2. Einstein–Maxwell equation

To study the gravitational dynamics of a weakly rotating self-gravitating fluid, we need to explore the linearized limit of the Einstein equation. When the space–time metric is almost Minskowskian, $g_{\mu \nu }=\eta _{\mu \nu }+\bar {h}_{\mu \nu }$, and terms of $\mathcal {O}(c^{-4})$ or higher are neglected, we can write the Einstein equation

(2.1)\begin{equation} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={-}\frac{16{\rm \pi} G}{c^4}T_{\mu\nu} \end{equation}

as a set of linearized equations almost identical to the electromagnetic Maxwell equation as (Braginsky, Caves & Thorne Reference Braginsky, Caves and Thorne1977; Thorne Reference Thorne, Fairbank, Deaver, Everitt and Michelson1988; Manfredi Reference Manfredi2015)

(2.2)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{E}_g={-}4{\rm \pi} \rho, \end{gather}

(2.3)\begin{gather}\boldsymbol{\nabla}\times \boldsymbol{E}_g=0, \end{gather}

(2.4)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{B}_g=0, \end{gather}

(2.5)\begin{gather}\boldsymbol{\nabla}\times \boldsymbol{B}_g={-}\frac{16{\rm \pi} G}{c} \rho\boldsymbol{v}+\frac{1}{c}\frac{\partial \boldsymbol{E}_g}{\partial t}, \end{gather}

where $T_{\mu \nu }$ is the stress energy tensor, $\boldsymbol {E}_g=-\boldsymbol {\nabla } \phi$ is the Newtonian gravitational field, $\phi$ is the gravitational potential and $\boldsymbol {B}_g=\boldsymbol {\nabla }\times \boldsymbol {A}_g$ is the gravitomagnetic field, with $\boldsymbol {A}_g$ being the corresponding vector potential. Also, $G$ is the gravitational constant and $\rho$ is the matter density. It should be noted that the right-hand side of (2.3) does not have any $\partial \boldsymbol {B}_g/\partial t$ to this order (Thorne Reference Thorne, Fairbank, Deaver, Everitt and Michelson1988).

3. Vortical fluid dynamics

In the weak-field, slow-velocity limit of general relativity, the Euler hydrodynamics equation for a rotating ideal fluid can be written as (Thorne, Price & MacDonald Reference Thorne, Price and MacDonald1986)

(3.1)\begin{equation} \frac{{\rm d}\boldsymbol{v}}{{\rm d}t} ={-}\frac{\boldsymbol{\nabla} p}{\rho}+\boldsymbol{E}_g + \left(\boldsymbol{v}\times\frac{ \boldsymbol{B}_g}{c}\right), \end{equation}

where ${\rm d}/{\rm d}t=(\partial /\partial t+\boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {\nabla })$, $\boldsymbol {v}$ is the fluid velocity and $p$ is the fluid pressure.

The continuity equation for the corresponding fluid is

(3.2)\begin{equation} \frac{\partial \rho}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol{v})=0. \end{equation}

Now using the vector identity $(\boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {\nabla }) \boldsymbol {v}=\boldsymbol {\nabla }(v^2/2)-\boldsymbol {v}\times (\boldsymbol {\nabla }\times \boldsymbol {v})$ and the expressions $E_g=-\boldsymbol {\nabla } \phi$ and $\boldsymbol {B}_g=\boldsymbol {\nabla }\times \boldsymbol {A}_g$, we can rewrite (3.1) as

(3.3)\begin{equation} \frac{\partial \boldsymbol{v}}{\partial t} =\boldsymbol{v}\times \left(\boldsymbol{\nabla}\times \boldsymbol{v}+\frac{\boldsymbol{B}_g}{c}\right)-\boldsymbol{\nabla}\left(\frac{ p}{\rho}+\frac{v^2}{2}+\phi\right), \end{equation}

where we have assumed a barotropic equation of state for pressure, i.e. $p(\rho )$. Since the vector potential $\boldsymbol{A}_g$ does not depend on time, we can rewrite (3.3) as

(3.4)\begin{equation} \frac{\partial}{\partial t}\left(\boldsymbol{v}+\frac{\boldsymbol{A}_g}{c}\right) =\boldsymbol{v}\times \left(\boldsymbol{\nabla}\times \boldsymbol{v}+\frac{\boldsymbol{B}_g}{c}\right)-\boldsymbol{\nabla}\varPhi, \end{equation}

where we identify the quantity $\boldsymbol {P}_g = (\boldsymbol {v}+\boldsymbol {A}_g/c)$ as a new canonical momentum and $\varPhi$ contains all the potentials for the gradient forces.

Now, we take the curl of (3.4) and obtain the vorticity transport equation:

(3.5)\begin{equation} \frac{\partial}{\partial t}\left(\boldsymbol{\nabla}\times \boldsymbol{v}+\frac{\boldsymbol{B}_g}{c}\right)-\boldsymbol{\nabla}\times\boldsymbol{v}\times \left(\boldsymbol{\nabla}\times \boldsymbol{v}+\frac{\boldsymbol{B}_g}{c}\right)=0, \end{equation}

where we identify $\boldsymbol {\varOmega }_g=\boldsymbol {\nabla }\times \boldsymbol {P}_g= \boldsymbol {\nabla }\times \boldsymbol {v}+ \boldsymbol {B}_g/c$ as the generalized vorticity.

It should be noted here that (3.5) does not contain any source terms, which implies that if the vorticity is zero at any time, it remains so for all times in an ideal barotropic fluid.

4. Equilibrium state

In this paper, we are interested in large-scale equilibrium structures in a self-gravitating and weakly rotating fluid. The stationary solution of (3.5) can be written as

(4.1)\begin{equation} \boldsymbol{\varOmega}_g\equiv\boldsymbol{\nabla}\times\boldsymbol{v}+\boldsymbol{\omega}_g=\mu\frac{16{\rm \pi} G\rho}{c^2}\boldsymbol{v}, \end{equation}

where we have defined $\boldsymbol {\omega }_g=\boldsymbol {B}_g/c$ and satisfies the requirement of vanishing divergence of generalized vorticity and time-independent continuity equation, i.e. $\boldsymbol {\nabla }\boldsymbol {\cdot }(\rho \boldsymbol {v})=0$.

To fully solve the equilibrium state, we need to supplement the Beltrami condition with the time-independent gravitomagnetic Ampère law rewritten in terms of the new quantity $\boldsymbol {\omega }_g$, which has the following form:

(4.2)\begin{equation} \boldsymbol{\nabla}\times\boldsymbol{\omega}_g={-}\frac{16{\rm \pi} G\rho}{c^2}\boldsymbol{v}. \end{equation}

The separation constant $\mu$ in (4.1) can be identified as the Lagrange multiplier when the Beltrami condition is derived via the variational principle.

For (4.1) to be defined as the stationary solution of (3.5), we need to impose the Bernoulli constraint which is an expression of the balance of all remaining potential forces, i.e. $\boldsymbol {\nabla }\varPhi =0$.

Combining (4.1) and (4.2), we obtain the following equation:

(4.3)\begin{equation} \boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\boldsymbol{v}-4\boldsymbol{v}=\frac{\mu}{\tilde{\lambda}_J} \,\boldsymbol{\nabla}\times\boldsymbol{v}, \end{equation}

where we have normalized $|\boldsymbol {\nabla }|$ to the inverse of skin depth $\tilde {\lambda }_J=\tilde\alpha \lambda _J$ with Jean's length $\lambda _J=c_{s0}/\omega _J$, Jean's frequency $\omega _J=(4{\rm \pi} G \rho _0\hat {\rho })^{1/2}$ and $\tilde\alpha =c/c_{s0}$. Here we have defined sound speed $c_{s0}$ in terms of some ambient mass density $\rho _0$ and $\hat {\rho }$ is the density envelope which we take to be a constant of order unity for the rest of this paper. Equation (4.3) is known as the double curl Beltrami equation and has been studied thoroughly in the context of Hall magnetohydrodynamics (Mahajan & Yoshida Reference Mahajan and Yoshida1998; Mahajan et al. Reference Mahajan, Miklaszewski, Nikol'skaya and Shatashvili2001; Ohsaki et al. Reference Ohsaki, Shatashvili, Yoshida and Mahajan2001).

Next, (4.3) is written as

(4.4)\begin{equation} (\boldsymbol{\nabla}\times{-}\lambda_+)(\boldsymbol{\nabla}\times{-}\lambda_-)\boldsymbol{v}=0, \end{equation}

where

(4.5)\begin{equation} \lambda_{{\pm}}=\frac{1}{2}\left[\frac{\mu}{\tilde{\lambda}_J}\pm\left[\left(\frac{\mu}{\tilde{\lambda}_J}\right)^2+16\right]^{1/2}\right]. \end{equation}

It should also be noted that (4.4) is the combination of two Beltrami fields $\mathbb {G}_+$ and $\mathbb {G}_-$, i.e.

(4.6)\begin{equation} \boldsymbol{\nabla}\times\boldsymbol{\mathbb{G}}_{{\pm}}=\lambda_{{\pm}}\boldsymbol{\mathbb{G}}_{{\pm}}, \end{equation}

with the final solution

(4.7)\begin{equation} \boldsymbol{v}=C_+\boldsymbol{\mathbb{G}}_+ + C_-\boldsymbol{\mathbb{G}}_{-}, \end{equation}

where the constant amplitudes $C_{\pm }$ are determined from initial conditions and characterize the double curl Beltrami states along with their corresponding eigenvalues $\lambda _{\pm }$. The explicit solution for the Beltrami condition in (4.6) is provided by the Chandrashekar–Kendall function in cylindrical coordinates whereas it takes the form of the Arnold–Beltrami–Childress solution in Cartesian coordinates (Chandrasekhar & Kendall Reference Chandrasekhar and Kendall1957).

One can obtain the solution for gravitomagnetic field by using (4.1) which can be written as follows:

(4.8)\begin{equation} \boldsymbol{\omega}_g=\left(\frac{\mu}{\tilde{\lambda}_J^2}-\frac{\lambda_+}{\tilde{\lambda}_J}\right)C_+\boldsymbol{\mathbb{G}}_+\left(\frac{\mu}{\tilde{\lambda}_J^2}-\frac{\lambda_-}{\tilde{\lambda}_J}\right)C_-\boldsymbol{\mathbb{G}}_-. \end{equation}

Finally, it should be emphasized here that the coupling between gravitomagnetic and flow fields has enabled us to uncover a far richer equilibrium structure in the fluid compared with the traditional Trkalian flow.

4.1. Comparison with the double curl Beltrami states in plasmas

Though double Beltrami states emerge in both plasma and self-gravitating rotating fluid, the physical characteristics of these states might not be the same in both types of systems. In this section, we study possible differences between the two systems. For a single-species dynamic charged fluid with constant density in an appropriate neutralizing background, the generalized Beltrami condition can be written as

(4.9)\begin{equation} \boldsymbol{\varOmega}\equiv\boldsymbol{B}+\frac{mc}{q}\boldsymbol{\nabla}\times\boldsymbol{v}=\frac{4{\rm \pi} nq}{c}\boldsymbol{v}, \end{equation}

which, combined with the Ampère law, gives us the plasma counterpart of (4.3) (Mahajan Reference Mahajan2008):

(4.10)\begin{equation} \boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\boldsymbol{v}+\boldsymbol{v}=\frac{\mu}{\lambda_s} \,\boldsymbol{\nabla}\times\boldsymbol{v}, \end{equation}

with the following roots:

(4.11)\begin{equation} \lambda_{{\pm}}=\frac{1}{2}\left[\frac{\mu}{\lambda_s}\pm\left[\left(\frac{\mu}{\lambda_s}\right)^2-4\right]^{1/2}\right], \end{equation}

where species skin depth $\lambda _s=c/\omega _p$ and plasma frequency $\omega _p=\sqrt {4{\rm \pi} nq^2/m}$. The roots are real for $(\mu /\lambda _s)^2>4$ but form a complex conjugate pair when $(\mu /\lambda _s)^2<4$.

Apart from a factor of 4, there is a sign difference between the left-hand sides of (4.3) and (4.10) and the origin of this can be attributed to the fact that gravity is always attractive as reflected in (4.2). Contrary to the plasma equilibrium states, the roots $\lambda _{\pm }$ in (4.5) for double curl Beltrami states in self-gravitating fluid are always real. Moreover, the presence of an inherent length scale, i.e. Jean's length, has introduced a singular perturbation term $\boldsymbol {\nabla }\times \boldsymbol {\nabla }\times \boldsymbol {v}$ in (4.3). This implies equilibrium states in self-gravitating fluid are also endowed with two length scales in a gravitomagnetic field. This can have major consequences for the formation of large- and small-scale flow configurations in the fluid as demonstrated in charged fluid (Mahajan et al. Reference Mahajan, Miklaszewski, Nikol'skaya and Shatashvili2001; Kagan & Mahajan Reference Kagan and Mahajan2010).

Next, if we set $\mu =0$ and reverse the normalization of the gradients, (4.3) can be rewritten as

(4.12)\begin{equation} \nabla^2\boldsymbol{v}={-}\frac{\boldsymbol{v}}{\tilde{\lambda}^2_J}, \end{equation}

which can be compared to the corresponding limiting case of (4.10) in plasma:

(4.13)\begin{equation} \nabla^2\boldsymbol{v}=\frac{\boldsymbol{v}}{\lambda^2_s}, \end{equation}

which are nothing but the $\boldsymbol {\varOmega }=0$ solution of the vorticity transport equation for both plasma and fluid.

Though (4.12) and (4.13) have a similar structure, the physics exhibited by the two systems is completely opposite. Equation (4.13) is the superconducting limit in a plasma with a skin depth $\lambda _s$ beyond which the magnetic flux is completely expelled from the interior of the plasma; a complete antithesis to the behaviour of gravitomagnetic flux in a neutral fluid. Another way to interpret (4.12) and (4.13) is that electric current is restricted to the skin depth in a diamagnetic plasma equilibrium, whereas the mass current is not confined to Jean's length in a fluid equilibrium.

5. Helicity and energy

From our analysis in the previous section, we notice that the values of Lagrange multiplier $\mu$ determine the characteristics of equilibrium states in a fluid. First, we take the following definition of helicity as one of the invariants of the system:

(5.1)\begin{equation} H=\frac{c^2}{32{\rm \pi} G}\int {\rm d}^3\,x \,\boldsymbol{P}_g\boldsymbol{\cdot}\boldsymbol{\varOmega}_g, \end{equation}

which is a measure of different topological features of vortical field lines such as knottedness and twists (Moffatt Reference Moffatt1969).

The physical interpretation of the Lagrange multiplier can then be obtained by computing the helicity from (5.1):

(5.2)\begin{align} H & =\frac{c^2}{32{\rm \pi} G}\left\langle\left(\boldsymbol{v}+\frac{\boldsymbol{A}_g}{c}\right) \boldsymbol{\cdot}\left(\boldsymbol{\nabla}\times\boldsymbol{v}+\frac{\boldsymbol{B}_g}{c}\right) \right\rangle\nonumber\\ & =\frac{c^2}{32{\rm \pi} G}\left\langle\left(\boldsymbol{v}+\frac{\boldsymbol{A}_g}{c}\right) \boldsymbol{\cdot}\frac{\mu 16{\rm \pi} G\rho}{c^2} \right\rangle\nonumber\\ & =\frac{\mu c^2}{32{\rm \pi} G}\left\langle\frac{16{\rm \pi} G\rho v^2}{c^2}-\frac{\boldsymbol{A}_g \boldsymbol{\cdot}\boldsymbol{\nabla}\times\boldsymbol{B}_g}{c^2} \right\rangle\nonumber\\ & =\mu\left\langle\frac{1}{2}\rho v^2-\frac{B_g^2}{32{\rm \pi} G}\right\rangle=\mu E, \end{align}

yielding an expression for $\mu$ in terms of two invariants of the motion:

(5.3)\begin{equation} \mu=\frac{H}{E}, \end{equation}

where $\langle \rangle =\int {\rm d}^3\,x$. Therefore, the Lagrange multiplier is a measure of generalized helicity as a fraction of total energy. It should be noted that, gravity being an attractive force, the energy density of the gravitomagnetic field has a negative sign in front of it (Sebens Reference Sebens2020).