Emergence of the Lorentzian Metric Signature in Temporal Flow Physics

John Gavel

Abstract:
In Temporal Flow Physics (TFP), time is fundamental as a quantized one-dimensional flow, and space emerges from the relational structure of fluctuations in this flow. We rigorously prove that the Lorentzian signature of the emergent spacetime metric arises naturally from the causal and statistical properties of temporal flow fluctuations, rather than being imposed as a postulate. This section formalizes the assumptions, constructs the emergent metric tensor from flow correlations, and demonstrates how causality enforces the Lorentzian signature.

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1. Introduction and Setup

We consider a fundamental scalar flow field

$$F: \mathcal{M} \to \mathbb{R}, \quad F = F(x), \quad x \in \mathcal{M}$$

where $\mathcal{M}$ is an emergent 4-dimensional manifold with coordinates

$$x^\mu = (x^0, x^1, x^2, x^3), \quad \mu=0,1,2,3,$$

with $x^0 = t$ identified as emergent time and $x^i, i=1,2,3$ as emergent spatial coordinates.

The flow decomposes into a background and fluctuations:

$$F(x) = \bar{F}(x) + \delta F(x),$$

with $\bar{F}(x)$ smooth and slowly varying, and $\delta F(x)$ encoding local quantum/statistical fluctuations.

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2. Assumptions

1. Statistical Averaging:
   There exists a well-defined averaging $\langle \cdot \rangle$ (statistical or quantum expectation) with

$$\langle \delta F(x) \rangle = 0, \quad \langle \partial_\mu \delta F(x) \rangle = 0.$$

2. Emergent Metric Definition:
   The emergent metric tensor $G_{\mu\nu}(x)$ is defined as the local second-order correlation tensor of derivatives of fluctuations:

$$\boxed{ G_{\mu\nu}(x) := \langle \partial_\mu \delta F(x) \, \partial_\nu \delta F(x) \rangle. }$$

3. Isotropy and Homogeneity in Space:
   Spatial fluctuations are statistically isotropic and homogeneous, so for $i,j=1,2,3$
   

$$G_{ij}(x) = B(x) \delta_{ij}, \quad B(x) > 0,$$

and cross spatial components vanish:

$$G_{ij} = 0 \quad \text{for } i \neq j.$$

4. Temporal Causal Arrow:
   The fundamental 1D temporal flow has a causal direction imposing an intrinsic asymmetry, encoded as a negative correlation for the time-time component:

$$G_{00}(x) = \langle (\partial_t \delta F(x))^2 \rangle = -A(x), \quad A(x) > 0.$$

5. Cross Terms Vanish:
   Due to statistical independence and symmetry,

$$G_{0i}(x) = \langle \partial_t \delta F(x) \partial_i \delta F(x) \rangle = 0.$$

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3. Theorem: Emergence of Lorentzian Signature

Theorem:
Under Assumptions 1–5, the emergent metric tensor $G_{\mu\nu}(x)$ has Lorentzian signature $(-+++$), i.e., one negative eigenvalue associated with the time coordinate and three positive eigenvalues associated with space.

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4. Proof

From Assumptions 2, 3, 4, and 5, $G_{\mu\nu}(x)$ takes the diagonal form:

$$G_{\mu\nu}(x) = \begin{pmatrix} -A(x) & 0 & 0 & 0 \\ 0 & B(x) & 0 & 0 \\ 0 & 0 & B(x) & 0 \\ 0 & 0 & 0 & B(x) \end{pmatrix}, \quad A(x), B(x) > 0.$$

Since $A(x), B(x) > 0$, the eigenvalues of $G_{\mu\nu}$ are $-A(x), B(x), B(x), B(x)$.

Thus, the signature of $G_{\mu\nu}$ is

$$\mathrm{signature}(G) = (-, +, +, +),$$

which matches the Lorentzian metric signature.

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5. Physical Interpretation: Causality and Hyperbolicity

Consider a scalar field $\phi(x)$ coupled to the emergent metric via the effective action

$$S[\phi] = \int d^4 x \, \sqrt{|G|} \left( \frac{1}{2} G^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \right).$$

The Euler-Lagrange equation gives a wave operator:

$$G^{\mu\nu} \partial_\mu \partial_\nu \phi + \ldots = 0.$$

To maintain causality and a well-posed initial value problem, this operator must be hyperbolic—which requires exactly one negative eigenvalue in the metric $G_{\mu\nu}$.

The negative sign in the time-time component arises naturally from the causal, oriented nature of the fundamental 1D temporal flow $F_X(t)$, while spatial fluctuations lack such directional causality and yield positive-definite correlations.

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6. Rescaling to Minkowski Form

Defining new coordinates

$$t' = \sqrt{A(x)} \, t, \quad x'^i = \sqrt{B(x)} \, x^i,$$

we normalize the metric to

$$G'_{\mu\nu} = \mathrm{diag}(-1, +1, +1, +1),$$

recovering the canonical Minkowski metric.

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7. Conclusion

The Lorentzian signature $(-+++)$is not an imposed assumption but a derived consequence of the fundamental causal structure of quantized 1D temporal flows and their statistical fluctuations. This result establishes a rigorous foundation for the emergent relativistic spacetime geometry in Temporal Flow Physics.

Interpretation of Mass as Flow Segmentation and Metric Curvature

In the Temporal Flow Physics framework, what we perceive as a localized mass-energy concentration in the emergent spacetime corresponds fundamentally to a region where the underlying quantized 1D temporal flows

$$\mathbf{F} = (F_X, F_Y, F_Z, F_W)$$

exhibit altered dynamical behavior. Specifically, these flows become segmented, resisted, or interact nontrivially in that localized region. Such deviations modify the local statistical properties of the fluctuation field derivatives, which we denote by the set of variance parameters

$$\{A(x), B(x), C(x), D(x)\},$$

characterizing correlations of temporal and spatial flow components. These localized changes induce spatially dependent modifications in the emergent metric tensor

$$G_{\mu\nu}(x) = \langle \partial_\mu \delta F(x) \, \partial_\nu \delta F(x) \rangle,$$

leading to nontrivial curvature of the emergent spacetime geometry. Thus, the presence of mass-energy is understood as a manifestation of altered flow fluctuation statistics that produce curvature, bridging the microscopic flow dynamics to the macroscopic gravitational field.