Hybrid electrostatic waves in linearized gravity

Abstract

Linearized gravity around a rotating black hole or compact object introduces the concept of a gravitomagnetic field, which originates from the matter–current in the rotating object. Plasma in proximity to this object is subsequently subjected to motion guided by this gravitomagnetic term (where mass serves as the effective charge) in addition to the conventional magnetic field. Such an interplay of fields complicates the accessible plasma waves of the system and thus merits exploration to delineate this interplay, to identify any observable signatures of the gravitomagnetic field in weakly magnetized systems, and to motivate future numerical work in a fully relativistic setting where the effects may be stronger. In this work we analyse the dispersion of the upper and lower hybrid electrostatic waves in a plasma immersed in both magnetic and gravitomagnetic fields. In particular, we discuss the effective augmentation or cancellation of the two fields under the right conditions for the upper hybrid wave. In contrast, the lower hybrid wave experiences a frequency up-shift from the gravitomagnetic field regardless of whether it is parallel or antiparallel to the magnetic field for the studied field strengths.

1. Introduction

When the spacetime near a black hole or compact star is stationary, one can slice the spacetime into three-dimensional space plus one-dimensional time. This approach is dictated by the stationarity of background spacetime and is known as ‘$3+1$’ splitting (Thorne & Macdonald 1982; Thorne, Price & MacDonald 1986). The three-dimensional space is considered as curved rather than Euclidean and is the spatial part of background metric $g_{\mu \nu }$ in an appropriately chosen coordinate system. The metric of a stationary spacetime in the Boyer–Lindquist coordinate can be written as

(1.1)\begin{equation} {\rm d}s^2={-}\alpha^2 c^2\,{\rm d}t^2+g_{ij}({{\rm d}\,x}^j+\mathcal{N}^j\,{\rm d}t)({{\rm d}\,x}^i+ \mathcal{N}^i\,{\rm d}t), \end{equation}

where $\alpha$ is the lapse function, $\mathcal {N}^i$ is the shift vector and $g^{ij}$ is the spatial metric. In the weak gravity and low velocity ($v\ll c$) limit, one can define two gravitational potentials in this space: the traditional Newtonian gravitational scalar potential $\varPhi _g$ and a vector potential $\boldsymbol {A}_g=c \,\mathcal {\boldsymbol {\mathcal N}}$ known as the gravitomagnetic potential.

These two potentials can be used to define the following two vector fields:

(1.2a,b)\begin{equation} \boldsymbol{g}={-}\boldsymbol{\nabla} \ln\alpha, \quad \boldsymbol{B}_g=\boldsymbol{\nabla}\times\boldsymbol{A}_g, \end{equation}

where $\boldsymbol {g}$ is the Newtonian gravitational field (gravitoelectric) and $\boldsymbol {B}_g$ is a ‘magnetic’-type gravity known as the gravitomagnetic field, respectively (Braginsky, Caves & Thorne 1977; Hobson, Efstathiou & Lasenby 2006; Bhattacharjee & Stark 2021). The analogy of these two fields with the electromagnetic ones becomes more obvious when the $3+1$ split is applied to a weakly gravitating, rotating body. It can be shown that in the weak gravity limit, the Einstein field equations for this object can be cast into a form similar to the time-independent Maxwell's equations,

(1.3a,b)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{g}={-}4 {\rm \pi}\rho_m ,\quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B}_g=0, \end{gather}

(1.4a,b)\begin{gather}\boldsymbol{\nabla}\times \boldsymbol{g} =0, \quad \boldsymbol{\nabla}\times\boldsymbol{B}_g={-} \frac{4{\rm \pi} G}{c} \boldsymbol{J}_m, \end{gather}

where $\rho _m$ and $\boldsymbol {J}_m$ are the mass and mass-current density of the gravitating body, respectively. It should be noted that the minus sign indicates gravity is attractive.

One of the major predictions of gravitomagnetism is the precession of the gyroscope near a large rotating mass, also known as the frame-dragging effect. This notably has been confirmed by the Gravity Probe B experiments in 2005 (Everitt et al. 2011). Recently, the pulsar PSR J1141-6545 circling a white dwarf has been detected to show a long-term orbital drift by measuring the pulses emanated from the pulsar to an accuracy to within $100~\mathrm {\mu }$s over a period of nearly 20 years (Krishnan et al. 2020). In the context of black hole accretion disk theory, the gravitomagnetic force is predicted to cause differential precession of the disk. This force acting in conjunction with the viscous force can cause the accretion disk to have a warped structure, a phenomenon known as the Bardeen–Petterson effect (Bardeen & Petterson 1975; Kumar & Pringle 1985; Thorne et al. 1986; Nelson & Papaloizou 2000; Lei, Zhang & Gao 2012; Nealon, Price & Nixon 2015). Observationally, this effect can cause the astrophysical jets emanating from the black holes to change their orientations rapidly (Miller-Jones et al. 2019).

In this article, we study the impact of the gravitomagnetic field on the plasma waves in the accretion disk around a rotating black hole or compact object. Plasmas in the accretion disk are often magnetized, but the presence of a background gravitomagnetic field can change the characteristics of the plasma wave modes. The traditional approach is to solve the fully general relativistic (GR) system including the Einstein's field equation and relativistic plasma equation of motion. The analytical complexities of such a calculation makes it impossible to understand the results in terms of classical plasma wave modes. Here, we are primarily motivated to uncover possible first-order corrections to the classical plasma wave modes, which can serve as a benchmark and motivation for a much more detailed GR calculation in stronger gravity in the future. As a part of this scheme, we investigate two electrostatic wave modes important to astrophysical plasma phenomena: upper hybrid and lower hybrid modes in the presence of both magnetic and gravitomagnetic fields. In a magnetized plasma, these electrostatic modes travel perpendicular to the background magnetic field, which is often oriented along the axis of rotation. In this study, we allow the gravitomagnetic field to be parallel or antiparallel to the background field to best elucidate the interplay of the two fields. One can expect that the presence of a gravitomagnetic field in a magnetized plasma can alter the characteristics of these two modes and potentially have observational effects on the electromagnetic spectrum, plasma heating and magnetic reconnection processes in the accretion disk near a compact object under the right conditions (Roy & Lakhina 1985; Bell & Ngo 1990; Lesch 1991; Pécseli et al. 1996; Cairns & McMillan 2005; Wallace et al. 2010; Liang et al. 2013; Chen et al. 2020).

We present the background on plasma dynamics in a gravitomagnetic field in the first section. Next, we derive the dispersion relations for both modes the upper and lower hybrid waves. We conclude with a discussion of the application of these results to accretion disk plasmas.

2. Plasma dynamics

In this article, we consider the plasma dynamics in an accretion disk near a slowly rotating compact object such as a black hole or neutron star. The slowly rotating object can be characterized by its mass $M$ and spin parameter $a=J/Mc \ll 1$ (in the units of half-Schwarzschild-radius $r_g=GM/c^2$), in which $G$ is the gravitational constant, $c$ is the speed of light and $J$ is the angular momentum (Misner, Thorne & Wheeler 1973; Thorne et al. 1986; Hobson et al. 2006). Moreover, the plasmas we consider here are (i) located at larger radii from the Schwarzschild radius ($r\gg 2r_g=GM/c^2)$ and (ii) moving with a bulk velocity much lower than the speed of light ($v\ll c)$.

The accretion disk considered here is assumed to be unmagnetized initially, implying the magnetorotational instability is not operational yet. It has been shown in the pure hydrodynamical set-up that angular momentum transport can still be sustained in these types of disks (Fragile & Anninos 2005; Paoletti et al. 2012; Ghosh & Mukhopadhyay 2021). Instead of investigating the plasma dynamics of the accretion disk which has settled to its equilibrium structure, we concentrate on the time scale when seed magnetic fields are being generated in the disk by mechanisms such as Biermann battery, spacetime curvature drive, gravitomagnetic drive, etc. (Mahajan & Yoshida 2011; Bhattacharjee, Das & Mahajan 2015; Bhattacharjee & Stark 2021). Therefore, our analysis of wave modes will be limited to the range of seed magnetic field values in the accretion disk or where there are relatively unmagnetized regions of a larger structure.

Under these approximations, the plasma dynamics of individual species (labelled $s$) can be represented by the equation of motion and the continuity equation, respectively (Thorne et al. 1986; Bhattacharjee & Stark 2021),

(2.1)\begin{align} m_sn_s\left(\frac{\partial}{\partial t}+\boldsymbol{v}_s\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{v}_i & ={-}\boldsymbol{\nabla}p_s+q_sn_s\boldsymbol{E}+\frac{q_sn_s}{c}\left(\boldsymbol{v}_s +\boldsymbol{\mathcal{N}}\right)\times\boldsymbol{B} +\frac{m_sn_s}{c}(\boldsymbol{v}_s\times\boldsymbol{B}_g) \nonumber\\ & \quad +m_sn_s\boldsymbol{g}, \end{align}

(2.2)\begin{align} \frac{\partial n_s}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(n_s\boldsymbol{v}_s)& =0,\end{align}

where $m_s$ is the species mass, $n_s$ is the plasma number density, $\boldsymbol {v}_s$ is the velocity, $p_s$ is the pressure, $\boldsymbol {E}$ is the electric field, $\boldsymbol {B}$ is the magnetic field and $\boldsymbol {g}=-\boldsymbol {\nabla }\varPhi _G$ is the Newtonian gravitational acceleration. It should be noted here that the two new terms containing $(\boldsymbol {\mathcal {N}}\times \boldsymbol {B})$ and $(\boldsymbol {v}_s\times \boldsymbol {B}_g)$ on the right-hand side of (2.1) are corrections due to rotating spacetime with $\boldsymbol {B}_g$ representing the ‘magnetic’-type gravitomagnetic field and its corresponding vector potential $\boldsymbol {A}_g=c \, \boldsymbol {\mathcal {N}}$. The shift vector is divergence-free $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\mathcal {N}}=0$ in the Boyer–Lindquist coordinate (Thorne et al. 1986) and can be interpreted as the angular velocity of zero angular momentum observers (also known as fiducial observers) in a stationary background spacetime. It should be noted here that the Lorentz force term in (2.1) has a coupling between the magnetic field and gravity through the term $\boldsymbol {\mathcal {N}}\times \boldsymbol {B}$. The $m_sn_s(\boldsymbol {v}_s/c\times \boldsymbol {B}_g)$ term on the right-hand side of (2.1) is a new force that can be attributed to the frame-dragging caused by the spacetime's intrinsic rotation. The consequence of this new force has been studied quite extensively in the context of fluid dynamics in black hole accretion disks and is known as the Bardeen–Petterson effect (Bardeen & Petterson 1975; Nelson & Papaloizou 2000).

In an axisymmetric stationary spacetime near a weakly rotating object, the gravitomagnetic field takes the following form:

(2.3)\begin{equation} \boldsymbol{B}_g=\frac{2G}{c}\left(\frac{\boldsymbol{J}}{R^3}-\frac{3(\boldsymbol{J}\boldsymbol{\cdot} \boldsymbol{r})\boldsymbol{r}}{R^5}\right) \end{equation}

with its corresponding vector potential

(2.4)\begin{equation} \boldsymbol{A}_g=\frac{2G}{c}\frac{\boldsymbol{J}\times \boldsymbol{r}}{R^3}. \end{equation}

Here, $R=\sqrt {r^2+z^2}$ is the distance to the plasma element from the central mass, and $\boldsymbol {r}$ is associated with the cylindrical coordinate system $(r,\phi, z)$. If we take the angular momentum of the object $\boldsymbol {J}$ along the $z$-axis and assume the disk to be thin ($z\ll r$), then (2.3) becomes

(2.5)\begin{equation} \boldsymbol{B}_g=\frac{2G}{c} \frac{\boldsymbol{J}}{r^3}, \end{equation}

which can be expressed in the following normalized form:

(2.6)\begin{equation} \boldsymbol{B}_g=\frac{2 \tilde{a}c^2}{r_g}\frac{1}{\tilde{r}^3}\hat{z}. \end{equation}

Here we have used the following normalizations: spin parameter $\tilde a=J/Mcr_g$ and radial distance $\tilde {r}=r/ r_g$. The spin parameter $\tilde a$ can have a maximum value of $1$. However, in order for the gravitomagnetic approximation to be valid, we consider cases where spin parameter $|\tilde {a}|<0.3$. This specific range has also been confirmed to produce the frame-dragging effect when modelling the accretion disk near a rotating object by means of Newtonian physics (Chakrabarti & Khanna 1992).

3. Electrostatic wave modes in electron–ion plasma

To obtain the electrostatic wave modes in an incompressible electron–ion plasma, we need (2.1), (2.2) and Poisson's equation,

(3.1)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E}=\sum_s n_sq_s. \end{equation}

First, we assume the background electric field $\boldsymbol {E}_0=0$ and a uniform background plasma density, i.e. $\boldsymbol {\nabla }n_{0s}=0$. Next, we look for wave modes in the electron–ion plasma with wavevector $\boldsymbol {k}=k\hat {x}$ in a background magnetic and gravitomagnetic fields ($\boldsymbol {B}_0,\boldsymbol {B}_g$) $|| \hat {z}$. Then, using the following perturbations: $n=n_{0s}+\delta n_s$, $\boldsymbol {E}=\delta \boldsymbol {E}|| \boldsymbol {k}$ and $\boldsymbol {v}_s=\boldsymbol {v}_{0s}+\delta \boldsymbol {v}_s$, we linearize (2.1), (2.2) and (3.1) and solve for the dispersion relation of the system, yielding

(3.2)\begin{align} & 1-\frac{\omega_{{\rm pe}}^2}{(\omega-\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{v}_{0e})^2-\gamma_e k_b T_ek^2/m_e-\tilde{\omega}_e^2/m_e^2}\nonumber\\ & \quad -\frac{\omega_{{\rm pi}}^2}{(\omega-\boldsymbol{k}\boldsymbol{\cdot} \boldsymbol{v}_{0i})^2-\gamma_i k_bT_ik^2/m_i-\tilde{\omega}_i^2/m_i^2}=0, \end{align}

where $T_e$, $T_i$ stand for electron and ion temperature, respectively; electron plasma frequency $\omega _{{\rm pe}}=\sqrt {4{\rm \pi} n_eq_e^2/m_e}$, ion plasma frequency $\omega _{{\rm pi}}=\sqrt {4{\rm \pi} n_iq_i^2/m_i}$, $\tilde {\omega }_s=(q_s/c B_0+m_s \omega _g)$ is the modified cyclotron frequency for both species, gravitomagnetic frequency $\omega _g=B_g/c$ and $\gamma$ is the adiabatic index for each species.

For simplicity in isolating the new physics, for the rest of this article we assume background flow $v_{0s}\approx 0$ for both species and write (3.2) as

(3.3)\begin{equation} 1-\frac{1}{\tilde{\omega}^2-\beta_e^2-\tilde{\omega}_{{\rm ce}}^2(1-\epsilon_e)^2}- \frac{1}{Z^2}\frac{\alpha}{{\tilde\omega}^2-\tilde{\gamma}\alpha\beta_e^2-\alpha^2 Z^2\tilde{\omega}_{{\rm ce}}^2\left(1+\dfrac{1}{\alpha Z}\epsilon_e\right)^2}=0. \end{equation}

In (3.3), we have used the following normalizations: $\tilde {\omega }=\omega /\omega _{{\rm pe}}$, $\tilde {\omega }_{{\rm ce}}=\omega _{{\rm ce}}/\omega _p$, ${\alpha =m_e/m_i}$, $\beta _e=\sqrt {\gamma _e}(k v_{{\rm th}(e)}/\omega _{{\rm pe}})$, $\tilde {\gamma }=\gamma _i/\gamma _e$, $q_i=Ze$, $\epsilon _e=\omega _g/|\omega _{{\rm ce}}|$, $\epsilon _i=1/Z\alpha \,\epsilon _e$ where thermal velocity for both species $v_{{\rm the}(s)}=\sqrt {k_bT_s/m_s}$, $Z$ is the atomic number, and electron cyclotron frequency $|\omega _{{\rm ce}}|=eB/m_ec$.

In (3.3), it should be noted that both the second and third terms on the left-hand side are modified by the factors $(1-\epsilon _e)^2$ and $(1+({1}/{\alpha Z})\epsilon _e)^2$, respectively. These two terms are explicit contributions from gravitomagnetism with the factor $\epsilon _e=\omega _g/|\omega _{{\rm ce}}|$ measuring the relative strength between the gravitomagnetic and magnetic field. In the absence of a gravitomagnetic field, (3.3) becomes the standard dispersion relation for electrostatic waves in a magnetized electron–ion plasma.

Figure 1(a) plots the dispersion relation for different values of $\epsilon _e$, where the traditional upper hybrid oscillation is represented by a black dotted line. For the rest of the article, we assume a hydrogen plasma for simplicity. For different values of $\epsilon _e$, we notice different vertical shifts in the frequency plot from the $\epsilon _e=0$ case (black line). As $|\epsilon _e|$ increases from 0, the frequency at $\beta _e=0$ shifts from the classical upper hybrid oscillation in a direction depending on the sign of $\epsilon _e$. For $\epsilon _e < 0$, the frequency is pushed above the classical value, whereas for $\epsilon _e > 0$, the frequency is pushed below it. In the latter case, we see that setting $\epsilon _e=1.0$ results in a starting frequency at $\omega _{{\rm pe}}$. In this case, there is an effective cancellation of the magnetic and gravitomagnetic fields, resulting in $\omega _s=0$. Here the wave behaves as a standard unmagnetized electron plasma wave. We note, however, that for $\epsilon _e>1.0$, the gravitomagnetic field will then no longer be cancelled by the magnetic field and the frequency will once again rise above the plasma frequency. At $\epsilon _e=2.0$, the mode will match the case of $\epsilon _e=0$, but now the mode is primarily influenced by the gravitomagnetic field as opposed to the magnetic field.

Figure 1. (a) Plot of $\omega$ versus $\beta$ for the upper hybrid wave with modifications due to different $\epsilon _e$ values. The classical upper hybrid oscillation is given in the dashed black line. (b) Plot of the cutoff frequency $\omega (\beta =0)$ in $\omega _{{\rm ce}}$, $\omega _g$ phase space. Contours are given in black and additional annotation shows the relative contributions of the gravitomagnetic and magnetic fields.

The interplay between the magnetic and gravitomagnetic fields can further be visualized by constructing a $\omega _{{\rm ce}}, \omega _g$ phase space plot showing the cutoff frequency $\omega (\beta _e=0)$ for each combination. Figure 1(b) gives $\omega _g$ and $\omega _{{\rm ce}}$ each in the range of $-1$ to 1 (still normalized to $\omega _{pe}$) and the colour coding is $\omega (\beta _e=0)$ using the same normalization. The diagonal blue region is where both fields are parallel and equal in magnitude in the plasma ($\epsilon _e=1)$; however, because the electron has a negative charge in the magnetic field and a positive effective charge (its mass) in a gravitomagnetic field, the effect of the fields on the electron will cancel. This implies that a purely magnetized plasma near a weakly rotating object can sustain a Langmuir wave perpendicular to the background field under the right conditions, which is possible because both magnetic and gravitomagnetic fields cancel in their impact on electrons.

In contrast, the yellow regions at the upper left-hand and lower right-hand corners indicate that magnetic and gravitomagnetic fields are colinear ($\epsilon _e=-1$) in the plasma. Here the force on an electron in the gravitomagnetic field will be in the same direction as that from the magnetic field, so the effective magnetic field or cyclotron frequency will increase. Contours of $\omega (\beta _e=0)$ are denoted in black. The remaining four regions in the phase space can be distinguished based on the relative magnitudes of the background magnetic and gravitomagnetic fields within the plasma. This implies unmagnetized or weakly magnetized plasmas near the object can sustain electrostatic wave modes mimicking those seen in magnetized plasma.

Figure 2(a) shows the lower hybrid wave dispersion for different values of $\epsilon _e$. In contrast to the high-frequency modes, we do not see any significant differences in the dispersion relations between $\epsilon _e$ values of the same magnitude but opposite sign. As the classical lower hybrid oscillation frequency depends on both the electron and ion cyclotron frequencies, there will always be one species where the gravitomagnetic field effectively increases the cyclotron frequency and one species where it effectively reduces this frequency. Under the assumption of $\epsilon _e \gg \alpha$, the lower hybrid oscillation (setting $\beta _e=0$) is

(3.4)\begin{equation} \omega_{{\rm LH}}^2 \sim \frac{\alpha \omega_{{\rm ce}}^2}{1+\omega_{{\rm ce}}^2}+\epsilon_e^2 \omega_{{\rm ce}}^2 \end{equation}

to lowest order in $\alpha$ and $\epsilon _e$. The dashed horizontal line shows the classical lower hybrid oscillation frequency, and all tested values of $\epsilon _e$ cause an increase from this value due to the $\epsilon _e^2$ dependence in the equation.

Figure 2. (a) Plot of $\omega$ versus $\beta$ for the lower hybrid wave with modifications due to different $\epsilon _e$ values. The classical lower hybrid oscillation is given in the dashed black line. (b) Plot of the cutoff frequency $\omega (\beta _e=0)$ in $\omega _{{\rm ce}}$, $\omega _g$ phase space. Contours are given in black.

In figure 2(b), we once again plot the phase space of the gravitomagnetic and cyclotron frequencies and colour code by the cutoff frequency $\omega (\beta _e=0)$, this time for the lower hybrid wave. When the magnitude of the magnetic field increases, the relative gravitomagnetic contribution diminishes. We note that no field cancellation is observed in this lower frequency mode in the plasma under this set of plasma conditions. The contour lines (black) form a unique island-type structure for lower values of both magnetic and gravitomagnetic fields, but the contours become mostly vertical in the case of $|\omega _{{\rm ce}}| \gg |\omega _g|$.

4. Discussion and conclusion

We have presented the dispersion relation of electrostatic waves in a warm electron–ion plasma travelling perpendicular to background magnetic and gravitomagnetic fields. The plasma is assumed to be non-relativistic and located in an accretion disk near a weakly rotating astrophysical object. Though the system is non-relativistic, the first-order effect from GR in the plasma dynamics is in the form of a gravitomagnetic force. As a result, the combination of both magnetic and gravitomagnetic fields alters the standard cyclotron frequency, and this results in modifications to the classical upper and lower hybrid waves. The relative orientation of magnetic and gravitomagnetic fields permits the existence of pure Langmuir waves in this magnetized plasma under the conditions of gravitomagnetic cancellation. The existence of these modes can potentially be verified in observations during the propagation of electromagnetic transmissions in the accretion disk (Chen 2012). This prediction becomes obvious when we ignore the ion dynamics and consider the cold plasma limit. In this limit, (3.2) can be rewritten as

(4.1)\begin{equation} \frac{(\omega_{{\rm ce}}+\omega_g)^2}{\omega^2}=1-\frac{4{\rm \pi} nq^2}{m\omega^2}, \end{equation}

where $\omega _g=B_g/c$ is the gravitomagnetic frequency. As the plasma density varies in the disk, the electromagnetic transmission through the plasma varies as well based on the frequency spectrum. This profile should be different from the standard upper hybrid frequency – to which the electromagnetic waves can couple – because of the existence of gravitomagnetism and depends on the strength and relative orientation of the background magnetic and gravitomagnetic fields. This can serve as one of the tests for the GR prediction of frame dragging in the non-relativistic limit, in additional to serving as a potential indirect diagnostic of the rotation of the object.

However, since the regime in which the approximations are valid necessitates a relatively weak gravitomagnetic field, the objects and structures where this can be useful requires careful consideration. We can estimate $\omega _g$ as

(4.2)\begin{equation} \omega_g \sim \frac{\tilde{a}\times10^5 }{\tilde{M}\tilde{r}^3}, \end{equation}

introducing $\tilde {M}$ as the mass of the object in units of the solar mass. From this we see that increasing $\omega _g$ for a fixed $\tilde {r}$ requires smaller masses, so stellar mass black holes or neutron stars are more likely to lead to something observable. Looking then at $\omega _c$, we find

(4.3)\begin{equation} \omega_c \sim 1.76 \times 10^8 \beta, \end{equation}

in which $\beta$ is the magnetic field (in gauss) at the corresponding location $\tilde {r}$ from the object. To have $\omega _g \sim \omega _c$ (or $\epsilon _e \sim 1$), therefore

(4.4)\begin{equation} \beta \sim \frac{1}{176} \frac{\tilde{a}}{\tilde{M} \tilde{r}^3}. \end{equation}

Even with optimal gravitomagnetic parameters of $\tilde {a}=0.3$, $\tilde {M}=5$ and $\tilde {r}=40$ – while still remaining in the regime of validity of the gravitomagnetic approximations – we find that the magnetic field for $\epsilon _e\sim 1$ is ${\sim }10^{-9}$ G, which may be most appropriate for primordial black hole systems (Safarzadeh 2018; Araya et al. 2021; Papanikolaou & Gourgouliatos 2023) or other seed-field systems discussed in § 2. Since smaller $\epsilon _e$ values still alter the dispersion of the hybrid modes, more modern systems with larger magnetic fields may potentially have measurable gravitomagnetic fields. However, we would still be limited to relatively unmagnetized regions of accretion disks, which exist in such systems with complex structures (Riffert 1980), and in particular when there are oscillatory field structures (Bhattacharjee, Feng & Stark 2018). Furthermore, extended observation times would be required to reach frequency resolution necessary to observe any gravitomagnetic shifts.

This derivation was performed in the post-Newtonian limit, but if this were done in a full GR setting, one would be able to look at conditions closer to the object where these effects would be considerably stronger. While this calculation does not directly apply in such a regime, we hope that it serves as a motivation and benchmark for future simulation studies (likely requiring a GR particle-in-cell code), where greater specifics of observable signatures can be better explored.

When ion dynamics plays an important role, we found that the lower hybrid waves also experience modifications from gravitomagnetism. However, these modes change in a fundamentally different way compared with the upper hybrid modes, as we do not see field cancellation for our plasma conditions, even if the relative orientation between these two fields changes. It should be noted that as the relative strength of the gravitomagnetic field increases, the cutoff frequency increases as well. This, in turn, alters the spectrum of waves with which the lower hybrid waves can interact, on both electron and ion scales (Liu, Chen & Ma 2021). Lower hybrid waves can efficiently transfer energy from the perpendicular motions of ions to the parallel motions of electrons or vice versa, either accelerating particles or heating them (Verdon et al. 2008). What our results indicate is that gravity (through gravitomagnetic contributions to the lower hybrid wave mode) can play a non-negligible role in this energy transfer mechanism in ion and electron length scales, which will be explored later in detail. Moreover, the behaviour of these modes in the small magnitude field limit is quite different from their higher frequency counterparts. A further analysis is necessary to explore this behaviour in the low-frequency regime, which will be addressed in future work.