Harmonic oscillations of neutral particles in the metric
Abstract
We consider a well known static, axially symmetric, vacuum solution of Einstein equations belonging to Weyl’s class and determine the fundamental frequencies of small harmonic oscillations of test particles around stable circular orbits in the equatorial plane. We discuss the radial profiles of frequencies of the radial, latitudinal (vertical) and azimuthal (Keplerian) harmonic oscillations relative to the comoving and distant observers and compare with the corresponding ones in the Schwarzschild and Kerr geometries. We show that there exist latitudinal and radial frequencies of harmonic oscillations of particles moving along the circular orbits for which it is impossible to determine whether the central gravitating object is described by the slowly rotating Kerr solution or by a slightly deformed static spacetime.
I Introduction
The ZipoyVorhees spacetime, also known as metric is an asymptotically flat vacuum solution of Einstein’s equations which belongs to the Weyl class of static, axially symmetric spacetimes Zipoy (1966); Voorhees (1970). The metric is completely characterized by two parameters, namely which is related to the gravitational mass of the source and the deformation parameter . In the case of the Schwarzschild spacetime is recovered. On the other hand the cases of and correspond to oblate and prolate spheroidal sources, respectively, thus, showing that parameter can be considered as a deformation parameter and that for the coordinates are not spherical.
From the nohair theorem it is obvious that for the line element does not describe a black hole. In fact it can be shown that the surface corresponds to a genuine curvature singularity for every value of different from one Virbhadra (1996); Papadopoulos et al. (1981). The singular surface must be regarded are as infinitely redshifted surface which observationally may present features similar to the Schwarzschild event horizon. However, for certain values of the spacetime presents some unique features that allow it to be distinguished from the Schwarzschild solution.
For these reasons the metric can be considered as a black hole “mimicker” and, being an exact solution of Einstein’s equations, constitutes an excellent candidate to study possible astrophysical tests of black hole spacetimes.
The geometrical properties of the metric have been studied in Herrera et al. (1999); Herrera and Pastora (2000); HernándezPastora and Martín (1994); Bonnor (1992) while interior solutions have been found in Hernandez (1967); Stewart et al. (1982); Herrera et al. (2005). The motion of test particles and light rays in the metric has been studied in Herrera et al. (1999); Chowdhury et al. (2012); BenavidesGallego et al. (2019); Abdikamalov et al. (2019).
The frequencies of quasiperiodic oscillations in the Schwarzschild metric were studied for example in Abramowicz and Kluźniak (2005); Kološ et al. (2015), while for the Kerr metric they have been studied in various physical scenarios, for example in Török and Stuchlík (2005); Kološ et al. (2017); Tursunov et al. (2016). The application to astrophysics and the study of astrophysical black holes was discussed in Török et al. (2005); Bambi (2012); Bambi and Nampalliwar (2016); Stuchlík and Kološ (2015, 2016).
In the present article we study the frequencies of small harmonic oscillations of test particles about stable circular orbits in the metric and compare them with the corresponding frequencies in the Schwarzschild and Kerr spacetimes. We find that differences appear at small radii (either approaching the Innermost Stable Circular Orbit (ISCO) or approaching the infinitely redshifted surface) and a combination of measurements for epicyclic frequencies at small radii around compact objects could be used in principle to determine the nature of its geometry.
The paper is organized as follows: In section II we recap the equations describing the motion of test particles in the metric and derive the values of that separate different behaviours. Section III is devoted to the study of small harmonic oscillations for test particles about the circular geodesics. Finally in section IV the results are summarized and put in the context of possible future astrophysical observations of black holes. Throughout the paper we make use of natural units setting .
Ii Dynamics of test particles
In ErezRosen coordinates Erez and Rosen (1959) the metric is represented by the line element
(1)  
where
(2)  
From the asymptotic expansion of the gravitational potential it is easy to see that the total mass of the source as measured by an observer at infinity is HernándezPastora and Martín (1994). Also from evaluation of the Kretschmann scalar it is possible to see that the surface is a true curvature singularity for all values of Virbhadra (1996). ^{1}^{1}1However it is also an infinite redshift surface which means any signal emanating from it would be infinitely redshifted. It would therefore be rather innocous for an external observer. Therefore the radial coordinate in the spacetime takes values . If we understand the singularity as the regime at which the classical description fails, then we can interpret the surface as the boundary of an exotic compact object that is intrinsically quantumgravitational in nature. Our purpose is to investigate the properties of test particles orbiting around such an exotic compact object and determine whether they can in principle be distinguished from the corresponding cases around a black hole.
ii.1 Equations of motion
Since the spacetime under study does not depend explicitly on time, the Hamiltonian plays the role of the total energy of the system and governs the dynamics of neutral test particle. Such Hamiltonian for test particles in curved spacetime can written as
(3) 
where is fourmomentum that is defined as with and being mass and fourvelocity of the test particle, respectively. From Noether’s theorem, as the system does not depend explicitly on the coordinates and , we know that the associated conjugate momenta, namely the energy and angular momentum are conserved and
(4) 
From Eq. (4) one can find the and components of the fourvelocity of the test particle as
(5) 
and thus we obtain
(6) 
with
(7) 
The normalization condition leads to . Then, from Eq. (6) one finds that
(8) 
If one considers a particle moving on a plane with , as is the case for particles in accretion disks that are confined near the equatorial plane , then the equation of motion (8) takes the form
(9) 
which radial function can be written as
(10) 
with
(11) 
If, on the other hand, the particle is moving along a circular orbit on with , then the equation of motion (8) takes the form
(12) 
By combining the two conditions above, for a particle on circular orbit in the plane , we obtain
(13) 
and
(14) 
ii.2 Circular orbits
In the following we shall focus on the radii of characteristic circular orbits in the equatorial plane of the spacetime, since these are the most relevant orbits for astrophysical purposes, as they describe the motion of particles of gas in accretion disks around compact objects.
By solving Eqs. (13), simultaneously, one finds the specific energy and the specific angular momentum of test particles moving along circular orbits as
(15)  
(16) 
where is the angular velocity of the test particle as measured by distant observers and it is given by
(17) 
The above expressions (15)(17) rewritten for the metric read
(18)  
(19)  
(20) 
From the above expressions one can see that the regions of the spacetime where test particles can have circular orbits are given by:
(21) 
Moreover, from the fact that at the light ring (photon’s capture orbits) the specific energy (18) diverges because photon’s mass is zero and the relevant parameter is the ratio, , one can easily find the location of photon capture orbit as
(22) 
while for no photon capture orbit is present. By comparing (II.2) with (22) one can say that innermost positions of circular orbits of test particles are limited by the circular geodesics of massless particles. That is, photon circular orbit defines the existence threshold, , for timelike circular orbits. Further note that is always preceded by unstable circular orbits.
So circular orbits are therefore not stable or bound for . Unbound circular orbits are separated from the bound orbits by a critical geodesic called the marginally bound circular orbit that is found from the zero binding energy Bardeen et al. (1972); Hod (2013). Since the spacetime is asymptotically flat, therefore, gives
A particle with energy has unbound circular orbit. This means that the orbit is unstable in such a way that with an infinitesimal small outward perturbation, the particle will escape to infinity on an asymptotically hyperbolic trajectory. On the other hand for , perturbation of unstable circular orbit would however remain bound but it would climb to larger radius to have a stable circular orbit. In the case of , by solving Eq. (II.2), one obtains the marginally bound circular orbit in the Schwarzschild spacetime, .
Now we find the marginally stable circular orbits, also called innermost stable circular orbits (ISCO). All the stable circular orbits of test particles (with radius ) satisfy the following condition:
(24) 
with the equality holding for the smallest allowed value for stable circular orbits, namely the ISCO. By solving Eq. (24) one finds two roots for the ISCO given by
(25) 
Once again, for we retrieve which is the value of the ISCO for the Schwarzschild geometry. Here it is easy to notice that for , the two values of the ISCO coincide. Moreover, for the value of is always greater than the bounds imposed by conditions (II.2) and therefore there is always at least one marginally stable circular orbit. However, does not satisfy the bounds imposed by conditions (II.2) for values . Although it is natural to expect that stable circular orbits will be allowed at great distances and cease to exist at a certain distance from the source, the above discussion shows that in the range there exist a second range of stable circular orbits closer to center (see Fig. 1). We can understand better the reason for this behaviour by analyzing the effective potential in eq. (11) term by term. It is easy to see that at small radii the term proportional to will dominate over the other terms. Then as the term will go at the denominator causing the change in the behaviour of the effective potential. This new region where stable circular orbits are allowed extends from until a finite distance, determined by the second root of the ISCO equation. However, for the two values of the ISCO radii coincide and for stable circular orbits are allowed at any distance from the source, similarly to the Newtonian case.
In order to extract more information on the region of the stable circular orbits we can plot the radial function in eq. (10) for different values of (see Fig. 2).
Depending on the value of the effective potential can have up to two zeros. At these zeros the particle’s radial velocity vanishes and therefore, they represent the turning points of the particle’s trajectories. In the regions where the effective potential is positive no stable orbits are allowed for the test particle’s (orbits are unstable). If , the effective potential has two zeros. Outside these zeros, and , stable circular orbits exist. However, in the region between these zeros, , no stable circular orbits can exist; i.e. in this region only unstable circular orbits can exist. For , the behaviour is similar to the Schwarzschild case, where stable circular orbits are possible only for . For the outer and inner ISCOs coincide at and in both sides of this radius the radial function is negative, so the stable orbits can exist. Finally for the behaviour of the effective potential resembles the Newtonian case and all circular orbits are stable. This suggests that the term responsible for the angular momentum interaction with the mass (the term in the Schwarzschild case) behaves qualitatively similar to the Schwarzschild case for , namely it is attractive causing the stable circular orbits to cease to exist at a certain radius, while it behaves in the opposite way for . The interaction between angular momentum and mass turns from attractive to repulsive as goes from to . This can be clearly seen by studying the term
(26) 
in . It is immediately seen for we have and therefore the relativistic correction to the Newtonian behaviour vanishes. Then from the fact that changes sign at we see that changes from increasing (attractive) for to decreasing (repulsive) for .
Thus, taking into account the conditions (II.2), we can conclude that the stable circular orbits can exist in following two different regions depending on the parameter :
It is interesting to notice that through the behaviour of the motion of test particles we can gain some insight on the nature of the curvature singularity in the spacetime. In the case of Schwarzschild spacetime, the singularity located at is attractive and particles are crushed in it by diverging tidal forces. Similarly, in the case of the metric we can see that the same behaviour occurs at only when . On the other hand, when the interaction term between the mass and the angular momentum of the test particle is repulsive and blows up at , as in the case of the Monopole–Quadrupole solution of the Weyl solutions which describe small deviations from spherical symmetry Herrera (2005). Therefore in this case the singularity is “repulsive” and test particles will be ejected rather than crushed. Finally in the case when we see that goes to zero as goes to suggesting that the singularity in this case is milder, namely a particle will not be crushed by tidal forces at , while the curvature will still diverge.
Iii Harmonic oscillations of neutral test particle
In this section we will study the motion of test particles that slightly depart from circular orbit. As we mentioned in the previous section, stable circular orbits for test particles in the spacetime are located at a radius and latitudinal angle corresponding to the minimum of the function (or and , respectively) given in Eq. (7). If a test particle deviates slightly from the stable circular orbit it will start to oscillate around its equilibrium value performing an epicyclic motion. This oscillating motion is governed by the epicyclic frequencies.
iii.1 Epicyclic frequencies
For the sake of clarity we shall restrict the analysis to the linear regime, and consider separately the case of purely radial, i.e. , , and purely vertical,i.e. , , epicyclic oscillations about circular orbits in the equatorial plane . If we write the Taylor expansion of the functions and in powers of and , respectively as
(27)  
(28) 
and apply the circularity conditions (13) and (14), we arrive at
(29)  
(30) 
Considering that the total energy of the orbit is conserved, one finds that
(31)  
(32) 
Where and are the trivial solutions corresponding to no oscillations. It is immediately apparent that the parts within the square brackets of the above equations are in the form of harmonic oscillators as Abramowicz and Kluźniak (2005); Toshmatov et al. (2017)
(33)  
(34) 
with
(35)  
(36) 
where the specific energy and angular momentum of the test particles on circular orbits are given by Eq. (18) and Eq. (19), respectively. Therefore we can obtain the explicit forms of the frequencies as
(37)  
(38) 
By inspecting expression (37) one can notice that zeros of expression in the square bracket correspond to the radii of the ISCO as given in Eq. (25). Thus, we see that radial epicyclic frequency vanishes at the ISCO, . This is reasonable, considering the fact that the ISCO is a marginally stable orbit and that below ISCO, at least for values of , the particles start to fall towards the central object, and no radial oscillations occur.
Moreover, there is another very important angular frequency of circular epicyclic motion of the particle, namely the azimuthal (or Keplerian) frequency of the circular motion in the equatorial plane, defined by the relation
(39) 
Note that the frequencies , and are measured with respect to the proper time of a comoving observer. To get the observed frequencies by a observer at infinity, one needs to divide them by the square of the redshift factor as
(40) 
where squared redshift factor for the metric is given by
(41) 
Thus, frequencies (37), (38) and (39) take the following form when they are measured by observers at infinity:
(42)  
(43)  
(44) 
In Fig. 3 we show the dependence of the epicyclic frequencies for neutral test particles about the radii of stable circular orbits for different values of . One can see that the azimuthal frequencies are monotonically decreasing functions of the radius. More in detail, depending on the value of the parameter the epicyclic frequencies as measured by observers at spatial infinity behave as follows:

If , the latitudinal and azimuthal epicyclic frequencies diverge, , at ; the radial epicyclic frequency vanishes at the radius of the outer ISCO, namely at (see first panel of Fig. 3).

If , the latitudinal and azimuthal epicyclic frequencies diverge, , at ; the radial epicyclic frequency vanishes at the radius of ISCO, namely at . These are the known results for the Schwarzschild spacetime.

If , the latitudinal and azimuthal epicyclic frequencies vanish, , at ; the radial epicyclic frequency vanishes at the radius of ISCO, namely , at (see second panel of Fig. 3).

If , the latitudinal frequency vanishes, at , while the azimuthal frequency diverges, at ; the radial frequency vanishes at the extremal ISCO, namely at , and then monotonically increases with smaller radii to diverge at the singularity, namely at (see third panel of Fig. 3).

If , the latitudinal epicyclic frequency vanishes, , at , while the radial and azimuthal epicyclic frequencies diverge, , at . Notice that there is no ISCO in this range of values for (see fourth panel of Fig. 3).
Small changes in value of the parameter do not affect much the azimuthal, , and latitudinal, , frequencies, however, the effects on the radial frequency, , are more significant. However, a striking difference appears in the longitudinal and azimuthal frequencies, when we compare with the Schwarzschild case. If fact, for both and have finite, non vanishing value at , while for they either diverge or vanish.
The local extrema of the epicyclic frequencies are given by the condition
(45) 
where . For the values radial frequency always has local maximum close to the outer ISCO radius, , and this maximum value increases with decreasing value of . Thus, when , the radial epicyclic frequency of the spacetime reaches the possible highest maximum value, at .
It is well known that at large distances the Schwarzschild spacetime has Newtonian behaviour and for Newton all the three components of epicyclic frequencies have the same value given by
(46) 
As expected, at large distances the epicyclic frequencies in the metric exhibit Newtonian behaviour as
(47) 
which can be interpreted as a further confirmation that the total mass of the spacetime as measured by distant observers is .
In order to estimate the effects due to the departure from spherical symmetry of the spacetime, we can study how small deviations from the Schwarzschild metric affect the epicyclic frequencies. By considering with , the epicyclic frequencies take the form
(48)  
(49)  
(50) 
where
(51)  
(52) 
Thus, one can see from Eqs. (48), (49), (50) that oblate (prolate), i.e. values of (), deviations from spherical symmetry, i.e. Schwarzschild, increase the values of all components of the epicyclic frequency of test particles.
iii.2 Epicyclic frequencies in Kerr spacetime
In order to compare epicyclic frequencies of neutral test particles moving along circular orbits in the spacetime with the ones in the Kerr spacetime, we briefly review the epicyclic frequencies around a Kerr black hole. In BoyerLindquist coordinates the line element for the Kerr spacetime is given by:
(53)  
where
(54) 
The comparison between the and the Kerr spacetimes is justified by the fact at large distances both metrics tend to become asymptotically flat and so ErezRosen as well as BoyerLindquist coordinates tend to become the usual spherical coordinates. Therefore an observer at infinity, measuring some features of the motion of test particles in the accretion disk around a Kerr black hole would be able to compare, at least in principle, the obtained measurements with the expected observations of the corresponding situation in the spacetime. Similarly to the static axially symmetric case, in the stationary case, due to rotation of the source, circular orbits are located on the equatorial plane, thus allowing us to restrict the analysis to the case . The specific energy, angular momentum and angular velocity of the particle moving on a circular orbit around a Kerr black hole are derived from relations (15), (16) and (17), respectively as
(55)  
(56)  
(57) 
Where the and signs correspond to corotating and counterrotating particles with respect to the direction of rotation of the black hole. Again one can find the radii of circular photon orbits, marginally bound orbits and ISCO from Bardeen et al. (1972)
(58)  
(59)  
In Fig. 4 we show the degeneracy between the metric and the Kerr spacetime resulting from measurements of the ISCO or of the photon capture radius. In fact each given measured value of corresponds to one value of and one value on showing that the measurement of the ISCO radius alone is not enough to distinguish between the two spacetimes. Similar arguments hold for the photon capture radius. However, by observing that the degeneracy in the two cases is different we can conclude that a simultaneous measurement of and with enough precision would allow to distinguish between the two geometries around astrophysical compact objects.
The epicyclic frequencies, with respect to distant observers, of a test particle moving along circular orbits around a Kerr black hole are given by Török and Stuchlík (2005); Török et al. (2005)
(61)  
(62)  
(63) 
In the slow rotation limit, , the above expressions take the form
(64)  
(65)  
(66) 
Thus, one can see from (64), (65), (66) that rotation of the spacetime increases (decreases) radial frequency of the harmonic oscillations and decreases (increases) latitudinal and azimuthal frequency of the harmonic oscillations of the corotating (counterrotating) particle moving along the circular orbits.
This shows that the degeneracy between the Kerr metric and the metric due to measurements of the value of the ISCO can be broken if one is able to measure simultaneously and one between and .
To this aim, we consider the values of and for which the epicyclic frequencies of the neutral test particles have the same value. As we mentioned, if the circular orbits where the particles are moving are located far away from the central object then the two metrics can not be distinguished, since at large distances all frequencies become Newtonian. Therefore, we will focus our attention on relatively short distances. For simplicity we shall also consider slight deviations from the Schwarzschild spacetime for both cases, i.e. small deformations with , and slow rotation, . Thus, by using the expressions of epicyclic frequencies for the metric given by Eqs. (48), (49), (50) and for the Kerr black hole, given by Eqs. (64), (65), (66) we find the expressions for the coincidence of all components of the frequencies as
(67)  
(68)  
(69) 
In Fig. 5 we show the above expressions as functions of the dimensionless radial coordinate .
Since test particles are oscillating due to slight deviations from stable circular orbits in a spacetime which slightly deviates from spherically symmetry, it is useful to focus the attention on the region near , corresponding to the ISCO for the Schwarzschild metric. One can see from Fig. 5 that the three curves corresponding to same values for the three frequencies in Kerr and in the metric intersect in three different points. Therefore, if one measurement was in principle able to measure all three frequencies, the results could easily distinguish the spacetime from Kerr. On the other hand, a measurement of only two frequencies, may not be enough if performed in the vicinity of the point where two curves overlap. For example, the curves obtained from and overlap at with . Thus, a measurement of these two frequencies in the vicinity of this radius would not be able to distinguish the spacetime from Kerr.
From the above considerations we can derive some conclusion on which kind of objects may or may not be distinguished from a Kerr black hole.

If test particles are corotating relative to the Kerr black hole, i.e. , and the source of the spacetime is oblate, i.e. , then the two geometries are distinguishable.

If test particles are corotating relative to the Kerr black hole, i.e. , and the source of the spacetime is prolate, i.e. , then the two geometries may be indistinguishable.

If test particles are counterrotating relative to the Kerr black hole, i.e. , and the source of the spacetime is oblate, i.e. , then the two geometries may be indistinguishable.

If test particles are counterrotating relative to the Kerr black hole, i.e. , and the source of the spacetime is prolate, i.e. , then the two geometries are distinguishable.
Finally, if the latitudinal and radial frequencies of harmonic oscillations of a particle moving along the circular orbit with radius are detected, one will not be able to distinguish if the central gravitating object is described by the slowly rotating Kerr (and which rotational direction relative to central object is particle moving) or slightly deformed (and type of deformation) spacetimes. Since for all three frequencies the ratio is negative, similar considerations as the ones above apply also to the simultaneous measurement of radial and azimuthal frequencies and to the simultaneous measurement of azimuthal and latitudinal frequencies.
Iv Conclusion
The recent observation of the ‘black hole shadow’ by the Event Horizon Telescope, has opened the door to precise measurements of the geometry in the vicinity of black hole candidates Event Horizon Telescope Collaboration et al. (2019). To this day, astrophysical observations of black hole candidates have shown no departure from the relativistic description (see for example the recent measurement of the gravitational redshift of stars orbiting the supermassive black hole candidate at the center of the Milky Way Gravity Collaboration et al. (2018)). However such measurements have not been able yet to test whether the geometry in the vicinity of black hole candidates is well described by the Kerr metric. For this reason it is useful to study the predictions obtained from different geometries which, while being solutions of Einstein’s field equations in vacuum, do not describe a black hole. In this respect the metric is an ideal candidate due to its simplicity and to its immediate connection to the Schwarzschild spacetime.
The metric describes a deformed Schwarzschild object, it is prolate for and oblate for . It turns out that for the former, particle dynamics continues to have the same qualitative features as the Schwarzschild’s while for the latter it has much richer structure. In particular, there are two disjoint regions for existence of stable circular orbits for the range while for , there exist only stable circular orbits. This is exactly like the Newtonian case.
In the present paper we have studied some properties of characteristic circular orbits, i.e. circular null geodesics, marginally bound orbits, stable circular orbits, for massive and massless test particles around the metric. In particular we focused on the frequencies of harmonic oscillations around stable circular orbits for massive test particles.
We have shown that small deviations from equatorial circular orbits of the particle in the spacetime are described by harmonic oscillations in the radial, vertical, and latitudinal directions. By solving the harmonic oscillator equations we have found the analytical expressions for the frequencies and studied the epicyclic frequencies of particles in the uncoupled orthogonal (radial), vertical (latitudinal) and axial (azimuthal) oscillatory modes relative to comoving and distant observers. The frequencies as measured by distant observers are particularly important since they could, in principle, be measured from the observations of accretion disks around black hole candidates.
We have shown that radial oscillations vanish at the ISCO of the spacetime, and below the ISCO the particles fall towards the central object, and no radial oscillations occur. Since at large distances the spacetime approaches the Newtonian limit, all epicyclic frequencies tend to the same limit with . For this reason, we have studied the behaviour of the frequencies in the relativistic regime for slight deviations from spherical symmetry and compared the results with the corresponding frequencies in the slowly rotating Kerr black hole geometry.
We have shown that the two metrics are distinguishable if a simultaneous measurement of all three components of the epicyclic frequencies is available. However, if only two frequencies of oscillations for the test particles are obtained (for example near for the radial and vertical oscillations) then it may not be possible to distinguish if the geometry around the central object is described by the or Kerr spacetimes. Similarly we have shown that a simultaneous measurement of the radii of the ISCO and the photon capture orbit may be able to distinguish between the two geometries, while the measurement of one radius only is not enough.
As new an better observations of astrophysical black hole candidates become available, we will soon be able to test the nature of the geometry around such candidates and answer the question whether all such objects must necessarily be described by the Kerr spacetime or if nature allows for some other possibility.
Acknowledgments
The work was developed under the Nazarbayev University Faculty Development Competitive Research Grant No. 090118FD5348. The authors acknowledge the support of the Ministry of Education of Kazakhstan’s target program IRN: BR05236454 and Uzbekistan Ministry for Innovation Development Grants No. VAFAF2008 and No. YFAFtech20188. N.D. would like to acknowledge Nazarbayev University, Astana, Kazakhstan for the hospitality. D.M. and N.D. wish to express their gratitude to Bobomurat Ahmedov for useful discussion on the properties of the gamma metric and acknowledge the Ulugh Beg Astronomical Institute, Tashkent, Uzbekistan for the hospitality through Abdus Salam International Centre for Theoretical Physics through Grant No. OEANT01.
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