Self-force on a scalar charge in a circular orbit about a Reissner-Nordström black hole
Motivated by applications to the study of self-force effects in scalar-tensor theories of gravity, we calculate the self-force exerted on a scalar charge in a circular orbit about a Reissner-Nordström black hole. We obtain the self-force via a mode-sum calculation, and find that our results differ from recent post-Newtonian calculations even in the slow-motion regime. We compute the radiative fluxes towards infinity and down the black hole, and verify that they are balanced by energy dissipated through the local self-force – in contrast to the reported post-Newtonian results. The self-force and radiative fluxes depend solely on the black hole’s charge-to-mass ratio, the controlling parameter of the Reissner-Nordström geometry. They both monotonically decrease as the black hole approaches extremality. With respect to an extremality parameter , the energy flux through the event horizon is found to scale as as .
The self-force acting on a particle moving in a curved spacetime has been a fascinating subject for some time, principally motivated by the prospect of detecting low-frequency gravitational waves with future space-based missions such as LISA Amaro-Seoane et al. (2012). While self-force-based gravitational waveforms remain elusive, progress in self-force research has been steady, and has made key contributions to a fuller understanding of the strongly-gravitating two-body problem Poisson et al. (2011); Barack (2009); Wardell (2015). Beyond direct applications to gravitational-wave astronomy, the self-force has proven useful as a theoretical probe of the nonlocal features of the spacetime in which the particle moves. More specifically, the static self-force has been shown to be sensitive to central and asymptotic structure Drivas and Gralla (2011); Taylor (2013); Kuchar et al. (2013). At the frontier of self-force research there remains strong momentum for the calculation of fully self-consistent gravitational waveforms from extreme-mass-ratio inspirals (to second order in the mass ratio), in addition to active research programs pushing to develop the formalism to higher dimensions Taylor and Flanagan (2015); Harte et al. (2016, 2017) and alternative theories of gravity Zimmerman (2015). For the latter, it is of strong interest to understand how the extra gravitational degrees of freedom and their coupling might impact self-forced dynamics.
The work reported in this paper began with a consideration of self-forces in the context of alternative theories of gravity, particularly in scalar-tensor theories, as inspired by the seminal work of Zimmerman Zimmerman (2015). One tantalizing result of this work was the possibility of “scalarization” of a compact object by the action of the self-force. However, a close inspection of Zimmerman (2015) quickly reveals that this effect requires a non-trivial scalar field residing on a curved spacetime background. No-hair theorems for black holes in scalar-tensor theories then severely limit the possible realizations for scalarization via self-force. To be sure, there are loopholes to these theorems; certain scalar-tensor theories do admit hairy black hole solutions Torii et al. (2001); Kanti et al. (1996); Sotiriou and Zhou (2014); Antoniou et al. (2018, 2017). But these solutions are obtained mostly with numerical integration Sotiriou and Zhou (2014), making them difficult to study as backgrounds for concrete self-force calculations. (See, however, Babichev et al. (2017) for some simple hairy black hole solutions in Horndeski theory.)
A well-known solution in scalar-tensor theory is the so-called Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) solution Bocharova et al. (1970); Bekenstein (1974, 1975) in conformal scalar-vacuum gravity. This theory is defined by the action
and the BBMB solution reads
The metric in this solution is the extremal Reissner-Nordström solution, and the scalar field is non-trivial, though it clearly diverges at the putative event horizon at . This divergence muddles the interpretation of as a true event horizon and, correspondingly, of the BBMB solution as a legitimate black hole solution. But this interpretational issue can be eschewed when one’s primary concern is the impact of the scalar degree of freedom on the gravitational dynamics. This is the viewpoint espoused by our research program, of which this work is an initial step. Our proposal is to use the BBMB solution as a theoretical playground for studying scalar-tensor self-force effects.
Apart from this broader goal, the Reissner-Nordström spacetime is in itself an interesting spacetime on which to study self-force effects. As the unique, spherically-symmetric and asymptotically flat solution to the Einstein-Maxwell equations, it describes the spacetime outside a charged spherically-symmetric mass distribution. It is characterized by its mass and charge , and the spacetime is described by the metric
where and . Note that the coordinate is connected to Q. If the object in question is a black hole, then it will have the following features:
There are 2 horizons, an inner horizon at , and an outer horizon at , which happens to be the event horizon of the black hole.
In the case where , the black hole becomes extremal, with a degenerate horizon (and thus zero temperature).
Despite these interesting properties, astrophysical considerations preclude significant charge build-up, and thus the self-force on a Reissner-Nördstrom background has been largely neglected. The notable exception is a recent work by Bini et al. (Bini et al., 2016) in which they produced a 7 post-Newtonian (PN) order calculation of the scalar self-force on a circular geodesic. In this paper, we go beyond the PN approximation and present the first mode-sum calculation of the full strong-field scalar self-force on a circular geodesic. In the process of doing so, we found that our results for the self-force and energy flux is in disagreement with the slow-motion formulas presented by Bini et al. (Bini et al., 2016). This is surprising, as one would expect a numerical calculation to agree with a PN calculation up to the order reported. We have yet to establish a reason for this discrepancy. Nevertheless, we present several consistency checks on our results, to show that this disagreement is not from an error in the numerical calculation.
The paper is organized as follows: In Section II we provide a brief review of circular geodesics in Reissner-Nordström spacetime. In Section III we provide a derivation of the scalar field generated by a scalar point charge in a geodesic circular orbit, subject to ingoing wave conditions at and outgoing wave conditions at . In Section IV we briefly review the mode-sum regularization scheme, as well as providing a brief derivation of the regularization parameters for geodesic circular orbits in Reissner-Nordström spacetime. In Section V we provide numerical results computed from a frequency-domain calculation and compare it with analytical results obtained from (Bini et al., 2016).
Ii Circular geodesics
Considered as a test particle, the scalar charge will move along a geodesic of the Reissner-Nordström spacetime. Two Killing vectors of spacetime, and , provide the conserved quantities
where is the proper time along the orbit and is the particle’s four-velocity.
Due to the spherical symmetry of the spacetime, the test particle will move along a fixed plane. We can always choose our coordinates so that this plane is described by . Combining these with the normalization, , we arrive at the radial equation
where is the effective potential. For circular orbits (), the four-velocity reads
is the angular velocity of the particle with respect to an asymptotic observer. Note that this quantity, as an observable, is invariant to coordinate transformations. Circular orbits also require , which gives the condition
while in Eq. (7) gives
which can be combined to give
Putting this into Eq. (9) we finally get
Normalization of the four-velocity then gives
This completes the determination of the four-velocity for a particle in a circular geodesic.
Iii Field equations
iii.1 Multipole decomposition
We assume that the scalar field is a small perturbation of the fixed Reissner-Nordström spacetime, and that it satisfies the minimally coupled scalar wave equation
sourced by a scalar charge density . We model this scalar charge density as a -function distribution on the worldline, written as
which for a circular orbit becomes
Using the spherical harmonic completeness relations, we can further rewrite as
A similar decomposition for the scalar field into spherical harmonics and Fourier modes yields the form
We now want to impose boundary conditions.
iii.2 Boundary conditions
The wave equation can also be rewritten in terms of the so-called tortoise coordinate
Defining (where ) we get
where is the Laplacian on the unit two-sphere. Decomposing into its spherical-harmonic components
Eq. (25) becomes
The homogeneous part of this equation appears like a flat-space wave equation [in (1+1) dimensions] with a potential . This potential vanishes as (or as ) and as ().
The appropriate boundary conditions are ingoing waves at the event horizon and outgoing waves at infinity. Since (where for circular orbits), we shall then impose that as and as . Correspondingly, for the boundary conditions of interest are
These boundary conditions serve as initial data in the integration of Eq. (23). In practice, the integration cannot begin exactly at the horizon because vanishes and the potential term in Eq. (23) blows up. [The potential term of Eq. (27) is regular, but the horizon in these coordinates is inaccessible at .] Instead, we then begin the integration slightly away from the horizon, at for . An asymptotic solution as can be obtained by inserting the ansatz
into Eq. (23). This gives a recurrence relation for the coefficients which reads
The same considerations apply to the boundary condition as . Again we work with the ansatz
and obtain a recurrence relation for using Eq. (23). This reads
iv.1 Mode-sum regularization
To obtain the scalar self-force, we must first subject the unregularized force to a regularization procedure. In our case, we use the mode-sum scheme Barack and Ori (2000); Burko (2000); Barack and Ori (2003), where the self-force is constructed from regularized spherical harmonic contributions. We start with full force derived from the retarded field
where is the -mode component (summed over ) of the full force at an arbitrary point in the neighborhood of the particle. At the particle location, each is finite, although the sided limits often produce different values (which we then label as ) and the sum over may not converge. We then obtain the self-force using a mode-by-mode regularization formula
where the regularized contributions no longer have the ambiguity and the sum over is guaranteed to converge. The regularization parameters (-independent) and have been obtained for generic orbits about a Schwarzschild black hole Barack et al. (2002), and a Kerr black hole Barack and Ori (2003). In the next subsection we present a derivation of and for circular orbits about a Reissner-Nordström black hole.
iv.2 Regularization parameters
The procedure for deriving mode-sum regularization parameters is by now well-established Barack et al. (2002); Barack and Ori (2003); Haas and Poisson (2006); Heffernan et al. (2012, 2014); Heffernan (2012). Here, we directly follow the approach of Heffernan et al. (2012, 2014); Heffernan (2012), extending it to the case of Reissner-Nordström spacetime as given by the line element, Eq. (4). Since the essential details remain the same, we refer the reader to Refs. Heffernan et al. (2012, 2014); Heffernan (2012) for an extensive discussion, and give here only the key equations and results.
We start with an expansion of the Detweiler-Whiting singular field (Detweiler and Whiting, 2003) through next-from-leading order in the distance from the worldline,
Here, we have already specialized to the case of circular, equatorial orbits, and have introduced and
The above expressions are given in terms of the Riemann normal coordinates and , the same as are described in Heffernan et al. (2014).
It turns out that for circular orbits the and componets of the self-force are purely dissipative, meaning that only the radial component of the self-force requires regularization. We thus compute the contribution from the singular field to the radial component of the self-force using . Doing so, taking , and keeping only terms which will not vanish in the limit we get
Here, , just as in Heffernan et al. (2014).
Next, we obtain the regularization parameters by decomposing this into spherical-harmonic modes (as usual, we only need to consider the case since the other -modes do not contribute). Doing so, and taking the limit , we find
from which we can immediately read off the and regularization parameters. Here,
are complete elliptic integrals of the first and second kind, respectively.
V Self-force calculation
v.1 Scalar energy flux
Global energy conservation dictates that the local energy dissipation, represented by the -component of the self-force, is accounted for by the energy flux carried by scalar field radiation. We numerically calculate the energy flux to infinity and down the black hole, and verify that the result is consistent with the energy lost through the local dissipative self-force.
We briefly review the relevant formalism used to calculate the energy flux. The stress-energy tensor of the scalar field is given by
With , we construct the differential energy flux over the following constant hypersurfaces: , represented by , and , represented by . The differential energy flux then takes the form
where is the unit normal vector of the hypersurface, and is the hypersurface element. We then rewrite Eq. (42) as
Integrating over the two-sphere, we then express the energy transfer as
We present sample numerical data for , and in Tables 1, and 2 respectively. We see that as the extremality parameter approaches zero, monotonically decreases. We also note that compared to , exhibits a dramatic decrease as . We then investigate the scaling behavior of with respect to , which we present in Fig. 1 and 2. We note that while exhibits no discernible scaling behavior, exhibits power law scaling as , which in Fig. 2 corresponds to . This behavior for has been previously observed for near-extremal Kerr black holes, with a power scaling of Gralla et al. (2015).
In the same figures, we compared our numerical data with the slow-motion analytic formulas for the energy fluxes derived by Bini et al. Bini et al. (2016). While the qualitative behavior of Bini et al.’s formula is similar to our numerical results for , that cannot be said for . The qualitative behavior exhibited by Bini et al.’s formula for is opposite to that of our numerical results, and the disagreement worsens as .
v.2 Dissipative component of the self-force
For circular orbits, the dissipative components of the scalar self-force are , and . We note that due to in the circular orbit case, there is a simple relationship between the dissipative components of the self-force:
This relationship indicates that we need only one component to calculate. In this work, we choose to calculate .
For our set-up, the local energy dissipation must be accounted for by the energy fluxes towards infinity, and down the black hole. This energy balance relation can be expressed in terms of the self-force
This allows us to test our computation of by verifying that our numerical results satisfy Eq. (47).
Sample numerical results for are presented in Table 4. As a check, we compared our Schwarzschild results with those of Warburton and Barack Warburton and Barack (2010), and we are in agreement to all significant figures presented. Looking at our results, we see that as the black hole approaches extremality , the dissipative self-force decreases. One concludes from this that the black hole charge suppresses local energy dissipation.
We also compared our results to the slow-motion formula for the dissipative self-force derived by Bini et al. Bini et al. (2016), presented in Fig. 3. We see that the qualitative behavior of our results completely differs from Bini et al.’s formula, which worsens as . We note that this discrepancy in qualitative behavior is also present for , as presented in Fig. 1.
|Numerical Result||PN Result|
As a consistency check, we then examined the energy balance relation exhibited by our numerical results and Bini et al.’s slow-motion formulas. Sample data are presented in Table 3. While it is expected that energy balance will be better satisfied by numerical calculations compared to PN calculations, we note that the energy balance error exhibit by Bini et al.’s formulas are of relative 3 PN order, one order higher than the expected error in their formulas.
v.3 Conservative component of the self-force
For circular orbits, the conservative component of the self-force is contained entirely in . The calculation of this conservative self-force is more complicated than the dissipative piece, as the mode-sum requires regularization. We then checked the effect of the regularization parameters on the mode-sum, as presented in Fig. 4. Looking at the high -mode components, we see that the regularization parameters work as expected, leaving a residual field which exhibits fall-off behavior.
Sample numerical data for is presented in Table 5. We compared our Schwarzschild results with those of Diaz-Rivera et al. Diaz-Rivera et al. (2004), and we are in agreement up to six significant figures. Looking at our results, we see that as the black hole approaches extremality , the conservative self-force decreases. This implies that the black hole charge suppresses the entire self-force.
We also compared our numerical results to the slow-motion formula for the conservative self-force derived by Bini et al. Bini et al. (2016). We present this in Fig. 3, and we see that while Bini et al.’s formula follow the same qualitative behavior of our results, we begin to deviate as .
In this work we presented the first mode-sum calculation of the self-force exerted on a particle in circular orbits about a Reissner-Nordström black hole. We also present in this work regularization parameters , and for circular orbits in Reissner-Nordström spacetime.
We tested the validity of our results in various ways. The results for the Schwarzschild limit was found to agree with the results found in the literature Warburton and Barack (2010); Diaz-Rivera et al. (2004). We confirmed numerically that the local energy dissipation is balanced out by the energy carried away by scalar waves towards infinity and down the event horizon. We also investigated the -mode fall-off of the conservative self-force, and found that after subtracting the and regularization parameters the modes of the residual field fall off as , as expected.
Our results indicate that as the black hole’s electric charge increases, the self-force decreases in magnitude. This dampening is notably drastic for the flux of scalar radiation towards the event horizon, where in near-extremal Reissner-Nordström black holes, the scalar radiation flux scales as . This behavior is also seen for near-extremal Kerr black holes Gralla et al. (2015), thus a more detailed calculation is recommended as a future study.
We also compared our results with the slow-motion formulas obtained by Bini et al. Bini et al. (2016). While our results agree in the Schwarzschild limit, they disagree for , and as the electric charge increases, the disagreement between the results increases. We have yet to establish the reason for this disagreement.
We expect some of our results to have some bearing on future self-force studies in black hole solutions of scalar-tensor theories. The BBMB solution of conformal scalar-vacuum gravity is exactly the extremal Reissner-Nordström geometry, so similar results might be obtained in situations where the scalar field in these alternative theories can be approximated as test fields. Other hairy black hole solutions have now been discovered in other scalar-tensor theories of gravity and some of these are also of Reissner-Nordström form Babichev et al. (2017). Self-force phenomenology in these theories remains completely uncharted, and so remains a promising area of future research. By exploring the scalar self-force from a minimally-coupled scalar field in the Reissner-Nordström spacetime, we hope to have provided a useful guide and some benchmark numerical results for future self-force calculations in alternative theories of gravity.
This research is supported by the University of the Philippines OVPAA through Grant No. OVPAA-BPhD-2016-13 and by the Department of Science and Technology Advanced Science and Technology Human Resources Development Program - National Science Consortium (DOST ASTHRDP-NSC).
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