On the analogy between spinning disks coming to rest and merging black holes Open Access

Many of us have observed a spinning coin coming to rest on a table and have probably been captivated by its relatively long-lasting dynamics accompanied by a ringing with progressively increasing frequency. A disk coming to rest on a solid surface typically exhibits both precessional and rotational motion. The precession frequency progressively increases as the angle the disk forms with the platform decreases until the disk reaches full contact with the platform. This phenomenon has received increased attention since the promotion of Euler's disk, an educational toy with minimal dissipation that can roll and spin for several minutes. In the last few decades, the dynamics of a spinning disk have been the object of theoretical^1,2 and experimental^3–5 research works.

The motion of a spinning disk coming to rest presents some analogies with another phenomenon that occurs on a completely different scale: the merging of two compact astrophysical bodies.⁶ The analogy between spinning disks and merging black holes was first mentioned by Bildsten² in his reply to Moffat's model¹ of Euler's disk. The scientific community's interest in merging events has significantly increased following the first detection of the gravitational waves generated by two merging black holes.⁷ This historic measurement confirmed once again Einstein's theory of general relativity by detecting the gravitational wave produced during the final stages of the merger.

While previous educational and research works investigated either the disk dynamics or the emitted sound, here we measure and analyze the vertical vibration of the platform on which the disk spins and rolls. The experimental configuration proposed here is especially simple, and relevant signals can be acquired with a smartphone. This configuration has been developed as a class project by a group of master's students in the context of an advanced course in experimental physics. The manuscript is organized as follows. We first lay out analytical descriptions of the spinning disk coming to rest and of black hole mergers and highlight the similarities and differences between the systems (Sec. II). We experimentally show how the platform vibration is related to the precession of the disk (Sec. III) and compare it to the gravitational signal created by the merging of two black holes (Sec. IV). This work could serve as a reference for physics instructors in mechanics and astrophysics and help the students develop the ability to recognize analogies.

This section presents the analytical descriptions of the mechanics of a spinning disk coming to rest and that of two merging black holes. The schematics of the two phenomena are presented in Fig. 1. A side-by-side comparison of the main quantities derived in this section can be found in Table I.

We describe how to derive expressions for the disk's precession angular velocity $Ω$ and for the black holes' orbital angular velocity $ωK$ as a function of time, which are at the core of our analogy. The descriptions are developed in terms of the variation of the systems' total energy as a function of time. The total energy of the disk during its motion, which combines rotation about its symmetry axis and precession, depends on the angle, $α(t)$⁠, it forms with the underlying surface, which decreases over time due to dissipation. This angle can be related to the disk's precession angular velocity. On the contrary, the total energy of the black hole system depends on the distance between them, $l(t)$⁠, which decreases over time due to the radiation of gravitational waves and is related to the system's orbital angular velocity. The power laws describing $Ω(t)$ and $ωK(t)$ exhibit singular behaviors corresponding to the moments when the disk comes to rest and the two black holes merge, respectively. The power laws are characterized by their exponents that depend on the kind of dissipation involved: for the case of the black hole merger it is very well described by the quadrupole mass moment approximation (see Ferrari et al.⁶) resulting in a tiny range of variation for the associated exponent, whereas different dissipation mechanisms could be involved with the disk dynamics, including friction at the contact point and air resistance, resulting in a much wider range of variation for the measured exponent. While the black holes emit gravitational waves, the disk produces vibrations on the underlying platform, the measurement of which we present in Sec. III.

The orbital angular velocity diverges when $t→tc$⁠. Since gravitational waves in vacuum are not dispersive, the time evolution of their strain measured on Earth reflects the time variation of the orbital separation distance during the black hole merger. It is worth noting that, since the emission frequency is twice the orbital frequency,⁶ one gravitational wavelength is emitted in half an orbiting period.

The comparison between the main physical parameters for both the spinning disk and the merger of two black holes is reported in Table I. The disk's precession angular velocity and the orbital angular velocity of the black holes exhibit similar scaling with singularities at $t=tc$⁠. The frequencies of both systems increase as the singularity is approached. While the scaling of the black hole merger is characterized by a well-known exponent (−3/8), the scaling of the disk generally depends on the type of dissipation considered via the parameter $γ$⁠. This exponent will be experimentally determined in our system.

Our experiments focused on the vertical vibration of the platform on which the disk spins. The experimental setup is represented in Fig. 2(a). A custom-made aluminum disk with mass $m=154.2 ± 0.1$ g, thickness $12.7 ± 0.1$ mm, and diameter $76.1 ± 0.1$ mm is released by hand on the central area of a black acrylic platform with thickness 0.51 cm, length 60.0 cm, and width 29.0 cm (see the Figshare repository of supplementary materialsupplementary material for a sample video). The platform lies on two planar supports placed underneath its short sides. The platform's vertical acceleration is measured with a smartphone via its accelerometer with a sampling frequency of 405 Hz, and the application phyphox.⁹ The smartphone lies on the platform as depicted in Fig. 2(a). Prior to our experiments, we measured the free oscillations of the platform by hitting it at different positions and consistently found damped oscillations with frequency $fp≃11$ Hz, which is significantly lower than the frequency range relevant for our study, $f ∈ [20,36]$ Hz.

A typical acceleration signal acquired while the disk was spinning on the platform is reported in Fig. 2(b).

To check if the platform acceleration is a good proxy for the disk motion, we simultaneously acquired a side-view video of the disk's dynamics and tracked the motion of the disk's contact point with the platform. This requires appropriate contrast between the disk, the platform, and the background, obtained by illuminating a white panel behind the disk. The video was acquired with a tablet at 240 fps. A custom-made code for image analysis was then used to process each frame and track the position of the contact point. In this code, the contact point is defined as the center of the contact region, which is the black region that is imaged as connecting the disk to the platform. The width of the contact region ranges from $∼$1 mm to $∼$1 cm depending on the stage of the disk motion. In Fig. 2(c), we present three snapshots of a sample video and the corresponding processed images. A comparison between the contact point and platform accelerations (Fig. 2(b)) shows that they exhibit the same periodicity and similar amplitude trends. A complete precession of the disk thus corresponds to one platform oscillation as a result of the variation in time of the excitation point's position, which induces a vibration of the platform. Furthermore, the amplitudes of both signals increase as the disk approaches rest. While measuring the motion of the contact point requires the appropriate recording settings and image analysis we described, measuring the platform vibration only requires the smartphone to be placed on the platform itself. The latter allows for faster data acquisition over a consistent number of runs and is also more adapted to classroom demonstrations.

In Fig. 3, we show a side-by-side comparison between the vibration of the platform on which the spinning disk comes to rest (a)–(c) and the gravitational waves emitted during the merger of a binary black hole system⁷ (d)–(f). Despite these systems involving substantially different spatial and energy scales, we note that the timescales are accidentally comparable. Moreover, the time evolutions of the platform acceleration (a) and the gravitational wave strain (d) exhibit interesting similarities. Initially, both the amplitude and the oscillation frequency of the signal increase toward the singularity. For the disk, this phase corresponds to spinning with decreasing contact angle until full contact with the platform is reached. For the binary system, this phase includes the spiraling of the black holes toward the merger. In the final part of the evolution, both systems exhibit a ringdown characterized by amplitude decreasing over time, a feature that is not described by the models in Sec. II. For the disk system, the platform still vibrates after the disk achieves full contact because the oscillations generated previously require some time to be damped out.

Scaling laws are presented in Figs. 3(b) and 3(e) for the disk and black hole system, respectively. The time interval that we analyze for each measurement is denoted $tc$⁠, an example of which is represented in Figs. 3(a) and 3(d). In general, $tc$ is the characteristic time for the disk to achieve full contact with the platform or for the black holes to merge, and depends on the values of contact angle $α0$ and separation distance $l0$⁠, respectively, which we do not measure. For the gravitational signal, the fit of the data yields the exponent $−0.35 ± 0.03$ with 95% confidence bounds, in agreement with the exponent $−3/8$ expected from Eq. (23). For the disk, the fit of the data in Fig. 3(b) yields the exponent $−0.13 ± 0.01$ with 95% confidence bounds. We repeated the experiment 30 times and obtained the mean exponent $−0.13 ± 0.03$ with 95% confidence bounds. From this value and Eq. (12), we obtain $γ=2.9 ± 0.9$⁠. This is in the same order of magnitude but generally larger than the values theoretically estimated by Moffatt¹ (⁠ $γ=2$⁠) and Bildsten² (⁠ $γ=5/4$⁠), which are derived in ideal conditions, i.e., no friction at the contact point is considered, which is not respected by our setup.

We have decomposed the energy content of the signal in time and frequency using a wavelet transform,¹⁰ a procedure commonly employed to analyze gravitational wave signals.^7,11 Compared to classical Fourier analysis, the wavelet transform retains the time information, allowing for a precise frequency characterization of the signal at the various stages of the dynamics. In Figs. 3(c) and 3(f), we show the wavelet coefficients of the signals in Figs. 3(a) and 3(d), respectively.

Here, we use a mother wavelet known as “Morlet,” defined as a plane wave modulated by a Gaussian function.¹² The wavelet coefficients represent the energy of the processed signals. Both wavelet transforms exhibit energy transfer to high frequencies as the singularity is approached. This energy transfer is more evident for the black hole merger, as it is expected from the larger exponent characterizing its scaling law. We note that noise appears in the wavelet transform of the disk, while it is absent for the gravitational signal. Indeed, background noise was removed from the raw gravitational signal,⁷ while for the disk, we considered the signal as acquired by the accelerometer.

We investigated an analogy between a disk spinning on a surface and a phenomenon occurring at a very different scale, i.e., the merging of two orbiting black holes. We presented a simple experimental configuration for measuring the vibration of the platform on which the disk spins. Our setup allows for straightforward signal acquisitions that do not require any post-processing. By simultaneously measuring the platform vibration and the disk contact point, we showed that the platform frequency corresponds to the precession frequency of the disk. As highlighted by the theoretical expressions presented in Sec. II, the disk precession frequency is expected to have a similar scaling to the frequency of the gravitational waves emitted during the merger. The platform perturbation was thus compared to the space-time perturbation measured by the LIGO detector. Both signals exhibit increasing frequency and amplitude and energy transfer to high frequency. The corresponding scaling laws are characterized by different exponents that are $≃−0.13$ in our system and $≃−0.35$ from the black hole data due to the different mechanics and dissipation laws involved.

This work originates from a class project developed by master's students in a Physics Department and can be proposed in advanced courses of experimental physics also by varying the experimental conditions. For the disk, the exponent depends upon the type of dissipation involved in the dynamics via the parameter $γ$⁠, which appears in the expression of the dissipated power $Pdiss=kα−γ$⁠. The scaling laws of the two systems would be equivalent if $γ=1/3$⁠. Students could thus be challenged to vary the experimental conditions to obtain an exponent closer to the scaling law of the black hole merger.

Please click on Figshare repository of supplementary materialthis link to access the supplementary material at the reviewer website. Video acquired at 240 fps and reproduced here at 30 fps (8 times slower). Print readers can see the supplementary material at DOI for figshare repositoryhttps://doi.org/10.60893/figshare.ajp.c.7829717.

The authors are grateful to Andrea De Luca, who accompanied us during a significant part of this project and for developing the code that tracks the motion of the contact point.

The authors have no conflicts to disclose.