Yet Another Modification of Relativistic Magnetohydrodynamic Waves: Electron Thermal Inertia

This study investigates the properties of waves in relativistic extended magnetohydrodynamics (RXMHD), which includes Hall and electron thermal inertia effects. We focus on the case when the electron temperature is ultrarelativistic, and thus, the electron thermal inertia becomes finite at near the proton inertial scale. We derive the linear dispersion relation of RXMHD and find that the Hall and electron thermal inertia effects couple with the displacement current, giving rise to three superluminous waves in addition to the slow, fast, and Alfv\'en waves. We also show that the phase- and group-velocity surfaces of fast and Alfv\'en waves are distorted by the Hall and electron thermal inertia effects. There is a range of scales where the group velocity of fast wave is smaller than that of the Alfv\'en and slow waves. These findings are applicable to a region near the funnel base of low-luminosity accretion flows where electrons can be ultrarelativistic.


Introduction
Magnetohydrodynamics (MHD) is undeniably the most widely used model for understanding large-scale dynamics in astrophysics. While it is a minimal extension of hydrodynamics to include electromagnetic elds, MHD can explain a variety of plasma phenomena. However, MHD is valid only in a certain parameter range in which any microscopic e ects are negligible. To interpret modern astronomical observations, one must carefully consider the microscopic e ects. For example, the black hole shadow image at M87 captured by the Event Horizon Telescope [1] can be theoretically explained in multiple possible scenarios depending on how the microscopic e ects are treated [2]. As such, it is important to extend the MHD theory so that it includes microscopic e ects self-consistently.
Such an attempt has been achieved in a non-relativistic regime by developing the extended MHD (XMHD) [3][4][5][6], which has two essential microscopic e ects: the Hall e ect appearing at the ion skin depth i and the electronrest-mass-inertia e ect appearing at even smaller electron skin depth e . For the last few years, XMHD has been used to study a variety of processes, such as magnetic reconnection [7], nonlinear waves [8,9], and turbulence [10][11][12].
Meanwhile, relativistic e ects are critical to understand high-energy astrophysical phenomena. The relativistic XMHD (RXMHD) 1 was formulated using an empirical approach [15,16] and a variational approach [17], and it has been actively used for the last few years [e.g., [18][19][20][21][22][23][24][25][26]. In RXMHD, the electron thermal inertia (ETI) e ect emerges when the electron thermal energy exceeds the rest mass energy. Remarkably, the scale at which ETI switches on increases as the electron temperature increases, and it can be signi cantly larger than e where the electron rest mass inertia becomes nite. One of the most important consequences of the increased electron inertial scale is collisionless reconnection happening at a larger scale than e [22,27].
Surprisingly, despite the obvious success of RXMHD, the most basic characteristic of RXMHD, namely the properties of linear wave propagation, has yet to be studied. Knowing the linear wave properties has wide implications, including the development of direct numerical simulation codes (see, for example, [28][29][30][31][32] for relativistic MHD codes and [33] for a non-relativistic XMHD code). It is known that the properties of linear waves in nonrelativistic and relativistic ideal MHD are quite similar [34] except that the phase speeds of the three MHD waves, Alfvén, slow-and fast-magnetosonic waves, are bounded by the speed of light for relativistic MHD. On the other hand, the non-relativistic Hall e ect changes the wave properties dramatically [35] For example, the phase and group diagrams (also known as Friedrichs diagrams) are distorted by the Hall e ect, and the fast wave becomes 1 One may confuse RXMHD with the model with the same name developed by Chandra et al. [13]. The latter model is essentially a relativistic version of Braginskii's equations [14] which take into account of weakly collisional e ects, such as anisotropic pressure and heat ux but does not include microscopic e ects. On the other hand, RXMHD used in this study has microscopic e ects, but no collisionless e ects are included. strongly anisotropic, i.e., it primarily propagates in the direction parallel to the background magnetic eld. In our previous work [36], we explored the wave properties in relativistic Hall MHD (RHMHD), in which the electron temperature is modestly-or non-relativistic, and thus ETI is negligible in ion inertial scales. We demonstrated that relativistic Hall e ect and non-relativistic Hall e ect change the wave properties di erently. This is a remarkable di erence considering that waves are almost the same in non-relativistic and relativistic ideal MHD. In RHMHD, the fast wave is more isotropic than that in nonrelativistic HMHD, and the group velocity surface of the fast wave overlaps with that of Alfvén wave. Furthermore, the scale at which the Hall e ect switches on is di erent from i when the relativistic e ect is present. This scale shrinks as the magnetic energy increases. In other words, RHMHD becomes relativistic ideal MHD when the magnetic eld is strong.
In this study, we additionally include ETI at ion inertial scales by considering electrons with ultrarelativistic temperature and investigate how ETI changes the wave properties of RHMHD. It is suggested by the global simulations of black hole accretion ows that such ultrarelativistic electrons may exist in the funnel base region of low-luminosity accretion ows [37]. 2 Hence, the wave properties revealed here will be useful to understand relativistic jets in active galactic nuclei.

Linear Dispersion Relation of RXMHD
We consider a proton-electron plasma in a at spacetime de ned by the Minkowski metric, diag(1, −1, −1, −1). The set of RXMHD equations [15][16][17] are the continuity equation the momentum equation the generalized Ohm's law 2 Note that the electron temperature calculated by the global simulations strongly depends on the prescription of energy partition between ions and electrons [38], which is determined physically by the collisionless mechanisms. While the electrons become ultrarelativistic when the turbulent heating model [39][40][41] is employed, the electron temperature becomes much lower with the reconnection heating model [42,43]. Even with the turbulent heating model, the electron temperature can be lower if there are compressive uctuations [44,45]. and Maxwell's equation: where , , ℎ , , and are the elementary charge, rest frame number density, thermal enthalpy, thermal pressure, and Levi-Civita symbol, respectively. The subscript = (i, e) denotes the species label with i for protons and e for electrons. We also use the total enthalpy ℎ = ℎ i + ℎ e and total pressure = i + e . The four vectors and tensor represent the four velocity = ( , ∕ ), the Faraday tensor that gives electric eld = 0 and magnetic eld = −(1∕2) , and four current = = ( , ∕ ), where = 1∕ √ 1 − 2 ∕ 2 is the Lorentz factor of bulk ow, and is the charge density.
One nds quite a few additional terms in (1b) and (1c) compared to the relativistic ideal MHD. The nal term in the right hand side of (1c) is a Hall term, while the terms multiplied by ℎ e and in (1b) and (1c) are originated from the electron thermal and rest mass inertiae. In our previous paper [36], we neglected the electron inertia e ects and focused on the Hall e ect by assuming that the electron temperature is modestly-or non-relativistic, i.e., e ∕ e 2 ≲ 1, and wavelength is in proton inertial scales, i.e., i ∼ 1, where i = ( i 2 ∕4 0 2 ) 1∕2 is the proton skin depth. In this study, we consider electrons with ultrarelativistic temperature while the wavelength is slightly shorter than i but much larger than the electron inertial length e = 1∕2 i . More speci cally, we impose the following ordering The rst, second, and fourth conditions are numerically supported to be valid around the funnel base region (see, for example, Fig. 4 in Ref. [46] for the rst condition, Fig. 1 in Ref. [37] for the second condition, and Fig. 6 in Ref. [47] for the fourth condition). These conditions lead to = 8 ( i + e )∕ 2 0 ≲ 1 and = 2 0 ∕4 0 ℎ i0 ≳ 1, both of which means that the plasma is magnetically dominated, and they are valid around the funnel base region.
We then linearize (1b) and (1c) by splitting the elds as = 0 +̃ where the subscript 0 denotes the spatio-temporary constant background eld and tilde denotes the perturbation. We also assume that there is no background ow, i.e., 0 = 0. The background part of Maxwell's equations (1d) gives 0 = 0, 0 = 0, and 0 = 0. Dropping the terms with the order higher than ( 0 ), the spatial parts of (1b) and (1c) are reduced to One nds that (3a) is identical to the momentum equation of the linearized relativistic ideal MHD. The electron inertia e ect appears only in the left hand side of (3b). Note that (3b) is akin to non-relativistic XMHD [3][4][5][6], but the left hand side in (3b) vanishes in the non-relativistic limit because we are focusing on proton inertial scales i ∼ −1∕4 . Next, we combine (3a)-(3b) with Maxwell's equations (1d) and Fourier transform the perturbed elds bỹ ∝ exp(i ⋅ − ). Below, we assume the equation of state for ideal gas for protons (namely, ℎ i0 = i 2 + [Γ∕(Γ − 1)] i0 where Γ = 4∕3 is a speci c heat ratio in ultrarelativistic case [48]). Nonetheless, the following results do not change for other choices of the equation of state. We also omit the entropy wave = 0.

Wave Properties
The most striking di erence between RXMHD dispersion relation (4) and the non-relativistic XMHD counterpart (6) is that (4) is six order with respect to 2 while (6) is third order. This means that RXMHD has three additional waves in addition to the standard MHD wave family. As we found in our previous work [36], one of the three additional waves is originated from the relativistic Hall e ect because the right hand side of (5) is fourth order with respect to 2 . Here, we call this a Hall wave. In our previous study, we showed that the Hall wave is superluminous; more speci cally, as i increases from 0 to in nity, the phase velocity ph = ( ∕ 2 ) decreases from in nity to , and the group velocity gr = ∕ increases from 0 to . This means that the Hall wave is indistinguishable from light at small scales as long as one views the propagation properties.
When ETI is nite, we have two more waves, which we call ETI (1) and ETI (2) waves in the order of phase speed. These new waves are generated because ETI couples with the displacement current, as is evident from the fact that ℎ e is always paired with ∕ in (4). To see the order relation of phase speed of six RXMHD waves and their asymptotic behavior against ℎ e , we plot the parallel component of phase velocity of all wave solutions vs. ℎ e for i = 8, = 4, and i0 ∕ i 2 = 1 in Fig. 1. We nd that the phase speed of two ETI waves are greater than the Hall wave. Both the Hall wave and two ETI waves are superluminous with the phase velocity decreasing from in nity to as ℎ e increases from 0 to in nity. We also nd that the phase velocity of fast magnetosonic and Alfvén waves decreases as ℎ e increases while that of the slow magnetosonic wave is almost constant.
To explore the wave properties in all directions, we draw the phase diagram (a trajectory of ph ) in Figure 2 with various i = (0, 1, 3, 5, 10) for RHMHD (ℎ e = 0) and RXMHD (ℎ e ∕ e 2 = 4 −1∕2 ), where and i0 ∕ i ∕ 2 are xed at 4 and 1, respectively (see also Supplementary material 1 for a ner interval of i ). First, we summarize the features of HMHD waves that were presented in our previous study [36]; the phase velocity of Hall wave is almost perfectly isotropic. Moreover, the phase diagram of the fast wave becomes nearly oval shape with ph ≃ 2 ph⟂ as i increases, while that in non-relativistic HMHD becomes dumbbell shape with ph ≫ 2 ph⟂ [35], meaning that the relativistic Hall e ect isotropize the fast wave. In the case of RXMHD, the phase diagram of the fast wave is even more isotropic at large i [ Fig. 2 (i)], and the shapes of fast and Alfvén waves at large i are akin to those of relativistic ideal MHD ( i = 0). One nds that the phase diagram of ETI (1) wave is slightly anisotropic with ph⟂ > ph‖ at i ≲ −1∕4 [ Fig. 2 (g)], but it gets isotopic at large i because ETI (1) wave converges to light.
Next, we show the group diagram (a trajectory of gr , also known as ray surfaces) with the same parameter setting as Fig. 2 (see also Supplementary material 3 for a ner interval of i ). Drawing a group diagram is important as it graphically represents the shape of the wave front and the direction of energy ow [see 35, 50, for more details]. As we found in our previous work [36], the way relativistic Hall e ect modi es the group diagram is vastly di erent from the way non-relativistic Hall e ect does [35]. In non-relativistic HMHD, the group velocity of the fast wave is much greater than that of Alfvén wave in all direction at large i . On the other hand, in RHMHD, the group velocity surface of the fast wave mostly overlaps with that of Alfvén wave (which we call "coalescence") [ Fig. 3 (c-e)]. The coalescence happens because gr of fast waves is limited by the speed of light, and thus it cannot be deviated from the group surface of Alfvén wave. Now, when ETI is nite, we nd that the group diagram is signi cantly changed. First, comparing Fig. 3 (c) and (g), one nds that the group velocity surfaces of fast and Alfvén waves become much more complicated when ETI is nite. Second, the ETI (1) wave propagates anisotropically in a certain wavenumber range, and it primarily propagates in the perpendicular direction to 0 [ Fig. 3 (g)]. Third, the group velocity of Alfvén wave is greater than the fast wave when i is not too large [ Fig. 3 (h)]. Lastly, when i is large, the coalescence of fast and Alfvén wave is broken; more speci cally, the group velocity of the fast wave is greater than that of Alfvén wave in all directions when [ Fig. 3 (i)]. Comparing Fig. 2 (i) and Fig. 3 (i), one also nds that the phase and group diagrams of RXMHD are almost identical when i is large. Note that the group velocity of the ETI (2) wave is almost zero, and thus it does not play any meaningful role in this parameter range (yet, it can be meaningful in even smaller scales, but the electron rest mass inertia would further change the wave properties). Finally, we explore even strongly magnetized case. In RHMHD, as one can see from (5), the scale at which the relativistic Hall e ects switches on is not i but i [36]. Since i gets smaller as the magnetic energy gets larger, the wave properties in RHMHD are almost the same as those of relativistic ideal MHD. However, this behavior does not apply to RXMHD because the terms containing ETI in (5) do not depend on . This is evident in Fig. 4 which shows the group velocity diagram for = 40 case. The diagrams of RHMHD [ Fig. 4 (b)-(d)] are almost the same as relativistic ideal MHD [ Fig. 4 (a)] unless i is su ciently large. On the other hand, in the case of RXMHD, the group diagram is signi cantly di erent from that of relativistic ideal MHD. Figure 4 (g) shows that the Hall wave propagates anisotropically in the same manner as the ETI (1) wave with = 4 [ Fig. 3 (g)]. Even more interestingly, Fig. 4 (i) shows that the fast wave group velocity can be smaller than that of slow wave for the = 40 case while the slow wave is always smallest in the case of = 4 [ Fig. 3 (i)].

Concluding remarks
In this paper, we have derived the linear dispersion relation of RXMHD when the electrons are ultrarelativistic (i.e., e ∕ e 2 ∼ −1∕2 ) and wavelength is slightly shorter than ion skin depth (i.e., i ∼ −1∕4 ). We have shown that ETI signi cantly modi es the properties of waves. One wave emerges due to the Hall e ect, and two waves emerge due to ETI, all of which are superluminous. These waves can propagate anisotropically in a perpendicular direction to 0 at certain value of i . Moreover, the phase and group diagrams of three standard MHD waves are    distorted by ETI; the fast wave is isotropized, and the coalescence of fast and Alfvén waves are broken at large i . Finally, there is a range of wavenumber where the group velocity of fast wave is smaller than that of the Alfvén wave for moderate value of and smaller than that of slow wave for high .