Gravitomagnetic Instabilities of Relativistic Magnetohydrodynamics

Recently we presented three formulations of the relativistic magnetohydrodynamics (MHD): (i) fully nonlinear and exact perturbation formulation of MHD in Einstein's gravity, (ii) special relativistic (SR) MHD with weak (Newtonian) gravity, and (iii) MHD with first-order post-Newtonian (1PN) corrections; formulation (ii) is derived in the maximal slicing (temporal gauge or hypersurface) condition whereas formulations (i) and (iii) are presented without imposing the temporal gauge condition (Noh et al. 2019; Hwang & Noh 2020). The SR MHD with weak gravity is complementary to the PN approximation: in the SR MHD with weak gravity the fluid and field are fully relativistic while the gravity is Newtonian, whereas in the PN MHD the fluid, field, and gravity are consistently weakly relativistic.

Previously we studied effects of 1PN corrections on the MHD waves in a static homogeneous medium with an aligned magnetic field (Hwang & Noh 2020). In this work we include the gravity for the same homogeneous medium and study the gravitomagnetic instability of the PN MHD and the SR MHD with weak gravity. Both formulations include the Newtonian (0PN) limit.

Gravitational instability is a major factor causing gravitational collapse to form celestial objects. Magnetic field ubiquitous in the universe affects the stability. The gravitomagnetic instability in Newtonian context was analyzed by Chandrasekhar & Fermi (1953) and Chandrasekhar (1954, 1961). Here our aim is to extend this Newtonian study to a couple of relativistic situations.

Besides (i) the 1PN destabilizing effects on the instability criteria stated in the abstract and (ii) waves and instability in the SR MHD with weak gravity, we address (iii) the issue of inconsistency surrounding the Poisson's equation in the static homogeneous medium and (iv) the dependence of instability criteria on magnetic field to 0PN order.

Sections 2 and 3 are summaries of the MHD formulation to 1PN order, and the SR MHD with weak gravity. Section 4 is the Newtonian (0PN) study of gravitomagnetic instability of homogeneous and static medium with a barotropic pressure and an aligned magnetic field. Sections 5 and 6 are instabilities of the same medium extended to 1PN order and the SR MHD with weak gravity, respectively. Section 7 is a summary of our results and Section 8 is a discussion. We take the cgs unit.

We follow the 1PN convention of Chandrasekhar (Chandrasekhar 1965; Chandrasekhar & Nutku 1969). The metric convention is

$$\begin{eqnarray}&&\begin{array}{l}{\widetilde{g}}_{00}=-\left\{1-\displaystyle \frac{2}{{c}^{2}}\left[U+\displaystyle \frac{1}{{c}^{2}}\left(2{\rm{\Upsilon }}-{U}^{2}\right)\right]\right\},\\ {\widetilde{g}}_{0i}=-\displaystyle \frac{1}{{c}^{3}}{P}_{i},\ {\widetilde{g}}_{{ij}}=\left(1+\displaystyle \frac{2}{{c}^{2}}V\right){\delta }_{{ij}},\end{array}\end{eqnarray} \tag{ 1 }$$

with V = U; x⁰ = ct and a tilde indicates the covariant quantity; ϒ is a pure 1PN order potential introduced in Chandrasekhar (1965), see our Equation (9). The energy momentum tensor is decomposed into fluid quantities as

$$\begin{eqnarray}&&{\widetilde{T}}_{{ab}}=\widetilde{\mu }{\widetilde{u}}_{a}{\widetilde{u}}_{a}+\widetilde{p}\left({\widetilde{g}}_{{ab}}+{\widetilde{u}}_{a}{\widetilde{u}}_{b}\right)+{\widetilde{\pi }}_{{ab}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\widetilde{\mu }\equiv \mu \equiv \varrho {c}^{2},\ \varrho \equiv \overline{\varrho }\left(1+\displaystyle \frac{1}{{c}^{2}}{\rm{\Pi }}\right),\ \widetilde{p}\equiv p,\ {\widetilde{\pi }}_{{ij}}\equiv {{\rm{\Pi }}}_{{ij}},\end{eqnarray} \tag{ 3 }$$

with fluid velocities v_i and ${\overline{v}}_{i}$

$$\begin{eqnarray}&&\begin{array}{l}{\widetilde{u}}_{i}\equiv \gamma \displaystyle \frac{{v}_{i}}{c},\ \gamma \equiv \displaystyle \frac{1}{\sqrt{1-\tfrac{1}{1+2\varphi }\tfrac{{v}^{2}}{{c}^{2}}}},\ {v}^{2}\equiv {v}^{i}{v}_{i};\\ \displaystyle \frac{{\overline{v}}^{i}}{c}\equiv \displaystyle \frac{{\widetilde{u}}^{i}}{{\widetilde{u}}^{0}}=\displaystyle \frac{{{dx}}^{i}}{{{dx}}^{0}}.\end{array}\end{eqnarray} \tag{ 4 }$$

For φ, see Equation (14). To 1PN order we have

$$\begin{eqnarray}&&{v}_{i}={\overline{v}}_{i}+\displaystyle \frac{1}{{c}^{2}}\left(3U{\overline{v}}_{i}-{P}_{i}\right).\end{eqnarray} \tag{ 5 }$$

We will use ${\overline{v}}_{i}$. We are not imposing the temporal gauge condition, which can be used as an advantage in handling problems. In our analysis of gravitational instability of PN MHD in Section 5, in fact, we do not need to take the gauge condition. Thus, our results are valid in any temporal gauge condition.

A complete set of MHD equation valid to 1PN order is derived in Equations (53), (43), (56), (50), (49), (55), (54), and (48), respectively, in Hwang & Noh (2020) without imposing the temporal gauge condition. The mass, energy, and momentum conservation equations are

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left[\overline{\varrho }\left(1+\displaystyle \frac{1}{2}\displaystyle \frac{{v}^{2}}{{c}^{2}}+3\displaystyle \frac{U}{{c}^{2}}\right)\right]\\ \ \ +{{\rm{\nabla }}}^{i}\left[\overline{\varrho }{\overline{v}}_{i}\left(1+\displaystyle \frac{1}{2}\displaystyle \frac{{v}^{2}}{{c}^{2}}+3\displaystyle \frac{U}{{c}^{2}}\right)\right]=0,\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\overline{\varrho }\left(\dot{{\rm{\Pi }}}+{\boldsymbol{v}}\cdot {\rm{\nabla }}{\rm{\Pi }}\right)+p{\rm{\nabla }}\cdot {\boldsymbol{v}}+{{\rm{\Pi }}}^{{ij}}{v}_{i,j}=0,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left\{\overline{\varrho }+\displaystyle \frac{1}{{c}^{2}}\left[\overline{\varrho }\left({\rm{\Pi }}+{v}^{2}+6U\right)+p\right]\right\}\left({\dot{\overline{v}}}_{i}+\overline{{\boldsymbol{v}}}\cdot {\rm{\nabla }}{\overline{v}}_{i}\right)+{p}_{,i}\\ +\ {{\rm{\Pi }}}_{i,j}^{j}+\displaystyle \frac{1}{{c}^{2}}\left[\dot{p}{v}_{i}+2{p}_{,i}U+{\left({{\rm{\Pi }}}_{{ij}}{v}^{j}\right)}^{\cdot }-{v}_{i}{\left({{\rm{\Pi }}}_{k}^{j}{v}^{k}\right)}_{,j}\right]\\ -\ \left\{\overline{\varrho }+\displaystyle \frac{1}{{c}^{2}}\left[\overline{\varrho }\left({\rm{\Pi }}+2{v}^{2}+2U\right)+p\right]\right\}{U}_{,i}-\displaystyle \frac{1}{{c}^{2}}2\overline{\varrho }{{\rm{\Upsilon }}}_{,i}\\ +\ \displaystyle \frac{1}{{c}^{2}}\overline{\varrho }\left[\left(3\dot{U}+4{\boldsymbol{v}}\cdot {\rm{\nabla }}U\right){v}_{i}-{\dot{P}}_{i}-{v}^{j}\left({P}_{i,j}-{P}_{j,i}\right)\right]\\ =\ \displaystyle \frac{1}{4\pi }{\left[\left({\rm{\nabla }}\times {\boldsymbol{B}}\right)\times {\boldsymbol{B}}\right]}_{i}\\ +\ \displaystyle \frac{1}{4\pi {c}^{2}}\left\{{\left[{\left({\boldsymbol{v}}\times {\boldsymbol{B}}\right)}^{\cdot }\times {\boldsymbol{B}}\right]}_{i}+{\left({\boldsymbol{v}}\times {\boldsymbol{B}}\right)}_{i}{\rm{\nabla }}\cdot \left({\boldsymbol{v}}\times {\boldsymbol{B}}\right)\right.\\ \left.+\ {v}_{i}\left({\boldsymbol{v}}\times {\boldsymbol{B}}\right)\cdot \left({\rm{\nabla }}\times {\boldsymbol{B}}\right)-{\left[{\boldsymbol{B}}\times \left({\boldsymbol{B}}\times {\rm{\nabla }}U\right)\right]}_{i}\right\}.\end{array}\end{eqnarray} \tag{ 8 }$$

Einstein equations are

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}U+4\pi G\overline{\varrho }=-\displaystyle \frac{1}{{c}^{2}}\left[2{\rm{\Delta }}{\rm{\Upsilon }}+3\ddot{U}+{\dot{P}}_{\,,k}^{k}\right.\\ \left.+\ 4\pi G\overline{\varrho }\left({\rm{\Pi }}+2{v}^{2}+3\displaystyle \frac{p}{\overline{\varrho }}+2U+\displaystyle \frac{{B}^{2}}{4\pi \overline{\varrho }}\right)\right],\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{P}_{i}-{\left({P}_{,k}^{k}+4\dot{U}\right)}_{,i}=-16\pi G\overline{\varrho }{v}_{i}.\end{eqnarray} \tag{ 10 }$$

The Maxwell's equations are

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left[\left(1+\displaystyle \frac{U}{{c}^{2}}\right){\boldsymbol{B}}\right]={\rm{\nabla }}\times \left[\left(1+\displaystyle \frac{U}{{c}^{2}}\right)\overline{{\boldsymbol{v}}}\times {\boldsymbol{B}}\right],\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left[\left(1+\displaystyle \frac{U}{{c}^{2}}\right){\boldsymbol{B}}\right]=0.\end{eqnarray} \tag{ 12 }$$

Equations (6)–(12) are the complete set of MHD equations valid to 1PN order without imposing the temporal gauge condition.

The general slicing condition to the 1PN order is

$$\begin{eqnarray}&&{P}_{\,,i}^{i}+n\dot{U}=0,\end{eqnarray} \tag{ 13 }$$

where n = 3 and 4 correspond to the Chandrasekhar (Standard PN) gauge (Greenberg 1971) and the harmonic gauge (Nazari & Roshan 2018), respectively (Hwang et al. 2008; Poisson & Will 2014).

Equation (10) applies only to the 1PN order. Spatial indices of the fluid (v_i , ${\overline{v}}_{i}$, Π_ij ), field (B_i ), and the metric (P_i ) variables are raised and lowered using δ_ij and its inverse metric.

Our metric convention is

$$\begin{eqnarray}&&{\widetilde{g}}_{00}=-\left(1+2\alpha \right),\ {\widetilde{g}}_{0i}=-{\chi }_{i},\ {\widetilde{g}}_{{ij}}=\left(1+2\varphi \right){\delta }_{{ij}},\end{eqnarray} \tag{ 14 }$$

where α, φ, and χ_i are functions of spacetime with arbitrary amplitudes. Fully nonlinear and exact perturbation formulation based on this metric is presented in the appendix of Noh et al. (2019).

Equations combining SR MHD with weak gravity are derived in Equations (3)–(5) and (10)–(14) in the same work, by assuming

$$\begin{eqnarray}&&\alpha \equiv \displaystyle \frac{{\rm{\Phi }}}{{c}^{2}}\ll 1,\ \varphi \equiv -\displaystyle \frac{{\rm{\Psi }}}{{c}^{2}}\ll 1,{\gamma }^{2}\displaystyle \frac{{t}_{{\ell }}^{2}}{{t}_{g}^{2}}\ll 1,\end{eqnarray} \tag{ 15 }$$

where ${t}_{g}\sim 1/\sqrt{G\varrho }$ and t_ℓ  ∼ ℓ/c ∼ 2π/(kc) are gravitational timescale and the light propagation timescale of a characteristic length scale ℓ, respectively; k is the wavenumber with Δ = − k².

The mass, energy, and momentum conservation equations, and the two Maxwell equations, respectively, in conservative forms and Einstein's equations are

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left(\begin{array}{c}D\\ E\\ {m}^{i}\\ {B}^{i}\end{array}\right)+{{\rm{\nabla }}}_{j}\left(\begin{array}{c}{{Dv}}^{j}\\ {m}^{j}{c}^{2}\\ {m}^{{ij}}\\ {v}^{j}{B}^{i}-{v}^{i}{B}^{j}\end{array}\right)\\ \ \ =\ \left(\begin{array}{c}0\\ -\overline{\varrho }{\left(2{\rm{\Phi }}-{\rm{\Psi }}\right)}_{,i}{v}^{i}\\ -\overline{\varrho }{{\rm{\Phi }}}^{,i}\\ 0\end{array}\right),\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{B}_{\,,i}^{i}=0,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Phi }}=4\pi G\left(\varrho +\displaystyle \frac{3p}{{c}^{2}}+\displaystyle \frac{2}{{c}^{2}}{ \mathcal S }\right)=4\pi G\displaystyle \frac{E+S}{{c}^{2}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Psi }}=4\pi G\left(\varrho +\displaystyle \frac{1}{{c}^{2}}{ \mathcal S }\right)=4\pi G\displaystyle \frac{E}{{c}^{2}},\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray}&&\begin{array}{l}D\equiv \overline{\varrho }\gamma ,\ \varrho \equiv \overline{\varrho }\left(1+\displaystyle \frac{{\rm{\Pi }}}{{c}^{2}}\right),\ \gamma =\displaystyle \frac{1}{\sqrt{1-\tfrac{{v}^{2}}{{c}^{2}}}},\\ E=\varrho {c}^{2}+{ \mathcal S },\ S=3p+{ \mathcal S },\ { \mathcal S }\equiv \left(\varrho +\displaystyle \frac{p}{{c}^{2}}\right){\gamma }^{2}{v}^{2}\\ +\,{{\rm{\Pi }}}_{{ij}}\displaystyle \frac{{v}^{i}{v}^{j}}{{c}^{2}}+\displaystyle \frac{1}{8\pi }\left[{B}^{2}+\displaystyle \frac{1}{{c}^{2}}{\left({\boldsymbol{v}}\times {\boldsymbol{B}}\right)}^{2}\right],\\ {m}^{i}\equiv \left(\varrho +\displaystyle \frac{p}{{c}^{2}}\right){\gamma }^{2}{v}^{i}\\ +\,\displaystyle \frac{1}{{c}^{2}}\left[{{\rm{\Pi }}}_{j}^{i}{v}^{j}+\displaystyle \frac{1}{4\pi }\left({B}^{2}{v}^{i}-{B}^{i}{B}^{j}{v}_{j}\right)\right],\\ {m}^{{ij}}\equiv \left(\varrho +\displaystyle \frac{p}{{c}^{2}}\right){\gamma }^{2}{v}^{i}{v}^{j}+p{\delta }^{{ij}}+{{\rm{\Pi }}}^{{ij}}\\ +\,\displaystyle \frac{1}{4\pi }\left\{\displaystyle \frac{1}{{\gamma }^{2}}\left(\displaystyle \frac{1}{2}{B}^{2}{\delta }^{{ij}}-{B}^{i}{B}^{j}\right)\right.+\displaystyle \frac{1}{{c}^{2}}\left[{B}^{2}{v}^{i}{v}^{j}\right.\\ \left.\left.+\,\displaystyle \frac{1}{2}{\left({B}^{k}{v}_{k}\right)}^{2}{\delta }^{{ij}}-\left({B}^{j}{v}^{i}+{B}^{i}{v}^{j}\right){B}^{k}{v}_{k}\right]\right\}.\end{array}\end{eqnarray} \tag{ 20 }$$

The remaining metric component χ_i is determined by

$$\begin{eqnarray}&&{\rm{\Delta }}{\chi }_{i}=-\displaystyle \frac{4\pi G}{{c}^{3}}\left(4{\delta }_{i}^{j}-{{\rm{\Delta }}}^{-1}{{\rm{\nabla }}}_{i}{{\rm{\nabla }}}^{j}\right){m}_{j}.\end{eqnarray} \tag{ 21 }$$

Indices of χ_i , m_i , and m_ij are raised and lowered using δ_ij and its inverse.

The SR MHD with weak gravity formulation is valid in the maximal slicing (the uniform-expansion gauge in cosmology), setting the trace of extrinsic curvature (expansion scalar of the normal frame with a minus sign) equal to zero, which corresponds to the harmonic gauge in the PN approximation. If we ignore gravity, the SR MHD is valid in the Minkowski background, thus independent of the gauge condition. We have derived the formulation from the fully nonlinear an exact perturbation formulation of Einstein's gravity by taking the limits in Equation (15). We note that when we consider the conservation equations, in the strict sense of the limit used in Equation (15) the gravity part is valid only to the Newtonian order, thus we have Ψ = Φ = − U and ${\rm{\Delta }}{\rm{\Phi }}=4\pi G\overline{\varrho }$ (Noh et al. 2019). We have ${v}_{i}={\overline{v}}_{i}$.

To the 0PN order, Equations (6)–(12) give

$$\begin{eqnarray}&&\dot{\overline{\varrho }}+{\rm{\nabla }}\cdot \left(\overline{\varrho }{\boldsymbol{v}}\right)=0,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\begin{array}{l}\overline{\varrho }\left(\dot{{\boldsymbol{v}}}+{\boldsymbol{v}}\cdot {\rm{\nabla }}{\boldsymbol{v}}\right)+{\rm{\nabla }}p+{{\rm{\nabla }}}_{j}{{\rm{\Pi }}}_{i}^{j}-\overline{\varrho }{\rm{\nabla }}U\\ =\ \displaystyle \frac{1}{4\pi }\left({\rm{\nabla }}\times {\boldsymbol{B}}\right)\times {\boldsymbol{B}},\end{array}\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}U=-4\pi G\overline{\varrho },\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{B}}}={\rm{\nabla }}\times \left({\boldsymbol{v}}\times {\boldsymbol{B}}\right),\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{B}}=0.\end{eqnarray} \tag{ 26 }$$

For our stability analysis we do not need the energy conservation Equation in (7) which is in fact 1PN order (Hwang & Noh 2013). In this work we ignore the anisotropic stress. The above equations also follow from Equations (16)–(19) by taking c-goes-to-infinite limit with Φ = Ψ = −U.

We consider a static homogeneous background with a uniform magnetic field (Chandrasekhar & Fermi 1953; Chandrasekhar 1954, 1961). To the background order, we have a solution with

$$\begin{eqnarray}&&\begin{array}{l}{\overline{\varrho }}_{0}=\mathrm{constant},\ {p}_{0}=\mathrm{constant},\ {{\boldsymbol{v}}}_{0}=0,\\ {{\boldsymbol{B}}}_{0}={B}_{0}{\boldsymbol{n}}=\mathrm{constant},\ | {\boldsymbol{n}}| =1,\ {\rm{\nabla }}{U}_{0}=0,\end{array}\end{eqnarray} \tag{ 27 }$$

but ${\rm{\Delta }}{U}_{0}=-4\pi G{\overline{\varrho }}_{0}$.

Notice the inconsistent relations involving U₀. The trouble is caused because we consider Minkowski (thus static) background despite the presence of self-gravity of an infinite homogeneous background. The inconsistency is naturally avoided in the relativistic study of dynamic background as in the case of cosmology: the background order density, ϱ₀, is absorbed into the background (Friedmann) equations, and the Poisson's equation is valid only to perturbed order, thus U = δ U without U₀, see Equations (85)–(88), (108), and (119) in Hwang et al. (2008). In the static background, as in the present case, however, the inconsistency (often known as a swindle) remains. Jeans has made an explicit choice of ignoring the inconsistency, see Section 46 of Jeans (1902). Although the U₀ term does not appear in perturbation analysis to 0PN order, it appears in the perturbation equations to the 1PN order, see Equations (51)–(58). See a paragraph below Equation (58) and Section 8 for further discussion.

Considering the linear order perturbation, we expand

$$\begin{eqnarray}&&\begin{array}{l}\overline{\varrho }={\overline{\varrho }}_{0}\left(1+\delta \right),\ p={p}_{0}+\delta p,\ {\boldsymbol{B}}={{\boldsymbol{B}}}_{0}+\delta {\boldsymbol{B}},\\ {\boldsymbol{v}}=\delta {\boldsymbol{v}},\ U={U}_{0}+\delta U,\end{array}\end{eqnarray} \tag{ 28 }$$

and introduce

$$\begin{eqnarray}&&\delta p\equiv {c}_{s}^{2}{\overline{\varrho }}_{0}\delta ,\ {c}_{A}^{2}\equiv \displaystyle \frac{{B}_{0}^{2}}{4\pi {\overline{\varrho }}_{0}},\ \delta {\boldsymbol{B}}\equiv {B}_{0}{\boldsymbol{b}},\end{eqnarray} \tag{ 29 }$$

with the adiabatic sound speed ${c}_{s}=\mathrm{constant};$ thus, we are considering a barotropic equation of state with $p=p(\overline{\varrho })$. Equations (22)–(26) give

$$\begin{eqnarray}&&\dot{\delta }=-{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{v}}}+{c}_{s}^{2}{\rm{\nabla }}\delta -{\rm{\nabla }}\delta U={c}_{A}^{2}\left[{\rm{\nabla }}\left({\boldsymbol{n}}\cdot {\boldsymbol{b}}\right)-{\boldsymbol{n}}\cdot {\rm{\nabla }}{\boldsymbol{b}}\right],\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\delta U=-4\pi G{\overline{\varrho }}_{0}\delta ,\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{b}}}={\boldsymbol{n}}\cdot {\rm{\nabla }}{\boldsymbol{v}}-{\boldsymbol{n}}{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{b}}=0.\end{eqnarray} \tag{ 34 }$$

Expanding the perturbations in plane waves proportional to e^{i( k · x −ω t)}, we have

$$\begin{eqnarray}&&\begin{array}{l}\left[{\omega }^{2}-{c}_{A}^{2}{\left({\boldsymbol{n}}\cdot {\boldsymbol{k}}\right)}^{2}\right]{\boldsymbol{v}}+\left[\left(\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{s}^{2}-{c}_{A}^{2}\right){\boldsymbol{k}}\right.\\ \left.+\,{c}_{A}^{2}{\boldsymbol{nn}}\cdot {\boldsymbol{k}}\right]{\boldsymbol{k}}\cdot {\boldsymbol{v}}+{c}_{A}^{2}{\boldsymbol{kk}}\cdot {\boldsymbol{nn}}\cdot {\boldsymbol{v}}=0.\end{array}\end{eqnarray} \tag{ 35 }$$

By introducing coordinates as (Shu 1992)

$$\begin{eqnarray}&&\ {\boldsymbol{k}}\equiv k\widehat{{\boldsymbol{x}}},\ {\boldsymbol{n}}\equiv \cos \psi \widehat{{\boldsymbol{x}}}+\sin \psi \widehat{{\boldsymbol{y}}},\ {\boldsymbol{v}}\equiv {v}_{x}\widehat{{\boldsymbol{x}}}+{v}_{y}\widehat{{\boldsymbol{y}}}+{v}_{z}\widehat{{\boldsymbol{z}}},\end{eqnarray} \tag{ 36 }$$

Equations (30), (33), and (34) give v_x  = (ω/k)δ, b_x = 0,

$$\begin{eqnarray}&&{b}_{y}=\displaystyle \frac{k}{\omega }\left(\sin \psi {v}_{x}-\cos \psi {v}_{y}\right),\ {b}_{z}=-\displaystyle \frac{k}{\omega }\cos \psi {v}_{z},\end{eqnarray} \tag{ 37 }$$

and Equation (35) gives

$$\begin{eqnarray}&&\begin{array}{l}\left[{\omega }^{2}+{k}^{2}\left(\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{s}^{2}-{c}_{A}^{2}{\sin }^{2}\psi \right)\right]{v}_{x}\\ +\,{k}^{2}{c}_{A}^{2}\sin \psi \cos \psi {v}_{y}=0,\end{array}\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{k}^{2}{c}_{A}^{2}\cos \psi \sin \psi {v}_{x}+\left({\omega }^{2}-{k}^{2}{c}_{A}^{2}{\cos }^{2}\psi \right){v}_{y}=0,\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&\left({\omega }^{2}-{k}^{2}{c}_{A}^{2}{\cos }^{2}\psi \right){v}_{z}=0.\end{eqnarray} \tag{ 40 }$$

For v_z  ≠ 0 ( v perpendicular to B ₀- k plane), Equation (40) gives

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}={c}_{A}^{2}{\cos }^{2}\psi .\end{eqnarray} \tag{ 41 }$$

For nonvanishing v_x and v_y ( v in B ₀- k plane), from Equations (38) and (39) we have a dispersion relation

$$\begin{eqnarray}&&\begin{array}{l}{\omega }^{4}+\left[4\pi G{\overline{\varrho }}_{0}-{k}^{2}\left({c}_{s}^{2}+{c}_{A}^{2}\right)\right]{\omega }^{2}\\ \ +\,{c}_{A}^{2}\left({k}^{2}{c}_{s}^{2}-4\pi G{\overline{\varrho }}_{0}\right){\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}=0,\end{array}\end{eqnarray} \tag{ 42 }$$

with two solutions

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\omega }^{2}}{{k}^{2}} & = & \displaystyle \frac{1}{2}\left\{{c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\pm \left[{\left({c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)}^{2}\right.\right.\\ & & \left.{\left.-4{c}_{A}^{2}\left({c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}\right]}^{1/2}\right\}.\end{array}\end{eqnarray} \tag{ 43 }$$

Equation (42) was presented in Equation (169) of Chandrasekhar & Fermi (1953). Behaviors of the solutions are presented in Figure 1.

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For k ∥ B ₀ (ψ = 0^o), we have

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}={c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}},\ {c}_{A}^{2},\end{eqnarray} \tag{ 44 }$$

with the faster (slower) mode the fast (slow) MHD waves in the absence of gravity (Shu 1992). In the presence of gravity one mode is unstable for k < k_J with the Jeans wavenumber given as

$$\begin{eqnarray}&&{k}_{J}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}},\end{eqnarray} \tag{ 45 }$$

which does not depend on the magnetic field as pointed out in Chandrasekhar & Fermi (1953) for example, Chandrasekhar (1954) mentions "Jeans's criterion for the gravitational instability of an infinite homogeneous medium is unaffected by the presence of a magnetic field."; however, this is not a generally valid conclusion as we show below.

For k ⊥ B ₀ (ψ = 90^o), we have

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}=\left({c}_{s}^{2}+{c}_{A}^{2}\right)-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}},\ 0,\end{eqnarray} \tag{ 46 }$$

with the fast mode the magnetosonic wave in the absence of gravity (Shu 1992) and the slow mode vanishing. In the presence of gravity one mode is unstable for k < k_B with the critical wavenumber modified as

$$\begin{eqnarray}&&{k}_{{\rm{B}}}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}+{c}_{A}^{2}},\end{eqnarray} \tag{ 47 }$$

and thus depends on the magnetic field with k_B < k_J (Pacholczyk & Stodółkiewicz 1960; Strittmatter 1966).

For ψ other than 90°, the stability criterion remains the same as the Jeans criterion in Equation (45). However, as Figure 1 shows, the instability for k_B < k < k_J is suppressed depending on ψ, and as ψ approaches 90° the stability criterion effectively becomes Equation (47).

We similarly consider a static homogeneous background without anisotropic stress. To the background order, we have a solution with

$$\begin{eqnarray}&&\begin{array}{l}{\overline{\varrho }}_{0}=\mathrm{constant},\ {{\rm{\Pi }}}_{0}=\mathrm{constant},\ {p}_{0}=\mathrm{constant},\\ {{\boldsymbol{v}}}_{0}=0={\overline{{\boldsymbol{v}}}}_{0},\ {{\boldsymbol{B}}}_{0}={B}_{0}{\boldsymbol{n}}=\mathrm{constant},\\ {\rm{\nabla }}{U}_{0}=0={\dot{U}}_{0},\ {\rm{\nabla }}{{\rm{\Upsilon }}}_{0}=0,\ {{\boldsymbol{P}}}_{0}=0,\end{array}\end{eqnarray} \tag{ 48 }$$

but we have

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}{U}_{0}+4\pi G{\overline{\varrho }}_{0}=-\displaystyle \frac{1}{{c}^{2}}\left[2{\rm{\Delta }}{{\rm{\Upsilon }}}_{0}\right.\\ \left.+\,4\pi G{\overline{\varrho }}_{0}\left({{\rm{\Pi }}}_{0}+2{U}_{0}\right)+12\pi {{Gp}}_{0}+{{GB}}_{0}^{2}\right],\end{array}\end{eqnarray} \tag{ 49 }$$

with ΔU₀ ≠ 0 ≠ Δϒ₀. Thus, the inconsistent relations involving U₀ and ϒ₀ continue to 1PN order, see the paragraph below Equation (58). We consider p₀ and Π₀ as constants in space and time.

To the linear order perturbation, we expand

$$\begin{eqnarray}&&\begin{array}{l}\overline{\varrho }={\overline{\varrho }}_{0}\left(1+\delta \right),\ {\rm{\Pi }}={{\rm{\Pi }}}_{0}+\delta {\rm{\Pi }},\ p={p}_{0}+\delta p,\\ {\boldsymbol{B}}={{\boldsymbol{B}}}_{0}+\delta {\boldsymbol{B}},\ {\boldsymbol{v}}=\delta {\boldsymbol{v}},\ \overline{{\boldsymbol{v}}}=\delta \overline{{\boldsymbol{v}}},\\ U={U}_{0}+\delta U,\ {\rm{\Upsilon }}={{\rm{\Upsilon }}}_{0}+\delta {\rm{\Upsilon }},\ {\boldsymbol{P}}=\delta {\boldsymbol{P}}.\end{array}\end{eqnarray} \tag{ 50 }$$

Equations (6)–(13) give

$$\begin{eqnarray}&&\dot{\delta }=-{\rm{\nabla }}\cdot \overline{{\boldsymbol{v}}}-\displaystyle \frac{3}{{c}^{2}}\delta \dot{U},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&\delta \dot{{\rm{\Pi }}}=-\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+6{U}_{0}\right)\right]\dot{\overline{{\boldsymbol{v}}}}+\left(1+\displaystyle \frac{2}{{c}^{2}}{U}_{0}\right){c}_{s}^{2}{\rm{\nabla }}\delta \\ -\ \left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+2{U}_{0}\right)\right]{\rm{\nabla }}\delta U\\ -\ \displaystyle \frac{1}{{c}^{2}}\left(2{\rm{\nabla }}\delta {\rm{\Upsilon }}+\dot{{\boldsymbol{P}}}\right)\\ =\ {c}_{A}^{2}\left[-{\rm{\nabla }}\left({\boldsymbol{n}}\cdot {\boldsymbol{b}}\right)+{\boldsymbol{n}}\cdot {\rm{\nabla }}{\boldsymbol{b}}\right]\\ +\ \displaystyle \frac{{c}_{A}^{2}}{{c}^{2}}\left[-\left(\dot{{\boldsymbol{v}}}-{\rm{\nabla }}\delta U\right)+{\boldsymbol{nn}}\cdot \left(\dot{{\boldsymbol{v}}}-{\rm{\nabla }}\delta U\right)\right],\end{array}\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}\delta U+4\pi G{\overline{\varrho }}_{0}\delta =-\displaystyle \frac{1}{{c}^{2}}\left\{2{\rm{\Delta }}\delta {\rm{\Upsilon }}\right.\\ +\ 4\pi G{\overline{\varrho }}_{0}\left[\delta {\rm{\Pi }}+2\delta U+\left({{\rm{\Pi }}}_{0}+2{U}_{0}\right)\delta \right]+12\pi G\delta p\\ \left.+\ 2{{GB}}_{0}^{2}{\boldsymbol{n}}\cdot {\boldsymbol{b}}+3\delta \ddot{U}+{\rm{\nabla }}\cdot \dot{{\boldsymbol{P}}}\right\},\end{array}\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{P}}-{\rm{\nabla }}\left({\rm{\nabla }}\cdot {\boldsymbol{P}}+4\delta \dot{U}\right)=-16\pi G{\overline{\varrho }}_{0}{\boldsymbol{v}},\end{eqnarray} \tag{ 55 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{b}}}={\boldsymbol{n}}\cdot {\rm{\nabla }}\overline{{\boldsymbol{v}}}-{\boldsymbol{n}}{\rm{\nabla }}\cdot \overline{{\boldsymbol{v}}}-\displaystyle \frac{1}{{c}^{2}}{\boldsymbol{n}}\delta \dot{U},\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{b}}=-\displaystyle \frac{1}{{c}^{2}}{\boldsymbol{n}}\cdot {\rm{\nabla }}\delta U,\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{P}}=-n\delta \dot{U}.\end{eqnarray} \tag{ 58 }$$

In Equations (51)–(58) the U₀ term appears in the 1PN order. To the 0PN order, as the U₀ does not appear in the stability analysis we can ignore the Poisson's equation in the background order, assuming Poisson's equation is valid only to the perturbed order (Jeans 1902). However, the situation becomes more ambiguous to the 1PN order as the U₀, if we keep it, appears directly in the perturbation equations. As mentioned in the paragraph below Equation (27), in the relativistic study as in cosmology, the background is governed by dynamic equations like the Friedmann equations, and we have U = δ U, thus U₀ ≡ 0 (Hwang et al. 2008). However, in a static (Minkowski) background the inconsistency cannot be resolved. In the following we will keep track of the U₀ term in our analysis so that we can either ignore the U₀ term (Jeans' choice) or use Equation (49) to 0PN order.

We expand the perturbations in plane waves proportional to e^{i( k · x −ω t)}. Equations (51)–(57) give

$$\begin{eqnarray}&&\begin{array}{l}\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+6{U}_{0}-4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)\right]{\omega }^{2}\right.\\ \left.-\ {c}_{A}^{2}{\left({\boldsymbol{n}}\cdot {\boldsymbol{k}}\right)}^{2}\right\}\overline{{\boldsymbol{v}}}+\left\{-{c}_{s}^{2}\left[1+\displaystyle \frac{2}{{c}^{2}}\left({U}_{0}-3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\right]\right.\\ -\ {c}_{A}^{2}\left(1-\displaystyle \frac{4}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)+\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}\right.\right.\\ \left.\left.\left.+\ 2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+4{U}_{0}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+\displaystyle \frac{{\omega }^{2}}{{k}^{2}}\right)\right]\right\}{\boldsymbol{kk}}\cdot \overline{{\boldsymbol{v}}}\\ +\ \left(1-\displaystyle \frac{2}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){c}_{A}^{2}{\boldsymbol{nn}}\cdot {\boldsymbol{kk}}\cdot \overline{{\boldsymbol{v}}}+{c}_{A}^{2}\\ \times \ \left[\left(1-\displaystyle \frac{2}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){\boldsymbol{kk}}\cdot {\boldsymbol{n}}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}{\boldsymbol{n}}\right]{\boldsymbol{n}}\cdot \overline{{\boldsymbol{v}}}=0.\end{array}\end{eqnarray} \tag{ 59 }$$

To derive this we start from Equation (53) for $\overline{{\boldsymbol{v}}}$ and replace all the other variables in terms of $\overline{{\boldsymbol{v}}}$ using the perturbation expansion to 1PN order. Notice that we have not imposed the gauge condition in Equation (58): i.e., no choice of n is needed for our stability analysis. Thus, our result in this section is valid independently of the temporal gauge condition. By taking a coordinate in Equation (36), Equation (59) gives

$$\begin{eqnarray}&&\begin{array}{l}\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+6{U}_{0}-3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}{\sin }^{2}\psi \right)\right]\right.\\ \times \ {\omega }^{2}-\left[1+\displaystyle \frac{2}{{c}^{2}}\left({U}_{0}-3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\right]{k}^{2}{c}_{s}^{2}\\ -\ \left(1-\displaystyle \frac{4}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){k}^{2}{c}_{A}^{2}{\sin }^{2}\psi +\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}\right.\right.\\ \left.\left.\left.+\ 2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+4{U}_{0}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\right]\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}{k}^{2}\right\}{v}_{x}\\ +\ \left[\left(1-\displaystyle \frac{2}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){k}^{2}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}\right]{c}_{A}^{2}\sin \psi \cos \psi {v}_{y}=0,\end{array}\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[\left(1-\displaystyle \frac{2}{{c}^{2}}\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){k}^{2}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}\right]{c}_{A}^{2}\sin \psi \cos \psi {v}_{x}\\ +\ \left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+6{U}_{0}-4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right.\right.\right.\\ \left.\left.\left.+\ {c}_{A}^{2}{\cos }^{2}\psi \right)\right]{\omega }^{2}-{c}_{A}^{2}{k}^{2}{\cos }^{2}\psi \right\}{v}_{y}=0,\end{array}\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+6{U}_{0}-4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)\right]{\omega }^{2}\right.\\ \left.-\ {c}_{A}^{2}{k}^{2}{\cos }^{2}\psi \right\}{v}_{z}=0,\end{array}\end{eqnarray} \tag{ 62 }$$

with ${\boldsymbol{k}}\cdot {\boldsymbol{n}}=k\cos \psi$.

For v_z  ≠ 0 ( v perpendicular to B ₀– k plane), Equation (62) gives

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\omega }^{2}}{{k}^{2}} & = & {c}_{A}^{2}{\cos }^{2}\psi \left[1-\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}-4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right.\right.\\ & & \left.\left.+6{U}_{0}\right)\right].\end{array}\end{eqnarray} \tag{ 63 }$$

For nonvanishing v_x and v_y ( v in B ₀- k plane), Equations (60) and (61) give

$$\begin{eqnarray}&&\begin{array}{l}{\left\{\displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\right\}}^{2}\\ -\ \displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\left\{{c}_{s}^{2}\left[1-\displaystyle \frac{1}{{c}^{2}}\left(4{U}_{0}+3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right.\right.\right.\\ \left.\left.+\ {c}_{A}^{2}\right)\right]+{c}_{A}^{2}\left[1-\displaystyle \frac{1}{{c}^{2}}\left(6{U}_{0}+\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)\right]\\ -\ \displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}-2{U}_{0}+2\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{A}^{2}\right)\right]\\ \left.+\ \displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\left({c}_{s}^{2}+4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\right\}\\ +\ \displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\left\{{c}_{s}^{2}\left[1-\displaystyle \frac{1}{{c}^{2}}\left(10{U}_{0}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)\right]\right.\\ \left.-\ \displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}-8{U}_{0}+6\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{A}^{2}\right)\right]\right\}\\ =\ 0.\end{array}\end{eqnarray} \tag{ 64 }$$

Using

$$\begin{eqnarray}&&\begin{array}{l}{ \mathcal A }\equiv -{c}_{s}^{2}\left(4{U}_{0}+3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)-{c}_{A}^{2}\left(6{U}_{0}+{c}_{A}^{2}\right)\\ -\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}-2{U}_{0}+2\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)\\ +\ \displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\left({c}_{s}^{2}+4\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right),\\ { \mathcal B }\equiv -{c}_{s}^{2}\left(10{U}_{0}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+{c}_{A}^{2}\right)\\ -\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}-8{U}_{0}+6\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{A}^{2}\right),\end{array}\end{eqnarray} \tag{ 65 }$$

we can write it as

$$\begin{eqnarray}&&\begin{array}{l}{W}^{2}-W\left({c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+\displaystyle \frac{1}{{c}^{2}}{ \mathcal A }\right)\\ +\ \displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\left({c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+\displaystyle \frac{1}{{c}^{2}}{ \mathcal B }\right)=0,\end{array}\end{eqnarray} \tag{ 66 }$$

with two solutions

$$\begin{eqnarray}&&\begin{array}{l}W\equiv \displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\\ =\ \displaystyle \frac{1}{2}\left\{{c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+\displaystyle \frac{1}{{c}^{2}}{ \mathcal A }\right.\\ \left.\pm \ \sqrt{{\left({c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)}^{2}-4\displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\left({c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)+\displaystyle \frac{2}{{c}^{2}}\left[{ \mathcal A }\left({c}_{s}^{2}+{c}_{A}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)-2{ \mathcal B }\displaystyle \frac{{\left({\boldsymbol{k}}\cdot {\boldsymbol{n}}\right)}^{2}}{{k}^{2}}{c}_{A}^{2}\right]}\right\}.\end{array}\end{eqnarray} \tag{ 67 }$$

Behaviors of the 1PN solutions compared with the Newtonian ones are presented in Figures 2 and 3 for U₀ = 0 and $4\pi G{\overline{\varrho }}_{0}/{k}^{2}$, respectively, for a rather strong relativistic situation with $R\equiv {c}_{s}^{2}/{c}^{2}=0.01$. Dependence on R is presented in Figure 4. As our analysis is valid to 1PN order the solutions in Equation (67) can be Taylor expanded to 1PN order. In the figures we plot these expanded solutions.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Notice that, in Figure 2 for vanishing U₀, the 1PN correction terms cause ω² to diverge for small k (large-scale) limit. For k → 0, including U₀, we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{2}}{4\pi G{\overline{\varrho }}_{0}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\\ =\ 0,\ -1-2\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}{c}^{2}}+2\displaystyle \frac{{U}_{0}}{{c}^{2}}.\end{array}\end{eqnarray} \tag{ 68 }$$

Thus, for vanishing U₀, for small enough k, the PN correction terms cause the system more unstable and the exponent i ω diverges. The diverging instability due to the 1PN correction may imply breakdown of the 1PN approximation; this was noticed by Nazari et al. (2017) in the absence of magnetic field. By demanding the 1PN correction terms to be smaller than the 0PN order, we have

$$\begin{eqnarray}&&{k}^{2}\gt 2\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}^{2}}.\end{eqnarray} \tag{ 69 }$$

In terms of wavelength, λ = 2π/k, the 1PN approximation demands

$$\begin{eqnarray}&&\lambda \lt c\sqrt{\displaystyle \frac{\pi }{2G{\overline{\varrho }}_{0}}}\sim {{ct}}_{g},\end{eqnarray} \tag{ 70 }$$

where ${t}_{g}\equiv 1/\sqrt{G{\overline{\varrho }}_{0}}$ is the gravitational timescale. The diverging instability for λ greater than ct_g (the light propagation distance during gravitational timescale) is consistent with the eventual instability of the background system (known and studied in cosmology by Friedmann 1922, 1924 and Bonnor 1957), which we have ignored here by taking the Minkowski background. In a conservative stance, however, this simply implies the limit of the 1PN approximation.

On the other hand, in Figure 3 for ${U}_{0}=4\pi G{\overline{\varrho }}_{0}/{k}^{2}$, the 1PN correction terms in Equation (68) disappear. In the k → 0 limit, to the next leading order we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{2}}{4\pi G{\overline{\varrho }}_{0}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]=-\displaystyle \frac{2}{{c}^{2}}{c}_{A}^{2}{\cos }^{2}\psi ,\\ \qquad -\ 1-\displaystyle \frac{1}{{c}^{2}}\left(7{c}_{s}^{2}+6{c}_{A}^{2}{\sin }^{2}\psi \right).\end{array}\end{eqnarray} \tag{ 71 }$$

For k ∥ B ₀ (ψ = 0^o), we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\\ =\ {c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{\left({c}_{s}^{2}-\tfrac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right){ \mathcal A }-{c}_{A}^{2}{ \mathcal B }}{{c}_{s}^{2}-{c}_{A}^{2}-\tfrac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}},\\ {c}_{A}^{2}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{c}_{A}^{2}\left(-{ \mathcal A }+{ \mathcal B }\right)}{{c}_{s}^{2}-{c}_{A}^{2}-\tfrac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}}.\end{array}\end{eqnarray} \tag{ 72 }$$

One mode is unstable and the critical (Jeans) wavenumber becomes

$$\begin{eqnarray}&&{k}_{J}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+5{c}_{s}^{2}+2{U}_{0}\right)\right].\end{eqnarray} \tag{ 73 }$$

This coincides exactly with the Jeans wavenumber in the absence of the magnetic field, see Equation (78); thus, the criterion does not depend on the magnetic field as emphasized by Chandrasekhar & Fermi (1953), now even to 1PN order, see below Equation (45). In the absence of gravity, we have

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]={c}_{s}^{2},\ {c}_{A}^{2}\left(1-\displaystyle \frac{{c}_{A}^{2}}{{c}^{2}}\right).\end{eqnarray} \tag{ 74 }$$

For k ⊥ B ₀ (ψ = 90^o), we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{2}}{{k}^{2}}={c}_{s}^{2}\left[1-\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}+3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+4{U}_{0}\right)\right]\\ +\,{c}_{A}^{2}\left[1-\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}+\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+6{U}_{0}\right)\right]\\ -\,\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}-{c}_{A}^{2}+2\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-2{U}_{0}\right)\right],\\ 0.\end{array}\end{eqnarray} \tag{ 75 }$$

The critical wavenumber of the unstable mode becomes

$$\begin{eqnarray}\begin{array}{rcl}{k}_{{\rm{B}}}^{2} & = & \displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}+{c}_{A}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right.\right.\\ & & \left.\left.+\ 3{c}_{A}^{2}+5{c}_{s}^{2}+2{U}_{0}\displaystyle \frac{{c}_{s}^{2}+2{c}_{A}^{2}}{{c}_{s}^{2}+{c}_{A}^{2}}\right)\right].\end{array}\end{eqnarray} \tag{ 76 }$$

Now the choice for U₀ can be made. We can either ignore the U₀ terms (Jeans' choice), or use Equation (49) to 0PN order, thus ${U}_{0}=4\pi G{\overline{\varrho }}_{0}/{k}^{2}$. In the latter choice we have ${U}_{0}={c}_{s}^{2}$ in Equation (73) and ${U}_{0}={c}_{s}^{2}+{c}_{A}^{2}$ in Equation (76).

Equations (73) and (76) show that the 1PN corrections of the internal energy, pressure, sound velocity, and the Alfvén velocity cause the decrease of critical (Jeans) wavelength, λ = 2π/k, and thus reduce the Jeans mass, $M\equiv (\pi /6)\overline{\varrho }{\lambda }^{3}$, Figures 2–4 more generally show that for all ψ values the 1PN effects tend to increase the critical wavenumber (and thus lower the critical wavelength) and make more negative ω² (thus making the system more unstable). Therefore, we conclude that, to 1PN order, all relativistic corrections tend to destabilize the system.

5.1. Limiting Cases

Here, we compare our results with previous studies.

(i) To 0PN order we recover Equation (42); Equation (169) in Chandrasekhar & Fermi (1953).

(ii) Ignoring gravity, thus setting ${U}_{0}\equiv 0\equiv 4\pi G{\overline{\varrho }}_{0}$, we recover the 1PN MHD waves in Section 7 of Hwang & Noh (2020).

(iii) For vanishing magnetic field with B  = 0, thus setting ${\boldsymbol{n}}=0={c}_{A}^{2}$, from Equation (59) we have

$$\begin{eqnarray}&&\begin{array}{l}\left\{\displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+3\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+4{U}_{0}\right)\right]\right.\\ -\ {c}_{s}^{2}+\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\\ \left.\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+5\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}+2{U}_{0}\right)\right]\right\}{\boldsymbol{k}}\cdot {\boldsymbol{v}}=0.\end{array}\end{eqnarray} \tag{ 77 }$$

By setting ω = 0 we have k = k_J with

$$\begin{eqnarray}&&{k}_{J}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left(2{{\rm{\Pi }}}_{0}+2\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+5{c}_{s}^{2}+2{U}_{0}\right)\right].\end{eqnarray} \tag{ 78 }$$

This also follows from Equation (67). By choosing ${U}_{0}=4\pi G{\overline{\varrho }}_{0}/{k}^{2}$, we have ${U}_{0}={c}_{s}^{2}$ in Equation (78). Therefore, the PN corrections of the internal energy, pressure and sound velocity lower the critical (Jeans) wavelength. Thus, as generally shown even in the presence of magnetic field, all relativistic corrections tend to destabilize the system.

Equation (78), setting U₀ = 0, differs from the result in Equation (56) of Nazari et al. (2017) in the coefficient of c_s ². In our approach of separating the 0PN and 1PN orders clearly, the difference is caused by mixed decomposition of the 0PN and 1PN orders, for example, in Equations (3) and (5) of Nazari et al. (2017) by moving the 1PN order terms in Equation (3) properly to Equation (5) one can recover our result.

In Nazari et al.'s (2017) approach, however, the 0PN and 1PN orders are not separated. Compared to our c_s ² based on $p=p(\overline{\varrho })$, thus ${c}_{s}^{2}\equiv \delta p/\delta \overline{\varrho }$, their c_s ² is based on p = p(ϱ*), thus ${c}_{s}^{2}\equiv \delta p/\delta {\varrho }^{* };$ ϱ* is the conserved density to 1PN order introduced by Chandrasekhar (1965)

$$\begin{eqnarray}&&{\varrho }^{* }\equiv \overline{\varrho }\left[1+\displaystyle \frac{1}{{c}^{2}}\left(\displaystyle \frac{1}{2}{v}^{2}+3U\right)\right].\end{eqnarray} \tag{ 79 }$$

With ϱ* Equation (6) can be written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}{\varrho }^{* }+{\rm{\nabla }}\cdot \left({\varrho }^{* }\overline{{\boldsymbol{v}}}\right)=0.\end{eqnarray} \tag{ 80 }$$

Notice that ϱ* is already 1PN order, and the same for c_s ² defined based on ϱ*. Using this difference in the definition of c_s ² we can also recover the 5 factor in the coefficient of our c_s ². In this way the standard Jeans wavenumber in Equations (26) and (35) of Nazari et al. (2017) already contains the 1PN order; similarly all results in that paper are presented with the 0PN and 1PN orders not clearly separated. Qualitatively, however, as the coefficient of c_s ² is changed from 2 to 5 without the sign change, the reducing effect on the Jeans wavelength due to the PN sound velocity is not changed. (We thank Professor Mahmood Roshan and Dr. Elham Nazari for communications concerning their work).

We consider the same background medium with an aligned magnetic field. To the background order, we have a solution with

$$\begin{eqnarray}&&\begin{array}{l}{\overline{\varrho }}_{0}=\mathrm{constant},\ {{\rm{\Pi }}}_{0}=\mathrm{constant},\ {p}_{0}=\mathrm{constant},\\ {{\boldsymbol{B}}}_{0}={B}_{0}{\boldsymbol{n}}=\mathrm{constant},\ {{\boldsymbol{v}}}_{0}=0,\ {\rm{\nabla }}{{\rm{\Phi }}}_{0}=0,\end{array}\end{eqnarray} \tag{ 81 }$$

but we have

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{\Phi }}}_{0}=4\pi G\left({\varrho }_{0}+\displaystyle \frac{3{p}_{0}}{{c}^{2}}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{1}{4\pi }{B}_{0}^{2}\right).\end{eqnarray} \tag{ 82 }$$

Thus, the inconsistency concerning Φ₀ remains. But as in the Newtonian case Φ₀ does not appear in the perturbation analysis, and we can ignore Equation (82), see the paragraph below Equation (27).

To the linear order, Equations (16)–(18) give

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\begin{array}{c}D\\ E\\ {m}^{i}\\ {B}^{i}\end{array}\right)+{{\rm{\nabla }}}_{j}\left(\begin{array}{c}{{Dv}}^{j}\\ {m}^{j}{c}^{2}\\ {m}^{{ij}}\\ {v}^{j}{B}^{i}-{v}^{i}{B}^{j}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ -\overline{\varrho }{{\rm{\Phi }}}^{,i}\\ 0\end{array}\right),\end{eqnarray} \tag{ 83 }$$

$$\begin{eqnarray}&&{B}_{\,,i}^{i}=0,\end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{\Phi }}=4\pi G\left(\varrho +\displaystyle \frac{3p}{{c}^{2}}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{1}{4\pi }{B}^{2}\right),\end{eqnarray} \tag{ 85 }$$

with

$$\begin{eqnarray}&&\begin{array}{l}D=\overline{\varrho },\ E/{c}^{2}=\varrho +\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{1}{8\pi }{B}^{2},\\ {m}^{i}=\left(\varrho +\displaystyle \frac{p}{{c}^{2}}\right){v}^{i}+\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{1}{4\pi }\left({B}^{2}{v}^{i}-{B}^{i}{B}^{j}{v}_{j}\right),\\ {m}^{{ij}}=p{\delta }^{{ij}}+\displaystyle \frac{1}{4\pi }\left(\displaystyle \frac{1}{2}{B}^{2}{\delta }^{{ij}}-{B}^{i}{B}^{j}\right).\end{array}\end{eqnarray} \tag{ 86 }$$

Equations (19) and (21) are not needed.

Equations (83)–(85) can be arranged to give

$$\begin{eqnarray}&&\dot{\delta }=-{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 87 }$$

$$\begin{eqnarray}&&\delta \dot{{\rm{\Pi }}}=-\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 88 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)\right]\dot{{\boldsymbol{v}}}-\displaystyle \frac{1}{{c}^{2}}{c}_{A}^{2}{\boldsymbol{nn}}\cdot \dot{{\boldsymbol{v}}}\\ +\ {c}_{s}^{2}{\rm{\nabla }}\delta +{\rm{\nabla }}\delta {\rm{\Phi }}={c}_{A}^{2}\left[{\boldsymbol{n}}\cdot {\rm{\nabla }}{\boldsymbol{b}}-{\rm{\nabla }}\left({\boldsymbol{n}}\cdot {\boldsymbol{b}}\right)\right],\end{array}\end{eqnarray} \tag{ 89 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\delta {\rm{\Phi }}=4\pi G{\overline{\varrho }}_{0}\left[\delta +\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}\delta +\delta {\rm{\Pi }}+3{c}_{s}^{2}\delta +2{c}_{A}^{2}{\boldsymbol{n}}\cdot {\boldsymbol{b}}\right)\right],\end{eqnarray} \tag{ 90 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{b}}}={\boldsymbol{n}}\cdot {\rm{\nabla }}{\boldsymbol{v}}-{\boldsymbol{n}}{\rm{\nabla }}\cdot {\boldsymbol{v}},\end{eqnarray} \tag{ 91 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\boldsymbol{b}}=0.\end{eqnarray} \tag{ 92 }$$

These can be compared with Equations (51)–(57) in the PN case. As mentioned in a paragraph below Equation (21), to be consistent we need δΦ only to the Newtonian order in Equation (90); thus, ${\rm{\Delta }}\delta {\rm{\Phi }}=4\pi G{\overline{\varrho }}_{0}\delta ;$ in the same spirit we ignore gravity terms combined with relativistic (PN) order, like $4\pi G{\overline{\varrho }}_{0}/({k}^{2}{c}^{2})$. Thus, we do not need Equation (88). Compared with Newtonian Equations (30)–(34) differences occur only in the coefficients of $\dot{{\boldsymbol{v}}}$ terms in Equation (89).

Expanding in plane waves we can derive

$$\begin{eqnarray}&&\begin{array}{l}\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)\right]{\omega }^{2}-{c}_{A}^{2}{\left({\boldsymbol{n}}\cdot {\boldsymbol{k}}\right)}^{2}\right\}{\boldsymbol{v}}\\ +\ \left[\left(\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}-{c}_{s}^{2}-{c}_{A}^{2}\right){\boldsymbol{k}}+{c}_{A}^{2}{\boldsymbol{nn}}\cdot {\boldsymbol{k}}\right]{\boldsymbol{k}}\cdot {\boldsymbol{v}}\\ +\ {c}_{A}^{2}\left({\boldsymbol{kk}}\cdot {\boldsymbol{n}}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}{\boldsymbol{n}}\right){\boldsymbol{n}}\cdot {\boldsymbol{v}}=0.\end{array}\end{eqnarray} \tag{ 93 }$$

By taking a coordinate in Equation (36), Equation (93) gives

$$\begin{eqnarray}&&\begin{array}{l}\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}{\sin }^{2}\psi \right)\right]{\omega }^{2}-{k}^{2}\left({c}_{s}^{2}+{c}_{A}^{2}{\sin }^{2}\psi \right)\right.\\ \left.+\ 4\pi G{\overline{\varrho }}_{0}\right\}{v}_{x}+\left({k}^{2}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}\right){c}_{A}^{2}\sin \psi \cos \psi {v}_{y}=0,\end{array}\end{eqnarray} \tag{ 94 }$$

$$\begin{eqnarray}&&\begin{array}{l}\left({k}^{2}-\displaystyle \frac{1}{{c}^{2}}{\omega }^{2}\right){c}_{A}^{2}\sin \psi \cos \psi {v}_{x}+\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\right.\right.\\ \times \ \left.\left.\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}{\cos }^{2}\psi \right)\right]{\omega }^{2}-{c}_{A}^{2}{k}^{2}{\cos }^{2}\psi \right\}{v}_{y}=0,\end{array}\end{eqnarray} \tag{ 95 }$$

$$\begin{eqnarray}&&\left\{\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)\right]{\omega }^{2}-{c}_{A}^{2}{k}^{2}{\cos }^{2}\psi \right\}{v}_{z}=0.\end{eqnarray} \tag{ 96 }$$

For v_z  ≠ 0 ( v perpendicular to B ₀- k plane), Equation (96) gives

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}=\displaystyle \frac{{c}_{A}^{2}{\cos }^{2}\psi }{1+\tfrac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\tfrac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)}.\end{eqnarray} \tag{ 97 }$$

For nonvanishing v_x and v_y ( v in B ₀- k plane), from Equations (94) and (95) we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{4}}{{k}^{4}}\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)\right]\\ -\ \displaystyle \frac{{\omega }^{2}}{{k}^{2}}\left\{\left({c}_{s}^{2}+{c}_{A}^{2}\right)\left[1+\displaystyle \frac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\displaystyle \frac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)\right]\right.\\ \left.+\ \displaystyle \frac{1}{{c}^{2}}{c}_{s}^{2}{c}_{A}^{2}{\cos }^{2}\psi -\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right\}\\ +\ {c}_{A}^{2}{\cos }^{2}\psi \left({c}_{s}^{2}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}}\right)=0,\end{array}\end{eqnarray} \tag{ 98 }$$

with two solutions for ω²/k².

For k ∥ B ₀ (ψ = 0^o), we have

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\omega }^{2}}{{k}^{2}}=\displaystyle \frac{{c}_{s}^{2}}{1+\tfrac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\tfrac{{p}_{0}}{{\overline{\varrho }}_{0}}\right)}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}},\ \\ \displaystyle \frac{{c}_{A}^{2}}{1+\tfrac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\tfrac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)}.\end{array}\end{eqnarray} \tag{ 99 }$$

One mode is unstable and the critical (Jeans) wavenumber becomes

$$\begin{eqnarray}&&{k}_{J}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}},\end{eqnarray} \tag{ 100 }$$

as the gravity part is valid only to the Newtonian order.

For k ⊥ B ₀ (ψ = 90^o), we have

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }^{2}}{{k}^{2}}=\displaystyle \frac{{c}_{s}^{2}+{c}_{A}^{2}}{1+\tfrac{1}{{c}^{2}}\left({{\rm{\Pi }}}_{0}+\tfrac{{p}_{0}}{{\overline{\varrho }}_{0}}+{c}_{A}^{2}\right)}-\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{k}^{2}},\ 0.\end{eqnarray} \tag{ 101 }$$

The critical wavenumber of the unstable mode similarly becomes

$$\begin{eqnarray}&&{k}_{{\rm{B}}}^{2}=\displaystyle \frac{4\pi G{\overline{\varrho }}_{0}}{{c}_{s}^{2}+{c}_{A}^{2}}.\end{eqnarray} \tag{ 102 }$$

As mentioned, we consider the gravity of the stability analysis to be consistent only to the Newtonian order, and thus Equations (100) and (102) are the same as in the Newtonian MHD. Whereas, considering gravity only to the Newtonian order, the wave solutions in Equations (97), (99), and (101), and the two general solutions of Equation (98) for general ψ are valid for the fully SR MHD. In the absence of gravity, the wave speeds in Equations (97), (99), and (101) include the 1PN results in Equations (63), (74), and (75) as the 1PN limit.

Sections 2 and 3 are summaries of two formulations of relativistic MHD: MHD valid to 1PN approximation and the SR MHD with weak gravity. The equations of PN MHD are presented without taking the slicing condition, whereas the equations of SR MHD with weak gravity are valid in the maximal slicing which corresponds to the harmonic gauge in the PN formulation. Section 4 is the MHD instability of the the homogeneous medium with an aligned magnetic field in Newtonian (0PN) limit, largely overlapping with previous studies in Chandrasekhar & Fermi (1953) and Chandrasekhar (1954, 1961). Here, we clarify some unclear remarks made in the previous works concerning effects of magnetic field on the gravitational instability, see below our Equations (45) and (47). We also address the inconsistency issue related to Poisson's equation to the homogeneous background medium, see paragraphs below Equation (27) and below Equation (58).

Section 5 presents our main analysis of the 1PN gravitomagnetic instability of the same medium. We show that the post-Newtonian corrections of the internal energy (Π₀), pressure (p₀), sound velocity (c_s ), and the Alfvén velocity (c_A) consistently lower the Jeans wavelength and the Jeans mass, see Equations (73) and (76), and tend to destabilize the system, see Figures 2–4. We note that we have not fixed the temporal gauge condition for the analysis, thus the results are valid independently of the gauge condition and are naturally gauge invariant, see Section 6.3 in Hwang & Noh (2020).

Section 6 presents another main analysis of the waves and instability of the SR MHD with weak gravity. The MHD waves of the SR MHD are presented in Equations (97)–(99) and (101). When we consider the gravity, however, the analysis is consistent to Newtonian order only, and the critical wavenumbers in Equations (100) and (102) are the same as in the Newtonian MHD.

Stability analysis of an infinite homogeneous medium is a textbook exercise in the Newtonian case; gravitational instability in the presence of aligned magnetic field is rarely presented though (Chandrasekhar 1961). The exponential instability for an imaginary ω indicates no lack-of-time problem in the gravitational collapse in a static medium: i.e., the over-dense region larger than Jeans scale may collapse immediately. This may have a negative side in that we can hardly find astrophysical situations where the conditions (static homogeneous medium, putting aside the infinity condition) are met in the interstellar space. Interstellar matter and molecular clouds, in fact, are known to be in the state of (compressible) turbulence (Elmegreen & Scalo 2004).

Although the infinite static homogeneous background medium has the internal inconsistency (which remains even in the relativistic situation as we have shown in this work) the Jeans stability criterion survives in a consistent relativistic analysis made in an infinite (not needed in cosmology though) homogeneous, but dynamic, background medium (Lifshitz 1946); in the static limit we recover the Jeans result! In a power-law expanding medium (like in the matter dominated era) the growth rate is suppressed from exponential to a decelerating power law; this was originally recognized as too slow to serve for galaxy or star formation by Lifshitz (1946). Nowadays, our observable universe provides a prime example with highly successful gravitational instability as the main driving engine of the large-scale structure with a long duration (due to slow growth rate) from its generation in the early universe until even today.

Lifshitz's stability analysis, however, is made in the absence of magnetic field. In the presence of aligned magnetic field, the proper relativistic analysis demands more than the Friedmann cosmology with the anisotropic cosmological principle (Thorne 1967). Current success in cosmology does not encourage such a study, and which part of our study could survive the consistent analysis (in the static limit) is currently unknown.

As the static homogeneous medium is not realistic (i.e., cannot or rarely be found) in known astrophysical environments, an additional aligned magnetic field is also apparently not realistic. Our motivation for choosing the situation is to extend the textbook analysis of the gravitomagnetic instability known in Newtonian MHD to a couple of relativistic situations. We have shown that the inconsistency noticed by Jeans is due to the static nature of the background medium. The relativistic analysis can cure the inconsistency by demanding dynamic background medium (Lifshitz 1946; Hwang et al. 2008).

More realistic astrophysical situations in which relativistic MHD is needed can be found in accretion disks, magnetospheres, the plasma winds, and astrophysical jets near compact astrophysical objects (like neutron stars and black holes), and active galactic nuclei. Such situations may require the spherical, cylindrical, or disk geometries (Thorne et al. 1986; Beskin 2010). Stability analysis in these weakly relativistic situations using our two new relativistic MHD formulations summarized in Sections 2 and 3 will be interesting. Being fully nonlinear, these two relativistic MHD formulations may be convenient to numerically handle the weakly relativistic (1PN approximation) or weak gravity situations.

We wish to thank the referee for constructive comments. J.H. was supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT and future Planning (No. 2018R1A6A1A06024970 and NRF-2019R1A2C1003031). H.N. was supported by the National Research Foundation of Korea funded by the Korean Government (No. 2018R1A2B6002466).