2. Elsasser formulation of general relativistic magnetohydrodynamics

The equations of general relativistic magnetohydrodynamics (GRMHD) (Anile Reference Anile1989) describe the motion of a perfectly conducting fluid under the influence of the electromagnetic fields and gravity,Footnote ¹ and may be derived assuming that the electric field vanishes in the local fluid rest frame. These equations are, first, the conservation of mass

(2.1)\begin{equation} \nabla_\nu (\rho u^{\nu}) = 0,\end{equation}

the stress–energy equation

(2.2)\begin{equation} \nabla_\nu T^{\mu\nu} =0,\end{equation}

and the induction equation

(2.3)\begin{equation} \nabla_\nu ( b^{\mu} u^{\nu} - b^{\nu} u^{\mu})=0.\end{equation}

In these equations, $\nabla _\nu$ denotes the covariant derivative, $\rho$ is the mass density, $u^{\mu }$ is the fluid 4-velocity, the GRMHD stress–energy tensor is

(2.4)\begin{equation} T^{\mu\nu} = \mathcal{E} u^{\mu} u^{\nu} - b^{\mu} b^{\nu} + \left( p+\frac{b^{2}}{2}\right)g^{\mu\nu},\end{equation}

where $g^{\mu \nu }$ is the metric tensor, the magnetic-field 4-vector is

(2.5)\begin{equation} b^{\mu} = \tfrac{1}{2} \epsilon^{\mu\nu\kappa\lambda}u_\nu F_{\lambda\kappa},\end{equation}

with $b^{2} = b^{\mu } b_\mu >0$, $F_{\lambda \kappa }$ the Faraday tensor divided by $\sqrt {4{\rm \pi} }$, $\epsilon ^{\mu \nu \kappa \lambda }$ the Levi-Civita tensor, and

(2.6)\begin{equation} \mathcal{E} = \rho + U + p + b^{2}, \end{equation}

where $U$ is the internal energy and $p$ is the thermal pressure. We use units such that the speed of light $c=1$. We use the notation

(2.7)\begin{equation} A^{2} = A^{\mu} A_\mu, \end{equation}

to denote the magnitude squared of any 4-vector $A^{\mu }$; for spacelike 4-vectors, we also write $A=\sqrt {A^{2}}$.

The 4-velocity satisfies

(2.8)\begin{equation} u^{2} ={-}1,\end{equation}

and (2.5) implies that

(2.9)\begin{equation} u_\mu b^{\mu} = 0.\end{equation}

Chandran et al. (Reference Chandran, Foucart and Tchekhovskoy2018) noticed that, just as in non-relativistic MHD, (2.2) and (2.3) may be cast in a useful pseudo-symmetric Elsasser (Reference Elsasser1950) form by multiplying (2.3) by $\pm \mathcal {E}^{1/2}$, adding to (2.2) and dividing the two resulting equations by $\mathcal {E}$. This results in

(2.10)\begin{equation} \nabla_\nu ( z_\pm^{\mu} z_\mp^{\nu} + \varPi g^{\mu\nu}) + \left( \frac{3}{4}z_\pm^{\mu} z_\mp^{\nu} + \frac{1}{4} z_\mp^{\mu} z_\pm^{\nu} + \varPi g^{\mu\nu}\right) \frac{\partial_\nu \mathcal{E}}{\mathcal{E}},\end{equation}

where

(2.11(a,b))\begin{equation} z_\pm^{\mu} = u^{\mu} \mp \frac{b^{\mu}}{\mathcal{E}^{1/2}}, \quad \varPi = \frac{2p + b^{2}}{2\mathcal{E}},\end{equation}

and $\partial _\nu$ refers to differentiation with respect to the coordinate $\nu$. Equation (2.10), along with (2.1) and an equation of state, comprise the Elsasser formulation of GRMHD. These equations have been used recently by TenBarge et al. (Reference TenBarge, Ripperda, Chernoglazov, Bhattacharjee, Mahlmann, Most, Juno, Yuan and Philippov2021) to study weak AW turbulence in the small-amplitude, magnetically dominated, anisotropic limit.

In the following, we restrict ourselves to the case of special relativity, for which the Minkowski metric may be written in Cartesian coordinates $g^{\mu \nu }=\mathrm {diag}(-1,1,1,1)$, and the covariant derivative reduces to the simpler 4-gradient operator $\nabla _\nu = \partial _\nu$. Thus, our results only apply when the length and time scales of the fluctuations are small compared with the scales over which the metric changes significantly.

3. Fluctuations on a uniform background

We take each quantity to be the sum of a background value plus a fluctuation

(3.1)\begin{equation} \left.\begin{array}{c@{}} \rho = \bar{\rho} + \tilde{\rho} \quad p = \bar{p} + \tilde{p} \quad U = \bar{U} + \tilde{U},\\ u^{\mu} = \bar{u}^{\mu} + \tilde{u}^{\mu} \quad b^{\mu} = \bar{b}^{\mu} + \tilde{b}^{\mu} \quad z_\pm^{\mu} = \bar{z}_\pm^{\mu} + \tilde{z}_\pm^{\mu}. \end{array}\right\} \end{equation}

We take the background quantities to be uniform in space and time,

(3.2)\begin{equation} \{ \bar{\rho}, \bar{\mathcal{E}}, \bar{\varPi}, \bar{u}^{\mu}, \bar{b}^{\mu} \} = \textrm{const.},\end{equation}

and consider fluctuations satisfying

(3.3(a,b))\begin{equation} \tilde{\rho} = \tilde{\mathcal{E}} = \tilde{\varPi} = 0, \quad \tilde{u}^{\mu} ={-} \frac{\tilde{b}^{\mu}}{{\mathcal{E}}^{1/2}}.\end{equation}

The final equation of (3.3a,b) implies that $\tilde {z}_-^{\mu } = 0$. We assume that the fluctuations are localized in space–time around a sequence of events $X^{\mu }$; at another sequence of events ${X'}^{\mu }$, $b^{\nu }(X'^{\mu }) \to \overline {b}^{\mu }$ and $u^{\mu }(X'^{\mu }) \to \overline {u}^{\mu }$ as $|(X-X')^{2}|\to \infty$.

It follows from (2.8) that

(3.4)\begin{equation} u^{\mu} u_\mu = \overline{u}^{2} + \tilde{u}^{2} + 2 \overline{u}^{\mu} \tilde{u}_\mu ={-}1.\end{equation}

The space–time localization of the fluctuations combined with the constancy of $\bar {u}^{\mu }$ implies that $\overline {u}^{2}=-1$, and hence

(3.5)\begin{equation} \tilde{u}^{2} ={-} 2 \overline{u}^{\mu} \tilde{u}_\mu.\end{equation}

Equation (2.9) further restricts the solution by requiring that

(3.6)\begin{equation} \frac{1}{\mathcal{E}^{1/2}}u^{\mu} b_\mu = \frac{1}{\mathcal{E}^{1/2}}\overline{u}^{\mu} \overline{b}_\mu - \tilde{u}^{2} - \overline{z}_+^{\mu}\tilde{u}_\mu = 0. \end{equation}

The localization of the fluctuations and the constancy of $\bar {u}^{\mu }$ and $\bar {b}^{\mu }$ imply that $\overline {u}^{\mu }\overline {b}_\mu =0$, so (3.6) becomes

(3.7)\begin{equation} \tilde{u}^{2} ={-}\overline{z}_+^{\mu}\tilde{u}_\mu.\end{equation}

Subtracting (3.5) from twice (3.7), we find that

(3.8)\begin{equation} \tilde{u}^{2} = \frac{2}{\mathcal{E}^{1/2}}\overline{b}^{\mu} \tilde{u}_\mu.\end{equation}

Finally, we calculate the scalar $b^{2}$. This is

(3.9)\begin{equation} b^{2}=b^{\mu} b_\mu = \overline{b}^{2} + \mathcal{E}\tilde{u}^{2} - 2\mathcal{E}^{1/2}\overline{b}^{\mu} \tilde{u}_\mu = \overline{b}^{2}, \end{equation}

a constant, where we have used (3.8) in the last equality. Thus, the wave packet has constant 4-magnetic-field magnitude, analogous to the constant-$B^{2}$ constraint for a large-amplitude AW in non-relativistic MHD (Barnes & Hollweg Reference Barnes and Hollweg1974; Goldstein et al. Reference Goldstein, Klimas and Barish1974).

At each point in space–time, we may boost into an accelerating frame moving with the instantaneous local fluid velocity $u^{\mu }$, the local fluid rest frame. In this frame, $b^{t}=0$ and, therefore, the magnetic-field 3-vector has magnitude squared $B^{2}=b^{2}$, which is a constant and therefore the same at each point; $B^{2}$ is not spatially constant in an arbitrary fixed inertial frame.

Equation (2.1) with $\rho$ constant and flat space–time gives

(3.10)\begin{equation} \partial_\nu \tilde{u}^{\nu} = \partial_\nu \tilde{z}_+^{\nu} =0,\end{equation}

and the $+$ Elsasser equation (2.10) then gives

(3.11)\begin{equation} \overline{z}_-^{\nu} \partial_\nu \tilde{z}_+^{\mu} = 0, \end{equation}

with the $-$ Elsasser equation vanishing by virtue of (3.10) and (3.3a,b). (3.11) is a linear wave equation for the evolution of $\tilde {z}_+^{\mu }$; thus, a three-dimensional Alfvénic wavepacket of (apparently; see (5.1)) arbitrary amplitude and arbitrary shape propagates without distortion on a homogeneous background.

4. Components in the background rest frame

We define a background rest frame (BRF)Footnote ² in which the homogeneous background (3.2) is at rest. In this frame $\overline {u}^{\mu } = (1,0,0,0)$, and $\overline {b}^{\mu } = (0, 0,0,b)$, where we have chosen to align the $z$ direction with the background magnetic field. Since $u^{\mu }$ is a future-directed 4-velocity, it is straightforward to show, working in the BRF and using (3.4), that $\tilde {u}^{2} \geq 0$, a relation that holds in all frames since $\tilde {u}^{2}$ is a scalar. (The equality $\tilde {u}^{2} = 0$ is obtained only when $\tilde {u}^{\mu } = 0$.) Calculating (3.5) and (3.8) in the BRF, the $t$ and $z$ components of the fluctuation are given by

(4.1(a,b))\begin{equation} \tilde{u}^{t} = \frac{1}{2}\tilde{u}^{2}, \quad \tilde{u}^{z} ={-} \frac{\mathcal{E}^{1/2}}{2 b} \tilde{u}^{2} \text{ (BRF)},\end{equation}

and the magnitude of the remaining (perpendicular) fluctuation components $\tilde {u}_\perp =\sqrt {(\tilde {u}^{x})^{2} + (\tilde {u}^{y})^{2}}$ is

(4.2)\begin{equation} \tilde{u}_\perp{=} \sqrt{\tilde{u}^{2} + (\tilde{u}^{t})^{2} - (\tilde{u}^{z})^{2}} =\tilde{u}\sqrt{1-\frac{\tilde{u}^{2}}{4\sigma}} \text{ (BRF)},\end{equation}

where $\sigma = b^{2}/(\rho +U+p)$. Providing $\tilde {u}^{2}$ thus gives us nearly all the information in the fluctuation 4-vector, apart from the direction in the $y$-$z$ plane that the perpendicular fluctuation points. To determine this, we must use (3.10) and a particular functional form for $\tilde {u}^{2}$. This amounts to solving a two-dimensional first-order quasilinear partial differential equation  with analytic coefficients for one of the components (say, $\tilde {u}^{x}$).

Evaluating the fluid 3-velocity $v^{i}=u^{i}/u^{t}$ and magnetic-field 3-vector $B^{i}=b^{i} u^{t} - b^{t} u^{i}$, it is clear that, in the BRF, $B^{2}$ is not constant, the 3-vector magnetic and velocity fluctuations are not related to each other by a constant of proportionality, and do not in general even point in the same direction.

5. Maximum amplitude

Equation (4.2) implies an upper limit on the magnitude of the fluctuations,

(5.1)\begin{equation} \tilde{u} \leq \tilde{u}_{\max} = \frac{2 b}{\sqrt{\rho+U + p}}.\end{equation}

In the non-relativistic case, this has been recently noticed in solar wind AWs by (Matteini et al. Reference Matteini, Stansby, Horbury and Chen2018). The observed magnitude of the fluid 3-velocity in the BRF is

(5.2)\begin{equation} v=\frac{\sqrt{u^{i}u_i}}{u^{t}} = \frac{\tilde{u}\sqrt{1+\tilde{u}^{2}/4}}{1+\tilde{u}^{2}/2} \text{ (BRF)}, \end{equation}

an increasing function of $\tilde {u}$; $v<1$ and $v\to 1$ as $\tilde {u}\to \infty$.

6. Alfvén velocity and wave frame

The propagation of the wave is controlled by the constant time-like 4-vector $\overline {z}_-^{\mu }$, with

(6.1)\begin{equation} \overline{z}_-^{2} = \frac{b^{2}}{\mathcal{E}} - 1 <0. \end{equation}

(In the limit $b^{2}\to \infty$ while keeping $\rho$, $U$ and $p$ constant, $\overline {z}_-^{2} \to 0$.) An observer moving with 4-velocity $\overline {z}_-^{\mu } / \sqrt {-\overline {z}_-^{2}}$ sees a time-independent structure; such an observer is in the wave frame (WF). The WF 3-velocity relative to another frame is $v_w^{i} = \overline {z}_-^{i}/\overline {z}_-^{t}$, and specifically, relative to the BRF is

(6.2)\begin{equation} v_w^{i} = v_A^{i} = \frac{b}{\mathcal{E}^{1/2}} (0,0,1) \text{ (BRF)}.\end{equation}

Thus, the three-dimensional relativistic AW propagates along the background field lines at this relativistic Alfvén velocity, just like the planar AW (Barnes & Suffolk Reference Barnes and Suffolk1971; Heyvaerts et al. Reference Heyvaerts, Lehner and Mottez2012). Note that this does not depend on $\tilde {u}^{\mu }$, so the wave does not steepen into a shock. If all of $b^{2}/\rho,U/\rho,p/\rho \ll 1$, we recover the usual non-relativistic Alfvén velocity. As $b^{2}\to \infty$, $v_A^{2}\to 1$, the ultra-relativistic limit of the AW previously studied by (for example) Thompson & Blaes (Reference Thompson and Blaes1998) and Heyl & Hernquist (Reference Heyl and Hernquist1999).

7. Structure in the WF

In the WF, the spatial components of $z_-^{\mu } = \overline {z}_-^{\mu }$ are zero and so

(7.1)\begin{equation} u^{i} ={-} b^{i}/\mathcal{E}^{1/2}\text{ (WF)}. \end{equation}

Using (2.9), we may obtain

(7.2)\begin{equation} b^{t} ={-}\mathcal{E}^{1/2}\gamma v^{2} \text{ (WF)},\end{equation}

where $v^{2}$ is the square of the 3-velocity $v^{i}=u^{i}/\gamma$ and $\gamma =u^{t}$. Then, we may calculate the magnetic-field 3-vector,

(7.3)\begin{equation} B^{i} = b^{i} u^{t} - b^{t} u^{i} ={-}\mathcal{E}^{1/2} v^{i} \text{ (WF)},\end{equation}

so in the WF the 3-velocity is parallel and proportional to the magnetic-field 3-vector, just like in the non-relativistic case. This also implies $\partial _i v^{i}=0$ in the WF. We may also calculate

(7.4)\begin{equation} b^{2} = b^{i} b_i - (b^{t})^{2} = \mathcal{E} v^{2}=B^{2} \text{ (WF)},\end{equation}

and so, since $b^{2}$ is constant, in the WF $v^{2}$ and $B^{2}$ are both constant, just like in the non-relativistic case (Matteini et al. Reference Matteini, Horbury, Pantellini, Velli and Schwartz2015).

Let us consider the components of the stress–energy tensor (2.4) in the WF. First, using (7.2) and (7.4),

(7.5)\begin{equation} T^{tt} = \rho+U+\frac{3b^{2}}{2}\text{ (WF)}, \end{equation}

a space–time constant for our solution. Applying (7.2) again,

(7.6)\begin{equation} T^{ti}=T^{it}= \mathcal{E} v^{i}\text{ (WF)}, \end{equation}

which from (7.3) has no spatial divergence, maintaining the constancy of $T^{tt}$ in the equation $\partial _\nu T^{t\nu } = 0$. Finally,

(7.7)\begin{equation} T^{ij} = \left( p+ \frac{b^{2}}{2}\right)\delta^{ij}\text{ (WF)}, \end{equation}

a space–time constant, thus enforcing the constancy in time of $T^{it}$ in the equations $\partial _\nu T^{i\nu } = 0$. The cancellation of the first two terms in the space–space components of (2.4) in the WF generalizes the result for the non-relativistic AW that the centrifugal force exactly balances the tension force in the magnetic field, keeping the fluid flowing exactly along the field lines in the WF.