120-fisher.md

Part 1: English Version

The Geometry of Liquidity Traps: Fisher’s Equation on a 2D Riemannian Manifold

Project EMIS Technical Note #120 Date: Oct 2023 Status: Draft / Pre-print Topic: Econophysics, Differential Geometry, Monetary Theory

Abstract

We propose a geometric reformulation of the classical Fisher Equation ($MV = PT$) within the framework of the Economic Manifold of Interacting Systems (EMIS). By treating the economy as a 2D Riemannian manifold rather than a flat Euclidean space, we demonstrate that the Quantity Theory of Money is a specific case of the Divergence Theorem. Furthermore, we derive the phenomenon of the Liquidity Trap not as a behavioral anomaly, but as a Gravitational Redshift effect. When wealth concentration creates significant curvature in the market geometry, the observed velocity of money ($V$) approaches zero relative to a flat background, neutralizing monetary injection ($M$).

1. The Geometric Fisher Equation

In classical economics, Fisher’s Equation is an algebraic identity. In EMIS, we treat Money as a conserved fluid on a 2D manifold $\mathcal{M}$. The equation represents the conservation of flux and energy dissipation.

1.1 The Integral Form (Flux Conservation)

Let $\Omega \subset \mathcal{M}$ be an economic region bounded by $\partial \Omega$. The macroscopic Fisher Equation is the conservation of flux:

\[\oint_{\partial \Omega} \vec{J} \cdot \vec{n} dl = \iint_{\Omega} \mathcal{W} d^2x\]

Where:

  • $\vec{J} = \rho \vec{u}$: Money Current Density Vector. ($\rho$ is money supply density, $\vec{u}$ is velocity vector).
  • $\vec{n}$: Unit normal vector to the boundary.
  • $dl$: Line element along the 1D boundary (e.g., market entry).
  • $\mathcal{W}$: Transaction Power Density ($P \times T$ per unit area).
  • $d^2x$: Area element on the 2D manifold.

Interpretation: The net flux of money entering a closed economic zone must equal the total value of transactions (work) performed within that zone.

1.2 The Differential Form

Applying the Divergence Theorem (Gauss’s Theorem), we obtain the local field equation:

\[\nabla \cdot (\rho \vec{u}) = \mathcal{W}\]

This connects the local divergence of liquidity to local economic activity.

2. Metric Induced Velocity Suppression (The Liquidity Trap)

Why does increasing $M$ (Central Bank Injection) fail to increase $PT$ (GDP) during crises? The answer lies in the geometry.

2.1 The Metric of Wealth

We assume the manifold $\mathcal{M}$ is curved by the stress-energy tensor of capital $T_{\mu\nu}$. In the static limit, the metric takes the form: \(ds^2 = -e^{2\Phi(x)} dt^2 + g_{ij} dx^i dx^j\) Where $\Phi(x)$ is the gravitational potential determined by wealth concentration. In high-concentration zones (Monopolies/Whales), $\Phi(x)$ is deep (negative).

2.2 Gravitational Redshift of Money Velocity

Velocity is a spatial displacement with respect to time: $v = dx/dt$. However, “Market Time” ($dt$) is relative. For an external observer (the aggregate market), the observed velocity $V_{obs}$ is related to the local proper velocity $V_{local}$ by the redshift factor:

\[V_{obs} = V_{local} \sqrt{-g_{00}}\]

2.3 The Trap Mechanism

As wealth creates a “Black Hole” (singular concentration, $T_{00} \to \infty$), the metric component $g_{00} \to 0$ near the event horizon.

\[\lim_{g_{00} \to 0} V_{obs} = 0\]

Conclusion: Even if local agents are trading actively ($V_{local}$ is high within the financial sector), the high curvature causes time dilation. From the perspective of the global economy (GDP), the money appears frozen.

  • Result: $\Delta M \uparrow \times (V \to 0) = \Delta (PT) \approx 0$.
  • The liquidity is trapped not by psychology, but by the geometry of the inequality.

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