Quasinormal modes as a distinguisher between general relativity and f(R) gravity: charged blackholes
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Abstract
The ringdown phase of blackhole perturbations is governed by the quasinormal modes (QNM) and offer valuable insight into the nature of the objects emitting them, raising an interesting question: whether QNMs can be used to distinguish between theories of gravity? We construct a consistency test of general relativity (GR) which enables one to distinguish between general relativity and a specific class of modified theories of gravity: f(R). We show that an energetic inequality between scalar (polar) and vector (axial) type gravitational perturbations will exist for Reissner–Nördstrom solutions of GR  using which we find a novel method of determining the charge of a nonspinning black hole in GR. We then show that there will be a further energetic difference for charged black holes in f(R). Finally, we utilize this extra difference to construct a parameter to quantify deviation from GR.
1 Introduction
General relativity has been an unprecedented success in describing the majority of astrophysical phenomena. Its recent successes include the direct detection of gravitational waves from the binary black hole and binary neutron star mergers by the LIGOVIRGO collaboration [1, 2, 3]. These detections have raised questions about the possibility of obtaining the severe constraints on the degree of validity of general relativity in the strong gravity regimes.
General relativity is far from being a complete theory. It predicts blackholes, with a singularity at their centers, and represents a breakdown of general relativity as the classical description cannot be expected to remain valid in the extreme condition near the singularity. Physically, the singularity of the stationary vacuum isolated blackhole solutions is connected with the infinite growth of the curvature invariants, such as the Kretschmann invariant.
Several modifications to general relativity have been proposed to remove the singularity. In general, the modifications contain higher derivative and nonlocality [4, 5, 6, 7, 8, 9, 10]. The question that naturally arises is, how to distinguish between general relativity and modified theories of gravity? Are there any unique signatures for the modified gravity theories that can potentially be detected in the terrestrial (like Einstein Telescope) or space based observations (like eLISA)?
Astrophysical blackholes that are in the centers of the galaxies or formed due to the collapse interact with the external surroundings. Thus, these blackholes are perturbed continuously compared to the exact solutions in general relativity or modified theories of gravity. The perturbed blackhole responds by emitting gravitational waves [11, 12, 13, 14, 15, 16]. More specifically, the response consists of a broadband burst, followed by the quasinormal mode ringing [17, 18, 19, 20, 21, 22]. Interestingly, the quasinormal modes  damped resonant modes of blackholes  are independent of the nature of the perturbation and, only depends on the blackhole parameters like mass M, charge Q and angular momentum a [11, 12, 13, 14, 15, 16].
Many modified theories of gravity predict an extra degree of freedom, besides the two transverse modes, in the gravitational waves [2, 23, 24, 25, 26, 27]. The question that raises is: whether the extra mode leaves any signatures in the quasinormal mode ringing and provide a new way of distinguishing general relativity from modified theories of gravity? Previous analysis of Reissner–Nördstrom black holes in f(R) theories were done in [28, 29, 30].
Recently the current authors showed explicitly while the two types of black hole perturbations  scalar (polar) and vector (axial)  share equal amounts of emitted gravitational energy in General relativity, in f(R) theories they do not share same amounts of emitted gravitational energy. The current authors also identified a parameter to distinguish between general relativity and f(R) [31].
It is important to note that the astrophysical blackholes probably neutralize their electric charge rather quickly and are expected to remain nearly neutral. However, the reasons for the doing the detailed analysis for Reissner–Nördstrom blackhole are: (1) like Kerr, unlike Schwarzschild, Reissner–Nördstrom has two parameters to describe the blackhole. (2) Unlike Kerr, Reissner–Nördstrom is spherically symmetric, and it gives critical insight into obtaining a quantifying tool for deviations from general relativity. (3) The timescales involved in charge neutralization is usually longer than the timescale in which blackhole forms in an NS–NS merger [3]. Parameter estimation of these black holes are done by matching the observed waveforms with available simulation templates. However, due to the immense computing power already required to estimate such parameters, currently there is no template for obtaining the charge these black holes might possess. In this work, we show that QNMs at the epoch of the formation of such blackhole will contain signatures of the charge of these blackholes. More specifically, we show that it is possible to estimate the charge of a black hole from the energetics of quasinormal modes. We also explicitly obtain a measure for deviations from general relativity using the energetics of quasinormal modes in f(R) theories.
In Sect. 2, we formulate the problem of a charged blackhole perturbations in general relativity and illustrate a novel method of finding the charge of a Reissner–Nördstrom blackhole  an unique method proposed in this work. In Sect. 3, we obtain an expression for the radiated energy–momentum of perturbation for a charged blackhole spacetime. In Sect. 4 we extend the gauge invariant analysis of Sect. 2 for a charged black hole in f(R) theory and obtain a new quantifying measure of the energetic difference due to the presence of the massive scalar mode.
In this work the metric signature we adopt is \((,+,+,+)\) and we set \( G=c=1 \), \( 4\pi \epsilon =1 \), implying that a point charge Q has a Coulomb potential \(\frac{Q}{r}\). We use Greek letters to refer to 4dimensional spacetime indices \((0 \ldots 3)\), lower Latin indices refer to the orbit space coordinates (0, 1), and upper Latin for the two angular coordinates (2, 3). The various physical quantities with the overline refer to the values evaluated for the spherically symmetric background, whereas superscript (n) represents the nth order perturbed quantity.
2 Perturbations of RN spacetime in general relativity
In this section, we discuss the formalism to obtain the linear order perturbations about Reissner–Nördstrom spacetime in general relativity (GR). The discussion in this section follows the formalism developed by Kodama and Ishibashi [38, 39, 40]. In Sect. 2.2, we show that for a nonspinning blackhole in GR, any energetic difference would imply that the blackhole possesses charge.
2.1 Dynamics of perturbation
2.2 Difference in radiated energy flux at infinity between scalar and vector perturbations
In the case of Schwarzschild, only gravitational (scalar and vector) modes exist. The scalar and vector modes are related. Also, the effective potentials they satisfy are associated. Thus, gravitational radiation from the perturbed Schwarzschild blackhole, as detected at asymptotic spatial infinity, have an equal contribution from the scalar and vector modes [15, 31].
In the case of RN, as mentioned above, along with the two gravitational modes, two types of electromagnetic perturbations are also present. As discussed in Appendix A, it is the superposition of the gravitational and electromagnetic perturbations that are related. Thus, the two perturbations cannot be treated separately; perturbation of one will affect the other [15, 38, 39, 40, 43]. In other words, an incident wave from radial infinity in the RN spacetime that is purely gravitational will result in a scattered wave which has both gravitational and electromagnetic component.
Gunther [43] quantified the fraction of incident gravitational radiation converted into electromagnetic radiation due to the scattering process. Defining the conversion factors \( C_{V/S} \) as the fraction of the incident gravitational energy flux of vector/scalar type, converted into electromagnetic energy flux, [43] shows that \( C_S\ge C_V \). The equality holds for \( Q=0 \) for which \( C_S=C_V=0 \). Hence, in RN, the scattered gravitational radiation due to a purely gravitational incoming wave will have less contribution from scalar perturbations compared to vector perturbations.
3 Perturbations in curved Ricci flat spacetimes for f(R) theories
3.1 Linear order perturbation equations
3.2 Energy–momentum pseudotensor of perturbation
4 Quantifying tool for RN blackholes
4.1 Modified dynamics
4.2 An extra difference in energy densities
As seen in our earlier study for the Schwarzschild spacetime [31], modification to the Einstein–Hilbert action and its corresponding perturbed equations of motion only modifies the scalar sector of the perturbation leaving the vector modes unchanged. In other words, the part of the perturbed energy density of the scalar sector leaks into this new massive mode, leading to a decreased energy density in the scalar modes. Due to the massive nature of the extra mode, excitations of the field around the black hole do not travel to asymptotic infinity and cannot be detected. However, the relative difference between the energy densities of scalar and vector modes can be used as an indirect probe to determine if such fields exist in nature.

Measure of the charge of a black hole obtained in general relativity from energetic difference of the massless modes and from the real and imaginary parts of the quasinormal frequencies are equal.

In f(R) theories, the extra massive field will lead to further energetic difference of the massless modes. Thus a measure of charge obtained from the energetic difference will be different from the one obtained from the quasinormal frequencies which only brings information about the three parameters mass, charge, and angular momentum  owing to nohair like theorems holding for f(R) theories [50] and for Lovelock theories [51].

The difference in the measured charge (the magnitude of difference depending on \( \alpha \)) from the two methods should in principle allow us to detect and constraint any deviations from general relativity. Note that expression (50) implies that deviation from GR is better detectable \( \left( \sim \frac{1}{r_H^2}\right) \) from the ringdown of smaller blackholes.
5 Conclusions and discussions
As we have shown, the energetic equality between scalar and vector type gravitational perturbation is broken in the presence of charge in a blackhole. Hence, scattered gravitational radiation of the scalar and vector types carry different energies with them to asymptotic infinity  the extent of the difference depends on the magnitude of the charge. In other words, scalar type perturbations carry less energy to an observer at infinity compared to the vector type. This is because of the coupled nature of gravitational and electromagnetic perturbations in a Reissner–Nördstrom spacetime and the fact that the degree of coupling depends on the perturbation type.
On a modification to gravity like f(R) , there comes an intrinsic extra massive degree of freedom in addition to the two massless ones. The massive mode couples itself only to the scalar type. We have shown explicitly that this massive mode can be excited relatively easily around a blackhole as compared to flat space. Hence, as seen in the RHS of Eq. (43), the scattered gravitational wave will have leakage of energy to the massive mode owing to the ease of its excitation and a coupling to the scalar type perturbation.
Using Isaacson’s prescription of calculating the energymomentum pseudotensor of blackhole perturbations, we have obtained an accurate and robust method of estimating the difference in the radiated energies of the scalar and vector modes due to modifications to general relativity. The difference in the scattered energies of the massless modes for a charged blackhole will then have contributions from both the charge and the presence of the massive mode  which will result in the measure of charge using energetic difference (following Fig. 2) to be larger than the actual charge. Using the real and imaginary parts of the quasinormal frequencies the charge of the blackhole can be obtained. Although astrophysical blackholes are expected to be chargeneutral, the timescales involved in charge neutralization is usually longer than the timescale in which blackhole forms in an NSNS merger. Thus, the presence of charge after the merger can also cause a difference in radiated energies of the massless modes. It is then possible that the charge of a blackhole is so small that \( \Delta _{GR} \) becomes comparable to \( \Delta _{mod} \). But even in the tiny charge case, there will always be a difference in the two measurements (frequency and energetics) of the blackhole charge for a modification to gravity in the form of an intrinsic extra degree of freedom. This difference, if any, will constrain deviations from general relativity. However, it is necessary to note that the estimation of charge from the real and imaginary parts of the quasinormal frequencies involve inclusion of the charge parameter in the template waveforms that are matched with the data to estimate all the parameters of the merging phenomenon. Currently, because of the immense computing power already required, templates taking charge into account are absent. However, better sensitivity and efficient algorithms of detection will help in expanding the parameter space for charge as well as modified gravity theories.
Although in this work, we have focused on spherically symmetric spacetimes with the matter, the analysis in Sect. 3.2 holds for arbitrary curved spacetimes, including Kerr and Kerr–Newmann. In the case of Kerr, last two terms in Eq. (37) vanishes. It is possible to obtain a measure of the energy leakage from the massless to the massive mode as like in Eq. (50). However, owing to the reduction in the symmetry of Kerr compared to Schwarzschild or Reissner–Nördstrom spacetimes, the breaking up of a general perturbation into scalar and vector types (which was possible under rotations in \( \mathscr {S}^2 \) in a spherically symmetric spacetime) becomes a nontrivial problem and has so far only been done in special cases like slow rotation approximation [52, 53]. We hope to address the perturbation of Kerr blackholes in f(R) theories in a later paper.
The analysis in the paper has focused on a specific form of modified theories of gravity, and we plan to extend the analysis to other theories of gravity for spherically symmetric spacetimes.
Notes
Acknowledgements
The authors would like to thank C. P. L. Berry and A. Nagar for clarifications through email. SB is financially supported by the MHRD fellowship at IISERTVM and would like to thank IIT Bombay for hospitality. The work is supported by DSTMax PlanckIndia Partner Group on Gravity and Cosmology.
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