Modelling gravitational-wave emission and detection on spandex using a high-speed camera

This paper demonstrates a variation of the rubber sheet experiment (Gravitational Waves Work Like This Drill on Spandex) for measuring the properties of modelled gravitational waves. Mechanically induced waves on the rubber sheet are observed by a high-speed camera and the slow-motion videos are analysed with the Tracker program. We describe the theoretical background and the execution of the measurement process. The measured displacements are suitable for modelling real gravitational-wave signals and determinating properties of the sources.


Introduction
One hundred years after Albert Einstein's prediction of the existence of gravitational waves [1,2], in 2015 the LIGO-Virgo Collaboration made the first detection [3].In 2017 three prominent members of the LIGO-Virgo Collaboration-Barry C Barish, Kip S Thorne and Rainer Weiss-were * Author to whom any correspondence should be addressed.
awarded the Nobel Prize in Physics for the discovery of gravitational waves.It has been the subject of much educational material but is not yet integrated into school curricula in most countries.
There are several current successful attempts on introducing general relativity [4][5][6] and gravitational-wave astronomy [7][8][9] on lower educational levels.Burko has formulated several ideas to introduce in high school [10].North made a complete hands-on workshop about data analysis [11].Douglas and Ingram developed a lab activity for building a Michelson interferometer [12,13].
The measurement experiment presented here was developed with two high school students, authors of this paper, and shows a new method to demonstrate the detection and data processing of gravitational waves by using a student-centred model.The setup provides the opportunity to measure the amplitude and frequency of the modelled waves, two of the most important parameters in the detection of gravitational waves.

Theory
General relativity fundamentally breaks with Newton's conception of gravity.According to this theory, no gravitational force occurs between bodies with mass, but the geometry of spacetime is altered and warped.This means that temporal and special distances around masses are distorted, so that other bodies move in the distorted fabric of space-time.We see this as a gravitational force affecting the motion of the bodies.
The curvature of space-time is not a static thing, it is constantly changing due to the movements of masses.Such perturbations illustrated in figure 1 propagate through space-time as waves, called gravitational waves.
When passing between two points in space, gravitational waves contract and stretch the distance between the points.Being transverse waves in a plane perpendicular to the direction of travel, they alternately cause stretching in a given direction and contraction perpendicular to it.This effect is what can be detected by interferometers.The longer the arms of an interferometer, the greater the change in length of a gravitational wave.The Laser Interferometer Gravitational-Wave Observatory (LIGO), the arms are 4 km long, with bundles of arms that bounce back and forth hundreds of times, increasing the effective arm length [14].The instrument can thus measure the change in the length of the arms relative to each other as a tenth of a thousandth of the size of an atomic nucleus!Such sensitivity is necessary, since the distorting effect of incoming waves is so small even for the near-light-speed motions of bodies with masses comparable to that of the Sun.The precision of the measurement makes noise filtering extremely important, since any small ripple in the environment of the detectors can have a stronger effect than a passing gravitational wave.Sources of the waves that can potentially be detected today include, for example, spiralling compact binaries shown in figure 2 (e.g. two black holes orbiting each other and then merging), nonrotationally symmetric neutron stars and highenergy explosions such as gamma-ray bursts or supernova explosions [15].

Setup and procedure
The rubber sheet is a tool often used to demonstrate the basic idea of general relativity.Although the concept of the model has its drawbacks and the rubber table itself does not model the laws of general relativity well [16], it is a good demonstration of the concept and can be used to model phenomena such as gravitational waves.Some of the limitations of the model are: the speed limit of the spandex can be broken [17]; dissipation and edgereflections change the wave-pattern; motions on the spandex fabric are not identical to motions according to Newton's law of gravity or general relativity [16].
The rubber sheet represents space-time, and the waves generated on it represent gravitational waves.The material of the sheet should be flexible, we used spandex with 10% elastane on a table with a diameter of 1.4 m.The table itself was folded from 2 cm diameter Pex-Al-Pex tubing, but you can find different methods of assembly on the internet.
The source of the waves is a drill head consisting of two rollers 10 cm apart attached to a wooden plate shown in figure 3. Rotated with the drill, this models a compact binary, in the constant phase before the merger.The drill was used with the sheet lightly pressed against the drill bit at 1200 RPM.The resulting waves are visible to the naked eye, shown in figure 4.
Choosing the right rotational frequency is essential for a successful experiment.If the speed is too law, waves are not generated, while a rotation too fast may rip the fibre.The tightness of the sheet has also a significant effect on the waves.In our set-up, the ideal tightness can be described as a fitted sheet stretched as usual.
The set-up can be improved by a bigger table and some reflection damping to make the pattern and the data clearer.Also, our table was built mobile so we can show it at different locations.A more stable (e.g.wooden or metal) frame makes the measurements easier and more consistent.More details about the set-up can be found in the video of Steve Mould [18].
The resulting waves were analysed in slow motion (see in supplementary material 1) 5 .For the measurement a high-speed camera is needed.According to the available settings of our camera, we recorded at 500 and 1000 fps.It is worth to choose the higher fps-rate, even though it increases the processing time.The waves need to be measured in some way.Since the amplitude is small, it is hard to measure the waves from a side view by observing the height of the waves.Therefore, we set the camera in a top view position and used the principle that if an object is closer to the camera, it will be larger in the image.Two points were drawn on the sheet with a marker at a distance of L = 8 cm, from each other, which is the object size.This 'detector' was 18 cm away from the centre of mass of the source.As the wave crest reaches these two points, they move closer to the camera, so the apparent size of the distance in the image increases.As the wave trough reaches them, they move away from the camera, at which point the apparent distance between the two points is at a minimum on the picture.The geometry of this stage and the one at rest is illustrated in figure 5.The apparent distance is i(t) determined by the quotient of the object size and the size of the field of view: f 0 at rest and f(t) in motion, at rest and during the measurement process.Thus Because of the similarity of two triangles, where z (t) is the distance between the camera and the rubber sheet.Written into equation ( 3), The displacement is the difference of the instantaneous and original camera distance: Compared with equation ( 5) and doing some transformation, This means, that by measuring the apparent distances on the frames and knowing the original camera distance, the instantaneous displacement of the rubber sheet at the measuring points is determinable.Measuring the length of the distance between the two points is modelling the length change of one arm of the laser interferometer.
It is important to clarify to the students, that the deformation of the rubber sheet is vertical, but the measured length-change is horizontal, which is only a virtual change.It is also worth to consider introducing only the apparent-distance change, as a fairer analogy to gravitational waves, without speaking about how it is caused by the cameradistance change.It also makes the theoretical explanation easier.The downside of this way is that the process gets less transparent, and we are not measuring the very amplitude of the tangible wave.
The frequency can then be easily read off the displacement-time graph.Of course, we know the speed of the drill, but students can measure this as well.Figure 6 shows the experimental layout in real life.
The propagation speed was not measured in this setup, as gravitational waves propagate at the speed of light, so it is only a theoretical check, and no new information is expected in the model.That said, it can be easily measured by plotting two points in the radial direction.The propagation velocity of a wave crest can be determined from the time it takes to register two points in succession, given the distance between the points.Knowing frequency, the wavelength can be calculated as well.The wavelength is also measurable from a single frame.In our case, it was 13.6 cm.
When choosing the distance of the two measurement points, the wavelength may be a good reference.The LIGO and the Virgo detectors are optimized to frequencies between 10-10 000 Hz which equals to 30-30.000km of wavelength [19].The effective length of the LIGO detector arms is 1600 km, so the 8 cm separation and 13.6 cm wavelength ratio are comparable to the gravitational wave detections.A smaller separation may be considered, but it requires a better resolution image we could reach.
The source localisation in LIGO-data needs at least two detectors (Hanford and Livingston, separated by 3002 km).If the wave reaches one of the detectors earlier, that means the source is closer to that detector.And the model allows to make this very similar direction determination as well.If the wave reaches of the measurement point earlier, it means the source is closer to that point.Nevertheless, this breaks the analogy, since now one point is representing one detector, not one end of a detector-arm.

Data analysis
The videos were analysed using the Tracker software.By selecting the two measurement points, their positions can be automatically tracked to obtain the x and y coordinates of the two points as a function of time in tabular form.These were exported and analysed in Excel.
The distance between the two points can be determined using simple coordinate geometry.
If the wave crest does not reach the two points simultaneously, one of the points' data series must be shifted by as many frames as the delay.We established the shift from the video, which was typically between 0 and 2 frames.Our time shifting is like when the two (or more) detector datasets are shown on the same diagram, time-shifted (see figure 8).This can be avoided with some attention if needed, or by choosing a smaller value for L. On the dataset we show in this study, there was no time-shifting needed.
From the resulting distance data, the variation of the sheet deflection with time is obtained using the previous formula.In a measurement, a time interval of one second at most is analysed, which at 1000 fps means one thousand data points.At 1200 RPM, this means 20 complete rotations a second (since both rollers of the head create a surge, this will result in 40 complete wave periods).

Sensitivity
A theoretical sensitivity of the experiment can be determined from the pixel size of L, which was 966 pixels in our case.If we assume that we can measure a displacement of a measurement point as big as 1 pixel, equation (7) gives the following result: The largest length-change measured was about 18 pixels, which means 9 pixels displacement of one point.

Results
Figure 7 shows the result of a measurement using the methods described.
The data clearly resemble a sine curve.After taking many measurements, and eliminating various possibilities for error, we have concluded that there will always be anomalies in the data.For example, the reason for the alternating smaller and larger minima is likely caused by unintended eccentric orbits [20].
If we look at the real gravitational-wave detections in figure 8, we also find large variations due to different sources of noise.The pale yellow and blue lines show the theoretically predicted signal shape of a spiralling binary, while the darker orange and blue ones show the first real detection by LIGO's Hanford and Livingston Observatory.
We made further measurements to find the possibilities of the tool, to model parameter estimation (e.g.source-size, roller-separation, sourcedistance, source-type, asymmetry).The data is currently under evaluation and may be the subject of a forthcoming article focussing on parameter estimation and a student workshop based on this process.

Conclusion
Present paper demonstrates that a simple measurement experiment that uses elementary mathematics is suitable for modelling the process of gravitational-wave detection.The set-up is eligible for determining the frequency, amplitude and propagation velocity of the waves generated.The results and data processing can be compared with real processes, bringing the field closer to the students.
András Molnár is a PhD student at the Eötvös Loránd University in Budapest and teaches physics in high school since 2016.He is a member of the LIGO-Virgo-KAGRA Collaboration, which also records the first direct detection of gravitational waves in 2015.
He is interested in teaching gravitational-wave astronomy on high school level and researching the ways to promote physics.
Márk Czura is a student at the Tamási Áron High School in Budapest.He is interested in astrophysics and particle physics.
Bence A. Dercsényi is a student at the Városmajori High School in Budapest.He is a future astrophysics university student.

Figure 2 .
Figure 2. The theoretical waveform of a gravitational wave generated by a binary black hole.Reproduced from [3].CC BY 3.0.

Figure 3 .
Figure 3.The drill head and the drill used in the experiment.

Figure 4 .
Figure 4.The wave-pattern generated by the drill.The reflecting waves are visibly interfering with the original waves, making the middle region the best choice for measurements.

Figure 5 .
Figure 5.Because of the proportionality of the sides in the similar triangles, the camera distance determinates the apparent distance.

Figure 6 .
Figure 6.The rubber sheet with the markers and the camera.

Figure 7 .
Figure 7. Displacements determined according to the measurements on the rubber sheet.See raw data in supplementary material 2.

Figure 8 .
Figure 8.The first observed gravitational-wave signal by each LIGO detector.Credit: Caltech/MIT/LIGO Lab.