Can one hear the shape of an ultra compact object?

by Sebastian Völkel and Kostas Kokkotas

A journey from ultra compact objects to quasi-normal modes and back

Never before has gravitational wave research been more promising and attractive than nowadays. With the repeated detection of gravitational waves from binary black hole mergers by LIGO [1–3], not only the long-standing pursue for one of Einstein’s most challenging predictions was confirmed, but also a milestone for many future applications reaching from fundamental physics to astronomy was set. One of the many applications that could follow is addressed in our new CQG paper [4] and shall be broadly presented within a more general introduction to some recent developments in the following lines.

CQG+ format pictures

About the Authors (Left to right): Sebastian Völkel is a first-year PhD student in the Theoretical Astrophysics group of Professor Kostas Kokkotas at the University of Tübingen, located in the south of Germany. Among his research interests is the study of compact objects along with their gravitational wave properties.  Professor Kostas Kokkotas is leading the group of Theoretical Astrophysics at the University of Tübingen. The focus of his research is on the dynamics of compact objects (neutron stars & black-holes), gravitational waves and alternative theories of gravity. More information about the group can be found here.

Gravitational waves detected by LIGO, and more sensitive detectors in the future, will allow us to study compact objects like Earth’s structure via seismic waves. It is known for a long time that the gravitational perturbations of black holes and neutron stars lead to characteristic modes, whose frequencies and damping times are imprinted in the gravitational waves they emit, see [5, 6] for two classic reviews from that time. Identifying these so called quasi-normal modes in future detections will provide us with a unique tool to investigate the properties of the source or even test the underlying gravitational theory itself.

It can be shown that under simplified conditions (e.g. spherical symmetry and no rotation) the perturbation problem reduces to solving a one dimensional wave equation with an effective potential that characterizes the object. For a Schwarzschild black hole it is known as the Regge-Wheeler [7] (axial) or Zerilli [8] (polar) potential and describes a single potential barrier. Neutron stars on the other hand are qualitatively described by a reflective potential at the center that falls off with increasing distance. The quasi-normal mode spectra of both systems are genuinely different, which might be surprising since both type of objects are very compact compared to other astronomical systems. The key difference to understand this are the different type of potentials and boundary conditions being applied. For both systems one typically considers outgoing waves, but for black holes one demands purely ingoing radiation at the horizon while the different type of potential for compact stars demands regularity and thus a vanishing amplitude at their center.

Even though quasi-normal modes and the corresponding time evolution of perturbations from ultra compact stars have already been studied decades ago [9–14], they were recently revived under the name “echoes”, see [15] for the latest review. Inspired by exotic compact objects or modifications of the black hole horizon [16], a similar type of perturbation potential emerges. For horizonless objects, more compact than R/M≲3 the typical potential is shown in Fig. 1. For this kind of potentials different classes of modes appeared which are known for a long time as “trapped w-modes”. These modes can be understood in terms of quasi-bound states described by a generalized Bohr-Sommerfeld rule [17, 18]. Trapped in the potential well, their damping times are very long and depend on the shape of the potential barrier.


FIG. 1. The axial mode potential for constant density stars with varying compactness R/M and angular decomposition parameter l. Figure taken from [18] S. H. Völkel and K. D. Kokkotas, Classical and Quantum Gravity 34, 125006 (2017),

The time evolution of the perturbations shows some interesting new features which are not parts of the previously known spectrum. For example, if such an object gets perturbed by an ingoing pulse, the first scattering at the potential barrier reflects one part of the pulse, while another one gets transmitted [11]. The reflected pulse carries the information of the potential barrier, since it turns out to be well described by the fundamental black hole quasi-normal mode frequency. The transmitted pulse will be reflected back and forth inside the potential well and excite the trapped w-modes. The bouncing pulses in the potential well being partially transmitted and reflected are leading to continuous emission of wave packets carrying information about both the potential well and the barrier. These outgoing pulses have recently been called echoes and triggered a lot of attention, since there were claims of their detection in the LIGO data [19].

Let us assume for a moment that some kind of ultra compact object with this type of potential exists. What could we learn from the echoes and the trapped modes? It is clear that the times between pulses should correspond to the width of the potential, but can one reconstruct in detail the shape of the potential from the trapped modes? It turns out that this question is very similar to the famous question whether one can hear the shape of a drum [20]. In our recent CQG paper [4] we show how the spectrum of the trapped modes can be used to successfully reconstruct (approximately) the potential, see Fig. 2. The individual reconstruction of a single potential well from real eigenvalues or of a single potential barrier from the transmission function was already known in the literature [21, 22]. Within the realm of semi-classical methods, like the Bohr-Sommerfeld rule, we were able to reconstruct the combined potentials (barrier and well) for the complex valued spectra of ultra compact stars. It turns out that the reconstruction is unique, thanks to Birkhoff’s theorem, which implies the uniqueness of the perturbation potential outside the object. In our paper, we applied the inverse method to successfully reconstruct the axial mode potential of constant density stars and gravastars. It is not surprising that the reconstruction is more accurate the more modes the potential admits.


FIG. 2. The reconstruction works roughly in three steps. a) The width L1(E)= x1(E)-x0(E) between the first two turning points is determined. b) The width of the barrier L2(E)= x2(E)-x1(E) is determined. c) To find a unique solution the external turning point x2(E) is provided using Birkoff’s theorem. Figure is taken from [4]  S. H. Völkel and K. D. Kokkotas, Classical and Quantum Gravity. 34 175015 (2017), arXiv:1704.07517 [gr-qc].

As always, reality is more complicated and effects like coupling to polar perturbations and rotation will affect the picture. They will lead in general to coupled set of equations, which can not be easily studied with the same semi-classical methods. Addressing these challenging problems is the next step of our study, while the detectability of “echoes” with gravitational wave detectors is already partially addressed in [23].

[1] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
[2] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. Lett. 116, 241103 (2016).
[3] B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Phys. Rev. Lett. 118, 221101 (2017).
[4] S. H. Völkel and K. D. Kokkotas, Classical and Quantum Gravity. 34 175015 (2017), arXiv:1704.07517 [gr-qc].
[5] K. D. Kokkotas and B. G. Schmidt, Living Reviews in Relativity 2, 2 (1999), gr-qc/9909058.
[6] H.-P. Nollert, Classical and Quantum Gravity 16, R159 (1999).
[7] T. Regge and J. A. Wheeler, Physical Review 108, 1063 (1957).
[8] F. J. Zerilli, Physical Review Letters 24, 737 (1970).
[9] S. Chandrasekhar and V. Ferrari, Proceedings of the Royal Society of London Series A 434, 449 (1991).
[10] K. D. Kokkotas, Mon. Not. R. Astron. Soc. 268, 1015 (1994).
[11] K. D. Kokkotas, in Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, September 26-October 6, 1995 (1995) pp. 89–102, arXiv:gr-qc/9603024 [gr-qc].
[12] K. Tominaga, M. Saijo, and K.-i. Maeda, Phys. Rev. D 60, 024004 (1999).
[13] V. Ferrari and K. D. Kokkotas, Phys. Rev. D 62, 107504 (2000), gr-qc/0008057.
[14] J. Ruoff, Phys. Rev. D 63, 064018 (2001).
[15] V. Cardoso and P. Pani, (2017), arXiv:1707.03021 [gr-qc].
[16] V. Cardoso, E. Franzin, and P. Pani, Physical Review Letters 116, 171101 (2016), arXiv:1602.07309 [gr-qc].
[17] V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, Phys. Rev. D 90, 044069 (2014), arXiv:1406.5510 [gr-qc].
[18] S. H. Völkel and K. D. Kokkotas, Classical and Quantum Gravity 34, 125006 (2017), arXiv:1703.08156 [gr-qc].
[19] J. Abedi, H. Dykaar, and N. Afshordi, ArXiv e-prints (2016), arXiv:1612.00266 [gr-qc].
[20] M. Kac, The American Mathematical Monthly 73, 1 (1966).
[21] J. A. Wheeler, Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann (Princeton University Press, 2015) pp. 351–422.
[22] J. C. Lazenby and D. J. Griffiths, American Journal of Physics 48, 432 (1980).
[23] A. Maselli, S. H. Völkel, and K. D. Kokkotas, Phys. Rev. D 96, 064045 (2017)

Read the full article in Classical and Quantum Gravity:
Ultra compact stars: reconstructing the perturbation potential
Völkel et al 2017. Class. Quantum Grav. 34 175015

Sign up to follow CQG+ by entering your email address in the ‘follow’ box at the foot of this page for instant notification of new entries.

CQG papers are selected for promotion based on the content of the referee reports. The papers you read about on CQG+ have been rated ‘high quality’ by your peers.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.