by Sebastian Völkel and Kostas Kokkotas
Could you distinguish the sound of a wormhole from an ultra compact star or black hole?
Such an exotic, though quite fundamental question, could be asked to any physicist after the groundbreaking and Nobel Prize winning discoveries of gravitational waves from merging black holes and neutron stars. Gravitational waves provide mankind with a novel sense, the ability to hear the universe. This analogy, between sound waves and gravitational waves, will bring to the minds of many physicists Mark Kac’s famous question: “Can One hear the Shape of a Drum?” , and not just to the drummers amongst us. The possibility of this analogy is one of the ways in which gravitational waves are very distinct from the usual tool of astronomy, light.
To answer the question for our exotic instruments, we will rephrase it in a more technical form. In the simplest version one can describe linear perturbations of spherically symmetric and non-rotating models of wormholes and ultra compact stars. It is well known that the perturbation equations for these cases can simplify to the study of the one-dimensional wave equation with an effective potential. The solutions, which are usually given as a set of modes, represent the characteristic sound of the object. The so-called quasi-normal mode (QNM) spectrum is the starting point for our discussion.
In our recent work  we extend an inverse spectrum method that we already developed for the study of ultra compact stars last year in  to the problem of some particular wormhole models. The underlying idea is to use the knowledge of the QNM spectrum of an astrophysical object in order to unveil its properties. The method we use is rather general and not limited to the study of wormholes or stars alone. It is possible to recover significant information about two different types of potentials being drawn in Fig. 2. The perturbation potentials of ultra compact stars and some other exotic compact objects are qualitatively described by a potential well next to a single potential barrier, shown in the left panel. On the other hand, some type of wormholes, like the one proposed by Damour and Solodukhin [4, 5] in some range of parameters, are described by two symmetric potential barriers that also provide a potential well between them. This is shown in the right panel. The technical difference in the description of both potentials is that the first admits three classical turning points, while the latter admits four. Both types of potentials admit so-called “trapped” modes. These correspond to (semi-) bound states in the potential well that slowly decay and has led to the unsettled question of whether the so-called “echoes” have been heard by LIGO [6–8]. If such signals are present in the data, they are smoking guns that the final objects are not black holes, but at the same time more compact than any realistic neutron star model we know [9, 10].
By using the inverse spectrum method, which is based on a combination of the “inverted” Bohr-Sommerfeld rule and Gamow formula [11–15], we can approximately recover the perturbation potential once the trapped modes are provided. However, it is known from the inversion of the Bohr-Sommerfeld rule for pure potential wells (with two classical turning points), that every potential with the same “width” admits the same spectrum. So if one of the two turning points is not provided, there are infinitely many potentials that admit the same spectrum. In our work we arrive at a similar result for the two types of potentials being studied. We found that using the trapped modes alone, there can be infinitely many equivalent potentials, for both types of potentials. Therefore, just like the answer to Marc Kac’s question is that different shapes of a drum can produce the same sound, the nature of the astrophysical object cannot be resolved uniquely. This includes the type of object (star-like or wormhole-like) and what specifictype of star or wormhole itself.
Luckily, in our case we can use additional information about the potential by rather general assumptions, like the validity of Birkhoff’s theorem for the external space-time of stars or that the underlying wormhole should be symmetric. Using these rather general assumptions, it is possible to find a unique potential of each type. We thus end up with the situation that one might provide us with the trapped modes and we can recover two specific potentials, but we can not tell which of the two types of objects is the correct one.
However, there is at least one way out of the problem. The spectrum itself is a time-independent property, so how does the situation change if we include information from scattered waves on these objects? This brings us to the study of “echoes”, which are expected for both type of objects. The first scattering of an incoming pulse provides us with information about the width of the region around the peak of the potential barrier. This can, in principle, resolve the ambiguity and we can distinguish between the two otherwise equivalent potentials, since their potential barriers are of different widths.
A natural question is how this approach can be applied to black holes. It is true, that this question has been already addressed in various studies with conflicting answers [9, 16–21]. One approach proposes that wormholes can mimic black holes at all times, while another one suggests that wormholes emit similar signals at early times, but the late parts of the signals are different. A closer look to the relevant perturbation potentials, like the single barrier for a black hole (and some wormhole models!) and the double barrier of some of the wormholes, simply show that it depends on the type of wormhole model under consideration. Thus it remains an open question what a reasonable wormhole should be. The answer might be found in the future by the LIGO/Virgo observatories.
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