be understood as
a claim that all solutions within the analytic–gravity hull
have been classified or exhausted. On the contrary, the
hull admits a rich space of analytic branches whose de-
tailed study remains open. The point is rather that any
such results must arise from the same underlying princi-
ples; no additional geometric structure can be introduced
without modifying those principles.

At the same time, analytic completeness exposes a
structural boundary of the geometric description itself.
The analytic hull generates structures—such as global
phase information and multi-sheeted analytic continua-
tion—that cannot be assigned physical meaning within a
theory formulated solely in terms of spacetime geometry.
When physical phenomena are encountered that cannot
be represented within this hull, the difficulty lies not in
a lack of exploitation of the existing principles, but in a
limitation of what those principles can express.

a. *Structural closure of the analytic-gravity hull.*
The analysis presented here establishes a precise sense
in which the analytic-gravity framework is closed at
the level of spacetime geometry. Within the assump-
tions adopted—analytic completeness, diffeomorphism
covariance, tensorial field content, and a strict opera-
tional definition of observables via projection to a real
Lorentzian metric—no further independent geometric de-
grees of freedom can arise. In particular, the framework
admits at most scalar phase structure under symme-
try reduction and does not support spinorial, operator-
valued, or probabilistic structures without the introduc-
tion of additional principles. Any extension beyond this
hull must therefore modify the set of principles them-
selves, either by introducing new kinematic structure or
by revising the notion of physical observables.

C. Relation to other complex metric approaches

Complex or nonsymmetric metric formulations have
appeared previously in several approaches to gravita-
tional theory, including Euclidean quantum gravity [15]
and complex or nonsymmetric extensions of General Rel-
ativity [16, 17]. In many such frameworks, complex met-
ric components are interpreted as physical or dynamical
degrees of freedom, or are employed as calculational tools
to regularize curvature singularities.

The present approach differs in both intent and inter-
pretation. Complexified metric configurations are intro-
duced solely as an analytic envelope required by analytic
completeness of the gravitational configuration space.
Physical spacetime geometry is defined exclusively by a
real Lorentzian metric obtained through projection, and
no attempt is made to regularize or eliminate curvature
singularities. Instead, singularities are reclassified as pro-
jection boundaries: loci at which the Lorentzian space-
time description ceases to exist even though analytic con-
tinuation of the underlying configuration may persist.

In this sense, the complex structure plays an organi-
zational rather than a physical role. It encodes global
analytic information and branch structure without intro-
ducing new gravitational degrees of freedom or altering
the local dynamical content of General Relativity.

D. Outlook to further extensions

The methodological stance adopted in this study aligns
with Einstein’s distinction between principle theories and
constructive theories: one first formulates a minimal set
of structural principles that delimit admissible descrip-
tions, and then derives their maximal consequences be-
fore introducing additional structure [8, 9]. The present
work follows this discipline by preserving the principles of
General Relativity while removing an implicit limitation
on their maximal consequences through analytic closure.

From this perspective, further theoretical development
may proceed only in a limited number of ways. One may
introduce additional principles in order to represent phys-
ical structures not expressible within spacetime geome-
try alone, or one may revise or relax existing principles
whose domain of applicability has been exceeded. What
is not available is an unconstrained freedom to modify
the theory arbitrarily: any extension must alter the set
of principles—by addition or retraction—in a way that
 renders the new structures both expressible and testable.

The analytic–gravity hull therefore provides a well–
defined classical foundation. It delineates both what can
be said about spacetime geometry under the present prin-
ciples and where those principles cease to suffice. Exten-
sions involving matter representations, quantum states,
or probabilistic structure lie beyond the scope of the
present paper and will be addressed separately.

ACKNOWLEDGEMENTS

The author acknowledges the use of large language
models, including ChatGPT (OpenAI) and Grok (xAI),
as interactive tools for discussion, clarification of math-
ematical structure, and assistance in organizing and re-
fining the presentation of ideas. All conceptual devel-
opment, physical interpretation, and responsibility for
the content of this work remain entirely with the au-
thor. This work was carried out independently and does
not necessarily reflect the views of the author’s affiliated
institution.

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