Quasinormal modes of accelerating spacetime

The first detection of gravitational waves (GWs) from the merger of two black holes (BHs) [1], along with the release of the first image of the supermassive BH in the galaxy M87 [2], has provided strong evidence that BHs are real celestial objects rather than just theoretical concepts. When a BH is perturbed, the relaxation can be described by a superposition of exponentially damped sinusoids termed quasinormal modes (QNMs) [3–5]. Thus, during the ringdown stage of the coalescence of two astrophysical BHs, the GWs can be expressed as a superposed QNM of the remnant BH. According to the no-hair theorem, the frequencies and decay rates of these QNMs are uniquely determined by the final BH's physical parameters [6]. The measurement of the QNMs from GW observations facilitates the test of general relativity and provides insights into the nature of remnants formed in compact binary mergers. This program is known as BH spectroscopy [7]. Moreover, the QNMs can also be used to determine the linear stability of a perturbed BH. For example, an analysis of the QNMs of massless scalar fields in the exterior of Reissner-Nordström-de Sitter (RNdS) BHs can be used to determine whether the strong cosmic censorship conjecture is violated [8].

Recently, the QNMs of the C-metric have attracted increasing attention in various physical contexts [9–17]. The C-metric describes an axially symmetric and stationary spacetime containing two causally separated BHs that accelerate away from each other in opposite spatial directions under the action of conical singularities along the axis [18–20]. This model may be useful for understanding the behavior of moving and accelerating BHs, such as those resulting from a BH superkick or a cosmic string connecting two BHs [21, 22]. These studies on the QNMs of accelerating BHs rely on the C-metric's Petrov type D classification, which allows the separation of perturbation equations for various test fields [23–26]. Notably, QNMs of charged and rotating accelerating BHs can be classified into three families: photon sphere, near-extreme, and acceleration modes. Photon sphere modes correspond to peaks in the potential barrier, while near extreme modes relate to the BHs' near-horizon geometry. Acceleration modes depend solely on the acceleration horizon and are absent in non-accelerating spacetimes. This new mode types were first identified for scalar perturbations of charged accelerating BHs [10]. Later, a similar phenomenon was also verified for scalar [14] and gravitational perturbations [17] of rotating accelerating BHs.

In this study, we investigate the origin of the acceleration modes. Given that these modes have a weak dependence on the BH's charge or rotational parameter and depend solely on the surface gravity in the Minkowski limit rather than the BH's surface gravity, we speculate that the acceleration modes may originate from purely accelerating spacetime without a BH. Notably, pure de Sitter (dS) spacetime also exhibit dS modes [27–29], which become deformed when BHs appear [8, 30–32]. Given the significant similarity between accelerating spacetime, which features an accelerating horizon, and dS spacetime, which has a cosmological horizon, it is reasonable to associate acceleration modes with their counterpart in pure accelerating spacetime.

In this study, we calculate the QNMs of massless perturbations with spin $ s\leq2 $ in purely accelerating spacetime. By taking the Minkowski limit of the master equations describing perturbations of accelerating BHs − specifically by setting the mass, electric charge, or spin to zero − we can derive the equations that govern perturbations in the empty accelerating spacetime. Two methods are employed to solve the perturbation equations. The first method follows the approach in [27], where a direct coordinate transformation shows that the perturbation equations are expressed as a hypergeometric type. QNMs can then be easily obtained by imposing appropriate boundary conditions. This method works straightforward for scalar perturbations [10]. The second approach analyzes the properties of the Minkowski limit of the Teukolsky-like equations governing various massless field perturbations of spinning accelerating BHs [24]. This approach involves identifying all singular points and demonstrating that the equations are of the Fuchsian type, with all singular points being regular. If exactly three regular singular points are present, the equations can be transformed into the standard form of the hypergeometric differential equations (HDEs) [33, 34]. By imposing appropriate boundary conditions, we can derive the QNMs. To validate this approach, we apply it to calculate dS modes and compare our results with those in [27]. We confirm that our results for dS modes of scalar, electromagnetic, and gravitational perturbations are consistent with those of previous findings. An advantage of this method is that it does not require complex coordinate transformations to convert the equations into HDEs.

The remainder of this paper is organized as follows: In Sec. II, we explicitely show the derivation of the QNMs of massless scalar perturbations in purely accelerating spacetime by solving the Minkowski limit of the Klein-Gordon (KG) equation in the background of the C-metric, employing the two methods described above. In Sec. III, we calculate the QNMs of massless perturbations with spin $ s\leq2 $ by solving the Minkowski limit of the Teukolsky-like equations in the spinning C-metric spacetime. In Sec. IV, we discuss the application of the new method to calculate dS modes of scalar, electromagnetic, and gravitational perturbations in D dimensional dS spacetime and compare the results with those in [27]. Finally, in Sec. V, we summarize the study, and in Appendix A, we introduce the basics of Fuchsian equations relevant to this study. By convention, we employ geometric units $ c=G=1 $ and the metric signature $ (-,+,+,+) $.

Initially, we examine the QNMs of scalar perturbations in purely accelerating spacetime. We consider a massless neutral scalar filed minimally coupled to gravity living in the spacetime of a C-metric. The evolution of the scalar perturbations is described by the KG equation, which has been shown to be separable through conformal transformation, as demonstrated in [10]. The C-metric describing a single BH can be expressed in terms of spherical-type coordinates as given in [20]:

where

Here $ {\cal{A}} $ is the parameter describing acceleration, while M is the mass of the BH. As $ {\cal{A}}\to0 $, this metric asymptotes to Schwarzschild metric. As explained in [35], taking the Minkowski limit $ M=0 $, the metric (1) in the region $ r<1/ {\cal{A}} $ can be transformed into the uniformly accelerated metric (also known as Rindler metric) through an appropriate coordinate transformation. Notably, $ {\cal{A}} $ is interpreted as the acceleration of a test particle located at the origin $ r=0 $.

Ref. [10] found that under the conformal transformation $ \tilde{g}_{\mu\nu}\to\Omega^2g_{\mu\nu} $, $ \tilde{\Phi}\to\Omega^2\Phi $, where $ \Omega=1- {\cal{A}} r \cos \theta $, the KG equation $ \nabla^\mu\nabla_\mu\Phi=0 $ becomes separable by choosing the ansatz

where $ \omega_{lm} $ is the quasinormal frequency, and m is the magnetic (azimuthal) quantum number. Subsequently, we omit the subscript $ lm $ for simplicity. As seen from metric (1), conical singularities appear along the axis at both $ \theta=0 $ and $ \theta=\pi $. To eliminate the conical singularity at $ \theta=\pi $, the period of the azimuthal coordinate φ can be appropriately specified as $ 2\pi C $, where $ C=1/P(\pi) $. Consequently, the magnetic quantum number m would not be an integer; instead, m must be of the form $ m=m_0 P(\pi) $ with $ m_0 $ being an integer [24], given that one has $ \mathrm{e}^{\mathrm{i}m(\varphi+2\pi C)}=\mathrm{e}^{\mathrm{i}m\varphi} $.

This leads to the following set of separated equations:

where

and

Here $ r_* $ is the tortoise coordinate, and λ is a separation constant. Taking the Minkowski limit $ M\rightarrow0 $, the separated wave Eqs. (5) and (6) governing the scalar perturbations in purely accelerating spacetime are reduced to

and

Notably, no difference is observed between m and $ m_0 $ in the Minkowski limit. The angular Eq. (11) has the same form as the equation of the Laplacian spherical harmonics $ Y_{lm} $ in spherical coordinates. Therefore, we can write the angular equation as

from which the separation constant λ is related to the eigenvalue of the spherical Laplacian operator by

We follow the steps in [27] to show that the radial Eq. (10) can be transformed into the HDE by a change of variable. Initially, by introducing two dimensionless quantities

Eq. (10) can be written as

Then, by changing the variable $ y=x^2 $, the equation is transformed into

Furthermore, we make the ansatz

where the parameters A and B are chosen as

to find that the function $ \tilde{\phi} $ must be the solution of the HDE

with the parameters α, β, and γ equal to

One can verify that Eq. (19) indeed has three regular singular points $ y=0 $, $ 1 $, and $ \infty $, which correspond to the standard form of the HDE. By contrast, Eq. (15) has three regular singular points $ x=0 $, $ 1 $, and $ -1 $, while $ \infty $ is an ordinary point. The structure of the regular singular points changes with the variable transformation from x to y. The subsequent transformation (17) is employed to obtain the standard form of the HDE. There are several possible combinations for the values of A and B, and therefore of α, β, and γ. Thus, the solutions of Eq. (A12) can be expressed in several equivalent forms. In the following, we study in detail the case $ A=\dfrac{\ell+1}{2} $ and $ B=\dfrac{\rm i \tilde{\omega}}{2} $.

Similar to dS QNMs, the QNMs are solutions to the equations of motion that satisfy the following physical boundary conditions: (i) the field is regular at the origin, and (ii) the field is purely outgoing near the acceleration horizon. Depending on the parameter values, the HDEs have different solutions [27, 36]. The regular behavior at $ r=0 $ selects the solution of Eq. (15) to be

where $ { }_2 F_1(\alpha, \beta; \gamma; y) $ denotes the hypergeometric function. To examine the behavior of the solution at $ y\to1 $ (the acceleration horizon), we utilize the linear transformation formulas for the hypergeometric functions [36]. If the quantity $ \gamma-\alpha-\beta=-{\rm i} \tilde{\omega} $ is not an integer, then the relation between two hypergeometric functions with variables y and $ 1-y $ is

where $ \Gamma(y) $ denotes the Gamma function. We can verify that the first term on the right hand side of Eq. (22) represents an ingoing wave, while the second term represents an outgoing wave. To satisfy the boundary conditions of QNMs, the first term must be discarded. This is achieved by imposing ¹

such that $ \Gamma(\gamma-\alpha) $ or $ \Gamma(\gamma-\beta) $ becomes infinity. However, from Eq. (20), if $ \gamma-\alpha-\beta=-{\rm i} \tilde{\omega} $ is not an integer, then neither $ \gamma-\alpha=(\ell+2-{\rm i}\tilde{\omega})/2 $ nor $ \gamma-\beta= (\ell+1-{\rm i}\tilde{\omega})/2 $ can be an integer. Given that $ (\ell+2)/2 $ and $ (\ell+1)/2 $ are integers or half-integers, and $ -{\rm i}\tilde{\omega}/2 $ is neither an integer nor a half-integer, their sum cannot be an integer. Hence the condition (23) cannot be satisfied if $ \gamma-\alpha-b $ is not an integer.

Instead, we must assume $ \gamma-\alpha-\beta $ is an integer. If $ \gamma-\alpha-\beta=-n_1 $, where $ n_1=1,2,3,... $ ², we can express the radial function (21) as

where $ \psi(y)={\rm d}\Gamma(y)/{\rm d}y $. Notably, the first term in the curly brackets represents an outgoing wave, while the second term represents an ingoing wave. Thus, to satisfy the boundary conditions of QNMs, we must impose the condition

to retain the outgoing wave. By combining this condition with $ \gamma-\alpha-\beta=-n_1 $, we can find the accelerating QN frequencies are equal to

which can also be expressed as

The result shows that the previous assumption is self-consistent. We can verify that our results match those of the acceleration modes for a scalar field propagating in the spacetime of the charged [10] and spinning C-metrics [14] when taking the Minkowski limit. Therefore, we conclude that the acceleration modes found for accelerating BHs originate from the empty accelerating spacetime and become deformed when BHs are present in the spacetime.

In this subsection, we derive the QNMs of the scalar field in the empty accelerating spacetime by using the established connection between HDEs and Fuchsian equations. A linear differential equation where every singular point, including the point at infinity, is a regular singularity is called a Fuchsian equation or an equation of Fuchsian type. Particularly, a Fuchsian equation with three regular singular points reduces to the HDE. Appendix A provides a brief review of the basics of Fuchsian equations relevant to this study, referring to Refs. [33] and [34]. Understanding Fuchsian equations enables us to transform an equation with three arbitrary regular singular points into the standard form of the HDE in a systematic approach.

Initially, the radial Eq. (15) can be expressed in the form of Eq. (A1) as follows:

We can readily identify the three regular singular points, which are denoted by $ a_1=0 $, $ a_2=1 $, and $ a_3=-1 $. Using Eqs. (A2) and (A3), by expanding $ p(x) $ and $ q(x) $ around these singular points, we can determine the following coefficients:

Given that $ x=\infty $ is an ordinary point, the parameters $ A_r $, $ B_r $, and $ C_r $, with $ r=1, 2, 3 $, should satisfy the constraint (A8). We can verify from these expressions that the solution satisfies the required boundary conditions.

Subsequently, from the indicial Eq. (A5), we can determine the characteristic exponents of each regular singular point, given by

One can verify that these results satisfy the constraint given by Eq. (A10). This is achieved through the transformation described in Eq. (A11), that is

where the radial Eq. (15) can be transoformed into the standard form of HDE, Eq. (A12), with the parameters given by Eq. (A13)

As discussed in the previous subsection, one of the solutions of Eq. (15) that is regular at $ r=0 $ is given by

As previously described, we need to change the variable of hypergeometric function from y to $ 1-y $ to analyze the behavior of the solution at the acceleration horizon. If the quantity $ \gamma-\alpha-\beta=-{\rm i} \tilde{\omega} $ is not an integer, then we have

The second term on the right hand side of Eq. (37) represents an outgoing wave, which satisfies the boundary condition at the acceleration horizon. To achieve this, we impose

One can verify that this condition cannot be satisfied, indicating that we should instead assume that $ \gamma-\alpha-\beta $ is an integer. If $ \gamma-\alpha-\beta=-n_1 $, where $ n_1=1,2,3,... $, the solution can be expressed as

Notably, the first term in the curly brackets represents an outgoing wave, while the second term represents an ingoing wave. Thus, to satisfy the boundary conditions of QNMs, we must impose

Combining this condition with $ \gamma-\alpha-\beta=-n_1 $, the accelerating QN frequencies are equal to

which is equivalent to Eq. (27). Thus, we derive consistent results from two distinct methods, although the radial Eqs. (21) and (36) differ in these cases.

In this section, we calculate the QNMs of massless perturbations of any spin in the accelerating spacetime. The master equation describing massless perturbations of the spinning C-metric due to fields of any spin has been derived in [24] by following the approach of Teukolsky [37] within the context of the Newman-Penrose formalism [38]. Notably, similar to the Teukolsky equation for the Kerr BH, the master equation can be separated into its radial and angular parts because the spinning C-metric is of Petrov type D. By taking the Minkowski limit, we can obtain the equations capturing the dynamics of scalar, Dirac, electromagnetic, and gravitational perturbations in the empty accelerating spacetime.

The metric of spinning C-metric expressed in Boyer-Lindquist-type coordinates $ (t, r, \theta, \phi) $ is given by [39]

where a is the rotation parameter, and the functions Ω, Σ, P, and Q are defined by

The master equation describing the dynamics of a massless field with spin weight s in this spacetime is given by

where $ \Psi_2=-(1+{\rm i}a {\cal{A}})M\Omega^3/(r-{\rm i}a\cos \theta)^3 $ is the nonvanishing Weyl scalar in the spinning C-metric background, and a "connection vector" is introduced by

Ref. [24] demonstrated that the master Eq. (44) admits separable solutions of the form

where $ \omega_{lm} $ is the wave frequency, and m is the azimuthal number. Additionally, we omit the subscript $ lm $ for simplicity in the following discussions. The radial equation is then expressed as

with

where K is the separation constant, and $ \sigma_0=(1+a^2 {\cal{A}}^2) $. By introducing

and the "tortoise" coordinate $ r_* $, where

the radial equation can be transformed into the one-dimensional Schrödinger-like equation

with the potential

where

Moreover, the angular equation is given by

with

where $ Y(\theta)= \sqrt{P}S(\theta) $, $ z=a\omega+s M {\cal{A}} $, and $ w=-m+ 2sM {\cal{A}} $.

Taking the Minkowski limit, $ M=a=0 $, the angular equation reduces to

Comparing this equation to that of the spin-weighted spherical harmonics $ {}_sY_{\ell m} $ [40]

we can immediately find that

Thus, the separation constant of the angular equation in the Minkowski limit is known, and the QN frequencies can be obtained solely by considering the radial equation. In this limit, the radial equation (51) simplifies to

which in terms of the dimensionless quantities (15) can be expressed in the form of Eq. (A1)

In the following analysis, we show that Eq. (60) is a Fuchsian equation with three regular singular points $ 0 $, $ 1 $, and $ -1 $, which can be readily transformed into the standard form of the HDE. From Eq. (A1), we know

We can verify that the three points $ 0 $, $ 1 $, and $ -1 $ are the regular singular points of the equation, and the infinity is an ordinary point. Furthermore, we can expand $ p(x) $ and $ q(x) $ around each regular singular point as Eq. (A2) and Eq. (A3), respectively, with the expansion coefficients given by

and

The three regular singular points are labeled as $ a_1=0 $, $ a_2=1 $, and $ a_3=-1 $. Subsequently, from the indicial Eq. (A5), we can determine the characteristic exponents of each regular singular point, which are given by

Applying the transformation given in Eq. (A11), which is expressed as

the radial Eq. (60) can be transformed into the standard form of HDE Eq. (A12), with the parameters given by Eq. (A13):

The subsequent analysis is parallel to the scalar field case; therefore, we present only the main steps. Imposing regularity at the origin of the accelerating spacetime selects the hypergeometric function as the appropriate solution to the HDE. To examine the behavior of this solution at the acceleration horizon, namely, $ y=1 $, we need to change the variable in the hypergeometric function from y to $ 1-y $. Depending on the values of the parameter $ \gamma-\alpha-\beta $, the resulting expression can have different forms, corresponding to either Eq. (37) or (39). To satisfy the boundary condition that the field is purely outgoing near the acceleration horizon, condition (25) must also be imposed, leading to

We obtain the same QN frequency as that in the previous section. Notably, the spectra are purely imaginary and independent of the spin of the perturbations. Moreover, the result for $ s=-2 $ matches that of the acceleration modes of gravitational perturbations in the spinning C-metric [17] after taking the Minkowski limit.

Similar to the analysis of the scalar field in the previous section, the radial equation (60) can be converted into the standard form of the HDE through an appropriate transformation; however, this process is more complicated than the case of the scalar field. A direct approach is to make the change of variable $ y=x^2 $, which leads to

In the case of the scalar fields $ s=0 $, via the change of variable $ y=x^2 $, the regular singular points of the equation change from $ (0, 1, -1) $ to $ (0,1, \infty) $. The equation can then be converted into the form of the HDE. However, for $ s\ne0 $, due to the appearance of the $ \sqrt{y} $ term in the equation, the point $ y=0 $ is no longer a regular singular point, further complicating the situation. Nevertheless, the similar issue was encountered in dS spacetime and a coordinate transformation was proposed [41]:

which may be useful in this study. Applying this change of variable, the radial equation (60) becomes

Notably, the regular singular points for this equations are $ (0,1,-1) $, making it natural to transform the equation into the standard form of the HDE. However, with the new coordinate z, the positions of the origin of the accelerating spacetime and the acceleration horizon are interchanged, which complicates subsequent analysis. To address this issue, we introduce

such that the origin remains at $ y=0 $ and the acceleration is at $ y=1 $. Thus, Eq. (73) becomes

Consider the ansatz for $ H(y) $,

where

we get that the function $ \tilde{H}(y) $ must be a solution of the HDE

with the parameters given by

There are four equivalent choices for A and B, and the one with $ A=\ell+1 $ and $ B=\dfrac{1}{2}(-s+{\rm i} \tilde{\omega}) $ yields exactly the same parameters as those of the HDE in Eq. (68). Given that the two HDEs are identical, their QNM spectra are also expected to be identical.

In this section, we apply the transformation techniques of HDEs and Fuchsian equations with three regular singular points to calculate the dS modes, demonstrating the validity and convenience of this method. We consider scalar, electromagnetic, and gravitational perturbations in D dimensional dS spacetime. As shown below, our results are identical with those presented in [27].

For simplicity, we consider only a massless scalar field minimally coupled to gravity living in the pure dS spacetime. The QN frequencies are represented in the radial part of the master equation [42], which is given by

In this and in the subsequent subsections, $ x=r^2/L $ and $ \tilde{\omega}=\omega L $, where L is the radius of the dS space and D is the spacetime dimension. We can observe that Eq. (79) has three regular singular points, which are labeled by $ a_1=0 $, $ a_2=1 $, and $ \infty $. By rewriting the radian equation in the form of Eq. (A1), we can expand $ p(x) $ and $ q(x) $ around $ a_1 $ and $ a_2 $ as Eq. (A2) and Eq. (A3), with the expansion coefficients given by

and

From Eqs. (A5) and (A6), which are identical, we determine the characteristic exponents at each regular singular point, which are given by

Moreover, through the transformation, Eq. (A11), the radian equation can be converted into the standard HDE with the parameters given by Eq. (A13). Comparison with the scalar field analysis in dS spacetime [27] shows that both the transformations connecting the radial equation to the HDE and the resulting HDEs are identical (see Eqs. (46)–(48) of [27]). This clearly implies that the QNM spectra in both cases are identical.

For electromagnetic fields moving in the D dimensional dS spacetime, the separation of the equations of motion has been studied in [43]. Depending on the choices of gauge for the electromagnetic field, two sets of physical solutions to the field equations arise, referred to as physical modes I and II. Correspondingly, there are two sets of radial equations and QNM spectra [27].

The radial equation of physical mode I is given by

Notably, this equation also has three regular singular points $ a_1=0 $, $ a_2=1 $, and $ \infty $. Consistent with the previous approach, we can express this equation in the form of Eq. (A1) and expand $ p(x) $ and $ q(x) $ around $ a_1 $ and $ a_2 $ as Eqs. (A2) and (A3), with the expansion coefficients given by

and

From indentical Eqs. (A5) and (A6), the characteristic exponents at each regular singular point are given by

We compare these results with those in [27], demonstrating that both the transformations connecting the radial equation to the HDE and the resulting HDEs are identical (see Eqs. (7)–(10) of [27]). This implies that the resulting QNM spectra are also identical.

The radial equation of physical mode II is given by

Similar to physical mode I, this equation has three regular singular points $ a_1=0 $, $ a_2=1 $, and $ \infty $. Consistent with the previous approach, we can express the equation in the form of Eq. (A1) and then expand $ p(x) $ and $ q(x) $ around $ a_1 $ and $ a_2 $ as Eqs. (A2) and (A3), with the expansion coefficients given by

and

From indicial Eqs. (A5) and (A6), the characteristic exponents at each regular singular point are given by

Similar to the physical mode I, both the transformations connecting the radial equation to the HDE and the resulting HDEs are exactly the same as those in [27] (see Eqs. (22)–(24) therein). This implies that the resulting QNM spectra would be identical.

In dimensions higher than four, the gravitational perturbations of D dimensional dS spacetime are classified into three types based on their tensorial behavior on the $ (D-2) $ sphere: tensor, vector, and scalar types [44]. By contrast, in four dimensions, gravitational perturbations consist of only the tensor type. The radial equations of these perturbations in this background can be uniformly expressed as [45]

where $ \tilde{\alpha} $ and $ \tilde{\beta} $ are introduced by

and

This equation has three regular singular points $ a_1=0 $, $ a_2=1 $, and $ \infty $. Consistent with previous approaches, we can express the equation in the form of Eq. (A1) and then expand $ p(x) $ and $ q(x) $ around $ a_1 $ and $ a_2 $ as Eqs. (A2) and (A3), with the expansion coefficients given by

and

From indicial Eqs. (A5) and (A6), the characteristic exponents at the three regular singular points are given by

Additionally, both the transformations connecting the radial equation to the HDE and the resulting HDEs are exactly the same as those in [27] (see Eqs. (34)–(36) therein). Therefore, the QNM spectra in these two cases would also be identical.

This study investigated the QNMs of perturbations with spin $ s\leq2 $ in purely accelerating spacetime to examine the origin of the acceleration modes recently discovered for scalar and gravitational perturbations of accelerating BHs [10, 14, 17]. The perturbation equations were obtained by taking the Minkowski limit of the master equations describing various types of perturbations of accelerating BHs. We employed two methods to obtain the QNM spectra by solving the radial part of the perturbation equations. The first method involves a change of variable to convert the radial equation into the form of the standard HDEs. This approach is effective for scalar perturbations, such as those describing dS mode in pure dS spacetime [27]. However, for other types of perturbations, the operation becomes more complicated. The second method utilizes the simple connection between Fuchsian equations with three regular singular points–of which the radial perturbation equations are an example– and HDEs. Both methods were shown to yield identical QNM spectra.

Notably, the resulting QNM spectra of the purely accelerating spacetime are imaginary and independent of the spin of the perturbations and match those of the acceleration modes of accelerating BHs after taking the Minkowski limit. This correspondence implies that the acceleration modes originate from the purely accelerating spacetime and become deformed in the presence of BHs. A similar situation arises in the case of dS modes [8, 30–32], which were first calculated in pure dS spacetime and then identified within the scalar QNMs of Schwarzschild-dS BHs [30]. Additionally, we applied the second method to compute the QNMs of scalar, electromagnetic, and gravitational perturbations in D dimensional dS spacetime and obtained results identical to those reported in [27], ultimately verifying the validity of this method.

Given that acceleration modes are closely linked to the acceleration parameter and exhibit robustness to BH parameters, investigating their observability may reveal signatures of acceleration of BHs within GW signals, potentially supporting the detection of moving or accelerating BHs.

We gratefully acknowledge Wei Xiong for his valuable assistance during the initial stages of this study.

Given a second-order linear ordinary differential equation

where $ a_r $ $ (r=1,2, \cdots, n) $ and $ \infty $ are regular singular points. Moreover, given that $ a_r $ is a regular singular point, it must also be a first-order pole of $ p(x) $. Therefore, we have

where $ A_r $ is the residue of $ p(x) $ at $ x=a_r $, and $ \varphi(x) $ is analytical in complex plane. When $ x\rightarrow \infty $, $ \varphi(x) $ approaches zero, as $ x=\infty $ is the regular singular point of the equation. This implies that $ \varphi(x) $ can be set to zero.

Based on similar analysis, $ q(x) $ can be expanded as

with

When considering the series solution of Eq. (A1), the coefficient of the lowest power of x satisfies the indicial equation. The roots of this equation determine the characteristic exponents, which in turn specify the form of the solution. For each regular singular point $ a_r $ ($ a_r\ne\infty $) of Eq. (A1), the corresponding two characteristic exponents satisfy the following indicial equation:

Moreover, if the point at infinity $ x=\infty $ is a regular singular point, its characteristic exponents obey the equation

From these two equations, we can obtain a key constraint in this case:

For example, for the HDE, the sum of all characteristic exponents is one, as $ n=2 $.

However, if $ x =\infty $ is an ordinary point rather than a singular point of Eq. (A1), the parameters $ A_r $, $ B_r $ and $ C_r $ satisfy the following equations:

For Fuchsian equations with three regular singular points, denoted by a, b, and c ($ \neq\infty $, thus $ \infty $ is an ordinary point), the equations can expressed in the following form based on the preceding relations:

where ($ \alpha_1 $, $ \alpha_2 $), ($ \beta_1 $, $ \beta_2 $), and ($ \gamma_1 $, $ \gamma_2 $) are the corresponding characteristic exponents of the regular singular points a, b, and c. From Eqs. (A5) and (A8), we find

Through the following transformation:

Eq. (A9) can be converted into the standard form of HDE,

whose parameters are related to those of the Fuchsian equations, given by

The detailed derivation of these equations using Riemann's P-functions can be found in [33].