Gravitomagnetic Reynolds Number

• Gravitomagnetic Reynolds Number is a dimensionless parameter defining the ratio of advective to diffusive transport of gravitomagnetic fields, analogous to the magnetic Reynolds number in MHD.
• It governs the transition between frozen-in field lines and diffusive regimes, influencing the behavior of gravitational Alfvén waves and turbulence in strongly coupled, high-density fluids.
• High gravitomagnetic Reynolds numbers indicate efficient fluid-field coupling in early-universe scenarios, potentially leaving signatures in stochastic gravitational wave backgrounds.

The gravitomagnetic Reynolds number is a dimensionless parameter that quantifies the relative importance of advection ("freezing-in") versus diffusion of gravitomagnetic fields in relativistic fluid systems, playing a role closely analogous to the magnetic Reynolds number in classical magnetohydrodynamics (MHD). In the context of early-universe cosmology and relativistic fluid dynamics, the gravitomagnetic Reynolds number determines the efficiency with which spacetime's gravitomagnetic structure is dynamically coupled to matter, controlling phenomena such as gravitational Alfvén waves, nonlinear turbulence, and potential signatures in the stochastic gravitational wave background.

1. Definition and Formalism

The gravitomagnetic Reynolds number, denoted $R_{g}$, is defined by the ratio of advective to diffusive transport of the gravitomagnetic field in a gravitating relativistic fluid:

$R_{g} = \frac{u L}{\eta_{g}}$

where:

• $u$ is the characteristic velocity of the fluid,
• $L$ is a typical macroscopic length scale (e.g., bubble size during a cosmological phase transition),
• $\eta_{g}$ is the gravitomagnetic diffusivity.

The gravitomagnetic diffusivity in the gravitoelectromagnetic (GEM) formalism is given by:

$\eta_{g} = \frac{c^{2}}{4\pi G \sigma_{g}}$

where $c$ is the speed of light, $G$ is the gravitational constant, and $\sigma_{g}$ is the effective "gravito-conductivity" of the medium (Liang et al., 4 Oct 2025).

This definition is directly inspired by the classical MHD expression, where a large $R_{g}$ signifies that the gravitomagnetic field is advected with the fluid ("frozen-in" regime), while a small $R_{g}$ corresponds to rapid dissipation and diffusion of the field relative to the fluid motion.

2. Physical Interpretation and Regimes

The gravitomagnetic Reynolds number controls the dynamical regime of gravitomagnetic hydrodynamics (GMHD):

• If $R_g \ll 1$: Gravitomagnetic diffusion dominates, and the field lines are not tied to the fluid; gravitational induction effects are weak and rapidly washed out.
• If $R_g \gg 1$: The system enters the "ideal GMHD" regime, where the gravitomagnetic field lines are "frozen" into the fluid and evolve with it.

In this ideal regime, the conservation law

$$\frac{d}{dt} \oint_{S} \mathbf{B}_g \cdot d\mathbf{S} = 0$$

holds in direct analogy to Alfvén's theorem in MHD. Here, $\mathbf{B}_g$ is the gravitomagnetic field defined in the GEM framework (Liang et al., 4 Oct 2025). The high-$R_g$ regime is characterized by strong fluid-spacetime coupling, which is particularly relevant in the high-temperature, high-density conditions of the early universe, such as during the electroweak phase transition.

3. GMHD Turbulence and Gravitational Alfvén Waves

When $R_g$ is large, dynamic interactions between the relativistic fluid and the gravitomagnetic field give rise to novel collective modes:

• Gravitational Alfvén Waves: Small perturbations in the fluid velocity $\mathbf{u}_1$ and gravitomagnetic field $\mathbf{B}_1$ obey

$$\frac{\partial^2 \mathbf{u}_1}{\partial t^2} = \frac{(\mathbf{B}_0 \cdot \nabla)^2}{16\pi G \rho} \mathbf{u}_1$$

where $\mathbf{B}_0$ is the background gravitomagnetic field and $\rho$ is the mass density. The wave speed is

$V_B = \frac{B_0}{4\sqrt{\pi G \rho}}$

• Energy Cascades: GMHD turbulence in the high-$R_g$ regime supports a turbulent cascade, transferring energy from large to small scales (direct cascade) or potentially vice versa (inverse cascade). Dimensional analysis analogously to MHD yields

$E(k) \propto (\varepsilon V_B)^{1/2} k^{-3/2}$

for the turbulent energy spectrum, where $\varepsilon$ is the energy transfer rate, $V_B$ the gravitational Alfvén speed, and $k$ the wavenumber (Liang et al., 4 Oct 2025). This is reminiscent of the Iroshnikov–Kraichnan spectrum in classical MHD turbulence.

4. Early-Universe Implications

The effective gravito-conductivity $\sigma_g$ is extremely large in the primordial universe due to high temperatures and densities, implying a very small $\eta_g$ and hence $R_g \gg 1$ (Liang et al., 4 Oct 2025). In this context:

• The strongly coupled GMHD state is realized, with spacetime geometry and plasma intimately co-evolving.
• Gravitational Alfvén waves and nonlinear turbulence become dynamically important.
• Stochastic gravitational wave backgrounds produced by phase transitions or other cosmological events can carry imprints of GMHD turbulence, because the gravitational wave energy density

$$\rho_{\mathrm{GW}} = \frac{|\mathbf{B}_g|^2}{16\pi G}$$

is directly tied to the magnitude and structure of the gravitomagnetic fluctuations.

These effects are expected to leave signatures in the gravitational wave spectrum, serving as potential probes of nonlinear, strongly coupled dynamics in the early universe.

5. Analogies to Classical MHD and Related Parameterizations

The gravitomagnetic Reynolds number is structurally analogous to the magnetic Reynolds number in classical MHD turbulence:

$\mathrm{Magnetic:\quad} R_m = \frac{u L}{\eta}$

$$\mathrm{Gravitomagnetic:\quad} R_g = \frac{u L}{\eta_g}$$

with all corresponding physical parallels—frozen-in field lines, turbulent cascades, and threshold values for transitions to turbulence or magnetic field generation (Liang et al., 4 Oct 2025). In both cases, the size of Reynolds number-like parameters governs the relative dominance of advective and diffusive processes, scaling transitions, and the appearance of collective wave or turbulent phenomena.

6. Broader Context and Limitations

While the GMHD framework and the gravitomagnetic Reynolds number provide a rigorous and mathematically tractable analogy to electromagnetic turbulence, several features distinguish gravitomagnetic phenomena:

• The source of gravitomagnetic fields (mass currents) is fundamentally tied to spacetime geometry, not external magnetic fields.
• The coupling constant is $G$, not $e^2$ or the magnetic permeability, rendering absolute strengths vastly weaker in present-day conditions.
• Only under extremely high density and temperature (as in the early universe) does the effective coupling and Reynolds number become dynamically significant.

In lower-density astrophysical environments, alternative quantities (such as $J/(M c^2)$ in frame-dragging effects (Iorio, 2014) or parameters appearing in gravitomagnetic Love numbers (Landry et al., 2015, Poisson, 2020, Gupta et al., 2020)) measure gravitomagnetic influences but may not correspond directly to an advection/diffusion competition.

7. Summary Table: Gravitomagnetic Reynolds Number in Early-Universe GMHD

Quantity	Expression	Physical Role
$R_g$	$R_g = (u L)/\eta_g$	Measure of advection vs. diffusion of $\mathbf{B}_g$
$\eta_g$	$\eta_g = c^2/(4\pi G \sigma_g)$	Gravitomagnetic diffusivity
Alfvén Speed	$V_B = B_0/(4\sqrt{\pi G \rho})$	Speed of gravitational Alfvén waves
GW Energy Density	$\rho_{\mathrm{GW}} = |\mathbf{B}_g|^2 / (16\pi G)$	Gravitational wave background affected by GMHD turbulence

For detailed derivations and discussion of all expressions above, see (Liang et al., 4 Oct 2025).

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The gravitomagnetic Reynolds number $R_g$ serves as a controlling parameter for the strong-coupling limit of GMHD, determining when field lines are frozen into the plasma, supporting gravitational Alfvén waves and turbulence, and influencing primordial gravitational wave signals through turbulent energy cascades and field amplification in the early universe (Liang et al., 4 Oct 2025). This frames a novel link between general relativity, hydrodynamic turbulence, and cosmological observables.