Are black holes in alternative theories serious astrophysical candidates? The case for Einstein-Dilaton-Gauss-Bonnet black holes

Are black holes in alternative theories serious astrophysical candidates?
The case for Einstein-Dilaton-Gauss-Bonnet black holes

Paolo Pani paolo.pani@ca.infn.it Dipartimento di Fisica, Università di Cagliari, and INFN sezione di Cagliari, Cittadella Universitaria 09042 Monserrato, Italy
Currently at Centro Multidisciplinar de Astrofísica - CENTRA, Dept. de Física, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
   Vitor Cardoso vcardoso@fisica.ist.utl.pt Centro Multidisciplinar de Astrofísica - CENTRA, Dept. de Física, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal &
Department of Physics and Astronomy, The University of Mississippi, University, MS 38677-1848, USA
Abstract

It is generally accepted that Einstein’s theory will get some as yet unknown corrections, possibly large in the strong field regime. An ideal place to look for these modifications is around the vicinities of compact objects such as black holes. Our case study here are Dilatonic Black Holes, which arise in the framework of Gauss-Bonnet couplings and one-loop corrected four-dimensional effective theory of heterotic superstrings at low energies. These are interesting objects as a prototype for alternative, yet well-behaved gravity theories: they evade the “no-hair” theorem of General Relativity but were proved to be stable against radial perturbations.

We investigate the viability of these black holes as astrophysical objects and try to provide some means to distinguish them from black holes in General Relativity. We start by extending previous works and establishing the stability of these black holes against axial perturbations. We then look for solutions of the field equations describing slowly rotating black holes and study geodesic motion around this geometry. Depending on the values of mass, dilaton charge and angular momentum of the solution, one can have differences in the ISCO location and orbital frequency, relatively to black holes in General Relativity. In the most favorable cases the difference amount to a few percent. Given the current state-of-the-art, we discuss the difficulty to distinguish the correct theory of gravity from EM observations or even with gravitational wave detectors.

pacs:
04.70.Dy,04.70.Bw,04.50.+h,04.40.Nr

I Introduction

Einstein’s theory of General Relativity (GR) has passed numerous consistency and experimental tests in a spectacular way Will:2005va (). Nevertheless, it is a general consensus that GR will get modified at some scale, if only because GR has resisted all attempts at its quantization. Moreover, other theories of gravity also pass the experimental tests, some with a better “quantum behavior” and should be taken as serious candidates. Unfortunately, the majority of these alternative theories are vastly more complex than GR in their full-fledged form. It is thus not surprising that progress in understanding the exact differences between one and the other and specially differences one can measure experimentally have been slow and mostly focusing on the weak-, far-field behavior.

One of the candidates for a theory of quantum gravity is string theory polchinski (). Since it is still difficult to study geometrical settings in superstring theories, most analyses have been performed by using low-energy string-inspired effective theories Callan:1986jb (). Thus, quantum gravity predictions can be tested by studying modifications of GR due to some low-energy truncation of string theory Gross:1986iv (). Typically the effective theories are supergravities involving not only the metric but also a scalar field (the dilaton) as well as several gauge fields Zwiebach:1985uq (). One of such theories is the one-loop corrected four-dimensional effective theory of the heterotic superstrings at low energies and a simple particular case is known as Einstein-Dilatonic-Gauss-Bonnet (EDGB) theory (see for instance Ref. Moura:2006pz () and references cited therein for a nice and concise introduction to this theory). In EDGB theory the gauge fields are neglected and only the (spacetime-dependent) coupling between the dilaton and the gravity is considered, with the anomaly-canceling Chern-Simons term also neglected (see for instance Alexander:2007kv (); Smith:2007jm (); Alexander:2007zg (); Alexander:2007vt () for work taking this term into account). At the first order in the Regge slope, , higher-derivative gravitational terms such as the Gauss-Bonnet (GB) curvature-squared term are present in the action, hence the name. The GB terms avoid some pathological features, for example the theory is ghost-free. Since the equations of motions are still of second order, EDGB theory provides one of the simplest consistent high-energy modifications to GR. Even though it does not seem to be a viable cosmological model EspositoFarese:2003ze (), the Parametrized Post-Newtonian Will () expansion of this theory is identical (to lowest order) to that of GR EspositoFarese:2004cc (); Sotiriou:2006pq (), which means that it passes all Solar system-like experimental tests of gravity. Differences arise only from full nonlinear effects and the ideal place to look for these is near compact objects such as black holes.

Dilatonic Black Holes (DBHs) do exist in EDGB theory kanti (); Torii:1996yi (); Alexeev:1996vs (). They have a regular event horizon and the geometry is asymptotically flat at infinity (recently DBHs in higher dimensions Guo:2008hf () and with non-flat asymptotical geometries Guo:2008eq () have also been studied). In what follows we refer to DBHs in EDGB theory only. DBHs are interesting for many theoretical and practical reasons. DBHs can evade the classical “no-scalar-hair” theorem bek (), and be dressed with classical non-trivial dilaton hair. This is a direct consequence of the GB term kanti () and opens up the exciting possibility of ruling out such objects and theories by testing the no-hair theorem, either by gravitational-wave observations of ringdown Berti:2005ys () or by observations of highly eccentric orbits around supermassive black holes Will:2007pp (). The lack of experimental data and some problems on the theoretical side make it very difficult to explore string theory and other theories of quantum gravity. Since quantum gravity effects are expected to play a significant role in strong gravity regime such as in cosmology Bamba:2007ef () and in black hole physics, the investigation of BHs in EDGB theory can shed new light on some aspects of quantum gravity and/or be used to develop testable predictions of the theory.

i.1 Astrophysical implications

Our purpose here is to begin exploring differences between GR and alternative theories that could be experimentally tested through astrophysical observations, in the strong field regime. Thus, even though the theory with which we will work is only a first order truncation of the full action, we will elevate it to the status of a complete theory, therefore we place no restriction on the Regge slope. The first important problem concerns the stability of DBHs. Credible alternatives to the Schwarzschild and Kerr metric of GR must be stable spacetimes. It has been shown kanti-lin (); torii () that DBHs are stable against a small subset of all possible perturbations, linear radial perturbations. In this work we characterize completely half of the possible degrees of freedom, by studying general axial perturbations. We find that DBHs are stable also against these perturbations. By itself this is an interesting result confirming that high-energy contributions lead to viable alternatives to classical BHs arising in GR. The viability of DBHs poses the following question: can one devise observational tests to discern a DBH from a classical BH? In classical Einstein-Maxwell theory BHs are characterized by three parameters Hawking:1971vc (): mass , electric charge and angular momentum . Astrophysical BHs are likely to be electrically neutral because of the effect of surrounding plasma Blandford:1977ds (), and therefore tests of alternative theories of gravity can in general focus on rotation alone. The task is still highly non-trivial and as we mentioned earlier, strong-field effects must be searched for in theories that are not already ruled out by Solar-system experimental data Sotiriou:2006pq ().

Most if not all of present-day astrophysical observations related to compact objects, concern directly or indirectly the motion of matter. Thus, a study of geodesic motion around compact objects in alternative theories is of utmost importance. Geodesics convey very important information on the background geometry. In particular circular orbits whose radius is close to the horizon may be extremely useful, because they already probe strong field regions. They can be used to compute the “innermost-stable-circular-orbit” (ISCO), a notion which is very important for interpretation of the experimental data concerning astrophysical black holes. For the Schwarzschild spacetime, , while for an extremal rotating Kerr geometry, for co- and counter-rotating circular orbits respectively. Measurements of the ISCO are also useful to evaluate the angular momentum of Kerr BHs Narayan:2005ie (). Current methods to measure the ISCO include spectral fitting, quasi-periodic oscillations and relativistic iron line measurements Narayan:2005ie (). Here we show that differences in the ISCO of slowly rotating DBHs and classical Kerr black holes can be significant, depending on the coupling parameter. These differences, which may be detectable in near-future experiments, are likely to increase for highly spinning black holes (which are unfortunately out of the scope of this work). In fact, it seems an exciting possibility that the GRAVITY experiment Eisenhauer:2008tg (), designed to make precision measurements of orbits of stars in the neighborhood of the black-hole in the center of our galaxy, might already be able to discriminate between these black hole solutions and GR solutions, or otherwise impose stringent bounds on the coupling parameter.

Current techniques to evaluate the angular momentum of astrophysical compact objects are based only on ISCO measurements Narayan:2005ie (). One needs to assume a particular theory of gravity in order to evaluate . Thus discerning a DBH from a Kerr black hole by ISCO measurements is not an easy task. Future gravitational wave experiments will provide a viable method to measure and independently Berti:2004bd (). The analysis below suggests that a possible deviation from the expected ISCO in GR can be explained in term of dilatonic charged BH. Therefore in a near future, gravitational wave astronomy may offer a the possiblity to explore string theory-inspired modifications of GR.

Finally, null unstable geodesics are closely related to the appearance of compact objects to external observers podurets (); amesthorne () and have been associated with the characteristic, or quasinormal modes (QNMs) of BHs Cardoso:2008bp (); cardosotopical (). Quasinormal modes are very important in devising experimental tests of GR and for gravitational wave astronomy. Measuring QNM frequencies may provide a definitive proof of the existence of BHs in GR and it could be useful to study corrections to GR too. Thus the analysis of geodesic motion around a DBH can shed new light on various and important aspects of high-curvature corrections to gravity.

Other theories might also suggest a different ISCO location, different quasinormal modes, etc. Thus, a deviation in these quantities is not a verification of a particular theory, but is a first step in understanding what different theories and scenarios predict in the strong field regime, which could potentially discriminate GR from other alternatives.

The paper is organized as follows. In Section II we briefly review the main aspects of BHs in EDGB theory. In Section III we prove the linear stability of DBHs against axial perturbations. We proceed by studying slowly rotating DBHs in Section IV. We prove that such slowly rotating solutions do exist and we characterize them, including a discussion on the ergoregion in these spacetimes. Section V discusses geodesics in both spherically symmetric and slowly rotating DBHs as well as the possible experimental tests which can be performed to discern a DBH from a Kerr BH. We compute the ISCO dependence on the angular momentum and the QNM frequencies for a spherical symmetric DBH in the eikonal limit. Conclusions are discussed in Section VI. In our notation, we use the signature and the curvature tensor is defined by .

Ii Spherically symmetric BHs in Einstein-Dilaton-Gauss-Bonnet theory

We consider the following low-energy effective action for the heterotic string Zwiebach:1985uq ()

(1)

where

(2)

is the GB invariant, is the Regge slope and is some gauge coupling constant. We set for the rest of this paper. String-inspired corrections to Einstein’s gravitation are included in the action (1), while gauge fields and matter are omitted for simplicity. We also note that there is some arbitrariness in the coupling, depending on which frame we take as fundamental Guo:2008hf (), we choose to keep the coupling in line with previous works. We shall refer to the action (1) as EDGB theory.

The dilaton field and Einstein’s equations derived from (1) are

(3)
(4)

where is the Einstein tensor and

(5)

Here,

From the right-hand-side of the modified Einstein’s equation (4), one can construct a conserved “energy momentum tensor”, ,

(6)

In Ref.kanti () is shown that the time component of , which in Einstein’s gravity would correspond to the local energy density , can be negative. The reason is that, as a result of the higher derivative GB terms, there are contributions of the gravitational field itself to . The positiveness of is one of the main assumptions of the no-scalar-hair theorem bek () which can be (and indeed is) evaded in EDGB theory.

We now focus on BH solutions in EDGB theory, considering the following spherically symmetric ansatz for the metric

(7)

The equations of motion derived from (3) and from (4) can be found in Ref. kanti-lin (). In a static, asymptotically flat geometry, black hole solutions exist only if kanti ()

(8)

where and are quantities evaluated on the horizon. In particular black hole solutions may exist only for . From the asymptotic behavior of the fields one can extract the ADM mass, and the charge . As shown in Ref. kanti () and are not independent quantities, thereby leading to the secondary nature of the dilaton hair wilczek (). These black hole solutions are uniquely characterized by two parameters , which correspond to a unique choice of . The equations of motion remain invariant under a shift and a simultaneous radial rescaling . As a consequence of the radial rescaling, the two other asymptotic parameters, and , are also rescaled according to the rule and . Due to the above invariance it is sufficient to vary only one of and . Following kanti-lin () we choose to keep fixed and to vary . Typical background fields are shown in Fig. 1. Differences in the metric coefficients occur only very close to the horizon. We checked our numerical solutions reproducing results shown in the Table 1 of Ref. kanti ().

After the rescaling, equation (8) can be written in terms of the coupling constant

(9)

The maximum value corresponds to . For larger values of the coupling constant no DBH solution exists. The dilaton charge, , as a function of is shown in Figs. 2, where it is compared with the analytical solution in the limit, Mignemi:1992nt ().

Figure 1: Metric components and for a DBH (solid lines) compared with an equal mass Schwarzschild hole (dotted lines) (, which corresponds to ).
Figure 2: The dilaton charge, , as a function of (re-scaled) (solid line). The whole range of in which a black hole solution exists is shown. Numerical solution behaves as in the limit Mignemi:1992nt () (dashed line).

Iii Linear stability analysis

The framework for a linear stability analysis of black holes in theories with Gauss-Bonnet terms was laid down by Dotti and Gleiser Dotti:2004sh (); Dotti:2005sq (); Gleiser:2005ra () in higher dimensions (with no dilaton), and generalized by Moura and Schiappa Moura:2006pz () in the context of Riemann tensor, , corrections. Perturbations of four-dimensional DBHs were considered by Kanti et al kanti-lin () and Torii and K. i. Maeda torii (). Unfortunately these authors considered only a very specific type of perturbations, here we want to generalize their results. Consider therefore perturbing the spacetime in the following linear way

(10)

where , and are the background fields, while and are the perturbations. The background metric and dilaton field are given by the numerical static black hole solution described above kanti ().

iii.1 General formalism

To study this problem we use the approach first described by Regge and Wheeler regge (). After a decomposition in tensorial spherical harmonics zerilli (); mathews (), the perturbations fall into two distinct classes: axial (odd) with parity and polar (even) with parity , where is the angular momentum of the particular mode. The theory described by (1) is invariant under diffeomorphisms111Despite the covariant derivative in eq. (5), equations of motion do not contain higher derivatives of the metric because of the GB term (see equation (4) in torii ()). This allows one to use the same gauge transformations first proposed in Ref. regge ()., as Einstein’s theory is. We can then use the gauge freedom in order to simplify the elements . In the classical Regge-Wheeler gauge the canonical form for the metric perturbations is (see also Ref. vish ()):
- axial perturbations:

(11)

- polar perturbations:

Where is the Legendre polynomial with angular momentum and , , , and are unknown radial functions.

Perturbations of the dilaton field, , do not appear in the axial equations (see also holzhey ()). The linear stability analysis proceeds by mapping the system of the equations of motion for the perturbation fields under consideration to a stationary one-dimensional Schrödinger problem, in an appropriate potential well, in which the ‘squared frequencies’ are the “energy eigenvalues”. Instabilities, then, correspond to bound states, i.e. to negative energy eigenstates or equivalently to frequencies with a positive imaginary component.

Presumably because polar perturbations are extremely complex to analyse, Kanti et al. kanti-lin () focused on a certain subset, the radial perturbations. They found that the spacetime was stable against radial perturbations, their results being confirmed in Ref. torii (). Thus, there are good indications that the spacetime is stable under polar perturbations. We thus focus here on the other set of perturbations, axial perturbations, which as far as we know are not dealt with in the literature.

iii.2 Axial perturbations

For axial perturbations only 3 nontrivial Einstein equations can be obtained by substituting (11) and (10) into (4). The zeroth-order equations are identically zero, due to the background solution, while the equations for the perturbations, and , read:

(12)
(13)
(14)

where , and depend on the radial background function and found in kanti (). Their explicit form can be found in Appendix A. We observed numerically that equation (14) is automatically satisfied as a consequence of the other two equations and of background solutions. So we are left with a system of two ODEs for two unknown functions and . Eliminating from the first order equation (12) we obtain a second order differential equation for which can be recast (see Appendix A for details) in the following Schrödinger-like equation,

(15)

These functions are shown in Fig. 3.

The asymptotic behavior of equation (15) (see Appendix A for details) is

(16)
(17)

The asymptotic solutions are

(18)
(19)

Figure 4 shows the coefficients and as functions of . For a Schwarzschild BH, and . In the EDGB case one finds one finds and in the whole range (9). Thus the asymptotic solution (18) simplifies

(20)

Since is complex, the sign in equations (19) and (20) above has to be chosen so that the solution is regular on the horizon and at infinity. For unstable modes , thus the choice of the minus and the plus sign in eqs. (19) and (20) respectively guarantees that the corresponding eigenfunctions will vanish at infinity and at the horizon. We have searched for unstable modes using these boundary conditions. We integrated equation (15) outward starting from the horizon until we found the eigenfrequency corresponding to a vanish field at infinity. We used a Runge-Kutta 4th order method, considering and different values corresponding to different BH solutions kanti (). We systematically span the whole range (9) for , , . We also randomly span other values for and to no avail: no unstable modes were found. We checked the numerical accuracy of our results by changing numerical parameters such as . Our results strongly suggest that EDGB black holes are stable against axial perturbations. This completes the previous linear stability analysis kanti-lin (); torii () performed for radial perturbations.

Figure 3: Top Panel: The potential for the axial perturbations for a EDGB black hole. Different values of the angular momentum are shown for , which corresponds to . Bottom Panel: The function for the axial perturbations equation for a EDGB black hole for .
Figure 4: The coefficients (solid line) and (dashed line) as functions of in the whole range (9). As for a Schwarzschild BH, , whereas is always smaller than its Schwarzschild counterpart, .

It would be of great interest to understand also polar perturbations and reduce them to a single master ordinary differential equation. Due to the complexity of the equations involved we did not perform such an analysis. We note that the present analysis is only possible because axial and polar perturbations decouple in this theory, unlike for instance in Chern-Simons theory Yunes:2007ss ().

Iv Slowly rotating BHs in EDGB theory

Astrophysical black holes are expected to be highly spinning because of accretion and merger events Gammie:2003qi (); Merritt:2004gc (); Berti:2008af (); Kesdenreview (). Unfortunately the GB term is very hard to deal with for generic axisymmetric metrics and thus general rotating black holes are difficult to study without a full numerical integration of Einstein’s equations. Expansion in is another promising approach, which has been successfully implemented for non-rotating DBHs Mignemi:1992nt (). Here we use another method, searching for solutions describing rotating BHs at every order in , but in a slow rotation approximation. This method allow us to prove that such (slowly) rotating BHs do exist in EDGB theory. This strongly suggests that, in general, rotating DBHs exist. In alternative high energy modifications of GR, such as Chern-Simons modified gravity BHs, rotating black hole solutions are harder to find Grumiller:2007rv (); Konno:2007ze ().

We follow Hartle’s approach hartle () which is based on axisymmetric perturbations of a spherically symmetric equilibrium solution and on an expansion of the perturbed quantities in a power series of the angular velocity for frame dragging. The equilibrium solution is the one developed in Ref. kanti () and explored in the previous sections. To first order in the perturbed metric is

(21)

where is the angular velocity of an observer at freely falling from infinity. Since we consider only axisymmetric perturbations, we neglected the dependence and we can expand in terms of Legendre polynomials

(22)

Both the metric and dilaton perturbations are functions of even powers of the angular velocity , thus we can use the unperturbed metric and the unperturbed dilaton field to find -order corrections neglecting -order terms. Using the expansion (22) only the component of modified Einstein equations (4) is first order in and, interestingly enough, it gives a separable differential equation as follows

(23)

where

and , , are zeroth-order in , therefore given by the spherically symmetric background kanti (). For equation (23) is equivalent to the standard GR form (equation (43) in Ref. hartle () with vanishing stress tensor).

To solve the above equation we must specify the boundary condition for large . Since the scalar field vanishes asymptotically, the solution should approach flat space solution. Thus for large we can define the angular momentum of the BH via the following behavior,

(25)

It is worth to note that the dilaton field can introduce order corrections to the off-diagonal metric coefficients. In the slowly rotating approach we use, these corrections are neglected. The asymptotic behavior of equation (23) is

(26)

whose general solution is

(27)

where and are constant. With the asymptotic behavior (25), the equation above implies that only the mode is allowed. Thus the equation to solve is

(28)

The solution of the above equation is

(29)

where the constants and are fixed asking for the asymptotic condition at infinity, eq. (25). The analogous procedure to find slowly rotating Kerr solutions in classical GR is presented in Appendix B for completeness, which is basically a reproduction of the results of Hartle hartle (). There we show that this procedure leads to the Kerr metric in lowest order and furthermore that the term is the only possible term in the expansion. Applying the same procedure to a static boson star bosonstars (), one can prove that such slowly rotating solutions do not exist Kobayashi:1994qi ().

In what follows we shall drop the index, . The angular velocity for different slowly rotating BHs is shown in Figs. (5). In the limit , one recovers the results for a small rotating BH in General Relativity, obtained perturbing a Schwarzschild BH, as described in the section below. Interestingly, the angular velocity of a BH in the EDGB can be larger than the one for a slowly rotating Kerr BH (see the bottom panel in Fig.(5)) with the same angular momentum. A numerical relation which can be computed from Fig.(5) is

(30)

while for a slowly rotating Kerr BH the above relation is . Thus, a DBH is more “compact” than a BH with same mass and angular momentum.

Figure 5: Top Panel: Angular velocity as a function of for different slowly rotating BHs in EDGB theory. From top to bottom: decreasing values for . For or solutions approach the slowly rotating Kerr BH (solid line). Bottom Panel: Angular velocity of the BH, , for a DBH (dashed line) with compared to the same for a slowly rotating Kerr BH (solid line) and a Kerr BH (dotted line). Up to results for the slowly rotating Kerr BH reproduce the exact ones. The difference in the angular velocity between slowly rotating DBHs and slowly rotating Kerr BHs and can be as large as .

iv.1 Ergoregion and superradiance

Ergoregions can develop in rotating spacetimes. The ergoregion is found by computing the surface on which vanishes. An approximate equation to the location of the ergoregion schutzexistenceergoregion () is

(31)

which is expected to be a good approximation specially for very compact objects, such as BHs. The solution of Eq. (31) is topologically a torus. In the equatorial plane we have

(32)

The existence and the boundaries of the ergoregions can be computed from the above equations. The ergoregion width for a DBH compared to both a slowly rotating and a full rotating Kerr BH is shown in Figure 6 for different values. The ergoregion width, , for a Kerr BH on the equatorial plane is . The ergoregion width for a DBH can be larger than the width for a slowly rotating Kerr BH. Therefore BH superradiance in EDGB is expected to be stronger than in GR. As expected the difference between slowly rotating DBHs and slowly rotating Kerr BHs tends to zero in the limit.

Figure 6: Ergoregion width, , for a DBH (solid line), a Kerr BH (dashed line) and a slowly rotating Kerr BH (dotted line) as function of the angular momentum for . Up to results for the slowly rotating Kerr BH reproduce the exact ones. The ergoregion width for a rotating DBH can be larger than the one for a Kerr BH.

V BH Geodesics in EDGB theory

In this section geodesics in the exterior of both non-rotating and slowly rotating EDGB black holes are discussed. If we neglect back-reaction effects, which we do in the following, the geodesics in this spacetime correspond to paths followed by time-like or null particles. In principle back-reaction effects should be important for large bodies or strong dilaton fields. However, for small point particles around DBH black holes they should be negligible. From the analysis of the geodetic motion we compute orbital frequencies related to the ISCO for both non-rotating and slowly rotating DBHs and the QNM frequencies for spherically symmetric BHs in EDGB theory in the eikonal limit. These quantities can be directly measured and can be used as a promising tool to study high-energy modifications of GR.

The motion of test particles around a DBH is deeply related to the coupling between matter and the dilaton. However it turns out that the coupling depends on the choice between two different frames, namely the string frame and the Einstein frame (for a discussion in scalar-tensor theories see Ref. Faraoni:1999hp () and references therein). The two frames are inequivalent. In the string frame the coupling between matter and the dilaton field is minimal, but Newton constant, , depends on the coordinates and light bending is not correctly reproduced Bruneton:2007si (). Conversely, in the Einstein frame the light bending is correctly reproduced, since is constant, but particles non-minimally couple to the dilaton field. This leads to a violation of the equivalence principle. This violation is small and compatible with the available tests of the equivalence principle Faraoni:1999hp () and it is indeed regarded as an important test in discerning the two frames. Which of the two frames is the physically relevant is an open question, which will eventually be settled by experiments.

Following Refs. kanti () and Torii:1996yi () we always assume the Einstein frame. In this frame the total action reads

where is the EDGB action, eq. (1), is the mass of the test particle and is the action which describes some coupling between the test particle and the dilaton field. The constant in the equation above is the coupling between the matter and the dilaton field. Its particular value depends on the specific theory from which the low energy theory comes. For low energy modifications from heterotic string theory, . We shall discuss the motion of a test particle keeping a general value of and we only specialize to when we discussing numerical results. As already mentioned, in the equation above we neglect gauge fields, such as Maxwell fields. Furthermore we neglect any coupling between the dilaton field and the test particle, =0, as well as any back-reaction of the background fields.

The string frame and the Einstein frame are related by a Weyl transformation Casadio:1998wu (). Since null geodesics equations are Weyl invariant, the motion of massless particles is the same in both frames and it is described by the standard geodesics equations.

v.1 Geodesics: non-rotating case

Following Chandrasekhar chandra () and Ref. Cardoso:2008bp () we consider a four-dimensional stationary, spherically symmetric line element

(34)

We also take a spherically symmetric dilaton and set . Due to the symmetry of the above background fields, the trajectory of a particle is planar, say and the following conserved quantities can be defined: the (dimensionless) specific energy of the test particle, and the specific angular momentum, , where , and are the energy, the angular momentum and the mass of the test particle respectively. Let us restrict attention to circular orbits, for which the Lagrangian is

(35)

From the Lagrangian above, the equations of motion for the coordinates read

(36)
(37)

where for light-like and timelike geodesics respectively and the derivative is intended to respect with the proper time.

v.1.1 Time-like geodesics

We set in equation (36). The requirement for a circular orbit at is , thus

We choose the notation , with the subscript standing for timelike. Since the energy must be real we require

(38)

The condition for a stable circular orbit is

(39)

From equations (37) we define the orbital angular velocity

(40)

Equations above reduce to the usual ones for .

For the well known Schwarzschild case we have and . In this case the condition (38) reads and from condition (39) for stable orbits, we find . For only unstable orbits exist, hence the radius is known as “innermost-stable-circular-orbit” (ISCO). The orbital angular velocity at the ISCO is .

To compute the above quantities in EDGB theory we set , and , where , and represent the spherically symmetric BH solution found in Ref. kanti (). We also specialize to the case . Numerical results for a EDGB non-rotating BH are shown in Table 1 and Fig. 7. The comparison with Schwarzschild BH (with the same mass) and inclusion of null geodesics is discussed in the section below. For completeness we show results both for and . The effect of a non-vanishing coupling is leading and it makes the angular frequency for a DBH smaller than in the Schwarzschild case (conversely the ISCO is larger). We find that the difference between the orbital frequency in GR and in EDGB theory can be as large as , depending on the coupling constant . In the limit Schwarzschild results are recovered. The largest deviations occur when relation (8) is saturated by chosing the appropriate . This corresponds to the maximum value , or equivalently to . It is worth noticing that the same occurs for a Reissner-Nordström BH, for which we have a relation between the electrical charge and the mass, . Qualitatively, taking in account the coupling , in EDGB theory the ISCO is always larger than it is for a Schwarzschild BH and the orbital frequency for a timelike geodesic is always smaller. Results for null geodesics do not depend on the coupling , as explained in the section below. In Table 1 we also show the quantity

which is the binding energy per unit rest-mass at the ISCO. Because the accretion inside the ISCO is assumed to be in free-fall, this is equal to the integrated luminosity per unit rest-mass accreted, i.e. the radiative efficiency of accretion. This efficiency ranges from for a Schwarzschild black hole hole to for an extremal Kerr hole.

Finally, we searched for static equilibrium solutions (following Maki and Shiraishi Maki:1992up ()), i.e., a point particle at rest at a distance say, . For this, we set in Eq. 36 and ask for . We find no solution to these equations, meaning such a configuration does not seem to be possible for this theory.

Figure 7: Angular frequency for circular timelike, , for (solid line) and for (dashed line) and for null-like, (dotted line) geodesics, normalized to the Schwarzschild value.
Table 1: Some geodesics-related quantities for a spherically symmetric EDGB BH: orbital frequencies for timelike (), and null () circular orbits and the radiative efficiency, . We compare to the Schwarzschild case, for different values of covering whole spectrum . We also give the percentage () difference between EDGB and GR and we show results both for the coupling and . Null-like geodesics do not depend on the coupling .

v.1.2 Null geodesics

We now consider null circular geodesics, labeled by a “c” subscript. It is easy to show that, setting in equation (36), the dilaton field, , gives no contribution. Requiring we find

(41)

The condition for a circular orbit reads

(42)

whose sign does not depend on . The orbital angular velocity is

(43)

where is the impact parameter. In the Schwarzschild case the condition (41) restricts the null circular orbits to . The orbital angular velocity is . Results for a EDGB spherically symmetric BH are summarized in Table 1. In this case there is no effect from the coupling and the difference between EDGB and GR theory is of the order . From (42) we find that and therefore only unstable null circular orbits exist. Hence also in EDGB case null geodesics are always unstable aganist small perturbations.

v.1.3 QNMs in the large limit for spherically symmetric EDGB BHs

Table 2: Real and imaginary part of the QNM frequencies defined by the formula in the eikonal approximation Cardoso:2008bp (). Terms between parenthesis are the differences with respect to the Schwarzschild case, for which .

Particularly promising for gravitational-wave detection are the characteristic vibration modes of black holes Berti:2005ys (); cardosotopical (). These modes, called quasinormal modes (QNMs), are exponentially damped sinusoids and carry an imprint of the black hole, its features being independent on what exactly excited the modes. QNMs are excited to a large amplitude for instance in the inspiral and subsequent merger of black hole or neutron star binaries. In fact, the ringdown phase of supermassive black holes can have a larger signal-to-noise ratio than any other signal Flanagan:1997sx (); Berti:2005ys (); cardosotopical (); Berti:2007zu (), justifying dedicated searches for ringdown in current gravitational-wave detectors such as LIGO or TAMA Goggin:2006bs (); Tsunesada:2005fe ().

It was recently shown Cardoso:2008bp () that there is a simple relation between QNMs and circular null geodesics in general spacetimes in the eikonal, i.e. large- limit,

(44)

where is defined in eq. (43) and

(45)

is the Lyapunov exponent, describing the inverse of the instability timescale of the geodesic. For circular null geodesics the argument of the square root in eq. (45) is always positive. QNMs are easly computed from the equation above and again the difference between the real and imaginary part of the QNM frequency for a DBH and for a Schwarzschild BH is order , as shown in Table 2. We have not attempted to compute the least damped QNMs for these black holes, but we will assume they behave in the same way as the coupling constant varies cardosotopical (). Discriminating such small percentage differences is in principle doable with the gravitational-wave detector LISA Berti:2005ys (); Berti:2007zu (), though it may be very challenging due to systematic errors Berti:2007fi (); Berti:2007dg (); cardosotopical (); Berti:2005ys (); Berti:2007zu (). As far as we know, the QNMs of DBHs have not been considered in the literature, even though their computation is extremely relevant. Nevertheless, the computation of QNMs of black holes in higher-dimensional Gauss-Bonnet theories without a dilaton, or purely dilatonic black holes has been done Abdalla:2005hu (); Konoplya:2004xx (); Konoplya:2001ji (), we expect the four-dimensional case to proceed along similar lines.

v.2 Geodesics: rotating case

We will now consider geodesics in the the slowly rotating DBHs spacetime studied earlier in Section IV. For that, let’s take the general four-dimensional static axisymmetric spacetime on the equatorial plane

(46)

where we specialize to and . We also consider the dilaton field on the equatorial plane and set . Some interesting special cases of the metric above are listed in Table 3. Following Chandrasekhar chandra () we consider planar orbits, for which the Lagrangian is

(47)

The generalized momenta are

and the Hamiltonian is

(48)

where again , for time-like and null geodesics respectively. Using the integrals of motion and the equations of motion read

Axial Sym Sph. Sym Kerr SK Slow rot