Peter Daniel Swartz
Independent Researcher, Dayton, Ohio.
DOI: 10.4236/jhepgc.2025.114082   PDF    HTML   XML   84 Downloads   659 Views   Citations

Abstract

We propose a novel extension of general relativity and quantum field theory in which spacetime is treated as a complexified manifold containing both causal (real-time) and anti-causal (imaginary-time) components. Within this framework, black holes act as phase boundaries between forward-traveling photon states and backward-traveling tachyon states, facilitating a global time-symmetric flow of information and entropy. We construct a Lagrangian model in which tachyons arise as imaginary-mass solutions coupled to anti-causal curvature fields, and we derive modified Einstein field equations with both real and imaginary stress-energy sources. This formalism provides a geometric interpretation of entanglement collapse, avoids singularities via photon-tachyon phase transitions, and naturally accounts for dark matter and dark energy as anti-causal mass contributions flowing toward the Big Bang. We present specific observational predictions—including evolving features in the CMB, curvature anomalies near black holes, and entanglement decoherence at cosmological scales—and suggest avenues for testing this model via gravitational lensing, high-energy astrophysics, and entanglement interferometry. Our findings support the idea that the universe is globally time-symmetric, with causality and anti-causality jointly embedded in the complex geometry of spacetime.

Keywords

Black Holes, Big Bang, Quantum Entanglement, Wavefunction Collapse, Dark Matter, Dark Energy

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1. Introduction

The use of complex numbers in theoretical physics has long served as a powerful mathematical tool, yet the imaginary components of space and time are almost always treated as computational conveniences, not physically real quantities. In quantum mechanics, imaginary time appears in path integrals via Wick rotation [1], while in general relativity, complexification is central to techniques such as the Kerr-Newman metric and Penrose’s twistor theory [2]. However, these imaginary dimensions are seldom assigned a direct causal or physical role.

This paper proposes a shift in interpretation: imaginary components of spacetime coordinates may represent anti-causal geometry. That is, rather than viewing imaginary time and space as auxiliary constructs, we treat them as encoding propagation in directions orthogonal to causal time—directions corresponding to retrocausal phenomena. In this view, particles or fields that move along imaginary axes (e.g., tachyons or entangled state components) are not “unphysical” but instead describe structure outside the light cone of ordinary causality.

We call this framework anti-causal spacetime, and treat it as a complex extension of general relativity in which both real and imaginary components of the metric, curvature, and stress—energy tensor carry physical significance. In this model, photons and classical particles propagate along real null or timelike paths, while tachyons and retrocausal field components propagate along imaginary geodesics. The geometry thereby encodes time-symmetric and entanglement-preserving processes in a single, unified manifold.

This reconceptualization opens the door to multiple phenomena being given geometric explanations:

• Entanglement as a connection via shared imaginary-time geodesics.

• Wavefunction collapse as a projection from a complex state manifold onto a real-time slice.

• Dark matter as the real gravitational signature of particles with imaginary mass or anti-causal momentum.

• Black hole interiors as phase-transition boundaries where causal energy transforms into anti-causal structure.

This paper lays the mathematical foundation for such a theory. In Section 2, we revisit the geometric meaning of imaginary numbers. In Section 3, we explore the historical use of complex coordinates in physical theories. Sections 4 through 8 progressively build the formalism of a complex spacetime manifold, interpret its physical implications, and suggest testable consequences. A more physical application of this framework to cosmology is developed separately in a companion work.

2. The Algebraic Origins of Imaginary Geometry

The imaginary unit $i=\sqrt{-1}$ arose historically from attempts to solve polynomial equations with no real solutions, most notably $x^2+1=0$ . Yet even in its earliest applications, the square root of a negative number was not purely abstract—it was interpreted as encoding a quantity orthogonal to the real line. This interpretation deepened with the development of the complex plane, where multiplication by $i$ corresponds to a 90˚ rotation. This geometric role has since been codified into the algebra of complex analysis.

But there is a deeper origin still: early treatments of imaginary numbers sometimes described them as negative area, or even negative volume [3]. The real number line represents length; squaring it gives area. A negative square root implies an area that does not correspond to real spatial extent but to something rotated, inverted, or hidden. This idea was largely abandoned in favor of the now-standard vectorial representation on the Argand diagram, but it may yet carry physical meaning.

We propose restoring this geometric insight by interpreting imaginary values not just as rotated real numbers but as coordinates within a physically real complexified spacetime. If real values of length, time, or area describe quantities aligned with causal structure, then imaginary values naturally describe quantities aligned with anti-causal structure.

This interpretation leads to a fundamental shift: we are not merely adding mathematical degrees of freedom to the spacetime manifold; we are proposing that these imaginary dimensions describe orthogonal physical structure, just as the y-axis complements the x-axis in Euclidean space.

Multiplying a real physical quantity (like displacement or momentum) by $i$ rotates it out of the causal manifold and into the anti-causal one. From this perspective, tachyons—traditionally understood as faster-than-light particles with imaginary mass—may instead be viewed as orthogonal causal modes, propagating along imaginary directions in the same manifold.

In summary, imaginary quantities in physical equations may not be computational artifacts, but signatures of real physical processes operating outside the conventional arrow of time. The rest of this paper develops the geometry of this idea.

3. Complex Coordinates in Physics

Complex numbers have appeared throughout modern physics, often as indispensable tools in both classical and quantum formulations. Yet they are typically interpreted as intermediate constructs—useful for computation, but ultimately discarded in favor of measurable real quantities. In this section, we review key examples where complex coordinates emerge and examine the boundary between their utility and their interpretation.

3.1. Quantum Mechanics and the Schrödinger Equation

The Schrödinger equation itself is inherently complex:

$i\hslash \frac{\partial }{\partial t}\psi \left(\overrightarrow{x},t\right)=\widehat{H}\psi \left(\overrightarrow{x},t\right)$

The time evolution of the wavefunction is governed by the unitary operator ${\text{e}}^{-i\widehat{H}t/\hslash }$, which rotates the quantum state in complex Hilbert space. Despite this, the imaginary component of time is not assigned physical meaning—measurements yield real eigenvalues, and probabilities are constructed from modulus squares ${\left|\psi \right|}^2$ .

3.2. Wick Rotation and Imaginary Time

In quantum field theory and statistical mechanics, the technique of Wick rotation involves substituting $t\to -i\tau$ , mapping Minkowski spacetime to Euclidean space. This simplifies certain integrals and connects quantum mechanics to thermodynamics via path integrals:

$Z={\displaystyle \int \mathcal{D}\varphi {\text{e}}^{-S_E\left[\varphi \right]}}$

where $S_E$ is the Euclidean action. Notably, imaginary time here plays a calculational role, enabling convergence of functional integrals, but it is not usually interpreted as a real temporal dimension.

Stephen Hawking notably used imaginary time in cosmological models, proposing that near the origin of the universe, time could be treated as a spatial dimension [4]. Yet even in this context, imaginary time was employed to eliminate singularities—not to describe ongoing physical structure with causal significance.

3.3. Twistor Theory and Complex Geometry in GR

Penrose’s twistor theory [2] sought to recast spacetime physics in terms of holomorphic structures. In this formulation, fundamental objects are elements of complex projective space, and the light cone structure of spacetime emerges from complex geometry. While elegant, twistor theory remains largely disconnected from a direct physical ontology involving imaginary dimensions.

Similarly, in general relativity, many exact solutions—such as the Kerr and Kerr-Newman metrics—employ complex coordinate transformations. For instance, the complex shift $z\to z+ia$ underlies the derivation of rotating black hole solutions [5]. These manipulations are critical to obtaining correct metrics, but again, the imaginary components are not interpreted as physically real.

3.4. Limitations of Traditional Interpretation

Across these examples, a common theme emerges: complex coordinates are essential for formulation, yet are discarded during interpretation. Imaginary time is seen as a mathematical trick; imaginary spatial components are coordinate artifacts.

This suggests a fundamental tension. If complex structure is so deeply embedded in the mathematics of physics—so much so that the very existence of rotating black holes or quantum path integrals depends on them—why are we forbidden from assigning physical status to those imaginary terms?

This paper proposes resolving this tension by lifting the restriction: treating complex coordinates, especially imaginary time, as physically real but causally distinct. In this framework, fields can propagate in both causal (real-time) and anti-causal (imaginary-time) directions, and spacetime curvature can emerge from both real and imaginary stress-energy sources.

The next section formalizes this approach by defining a complexified spacetime manifold with a metric capable of describing both causal and anti-causal geodesics.

4. Complex Spacetime Manifold

Having motivated the physical interpretation of imaginary coordinates, we now define the mathematical structure of a spacetime manifold extended into the complex domain. This complexified geometry allows for both causal (real-time) and anti-causal (imaginary-time) propagation within a unified framework.

4.1. Complex Coordinates

Let spacetime points be elements of a complexified manifold ${ℳ}_{ℂ}$ , such that each coordinate has both real and imaginary parts:

$x^{\mu }=x_{\text{R}}^{\mu }+i{x}_{\text{I}}^{\mu },\mu =0,1,2,3$

where:

• $x^0=t+i\tau$ : real time $t$ , imaginary (anti-causal) time $\tau$ .

• $x^i=x_{\text{R}}^i+i{x}_{\text{I}}^i$ , $i=1,2,3$ : real and imaginary spatial coordinates.

4.2. Complex Metric Tensor

We define a complex-valued metric tensor:

$g_{\mu \nu }\left(x\right)=g_{\mu \nu }^{\left(R\right)}\left(x\right)+i{g}_{\mu \nu }^{\left(I\right)}\left(x\right)$

This tensor governs inner products in the complexified tangent space:

$\text{d}{s}^2=g_{\mu \nu }\text{d}{x}^{\mu }\text{d}{x}^{\nu }$

yielding a complex line element that mixes real and imaginary contributions.

Null geodesics now satisfy:

$g_{\mu \nu }\text{d}{x}^{\mu }d{x}^{\nu }=0$

which includes both real-null paths (for photons and causal particles) and imaginary-null paths (for tachyons or retrocausal propagation).

4.3. Causal and Anti-Causal Structure

We define two kinds of light cones:

1) Causal (real): trajectories where $\text{d}{s}^2=0$ and $\text{d}{x}^0=\text{d}t>0$ , corresponding to propagation along the real-time axis.

2) Anti-causal (imaginary): trajectories where $\text{d}{s}^2=0$ and $\text{d}{x}^0=i\text{d}\tau >0$ , corresponding to backward-time evolution in imaginary time.

These two classes of geodesics are orthogonal in the complexified manifold, yet both contribute to the curvature and topology of ${ℳ}_{ℂ}$ .

4.4. Physical Fields on the Complex Manifold

A field $\psi \left(x\right)$ defined on ${ℳ}_{ℂ}$ can likewise be decomposed:

$\psi \left(x\right)={\psi }^{\left(R\right)}\left(x\right)+i{\psi }^{\left(I\right)}\left(x\right)$

where the real component evolves along causal trajectories, and the imaginary component may represent an anti-causal or entangled evolution. The dynamics of such fields will be governed by complex generalizations of the Klein-Gordon and Einstein field equations, to be developed in later sections.

4.5. Example: Complexified Minkowski Metric

We begin with the standard Minkowski metric in real coordinates:

${\eta }_{\mu \nu }^{\left(R\right)}=\text{diag}\left(-1,+1,+1,+1\right)$

To complexify, we define:

$x^{\mu }=x_{\text{R}}^{\mu }+i{x}_{\text{I}}^{\mu },\text{with}{x}^0=t+i\tau ,x^i=x_{\text{R}}^i+i{x}_{\text{I}}^i$

We now compute the complex line element:

$\text{d}{s}^2={\eta }_{\mu \nu }^{\left(R\right)}\text{d}{x}^{\mu }\text{d}{x}^{\nu }$

Substituting the complex differentials:

$\text{d}{x}^{\mu }=\text{d}{x}_{\text{R}}^{\mu }+i\text{d}{x}_{\text{I}}^{\mu }$

The squared interval becomes:

$\text{d}{s}^2={\eta }_{\mu \nu }^{\left(R\right)}\left(\text{d}{x}_{\text{R}}^{\mu }\text{d}{x}_{\text{R}}^{\nu }-\text{d}{x}_{\text{I}}^{\mu }\text{d}{x}_{\text{I}}^{\nu }+2i\text{d}{x}_{\text{R}}^{\mu }\text{d}{x}_{\text{I}}^{\nu }\right)$

Thus, the complexified metric yields:

$\text{d}{s}^2=\text{d}{s}_{\text{R}}^2+i\text{d}{s}_{\text{I}}^2$

where:

$\text{d}{s}_{\text{R}}^2={\eta }_{\mu \nu }^{\left(R\right)}\left(\text{d}{x}_{\text{R}}^{\mu }\text{d}{x}_{\text{R}}^{\nu }-\text{d}{x}_{\text{I}}^{\mu }\text{d}{x}_{\text{I}}^{\nu }\right)$

$\text{d}{s}_{\text{I}}^2=2{\eta }_{\mu \nu }^{\left(R\right)}\text{d}{x}_{\text{R}}^{\mu }\text{d}{x}_{\text{I}}^{\nu }$

Interpretation

• The real part modifies the Minkowski interval by subtracting imaginary displacement, hinting that imaginary motion “reduces” real separation.

• The imaginary part encodes real-imaginary coupling, possibly governing how causal and anti-causal components of a field interact.

Even in flat spacetime, this complexification reveals a richer internal structure. Later sections will consider curved analogs of this metric where anti-causal mass and curvature sources play a more explicit role.

5. Physical Interpretation of Imaginary Components

The introduction of complex spacetime coordinates, fields, and metrics demands a reexamination of fundamental physical quantities. In this section, we assign tentative physical interpretations to imaginary mass, momentum, time, and energy, proposing that these components govern retrocausal processes and entangled information flow.

5.1. Imaginary Time as Anti-Causality

In the complexified coordinate $x^0=t+i\tau$ , we interpret $\tau$ as a proper time parameter governing anti-causal trajectories. A worldline moving forward in $\tau$ corresponds to a particle whose influence is directed backward along the real-time axis—i.e., a tachyonic or retrocausal mode.

Unlike Wick-rotated time used for convergence or thermodynamic arguments, this $\tau$ is proposed as a physically real temporal degree of freedom that governs anti-causal field evolution. In this view, ordinary causality (defined by light cones along $t$ ) coexists with anti-causality (defined by orthogonal cones along $\tau$ ).

5.2. Imaginary Mass and Tachyonic States

In special relativity, a tachyon is defined by the condition:

$m^2=-\frac{p^{\mu }{p}_{\mu }}{c^2}<0\Rightarrow m=i\mu ,\mu \in ℝ$

Traditionally, this imaginary mass has been a source of discomfort. But in the complexified framework, this is precisely the natural form for particles propagating along imaginary time. The field $\psi \left(x\right)={\psi }^T\left(x\right)$ , corresponding to a tachyonic particle, obeys the Klein-Gordon equation:

$\left(\square +{\mu }^2\right){\psi }^T\left(x\right)=0$

where ${\mu }^2=-m^2$ is now real and positive, reflecting propagation in $\tau$ , not $t$ .

Tachyons in this view are not faster-than-light in the usual sense; they are orthogonal to it—moving in the imaginary temporal direction, not violating causality but defining a complementary structure: anti-causality.

5.3. Imaginary Momentum and Retrocausal Flow

If a particle moves along an imaginary coordinate $x^{\mu }=x_{\text{R}}^{\mu }+i{x}_{\text{I}}^{\mu }$ , its canonical momentum acquires an imaginary component:

$p_{\mu }=\frac{\partial ℒ}{\partial {\dot{x}}^{\mu }}=p_{\mu }^{\left(R\right)}+i{p}_{\mu }^{\left(I\right)}$

The imaginary part $p_{\mu }^{\left(I\right)}$ represents the momentum associated with anti-causal flow. Conservation of total (complex) momentum implies that causal systems entangled with anti-causal ones may exhibit correlated behavior across spacetime.

This could underlie entanglement phenomena: a shared $p_{\mu }^{\left(I\right)}$ component between spatially separated systems would allow them to remain instantaneously correlated in real time, due to shared structure in the imaginary domain.

5.4. Imaginary Energy and Negative Entropy

From $E^2=p^2{c}^2+m^2{c}^4$ , if $m=i\mu$ , then:

$E=\sqrt{p^2{c}^2-{\mu }^2{c}^4}$

We propose that imaginary energy components are tied to negative entropy flow—a reverse in the statistical arrow of time. In anti-causal processes (e.g., black hole → Big Bang via tachyon conversion), energy may propagate with decreasing entropy—preserving information that appears to be lost from the causal perspective.

This directly links the imaginary sector to the resolution of the black hole information paradox, and perhaps to the low-entropy initial conditions of the early universe.

5.5. Physical Fields with Complex Mass

We now consider a field $\Phi \left(x\right)$ with a complex mass:

$m^2=m_{\text{R}}^2+i{m}_{\text{I}}^2$

Its Klein-Gordon equation becomes:

$\left(\square +m_{\text{R}}^2+i{m}_{\text{I}}^2\right)\Phi \left(x\right)=0$

Such a field propagates both causally and anti-causally, and its complex phase encodes interference between real and imaginary worldlines. This dual behavior could underlie phenomena like quantum tunneling, nonlocal collapse, or entanglement transport.

5.6. Conservation Laws in Complex Form

Energy—momentum conservation extends to the full complex form:

${\nabla }_{\mu }{T}^{\mu \nu }\left(x\right)=0\Rightarrow {\nabla }_{\mu }\left(T_{\text{R}}^{\mu \nu }+i{T}_{\text{I}}^{\mu \nu }\right)=0$

Both real and imaginary parts must be conserved independently. This implies that anti-causal (imaginary) stress-energy can affect the curvature of the manifold, even if it is not directly measurable by real-time observers. In particular, dark matter and dark energy may correspond to contributions from $T_{\text{I}}^{\mu \nu }$ .

5.7. Complex Mass in Field Theory: Lagrangian Formulation

To formalize the dynamics of fields on a complexified manifold, we consider a scalar field $\Phi \left(x\right)\in ℂ$ with complex mass:

$m^2=m_{\text{R}}^2+i{m}_{\text{I}}^2$

The natural Lagrangian density in flat spacetime becomes:

$ℒ=-{\partial }^{\mu }{\Phi }^{*} {\partial }_{\mu }\Phi -m^2 {\Phi }^{*}\Phi$

Substituting the complex mass:

$ℒ=-{\partial }^{\mu }{\Phi }^{*} {\partial }_{\mu }\Phi -\left(m_{\text{R}}^2+i{m}_{\text{I}}^2\right){\Phi }^{*}\Phi$

Splitting into real and imaginary parts:

${ℒ}_{\text{R}}=-{\partial }^{\mu }{\Phi }^{*} {\partial }_{\mu }\Phi -m_{\text{R}}^2{\Phi }^{*}\Phi$

${ℒ}_{\text{I}}=-m_{\text{I}}^2{\Phi }^{*}\Phi$

Equations of Motion

Applying the Euler-Lagrange equation to ${\Phi }^{*}$ , we obtain:

${\partial }_{\mu }{\partial }^{\mu }\Phi +m^2\Phi =0$

This becomes:

$\square \Phi +\left(m_{\text{R}}^2+i{m}_{\text{I}}^2\right)\Phi =0$

This complex Klein-Gordon equation describes propagation under both real and imaginary mass terms.

Physical Interpretation

• The real part $m_{\text{R}}^2$ governs causal wave propagation, particle mass, and inertial response.

• The imaginary part $m_{\text{I}}^2$ introduces anti-causal propagation, decay/growth behavior, or retrocausal interference.

• If $m_{\text{I}}^2<0$ , we obtain exponential growth in imaginary time-interpretable as forward-toward-Big-Bang tachyon flow.

Tachyon Field Lagrangian

For a purely tachyonic field $\psi \left(x\right)$ , where $m^2=-{\mu }^2$ , the Lagrangian becomes:

${ℒ}_T=-{\partial }^{\mu }{\psi }^{\text{*}} {\partial }_{\mu }\psi +{\mu }^2 {\psi }^{\text{*}}\psi$

Note the sign flip on the mass term—opposite that of conventional fields. This implies the field amplifies in directions not aligned with real causal propagation.

Implications for Anti-Causal Energy Flow

The Hamiltonian derived from this Lagrangian includes imaginary contributions:

$\mathscr{H}=\Pi \dot{\Phi }-ℒ\text{with}\Pi =\frac{\partial ℒ}{\partial \dot{\Phi }}={\dot{\Phi }}^{*}$

Leading to:

$\mathscr{H}={\left|\dot{\Phi }\right|}^2+{\left|\nabla \Phi \right|}^2+\mathrm{Re}\left(m^2\right){\left|\Phi \right|}^2+i\mathrm{Im}\left(m^2\right){\left|\Phi \right|}^2$

Thus, total energy density becomes complex, with the imaginary part representing anti-causal energy flux. If we couple this Hamiltonian to a complex stress-energy tensor, the curvature it induces includes both gravitational and “retrogravitational” effects.

6. Coupling Complex Fields to Spacetime Curvature

The gravitational influence of matter arises through the Einstein field equations:

$G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

In a complexified framework, both the Einstein tensor $G_{\mu \nu }$ and the stress-energy tensor $T_{\mu \nu }$ must admit complex structure. This extension leads to new gravitational behaviors originating from the anti-causal, or imaginary-time, sector.

6.1. Complex Stress-Energy Tensor

Let $\Phi \left(x\right)$ be a complex scalar field on a curved manifold with metric $g_{\mu \nu }\left(x\right)\in ℂ$ . Its energy-momentum tensor is:

$T_{\mu \nu }={\partial }_{\mu }{\Phi }^{*} {\partial }_{\nu }\Phi +{\partial }_{\nu }{\Phi }^{*} {\partial }_{\mu }\Phi -g_{\mu \nu }\left({\partial }^{\alpha }{\Phi }^{*} {\partial }_{\alpha }\Phi +m^2{\Phi }^{*}\Phi \right)$

If $m^2\in ℂ$ , this tensor is itself complex:

$T_{\mu \nu }=T_{\mu \nu }^{\left(R\right)}+i{T}_{\mu \nu }^{\left(I\right)}$

We interpret: $T_{\mu \nu }^{\left(R\right)}$ : causal energy-momentum flow; $T_{\mu \nu }^{\left(I\right)}$ : anti-causal stress-energy—e.g., tachyons, retrocausal fields, information inflow from the future.

6.2. Modified Einstein Field Equations

We propose that the geometry of spacetime responds to the full complex stress-energy tensor:

$G_{\mu \nu }=\frac{8\pi G}{c^4}\left(T_{\mu \nu }^{\left(R\right)}+i{T}_{\mu \nu }^{\left(I\right)}\right)$

As such, the Einstein tensor must also be complex:

$G_{\mu \nu }=G_{\mu \nu }^{\left(R\right)}+i{G}_{\mu \nu }^{\left(I\right)}$

We now interpret this equation component-wise:

$G_{\mu \nu }^{\left(R\right)}=\frac{8\pi G}{c^4}{T}_{\mu \nu }^{\left(R\right)}\left(\text{causal curvature}\right)$

$G_{\mu \nu }^{\left(I\right)}=\frac{8\pi G}{c^4}{T}_{\mu \nu }^{\left(I\right)}\left(\text{anti-causal curvature}\right)$

Physical Consequences

This formulation implies:

• Even unobservable (anti-causal) energy influences “observable curvature”

• “Dark matter” and “dark energy” may correspond to $T_{\mu \nu }^{\left(I\right)}$ , the imaginary components of retrocausal stress-energy.

• “Entanglement correlations” may be geometric effects of shared anti-causal curvature.

6.3. Example: Tachyon Field as Source of Curvature

Consider a tachyon field ${\psi }^T\left(x\right)$ , with Lagrangian:

${ℒ}_T=-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }{\psi }^{T\ast }{\partial }_{\nu }{\psi }^T+\frac{1}{2}{\mu }^2{\left|{\psi }^T\right|}^2$

This yields a stress-energy tensor:

$T_{\mu \nu }^{\left(T\right)}={\partial }_{\mu }{\psi }^{T\ast }{\partial }_{\nu }{\psi }^T+{\partial }_{\nu }{\psi }^{T\ast }{\partial }_{\mu }{\psi }^T-g_{\mu \nu }\left(g^{\alpha \beta }{\partial }_{\alpha }{\psi }^{T\ast }{\partial }_{\beta }{\psi }^T-{\mu }^2{\left|{\psi }^T\right|}^2\right)$

Because ${\psi }^T$ grows along imaginary time, the real part of this tensor may vanish or cancel, but the “imaginary part remains”:

$T_{\mu \nu }^{\left(I\right)}\ne 0\Rightarrow G_{\mu \nu }^{\left(I\right)}\ne 0$

So even if causal observers cannot detect the field directly, they observe its “gravitational influence”.

6.4. Dark Sector as Anti-Causal Curvature

We now reinterpret the gravitational effects currently attributed to unseen energy:

• Dark Matter: Persistent spatial curvature without visible mass → may be caused by anti-causal field configurations trapped in $T_{\mu \nu }^{\left(I\right)}$ .

• Dark Energy: Accelerating expansion → may reflect a time-varying component of imaginary curvature driving spacetime expansion from the future.

These phenomena need not involve exotic particles—they could emerge naturally from known fields when extended into the complex domain.

6.5. Conservation Laws in Complex Geometry

The Bianchi identities require:

${\nabla }^{\mu }{G}_{\mu \nu }=0\Rightarrow {\nabla }^{\mu }{T}_{\mu \nu }=0$

This applies separately to both parts:

${\nabla }^{\mu }{T}_{\mu \nu }^{\left(R\right)}=0\left(\text{causal conservation}\right)$

${\nabla }^{\mu }{T}_{\mu \nu }^{\left(I\right)}=0\left(\text{anti-causal conservation}\right)$

So anti-causal energy is not just a mathematical artifact—it must be conserved in its own right, providing a “symmetry partner” to causal energy and entropy flow.

6.6. Worked Example: Complex Perturbation of Schwarzschild Geometry

To explore how anti-causal (imaginary) stress-energy perturbs classical geometry, we consider a Schwarzschild-like background and introduce a small imaginary energy contribution from a tachyonic field.

Background Metric

Begin with the Schwarzschild metric:

$\text{d}{s}^2=-\left(1-\frac{2GM}{r}\right)\text{d}{t}^2+{\left(1-\frac{2GM}{r}\right)}^{-1}\text{d}{r}^2+r^2\text{d}{\Omega }^2$

Now allow the mass to be complex:

$M\to M+i\mu$

where $\mu$ corresponds to a small imaginary mass component representing tachyonic energy concentrated near the horizon.

The metric becomes:

$\text{d}{s}^2=-\left(1-\frac{2G\left(M+i\mu \right)}{r}\right)\text{d}{t}^2+{\left(1-\frac{2G\left(M+i\mu \right)}{r}\right)}^{-1}\text{d}{r}^2+r^2\text{d}{\Omega }^2$

Effect on the Horizon

The horizon radius is defined by $g_{tt}=0$ , so:

$1-\frac{2G\left(M+i\mu \right)}{r_H}=0\Rightarrow r_H=2G\left(M+i\mu \right)$

Thus, the horizon becomes complex:

$r_H=r_R+i{r}_I=2GM+2iG\mu$

The imaginary part shifts the “causal structure”: light cones near the black hole are no longer purely real—they now tilt into imaginary time. This could reflect “tachyonic escape” or retrocausal emission from the horizon.

Stress-Energy Tensor Perturbation

The corresponding perturbation to the stress-energy tensor:

$\text{Δ}{T}_{\mu \nu }=\text{diag}\left(\rho +i{\rho }_T,0,0,0\right)$

where ${\rho }_T\propto {\left|{\psi }^T\right|}^2$ is the local tachyonic field energy.

This yields a complex Einstein tensor:

$\text{Δ}{G}_{\mu \nu }=\frac{8\pi G}{c^4}\text{Δ}{T}_{\mu \nu }$

The imaginary component $G_{tt}^{\left(I\right)}~{\rho }_T$ contributes to gravitational curvature, even though ${\rho }_T$ is not visible through causal probes.

Interpretation

• A complexified Schwarzschild solution implies “retrocausal structure” near the black hole.

• “Tachyons ‘emerge’” from the interior by perturbing the metric into imaginary time.

• This explains how black holes can act as “sources of backward-flowing energy”, without violating classical thermodynamics or observable causality.

7. Entanglement Geometry and Anti-Causal Correlation

Entanglement is often described as a violation of classical locality, yet it may instead reveal an incomplete picture of spacetime geometry. In the complexified spacetime framework, we posit that entangled particles remain causally disconnected in real time but are linked via shared curvature in the imaginary-time dimension. This view eliminates the need for faster-than-light signaling or nonphysical collapse mechanisms.

7.1. The Standard Paradox of Entanglement

Entangled pairs are typically described by a non-factorizable wavefunction:

$\Psi \left(x_1,x_2\right)=\frac{1}{\sqrt{2}}\left[{\varphi }_{\uparrow }\left(x_1\right){\varphi }_{\downarrow }\left(x_2\right)+{\varphi }_{\downarrow }\left(x_1\right){\varphi }_{\uparrow }\left(x_2\right)\right]$

Measurements on either particle immediately determine the state of the other. This defies any explanation based solely on local hidden variables, as confirmed by the violation of Bell inequalities.

Yet this effect persists without observable energy transfer or signal propagation. In our framework, this apparent “nonlocality” is the projection of a deeper connection in the complex spacetime manifold.

7.2. Tachyon-Photon Pairing and Time-Symmetric Causality

We hypothesize that all entangled particles originate from a single causal–anti-causal pair:

$\text{Photon}:{\psi }^P\text{Tachyon}:{\psi }^T$

$\psi \left(x,t\right)={\psi }^P\left(x,t\right)+{\psi }^T\left(x,-t\right)$

The photon evolves forward in time at velocity $+c$ , while its paired tachyon evolves backward in time at effective speed $-c$ . The combined state evolves under time-symmetric boundary conditions, with both components influencing the joint outcome at measurement.

This time-reflection symmetry implies that measurement of one particle does not transmit information to the other, but instead “fixes” a global structure whose constraints propagate backward along ${\psi }^T$ and forward along ${\psi }^P$ , meeting at their common origin in imaginary-time space.

7.3. Field-Theoretic Representation of Entanglement Linkage

We propose a nonlocal, anti-causal term in the effective action:

$S_{\text{link}}={\displaystyle \int {\text{d}}^{4}x{\text{d}}^4{x}^{\prime }K\left(x,x^{\prime }\right){\psi }^{\text{*}}\left(x\right)\psi \left(x^{\prime }\right)}$

where the kernel $K\left(x,x^{\prime }\right)$ is localized in imaginary-time separation:

$K\left(x,x^{\prime }\right)\propto \delta \left({\left(x-x^{\prime }\right)}^2+{\tau }^2\right),\tau =\text{imaginary proper time}$

This effectively introduces a constraint surface in complexified spacetime that both entangled particles reside on, enforcing correlation at spacelike separation without requiring a causal connection.

7.4. Complex Entropy and Anti-Causal Information Flow

In conventional quantum theory, entanglement entropy is defined by tracing out part of a joint system:

$S=-\text{Tr}\left({\rho }_A\log {\rho }_A\right)$

In the complex geometry framework, we generalize this:

$S=S_R+i{S}_I$

where $S_I$ encodes the anti-causal entropy flux—a measure of information flowing “into the past”, or toward the Big Bang boundary. This reflects the tachyon component’s state-space constraint.

Prediction:

The existence of $S_I$ implies that black hole entanglement, early-universe inflationary modes, or even neutrino flavor oscillations may exhibit subtle departures from standard unitary evolution—traceable to anti-causal flux.

7.5. Geometric Picture: Shared Anti-Causal Curvature

We visualize entangled systems as causal worldlines linked by a curved path through imaginary time. If two particles are located at $x_A$ and $x_B$ , they are connected not via a null geodesic in real spacetime, but via a geodesic in the complexified manifold:

$\gamma \left(t\right)\subset {ℂ}^4:\mathrm{Re}\left[\gamma \right]=x_A\to x_B,\mathrm{Im}\left[\gamma \right]\ne 0$

This shared curvature (i.e. $G_{\mu \nu }^{\left(I\right)}\left(x_A\right)=G_{\mu \nu }^{\left(I\right)}\left(x_B\right)$ ) guarantees correlation between their stress-energy boundary terms, even at spacelike separation, as shown in Figure 1.

Figure 1. Two worldlines in Minkowski space, linked by an arc in the imaginary-time direction of complexified spacetime

7.6. Collapse as Geometric Boundary Fixing

In this model, wavefunction collapse is not a mysterious dynamical process but a redefinition of boundary conditions along the anti-causal channel. When one particle is measured, it defines a point on the anti-causal geodesic, constraining the entire path and retroactively fixing the correlation at the partner particle.

This is consistent with conservation laws in the complex manifold:

${\nabla }^{\mu }{T}_{\mu \nu }^{\left(I\right)}=0$

Collapse conserves total imaginary flux and imposes consistency on the global anti-causal structure.

7.7. Worked Example: Anti-Causal Link between Entangled Particles

To visualize how two entangled particles remain correlated through anti-causal curvature, we construct a toy model in a 1 + 1 dimensional complexified Minkowski spacetime.

Setup

Let particles $A$ and $B$ be entangled and emitted at the origin $x=0$ , $t=0$ . They move in opposite directions at the speed of light:

$x_A\left(t\right)=-ct$

$x_B\left(t\right)=+ct$

We assume no classical signal can travel between them once separated.

Now introduce an anti-causal tachyonic field ${\psi }^T\left(x,\tau \right)$ , where $\tau$ is imaginary proper time. Let both particles couple to this field through a conserved imaginary-time current $J_{\mu }^{\left(I\right)}\left(x\right)$ , localized along their worldlines.

Imaginary-Time Geodesic Connection

Define a geodesic path $\Gamma \left(\lambda \right)\subset {ℂ}^2$ connecting the two particles through imaginary time:

$\Gamma \left(\lambda \right)=\left(x\left(\lambda \right),t\left(\lambda \right)+i\tau \left(\lambda \right)\right)$

with endpoints:

$\Gamma \left(0\right)=\left(x_A\left(t\right),t\right),\Gamma \left(1\right)=\left(x_B\left(t\right),t\right)$

A natural path is:

$x\left(\lambda \right)=ct\left(2\lambda -1\right),\tau \left(\lambda \right)={\tau }_0\sin \left(\pi \lambda \right)$

So Γ curves through imaginary time, peaking at ${\tau }_0$ , and returns to real time at the opposite end, as shown in Figure 2.

Figure 2. Real-time trajectories of A and B diverging; an imaginary-time arc Γ connecting them through $\tau$ at mid-path.

Curvature Coupling

Now consider the complex Einstein equation with imaginary energy density localized on Γ:

$G_{\mu \nu }\left(x\right)=\frac{8\pi G}{c^4}\left(T_{\mu \nu }^{\left(R\right)}\left(x\right)+i{\delta }_{\Gamma }\left(x\right)J_{\mu \nu }^{\left(I\right)}\right)$

where ${\delta }_{\Gamma }\left(x\right)$ is a delta-function support along the imaginary-time arc. This curvature contributes no observable local field along $x_A\left(t\right)$ or $x_B\left(t\right)$ , but both particles “feel” the same imaginary curvature background:

$G_{\mu \nu }^{\left(I\right)}\left(x_A\right)=G_{\mu \nu }^{\left(I\right)}\left(x_B\right)$

Wavefunction Collapse via Boundary Constraint

At time $t=t_m$ , particle $A$ is measured. This imposes a boundary condition:

${\psi }^T\left(x_A,\tau \right)={\varphi }_m\left(\tau \right)$

The constraint propagates backward along Γ in imaginary time and forward to $x_B$ , collapsing the full state $\Psi \left(x_A,x_B\right)$ without real-time communication.

Implications

• No signal travels faster than light.

• Correlation arises from “anti-causal curvature symmetry” shared by both particles.

• Collapse is not a dynamical event but a “global consistency requirement” in the complex manifold.

7.8. Measurement Theory Implications

This geometric view offers a viable alternative to interpretations like:

• Copenhagen (wavefunction collapse).

• Many-worlds (branching universes).

• Hidden variables (pre-determined states).

Instead, entanglement is not a mysterious overlay on spacetime—it is a “manifestation of its deeper anti-causal geometry”.

8. Lagrangian Formulation and Field Equations

To encode the dynamics of photon-tachyon duality and its gravitational coupling, we now construct a field-theoretic action principle on a complexified pseudo-Riemannian manifold ${ℳ}_{ℂ}$ , where both the metric $g_{\mu \nu }$ and the fields possess real and imaginary components.

8.1. Total Action and Structure

We begin with the total action:

$S=S_{\text{gravity}}+S_{\text{fields}}+S_{\text{interaction}}$

Gravitational sector (complexified Einstein-Hilbert action):

$S_{\text{gravity}}=\frac{c^4}{16\pi G}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(R+i{R}^{\left(I\right)}\right)$

Here, $R$ is the Ricci scalar and $R^{\left(I\right)}$ is an imaginary curvature term sourced by anti-causal stress-energy.

Field sector

We model the photon and tachyon as two coupled complex fields:

$\psi \left(x\right)={\psi }^P\left(x\right)+{\psi }^T\left(x\right)$

with the photon field ${\psi }^P$ moving forward in time and the tachyon field ${\psi }^T$ propagating backward. The free field Lagrangian is:

${ℒ}_{\text{fields}}=-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }+\frac{i}{2}\left({\overline{\psi }}^T{\gamma }^{\mu }{\partial }_{\mu }{\psi }^T-{\partial }_{\mu }{\overline{\psi }}^T{\gamma }^{\mu }{\psi }^T\right)$

Here ${\psi }^T$ carries imaginary mass and obeys a time-reversed Dirac-like equation with complexified time $t\to t+i\tau$ .

8.2. Interaction and Duality Terms

We introduce a cross-term that encodes tachyon-photon duality and phase exchange at singularities:

${ℒ}_{\text{int}}=-g {\overline{\psi }}^T{\gamma }^{\mu }{\psi }^P {\nabla }_{\mu }\varphi +\text{h}.\text{c}.$

where $\varphi$ is a mediator field that couples real and imaginary components across a Kerr-like boundary (e.g. black hole core). This allows for phase transition:

${\psi }^P\to {\psi }^T\text{at}{\mathscr{H}}^{-}\left(\text{event horizon}\right)$

Symmetry structure

The total Lagrangian is invariant under:

• Local $U\left(1\right)$ gauge transformations.

• Time-reversal $\mathcal{T}$ : $t\to -t,{\psi }^P\leftrightarrow {\psi }^T$ .

• Complex conjugation symmetry in ${ℂ}^4$ .

8.3. Modified Einstein Equations with Tachyonic Source

Varying the total action with respect to the metric gives the field equations:

$G_{\mu \nu }^{\text{total}}=\frac{8\pi G}{c^4}\left(T_{\mu \nu }^{\left(P\right)}+T_{\mu \nu }^{\left(T\right)}\right)$

with

$G_{\mu \nu }^{\text{total}}=G_{\mu \nu }^{\left(R\right)}+i{G}_{\mu \nu }^{\left(I\right)},T_{\mu \nu }^{\left(T\right)}=i{\overline{\psi }}^T{\gamma }_{(\mu }{\nabla }_{\nu )}{\psi }^T+\text{h}.\text{c}.$

The imaginary part of the Einstein tensor $G_{\mu \nu }^{\left(I\right)}$ must be matched by anti-causal energy flux.

8.4. Boundary Conditions at Singularities

To prevent divergence at the black hole core, we impose the following phase-transition boundary conditions:

$\underset{r\to 0}{\lim }{\psi }^P\left(x\right)\to {\psi }^T\left(x\right),\text{with}\langle {\overline{\psi }}^T{\psi }^P\rangle \ne 0$

This ensures finite curvature and enforces that black holes act as converters—not terminators—of matter and information.

8.5. Wave Equations

The tachyonic field satisfies a modified Klein-Gordon equation in complexified spacetime:

$\left(\square -m_T^2\right){\psi }^T\left(x\right)=J^T\left(x\right)$

where $m_T^2<0$ and the box operator $\square$ acts over $t+i\tau$ . The source term $J^T$ is localized at collapsing events (e.g., black hole interiors).

8.6. Constraint Equations and Conservation Laws

Conservation of the total stress-energy tensor in complexified spacetime:

${\nabla }^{\mu }\left(T_{\mu \nu }^{\left(R\right)}+i{T}_{\mu \nu }^{\left(I\right)}\right)=0$

This implies that any perturbation in the real-time sector must be mirrored by compensating anti-causal adjustments elsewhere—consistent with our explanation of entanglement collapse and the CMB evolution.

9. Predictions and Experimental Signatures

A central test of any new physical theory lies in its ability to generate novel, falsifiable predictions. The complexified spacetime hypothesis—combining photon/tachyon duality, imaginary-time curvature, and global entanglement structure—offers a range of such predictions across cosmology, black hole physics, and quantum mechanics. These predictions may differ from standard theory either quantitatively or structurally.

9.1. Evolving Cosmic Microwave Background (CMB)

In standard cosmology, the CMB is a frozen relic from recombination. In our model, however, it reflects not a static emission, but an evolving inflow of anti-causal information from future tachyonic events.

• Prediction: Over time, fine structure in the CMB (e.g., multipole anomalies or polarization patterns) should shift in ways correlated with recent black hole formation events.

• Test: Time-series analysis of Planck/WMAP data or future high-sensitivity CMB missions may detect slow, non-thermal changes incompatible with standard cosmology.

9.2. Black Hole Anti-Causal Signatures

If black holes convert infalling matter into backward-traveling tachyons, they should generate subtle perturbations in nearby spacetime geometry that cannot be explained by classical GR alone.

• Prediction: Quasi-static, nonlocal metric perturbations or weak lensing asymmetries may appear near newly formed black holes.

• Test: High-resolution gravitational lensing surveys near active galactic nuclei or black hole merger remnants could reveal curvature anomalies consistent with imaginary stress-energy.

9.3. Entanglement Decorrelation under Cosmological Expansion

If entanglement is a geometric link through complex spacetime, extreme cosmic stretching (e.g., near the particle horizon) should weaken or break that link.

• Prediction: Quantum entanglement between particles emitted at large cosmological separations (e.g., early universe photons) may exhibit loss of coherence unaccounted for by standard decoherence.

• Test: Examine Bell-type correlations in cosmic neutrino backgrounds, distant gamma-ray burst photons, or primordial B-mode polarization.

9.4. Non-Local Collapse Imprints

If wavefunction collapse is driven by retrocausal constraints via imaginary curvature, then interference experiments involving spacelike-separated measurements may leave indirect, spatially extended imprints.

• Prediction: In nested or delayed-choice quantum eraser setups, statistical distributions may show subtle dependence on future boundary configurations.

• Test: Analyze distributions in double-slit or weak measurement scenarios where detector placement is dynamically altered post-emission.

9.5. Effective Negative Mass from Anti-Causal Matter

Dark matter in this framework consists of anti-causal flows of unobserved tachyonic or antimatter mass toward the Big Bang.

• Prediction: Inhomogeneous distributions of dark matter may correlate not only with past structure formation but with future mass infall or black hole growth rates.

• Test: Compare galaxy cluster lensing maps with black hole census surveys; search for future-predictive correlations in large-scale structure.

9.6. Thermodynamic Anomalies in High-Energy Environments

If imaginary entropy flux is real, then extreme matter/antimatter collisions or black hole information release should reflect it.

• Prediction: Unexpected entropy balance in gamma-ray bursts or Hawking radiation remnants.

• Test: Analyze high-energy transient events for nonthermal spectra or anomalous entropy-energy ratios.

9.7. Entangled Systems as Gravitational Probes

The curvature link between entangled systems may let them function as detectors of imaginary curvature flux.

• Prediction: Long-baseline entangled photon or atom interferometers may show phase drift or fidelity changes correlated with gravitational curvature.

• Test: Build space-based entanglement interferometry experiments to track violations of standard causal structure.

10. Conclusions and Future Work

In this paper, we have developed a geometric and field-theoretic foundation for an extended spacetime model in which photon-tachyon duality, anti-causal curvature, and complexified energy-momentum fields provide a new explanatory framework for long-standing cosmological and quantum phenomena.

By treating tachyons as imaginary-time counterparts to photons, and modeling black holes as conversion surfaces between real and imaginary components of mass-energy, we have shown that classical singularities may be replaced by phase transitions that preserve information and conserve entropy across time-symmetric boundaries. This allows black holes to act not as destructive endpoints, but as retro-causal emitters, linking future gravitational collapse with the origin of the observable universe.

Key predictions include evolving structure in the cosmic microwave background (CMB), the emergence of observable signatures in gravitational lensing and black hole surroundings, and novel perspectives on entanglement and wavefunction collapse. Our extended Einstein field equations incorporate both real and imaginary stress-energy sources and provide a concrete mathematical structure for testing anti-causal curvature in the presence of complexified mass-energy.

Several important theoretical questions remain open. Among them:

• How can the field dynamics of imaginary-time curvature be constrained from first principles, e.g., via a symmetry-breaking or phase-transition mechanism?

• Can tachyonic field dynamics be integrated into a larger quantum gravity or string-theoretic framework?

• Is it possible to derive the apparent “speed” of tachyons (e.g., $v_T=-c$ or $v_T=ic$ ) directly from reformulated Maxwell or Yang-Mills equations in complexified spacetime?

• What observational regimes (e.g., strong lensing, entanglement interferometry, gamma-ray bursts) are most sensitive to these predictions?

A major avenue of future work involves extending the formalism to include path integrals over complexified geometries, potentially allowing for a Wick-rotated formulation of anti-causal processes and boundary constraints in both directions of time. This could provide a rigorous connection between our model and the sum-over-histories interpretation of quantum mechanics, while offering a mechanism for the emergence of classical spacetime from complex topologies.

Furthermore, a dedicated simulation effort will be required to study black hole phase transitions and tachyonic emission numerically. This will demand simplification strategies in high-curvature regimes and a careful treatment of mass-antimatter conversion energy thresholds.

We conclude that the photon-tachyon duality model, formulated within a complexified manifold with anti-causal structure, offers a bold and testable framework with the potential to unify quantum nonlocality, cosmological dark phenomena, and black hole thermodynamics under a single geometric principle. In doing so, it may reveal that the structure of spacetime itself contains the seeds of time-symmetric causality, encoded not only in what we see, but in what is yet to come.

Epilogue: On the Fate of Mass and the Structure of Time

If the framework developed here is valid, then the universe as we experience it-the causal, time-forward domain of matter, light, and entropy—inherits its form from the anti-causal outflows of tachyonic mass through black holes. These outflows collectively form what we perceive as the Big Bang, not as an absolute beginning, but as the convergence point of all anti-causal information.

In this view, the ultimate fate of every particle of matter is not thermal decay into a cold, empty universe, but gravitational convergence into black holes, each acting as a local emitter of retrocausal energy. Every galaxy, star, and atom is bound by entropy and gravitation to a trajectory that culminates in black hole conversion. These black holes do not terminate information—They invert its arrow of time.

The implication is profound: the cosmos is not expanding from a single moment, but folding back into it. The apparent flow of time is merely one direction of a deeper, symmetric process in which information is conserved not by remaining in our future, but by returning to our past. In this light, the Big Bang is not the origin, but the attractor.

Thus, the real-valued universe is not itself inside a black hole, but destined—entirely—to pass through them. Each black hole is a tunnel in time, and their cumulative output constructs the anti-causal architecture of the cosmos. The final act of every mass-bearing particle is not to vanish, but to invert.

This is a universe not of endings, but of reversals.

Appendix. Derivation of Field Equations from the Action

We begin by varying the total action with respect to the metric $g_{\mu \nu }$ and the tachyon field ${\psi }^T$ . The total action is:

$S={\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left[\frac{c^4}{16\pi G}\left(R+i{R}^{\left(I\right)}\right)+{ℒ}_{\text{fields}}+{ℒ}_{\text{int}}\right]$

A1. Gravitational Variation

Varying with respect to $g_{\mu \nu }$ yields:

$\delta S=\frac{c^4}{16\pi G}{\displaystyle \int {\text{d}}^{4}x}\left(\delta \sqrt{-g}R+\sqrt{-g}\delta R+i \delta \sqrt{-g}{R}^{\left(I\right)}+i\sqrt{-g}\delta R^{\left(I\right)}\right)+\delta S_{\text{matter}}$

Recall:

$\delta \sqrt{-g}=-\frac{1}{2}\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu },\delta R=R_{\mu \nu }\delta g^{\mu \nu }+\left(\text{boundary terms}\right)$

Dropping boundary terms, we find:

$\delta S=\frac{c^4}{16\pi G}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left[\left(R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R\right)+i\left(R_{\mu \nu }^{\left(I\right)}-\frac{1}{2}{g}_{\mu \nu }{R}^{\left(I\right)}\right)\right]\delta g^{\mu \nu }+\delta S_{\text{matter}}$

Defining:

$G_{\mu \nu }^{\left(R\right)}=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R,G_{\mu \nu }^{\left(I\right)}=R_{\mu \nu }^{\left(I\right)}-\frac{1}{2}{g}_{\mu \nu }{R}^{\left(I\right)}$

We obtain:

$G_{\mu \nu }^{\text{total}}=G_{\mu \nu }^{\left(R\right)}+i{G}_{\mu \nu }^{\left(I\right)}=\frac{8\pi G}{c^4}\left(T_{\mu \nu }^{\left(R\right)}+i{T}_{\mu \nu }^{\left(I\right)}\right)$

A2. Variation with Respect to ${\psi }^T$

Now vary the action with respect to ${\overline{\psi }}^T$ from the Lagrangian:

${ℒ}_{{\psi }^T}=\frac{i}{2}\left({\overline{\psi }}^T{\gamma }^{\mu }{\nabla }_{\mu }{\psi }^T-{\nabla }_{\mu }{\overline{\psi }}^T{\gamma }^{\mu }{\psi }^T\right)-m_T{\overline{\psi }}^T{\psi }^T$

Using standard functional variation:

$\frac{\delta S}{\delta {\overline{\psi }}^T}=i{\gamma }^{\mu }{\nabla }_{\mu }{\psi }^T-m_T{\psi }^T=0$

Thus, we recover the tachyonic Dirac equation in curved complexified spacetime:

$i{\gamma }^{\mu }{\nabla }_{\mu }{\psi }^T=m_T{\psi }^T$

A3. Complexified Conservation Law

The divergence of the total stress-energy tensor must vanish:

${\nabla }^{\mu }\left(T_{\mu \nu }^{\left(R\right)}+i{T}_{\mu \nu }^{\left(I\right)}\right)=0$

ensuring that both the causal (real) and anti-causal (imaginary) energy-momentum sources are constrained by geometry.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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