On a Gödel-like Solution in Non-Relativistic Gravity

1. Problem Statement

The paper addresses the existence and causal structure of G¨odel-like solutions within the framework of Galilean gravity, a geometric formulation of non-relativistic gravitation.

• Context: In General Relativity (GR), G¨odel's 1949 solution describes a rotating universe containing Closed Timelike Curves (CTCs), which violate causality. While GR handles this by restricting analysis to causally well-behaved regions (e.g., behind horizons), the existence of CTCs is an intrinsic feature of the relativistic geometry.
• The Question: Does a similar breakdown of causality occur in a non-relativistic gravitational theory? Specifically, can a G¨odel-type solution exist in a covariant Galilean framework without generating CTCs?
• Motivation: Newtonian physics traditionally relies on absolute time, seemingly precluding causality violations. However, modern covariant formulations of non-relativistic physics (using 5D Galilean manifolds) allow for a geometric description of gravity analogous to GR. The authors investigate whether the "pathological" features of G¨odel universes are inherent to rotation or specific to the relativistic causal structure.

2. Methodology

The authors employ a 5-dimensional covariant formulation of non-relativistic physics, often referred to as the Bargmann or Galilean manifold approach.

• Geometric Framework:
  • The theory is defined on a 5D manifold with coordinates $x^\mu = (x, t, s)$, where $s$ is an extra coordinate required for covariance.
  • The metric tensor $g_{\mu\nu}$ has a specific signature allowing for Galilean transformations. The inner product is defined as $(x|y) = x \cdot y - x^4 y^5 - x^5 y^4$.
  • Dimensional Reduction: Physical 3+1 Newtonian dynamics are recovered via an "immersion" where the extra coordinate is fixed as $s = r^2/(2t)$.
• Matter Sector (Galilean Fluid):
  • Matter is modeled as a Galilean fluid derived from a variational principle.
  • The fluid is described by a scalar field $\phi$ and density $\rho$ on the 5D manifold.
  • The Lagrangian density is $\tilde{L} = -\frac{1}{2}\rho \partial_\mu \phi \partial^\mu \phi - V(\rho)$.
  • Upon dimensional reduction, this yields the Navier-Stokes equation and the continuity equation in 3+1 dimensions.
  • The authors choose a linear potential $V(\rho) = \rho$, implying constant pressure in the reduced dynamics.
• Field Equations:
  • Gravity is formulated analogously to GR: $G_{\mu\nu} = 8\pi T_{\mu\nu}$, where $G_{\mu\nu}$ is the Galilean Einstein tensor constructed from the 5D curvature.
  • A G¨odel-like metric ansatz is adopted:
    $$dS^2 = [dt + H(x)dy]^2 - dx^2 - D^2(x)dy^2 - F^2(s)dz^2 - ds^2$$
  • This leads to a system of highly nonlinear, coupled differential equations (Eqs. 24–29) involving functions $H(x)$, $D(x)$, $F(s)$, and matter profiles $\Theta(s)$, $\sigma(x)$.

3. Key Contributions

• Formulation of Non-Relativistic Gravity: The paper rigorously couples the geometric field equations of 5D Galilean gravity to a variational fluid source, demonstrating that Navier-Stokes dynamics emerge naturally from the 5D scalar field Lagrangian.
• Construction of Exact Solutions: The authors derive a specific class of exact solutions for the G¨odel-like ansatz by fixing the matter sector consistently (constant density $\Theta$, constant pressure $p=1$, and linear potential).
• Causal Analysis: Unlike previous studies that focused on relativistic G¨odel universes, this work explicitly analyzes the causal structure of the non-relativistic counterpart using numerical methods.

4. Results

The numerical analysis of the field equations yields the following results:

• Metric Functions: The spatial functions $D(x)$ and $H(x)$ are solved numerically. The results show that $D(x) > H(x)$ holds throughout the entire spatial domain.
• Causal Structure:
  • The metric component along the rotation axis ($y$-direction) is $g_{yy} = H(x)^2 - D(x)^2$.
  • Since $D(x) > H(x)$, it follows that $g_{yy} < 0$ everywhere.
  • Consequently, the Killing vector $\partial_y$ (associated with rotation) remains spacelike everywhere.
• Absence of CTCs: Because the Killing vector is spacelike, no Closed Timelike Curves (CTCs) arise in this solution. The rotating universe described by this non-relativistic model is fully causal.
• Matter Profiles: The spatial density profile $\sigma(x)$ is smooth and symmetric, and the pressure remains constant, consistent with the chosen potential $V(\rho)=\rho$.

5. Significance and Implications

• Causality is Relativistic, Not Rotational: The most significant finding is that the causality violation (CTCs) in G¨odel-type universes is not an intrinsic consequence of rotation or covariance itself. Instead, it is a direct consequence of the relativistic spacetime structure (specifically the Lorentzian metric signature and the finite speed of light).
• Physical Admissibility of Rotating Galilean Universes: In the non-relativistic regime, G¨odel-like solutions are physically admissible and do not suffer from the "pathological" time-travel paradoxes found in GR. This suggests that rotating cosmological models are viable in Newtonian gravity without requiring the exclusion of causally problematic regions.
• Cosmological Applications: The authors propose that Galilean G¨odel-like spacetimes could serve as alternative geometric settings for analyzing cosmological data exhibiting large-scale anisotropies. Unlike their relativistic counterparts, these models do not face conceptual difficulties regarding causality, making them potentially useful for testing anisotropic phenomena in the non-relativistic limit.
• Theoretical Insight: The work reinforces the utility of the 5D Galilean manifold as a tool for unifying the geometric description of non-relativistic physics with the mathematical rigor of relativistic field theories, while highlighting the fundamental differences in causal structure between the two regimes.

In conclusion, the paper demonstrates that while G¨odel's solution is a source of causality violation in General Relativity, its non-relativistic counterpart in Galilean gravity describes a causally consistent, rotating universe, thereby isolating the origin of CTCs to the specific geometry of relativistic spacetime.