Abstract
Over the last two decades, motivations for modified gravity have emerged from both theoretical and observational levels. f(R) and ChernSimons gravity have received more attention as they are the simplest generalization. However, f(R) and ChernSimons gravity contain only an additional scalar (spin0) degree of freedom and, as a result, do not include other modes of modified theories of gravity. In contrast, quadratic gravity (also referred to as Stelle gravity) is the most general secondorder modification to 4D general relativity and contains a massive spin2 mode that is not present in f(R) and ChernSimons gravity. Using two different physical settings—the gravitational wave energyflux measured by the detectors and the backreaction of the emitted gravitational radiation on the spacetime of the remnant black hole—we demonstrate that massive spin2 mode carries more energy than the spin0 mode. Our analysis shows that the effects are pronounced for intermediatemass black holes, which are prime targets for LISA.
Introduction
In quantum gravity, it is impossible to localize an event with a precision smaller than the Planck length (\(\ell _{\textrm{Pl}}\))^{1,2,3,4}. From the hoop conjecture and the uncertainty principle, we can deduce the existence of a Plancksized ball^{5}. Thus, the operational significance of the concept of spacetime points is lost^{6,7,8,9}. Most approaches to quantum gravity incorporate the Planck length by considering extended structures, rather than point particles, as fundamental blocks^{10,11}. The Generalized Uncertainty Principle (GUP) is a phenomenological approach that introduces modifications to the Heisenberg uncertainty principle in the ultraviolet regime and studies its consequences^{12,13}. Kempf et al. considered the following simplest modification to the canonical commutation relation^{12}:
\(\beta\) is a parameter characterizing the GUP whose value needs to be fixed by observations. Since the above GUP is nonrelativistic, it is impossible to compute GUP corrections in relativistic field theories. Recently, Todorinov et al.^{14} extended the above GUP to a generic class of covariant GUPs:
and studied the phenomenological features of such GUPs for scalar, spinor, and vector fields^{15,16}. \(\gamma _0\) is a positive numerical constant to be fixed by observations. There exists a correspondence between the GUPmodified dynamics of a massless spin2 field and quadratic gravity (QG) with suitably constrained parameters^{17}. Specifically, the authors showed that the 4D gravity action^{18,19}
is a classical manifestation of the imposition of a momentum cutoff at the quantum gravity level when \(\alpha =2\beta =\gamma /\kappa ^{2}\). QG or Stelle gravity—unlike f(R)—has extra massive spin2 and spin0 modes^{19}. Intriguingly, for this class of Stelle theories, the masses of the spin0 and spin2 modes coincide \((1/\sqrt{2 \gamma })\). The issue of unitarity in these theories is not completely settled yet^{20,21,22,23} and the massive spin2 mode may be of ghost nature. See, for instance, the fakeon approach for the same^{24,25}.
Given that the Stelle gravity is the most general QG in 4D, we ask the following questions: For identical masses, what is the role of the massive spin2 mode compared to the spin0 mode? What are the leading order corrections of QG compared to f(R) gravity? This work addresses these questions by evaluating the corrections to the radiation in QG.
Over the last two decades, motivations for modified gravity have emerged from both theoretical and observational levels^{26,27,28,29,30}. Since f(R) and ChernSimons gravity are the simplest generalization, they have received more attention. However, f(R) and ChernSimons gravity contain only the additional scalar degree of freedom and hence do not contain other modes of modified theories of gravity^{31}. We explicitly show that the massive spin2 mode carries more energy than the spin0 mode using two different physical settings: the energy flux of the gravitational waves, measured by an asymptotically placed gravitational wave detector, and the backreaction of the emitted gravitational radiation on the spacetime of the remnant black hole. Thus, we show that the leading order gravitational radiation correction in QG is linear in the coupling constant compared to f(R), where the corrections are quadratic in flat spacetime^{32}.
To show this explicitly, we start with the GUPinspired version of QG action (2), where^{17}
Varying action (2) with respect to the metric leads to:
Linearizing the field equation (3) about the Minkowski spacetime (\(\eta _{\mu \nu }\)), we get,
where \(h_{\mu \nu }\) is the metric perturbation, and \(\epsilon\) is a bookkeeping parameter. This leads to the following linearized equations:
where \({\bar{\Box }} \equiv \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\) and
As expected in higherderivative gravity theories, Eq. (5) contains fourthorder derivatives whose trace is given by:
To separate the contribution of the different spincomponents from linearized field equation (5), we use the following ansatz for the metric perturbations (See “Quadratic gravity as GR “minus” massive gravity” Section of the Supplementary material^{33} for a detailed calculation in Ricciflat spacetime):
where \(\psi = \eta ^{\alpha \beta } \psi _{\alpha \beta }\) and \(C_1, C_2\) are arbitrary constants, and \({\hat{R}}^{(1)}_{\mu \nu }=R^{(1)}_{\mu \nu }\frac{1}{4} \eta ^{\mu \nu } R^{(1)}\) is the traceless part of \(R^{(1)}_{\mu \nu }\). To our knowledge, this is the first time an explicit separation of metric fluctuations is used in this context, and we would like to emphasize the following points: First, in general relativity (GR), the constants \(C_1\) and \(C_2\) vanish because the linearized field equations only contain massless spin2 (graviton) mode. Thus, in GR, \(\psi _{\mu \nu }\) reduces to the familiar tracereversed metric perturbations. Second, in the case of Starobinsky model, the field equations contain additional contribution from the massive spin0 mode \((R^{(1)})\) only, hence \(C_1=6\gamma\) and \(C_2=0\)^{32,34}. Third, the existence of the massive spin2 ghost degrees of freedom may suggest a potential pathology due to Ostrogradsky instability^{18}, in the present context it is possible to avoid such a pathology by treating the massive and massless spin2 modes as a single structure^{35}. Lastly, like in GR, we use the deDonder gauge on the residual massless graviton mode (\(\psi _{\mu \nu }\)) to restrict the gauge freedom:
Substituting Eq. (9) in Eqs. (6,7) and using the gauge conditions (10), we get:
where we set \(C_1=2\gamma\) and \(C_2=4\gamma\). Taking cognizance of Eqs. (5, 11) and (8), we get the propagation equation for the graviton, the massive spin2 and spin0 modes as,
(See “Reduced equations of motion in Ricciflat spacetimes” Section of the supplementary material^{33} for a detailed derivation of the propagation equations in a Ricciflat background). Thus, the theory (2) is described by a massless graviton (\(\psi _{\mu \nu }\)); a massive spin2 (\({\hat{R}}^{(1)}_{\mu \nu }\)) and a massive spin0 (\({R}^{(1)}\))^{17,18,19,36} (The massive spin2 mode can be further decomposed into two tensor (helicity2) modes, two vector (helicity1) modes, and scalar (helicity0) mode, whereas the massless spin2 graviton gives rise two tensor (helicity2) modes and the massive spin0 particle has one scalar (helicity0) mode. The different helicity modes can, in principle, be mapped to the six metric degrees of freedom denoting the six polarization modes of the gravitational waves^{36,37,38,39}). The positivity of the parameter \(\gamma\) ensures that the massive modes are nontachyonic. Repeating the analysis for the Ricciflat background by using the following ansatz:
the equations of motion for the massless (12) and massive spin2 modes (13) in the transverse traceless gauge \(\left( {\bar{\nabla }}^\mu \psi _{\mu \nu }=0, ~{\bar{g}}^{\mu \nu } \psi _{\mu \nu }=0\right)\) are:
where \({\bar{g}}_{\mu \nu }\) is the background metric, \({\bar{R}}_{\mu \alpha \nu \beta }\) is the background Riemann tensor, \({\bar{\Box }}\equiv \bar{\nabla }_\sigma \bar{\nabla }^\sigma\) with \(\bar{\nabla }^\sigma\) being the covariant derivative due to the background spacetime and the traceless tensor \({\hat{R}}^{(1)}_{\mu \nu }\) is
The propagation of the massive spin0 mode is still governed by Eq. (14), with the abovedefined D’Alembertian operator. The mass degeneracy between the two massive (spin0 and spin2) modes demonstrates that they are not completely independent and are related by linearized Bianchi identities:
Having separated the metric fluctuations into massless and massive modes in Ricci flat background, we now evaluate the energy and momentum carried by the gravitational waves in degenerateStelle gravity. To go about that, we expand the field equation to secondorder in \(\epsilon\):
where \(\bar{{\mathscr {G}}}_{\mu \nu }\) represents the background quantity, \(\delta {\mathscr {G}}_{\mu \nu }\) are linear in perturbations (\(h_{\mu \nu }\)) and \(\delta ^2 {\mathscr {G}}_{\mu \nu }\) are quadratic in \(h_{\mu \nu }\). Through secondorder perturbations, Isaacson established a procedure to obtain an effective stressenergy tensor for gravitational radiation^{40,41}. Specifically, the effective stressenergy tensor of the emitted gravitational waves is obtained by averaging over a lengthscale l such that \(\lambda \!\!\!\!{}^{}\ll l \ll \Lambda\), where \(\lambda \!\!\!\!{}^{}\) is the wavelength of the fluctuations and \(\Lambda\) is the system size. The short wavelength components will be averaged out, yielding a gaugeinvariant measure of the effective gravitational wave (GW) stressenergy tensor^{42}:
In the Ricciflat background, we get
where \({\mathscr {A}}_{\mu \nu },{\mathscr {B}}_{\mu \nu },{\mathscr {C}}_{\mu \nu },{\mathscr {D}}_{\mu \nu }\) are related to the background Riemann tensor (and are explicitly given in “Coefficients in the effective GW stressenergy tensor” Section of the supplementary material^{33}). This is the first key result of this work, regarding which we would highlight the following points: First, in the Minkowski limit (as in the case of GW detectors), \({\mathscr {A}}_{\mu \nu },{\mathscr {B}}_{\mu \nu },{\mathscr {C}}_{\mu \nu },{\mathscr {D}}_{\mu \nu }\) vanish and \(t_{\mu \nu }^{\textrm{GW}}\) is proportional to partial derivatives of \(\psi ^{\rho \sigma }\), \({\hat{R}}^{(1)}_{\rho \sigma }\) and \(R^{(1)}\). Second, the first term within the triangular bracket gives the dominant contribution—the contribution of the graviton mode as in GR^{40}. However, the crucial difference is the dominant contribution of the massive spin2 mode. The massive spin2 mode contribution is proportional to \(\gamma\), whereas the massive spin0 mode contribution is proportional to \(\gamma ^2\). Thus, the above expression explicitly shows that the massive spin2 mode carries more energy than the massive spin0 mode. Third, while we have used Isaacson’s approach to obtain the stresstensor, other approaches also give similar results^{43}. Fourth, the leading order contribution of the massive spin2 mode is opposite to that of the graviton mode. Lastly, the corrections by QG to GR are much larger than f(R) gravity^{32,44}. Consequently, our study demonstrates that the f(R) theories overlook crucial information concerning the massive spin2 mode.
In what follows, we use the GW stressenergy tensor (22) to examine the effect of the massive spin2 mode under two distinct physical settings. First, we investigate the energy flux of GWs as measured by the GW detectors at asymptotic infinity. We then estimate the change in the spherically symmetric metric caused by the backreaction of the emitted GWs near the horizon.
Gravitational wave energy flux
The energy of the gravitational wave within a spatial volume V is^{45,46}
Using the stresstensor conservation equation (\(\partial _\mu t^{\mu \nu }_{\textrm{GW}} = 0\)) for the faraway observer, the power carried by the gravitational waves is^{43,46}
where S is the surface enclosing the volume V and \(n_i\) is the unit outward normal to S. The negative sign signifies that an outward propagating gravitational wave carries away energy from the source. Thus, plugging the Minkowskilimit of Eq. (22) in Eq. (24) and considering S to be a spherical surface at a large distance from the source, the gravitational wave energy flux passing through the detector is:
To make the calculations transparent, we assume that all the three (graviton, massive spin2, and spin0) modes of the following form:
where \(\zeta _{\mu \nu }, \theta _{\mu \nu }\) and \(\phi\) depend on r and t,
with \(k^0= w^0 = q^0 = \omega /c\), \({\textbf{k}}=\left( k^i\right)\), \({\textbf{w}}=(w^i)\), and \({\textbf{q}}= (q^i)\). \(\omega > c/\sqrt{2\gamma }\) and \(q > c/\sqrt{2\gamma }\) corresponds to oscillatory solution for \({\hat{R}}^{(1)}_{\mu \nu }\) and \(R^{(1)}\), respectively. A wave propagating radially outward \((\Phi _{\mu \nu })\) at large distances from the source can be represented to falloff radially^{46,47}:
where \(\chi _{\mu \nu }(t_r)\) is an arbitrary function of the retarded time \(t_r=tr/v\), and \(v=c/\sqrt{1\frac{1}{2 \gamma }\left( \frac{c}{\omega }\right) ^2}\) is the speed of the massive modes. Thus,
Substituting Eq. (26) in Eq. (25) and using Eq. (29) in the resultant expression leads to the energy flux on the GW detectors:
where dot denotes derivative w.r.t t. Here again, we note that the dominant contribution comes from the graviton mode with leading order corrections \((\gamma )\) from the massive spin2 mode; the massive spin0 mode contributes only in the second order. Hence, as expected, the measured energy energyflux in the case of QG is lower than that predicted by GR. In other words, the massive spin2 mode carries more energy than the spin0 mode. Since, this analysis is for the Minkowski background, \(\gamma \mathcal{B}_{\mu \nu }\) (in Eq. 22) is zero. However, in the case of curved geometry, \(\gamma \mathcal{B}_{\mu \nu }\) contribution might be significantly larger than the linear order term in the above expression.
In the case of GW detectors, the average \(<\ldots>\) is purely a temporal average^{46}, and the total energy flowing through the unit area of the detector is:
Note that the above analysis is strictly valid for \(\omega > c/\sqrt{2\gamma }\) and \(q > c/\sqrt{2\gamma }\) corresponding to oscillatory solutions for the two massive modes. In the second physical setting, we will relax this condition and obtain the contribution of the massive spin2 mode.
Backreaction of the emitted gravitational waves
To study the backreaction of the emitted GWs on the background black hole spacetime, we assume that the background spacetime is spherically symmetric Ricci flat and is an exact solution of the QG action (2). Though Schwarzschild solution in Stelle gravity is known to suffer from \(\ell =0\) mode instability (GregoryLaflamme instability) when the spin0 mode is nonpropagating^{48,49,50,51}, in the present work, we consider the spin0 mode to be dynamical and concentrate on the \(\ell =2\) modes. The backreaction on the background spacetime is obtained by evaluating the GW stressenergy tensor (\(t_{\mu \nu }^{\textrm{GW}}\)) w.r.t the Schwarzschild metric. Having obtained this, we then solve the effective Einstein’s equations:
where the Einstein tensor \({G_{\mu }^{\nu }}^{\textrm{mod}}\) is evaluated for spherically symmetric spacetime in dimensionless, EddingtonFinkelstein (EF) coordinates^{45}:
\(\nu \equiv \nu \left( V,\rho \right)\) and \(\lambda \equiv \lambda \left( V,\rho \right)\) encode the corrections from the emitted GWs, and \(d\Omega ^2\) is the metric on unit 2sphere. Note that V and \(\rho\) are dimensionless like \(\theta\) and \(\phi\). Regarding Eqs. (32, 33), the following points are in order: First, the ansatz (15) assumes that all the metric components are dimensionless. Hence, we have rescaled all the coordinates to be dimensionless. Second, \(V=constant\) hypersurfaces represent the ingoing null geodesics. As mentioned, the background metric is assumed to be Schwarzschild black hole; hence, \(e^{2\nu }=e^{2\lambda }=12M_0/\rho\) where \(M_0\) is the dimensionless mass parameter (setting \(c= G = 1\)). For M(V), the above metric gives the Vaidya line element. Third, the ingoing EF coordinates are smooth across the horizon for ingoing null geodesics and are suitable for analyzing the gravitational waveform close to the horizon^{52,53} as well as the shift in the horizon radius due to the backreaction. Finally, since \(t_{\mu \nu }^{\textrm{GW}}\) contains the contributions of graviton and massive modes, the LHS of Eq. (32) only contains the Einstein gravity. Note that the radiation from a remnant black hole decreases its energy content, inducing a change in the metric.
Since the dominant contributions to the GW stressenergy tensor (22) come only from the spin2 mode, to evaluate their effects on the metric, we use the following ansatz:
where \(o^{\mu \nu }\), \(\iota ^{\mu \nu }\) are constant, traceless (polarization) tensors, \(\psi (V,\rho )\), \(P(V,\rho )\) are scalar functions and \(Y^m_l(\theta , \phi )\) are spherical harmonics. The above assumptions essentially replaces the spin2 modes \(\psi ^{\mu \nu },~{\hat{R}}^{(1) \mu \nu }\) by scalar functions, where we ignored the nonlinear transformation among the components of the individual spin2 modes. To obtain the leading order corrections, we concentrate on the backreaction effect due to the \(\ell =2\) and \(m=0\) mode of the gravitational waveform and obtain the average contribution over the entire solid angle. Note that the \((\ell =2,m=0)\) mode contributes to the nonlinear memory of the GWs, which is otherwise difficult to observe in groundbased GW detectors^{54,55} and nontrivial to extract in numerical relativity simulations. We can trivially extend the analysis to other modes.
We assume the modified black hole to be described by the generalized spherically symmetric metric ansatz proposed by Johannsen and Psaltis (JP)^{56,57}. In the (dimensionless) EF coordinates (33), we have:
where \({\tilde{\epsilon }}_0=1\) and the first few coefficients of the expansion can be constrained from the PPNlike parameters^{56}. In the limit of \({\tilde{\epsilon }}_n=0,~(n>0)\), JP metric reduces to the Schwarzschild metric. The event horizon of the corresponding black hole is at \(\rho =2{\tilde{M}}\) and the (dimensionless) ADM mass is \(M_{\textrm{ADM}} ={\tilde{M}}(1{\tilde{\epsilon }}_1 /2)\)^{57}. As mentioned earlier, the remnant black hole decreases its energy content, inducing a change in the metric massfunction from the initial, dimensionless Schwarzschild value \((M_0)\)^{58,59}:
Assuming the mode functions to be regular and slowly varying in \(\rho\) close to horizon \((\Delta =\rho 2M_0<<1)\)^{52,53,60}, and expanding both sides of Eq. (32) for the \(\rho V\) component, we get (See supplementary material^{33} for details):
where \(\{C^{00},C^{23}\}=\{29(o^{00})^2,20(o^{23})^2\}\) and \('\) denotes derivative with respect to V. The expression is evaluated to the leading order in \(\Delta\) and \(M_0\). Integrating from some initial \(V=V_0\), when \(\Delta _M(V = V_0)=0\) to some final \(V=V_1\), we get,
This is the third key result regarding which we would like to stress the following: First, the massive and massless spin2 modes contribute oppositely to the change in the mass function, hence the shift in the horizon radius. Second, since \(\Delta _M\) is proportional to the sixth power of \(M_0\), it implies that the larger the mass of the perturbed black hole, the larger the corrections to the change in the mass. This assumes particular importance for intermediatemass black holes which are prime targets for LISA. Rezzolla and Zhidenko^{57} proposed an improved parametric metric in which, in the nearhorizon limit, the Taylor expansion is replaced by an expansion in continued fractions (CF). The RezzollaZhidenko metric has better convergence properties and can effectively reproduce any known solution even in scenarios where the JP parametrization fails. The CF expansion coefficients in the RezzollaZhidenko metric can be expressed in terms of the JP parameters; hence it is possible to extend our results to the RezzollaZhidenko metric^{57}.
Summary and discussions
In this work, we examined gravitational radiation in QG. We explicitly decomposed the GWs in Stelle gravity into massless and massive spin2 and spin0 modes. We demonstrated that the dominant contribution to the GW stressenergy tensor comes from the graviton mode, with leading order corrections coming from the massive spin2 mode, which is absent in f(R) gravity theories. We can ascertain this because the massive spin2 and spin0 have the same mass in the case of GUPinspired Stelle gravity. In contrast, the effects of the massive spin2 mode are the inverse of those of the GR mode. In the context of GW detectors, this results in a decreased energy flux measurement. This result is consistent with a recent finding that this Stelle gravity model reduces the amplitude of primordial gravitational waves produced by Starobinsky inflation^{61}.
We also provided an estimate of the backreaction effect of the GW emission on the background spacetime, where we once again observe that the massive spin2 mode decreases the rate of masschange and the rate of shift in the horizon radius. Our results indicate that intermediatemass black holes (prime targets for LISA) might be good candidates to test these aspects of modified gravity theories. Focusing on the \(\ell =2, m=0\) modes, our analysis suggests that the backreaction effect may play a crucial role in the study of nonlinear memory of GWs in modified gravity theories. These are currently under investigation.
The analysis in this work looks at the interesting possible signatures of strong gravity corrections in future GWs experiments. While the Starobinsky model and Stelle gravity are the lowenergy quantum gravity action, these are not exhaustive. For instance, we have not included \(R \ln \left( \Box R\right)\) and \(R_{\mu \nu } \ln \left( \Box R^{\mu \nu }\right)\)^{62}. It may be interesting to investigate the potential ringdown signatures of these terms as these terms may exceed the Stelle gravity contributions in low momenta. This is currently under investigation.
The decrease in measured energy flux and the decreased rate of horizon shift and mass change due to the massive spin2 mode indicate the existence of quasibound states of the massive spin2 modes surrounding the black hole^{63}. In the context of rotating black hole geometries, where this may lead to the formation of superradiantly induced spin2 boson clouds, the question assumes greater significance. A detailed analysis of the massive spin2 dynamics can resolve this question. However, such an analysis is beyond the scope of this paper so that we will leave it for future work.
Data availability
All data generated or analysed during this study are included in this published article and its supplementary information files.
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Acknowledgements
The authors thank Susmita Jana, S. Mahesh Chandran, and T. Parvez for the discussions. S.X. is financially supported by the MHRD fellowship at IIT Bombay. This work is supported by SERBMATRICS grant.
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A.C. contributed to the calculations and wrote the manuscript, S.X. contributed to the calculations, S.S. contributed to the conceptualization and wrote the manuscript. All authors reviewed the manuscript.
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Chowdhury, A., Xavier, S. & Shankaranarayanan, S. The dominating mode of two competing massive modes of quadratic gravity. Sci Rep 13, 8547 (2023). https://doi.org/10.1038/s41598023348028
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DOI: https://doi.org/10.1038/s41598023348028
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The dominating mode of two competing massive modes of quadratic gravity
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