Classroom Simulation of Gravitational Waves from Orbiting Binaries
James Overduin, Jonathan Perry, Rachael Huxford, and Jim Selway
Citation: The Physics Teacher 56, 586 (2018); doi: 10.1119/1.5080568
View online: https://doi.org/10.1119/1.5080568
View Table of Contents: http://aapt.scitation.org/toc/pte/56/9
Published by the American Association of Physics Teachers
Classroom Simulation of Gravitational
Waves from Orbiting Binaries
James Overduin, Jonathan Perry, Rachael Huxford, and Jim Selway, Towson University, Towson, MD
P
opular demonstrations commonly use stretched
spandex fabric to illustrate the way in which curved
spacetime mimics the force of gravity in general relativity.1-8 There are significant potential conceptual pitfalls to
such an approach. In particular, it obscures the fact that most
of what we ordinarily feel as gravity is due to the warping of
time rather than space, a concept that is admittedly harder
to demonstrate.9,10 Nevertheless, with appropriate caveats11
simulations of this kind can convey some of the wonder of
Einstein’s theory to non-specialists.
In this spirit, we wondered whether a similar model could
be used to illustrate gravitational waves from orbiting binaries, whose discovery was recognized with the 2017 Nobel
Prize in Physics. (See Refs. 12-17 for accessible discussions
of gravitational wave detection.) Our simple and inexpensive
demonstration reproduces the pattern of outgoing spiral ripples that has entered the public imagination through images
from numerical simulations.18 It should not be confused with
a demonstration of general relativity, although it does exhibit
some of the same features that gravitational waves share with
other forms of radiation in general.
Fig. 1. Hoop and support structure.
Demonstration
“Spacetime” in our demonstration is represented by a sheet
(60 in × 60 in or 1.5 m × 1.5 m) of polyester Lycra four-way
spandex fabric stretched over a 60-in (1.5-m) diameter hula
hoop.19 Hula hoops this large are hard to come by, so we made
our own using a 15-ft (4.6-m) length of stiff (160 psi or 1.1
MPa) plastic polyethylene pipe with diameter ¾ in (0.019 m).
We heated both ends of this pipe with a hair dryer until they
softened enough to allow the insertion of a plastic two-way
¾-in (0.019-m) barbed connector, which then held the pipe
together in an approximately circular shape when it cooled
back down. We raised this hoop off the ground on a circular
arrangement of eight 20-in (0.5-m) stands and clamps of the
kind that are found in most introductory physics laboratories
(Fig. 1). To ensure that the hoop remained circular, we found
it helpful to hold these stands apart using holes drilled in four
pieces of lumber (2 in × 2 in or 0.05 m × 0.05 m) arranged like
the spokes of a wheel. The spandex was then stretched over
the hoop and held in place with binder clamps.
For our “orbiting binary” we used a pair of 1½-in (0.038m) diameter rubber caster wheels mounted at either end of a
lightweight 0.3 m-long wooden crossbar (Fig. 2). We attached
a ¼-in (0.0064-m) hexagonal coupling nut firmly to the center
of mass of this assembly so that it could be inserted into the
chuck of an electric hand drill. To measure the orbital speed
v, we attached a PASCO photogate velocity sensor to the body
of the drill and fastened a lightweight trigger to the crossbar
(Fig. 2, inset). The sensor was connected to a DataStudio
586
Fig. 2. Setup with “orbiting binary” including photogate velocity
sensor strapped to drill (inset). DataStudio interface and strobe
light are visible in the background at left and right, respectively.
Fig. 3. Measurement of the “speed of light” in the spandex using
neodymium magnets and induction coils (inset) connected to
DataStudio interface.
THE PHYSICS TEACHER ◆ Vol. 56, December 2018
DOI: 10.1119/1.5080568
sound speed noted above. In fact, we obtained our best results at orbital speeds v ≈ 6 – 9 m/s, corresponding to strobe
frequencies fs ≈ 13 – 19 Hz (at R = 0.30 m). At greater speeds,
the waves become chaotic and “ragged” as they cannot keep
up with the disturbance that has created them. This situation,
of course, does not arise in the real world, where gravitational
waves travel at the speed of light c, which also sets a strict upper limit on the speed of the massive bodies. This fact alone
provides a useful way to remind students that the demonstration is in no sense a relativistic one. However, it also provides
an opportunity to make a connection with some of the physics
that gravitational waves do share with other forms of radiation in general.
Fig. 4. Typical wave pattern illuminated by the strobe light. Note
reflections at boundary (lower left).
interface. To model the effects of “inspiral” in a rudimentary way, we drilled four sets of holes into the crossbar so that
the caster wheels could be moved back and forth, allowing
for orbital radii R = 0.150 m, 0.125 m, 0.100 m, and 0.075 m.
Small offsets in the placement of the coupling nut, photogate
trigger, and caster wheels sometimes produced wobbles in the
motion. To mitigate this problem, we drilled one additional
hole near each end of the crossbar and added counterweights
in the form of metal washers, as needed.
We expected that the amplitude of our “gravitational
waves” would be largest for speeds v comparable to the characteristic wave speed (or speed of sound) cs, which is set by the
tension in the spandex. (In the same way, the amplitude of real
gravitational waves reaches a maximum as the orbital speeds
of the inspiraling compact objects become comparable to the
speed of light.) To test this idea, we placed two small neodymium magnets on the spandex a distance x apart (holding
them in place with paper clips on the underside of the fabric)
and suspended two wire coils immediately above them (Fig.
3). The coils were connected to the DataStudio interface via
voltage sensors. Flicking a finger sharply against the fabric
next to one magnet produced a voltage spike in the coil above
it (thanks to Faraday’s law of induction), which then propagated to the other coil after a time t. Trial and error showed
that best results were obtained for a relatively slack fabric with
cs = x/ t ≈ 3 m/s.20
This speed is, however, still sufficiently fast that the waves
can be hard to perceive in real time. To bring out the wave
pattern in a dramatic way, we used a strobe light. This also
allowed us to easily hold the rotational speed or frequency
constant at a desired value. For example, to achieve v ≈ cs ≈ 3
m/s with R = 0.15 m, we looked for a frequency f = v/2πR ≈ 3
Hz. This required a strobe frequency fs = 2f ≈ 6 Hz, because the
crossbar appears to return to the same angular position twice
during each rotation (as illuminated by the flash). Hence we
set fs ≈ 6 Hz and gradually increased the drill speed until the
crossbar appeared “frozen in time.” Typical results are illustrated in Fig. 4.
In our demonstration, it turned out to be quite possible
for the “orbiting bodies” to move more quickly than the mean
Theory
The amplitude of real gravitational waves from a coalescing binary of mass M and orbital radius R and frequency f is
given by
(1)
where r is distance. The physics behind this equation has been
elegantly explained in this journal by L. M. Burko.16 The factor of MR2 is the moment of inertia of the system. The factor
of f 2 arises because gravitational waves depend on the second
time derivative of this inertia. (An unchanging mass distribution cannot radiate, and a uniformly moving one cannot
either, because it appears static to a co-moving observer, and
the presence of radiation must be observer-independent.) The
factor of 1/r is common to any wave propagating in three spatial dimensions and arises because the luminosity L carried by
the wave must be proportional to h2 (to guarantee positivity
of energy), and must also fall off with distance as 1/r2 (to satisfy energy conservation as the spherical wavefront expands).
Thus h ~ 1/r. The factor of G/c4, finally, puts the result into
dimensionless form.
To appreciate the meaning of Eq. (1) in practical terms, we
can eliminate R using Kepler’s third law (GM ~ R3 f 2) so that
.
(2)
The first gravitational wave source detected by LIGO,
GW150914, consisted of a pair of black holes with combined
mass M ~70 M( at an estimated distance r ~ 300 Mpc.21 The
frequency of the strongest gravitational waves was fgw~150Hz,
corresponding to an orbital frequency f ~ 75 Hz. The dimensionless amplitude of these waves was h ~ 1×10-21, in
agreement with Eq. (2). This was the unthinkably tiny strain
measured by LIGO (i.e., the amount by which the arms of the
detector were stretched relative to each other as the gravitational wave passed through).
To connect Eq. (1) to our demonstration, we note that our
spandex waves propagate in two dimensions, not three. This
changes the dependence of amplitude on distance. The luminosity L (or flow of energy per time) carried by these waves is
still proportional to h2, but energy conservation now dictates
that L~1/r rather than 1/r2 (because the waves spread out over
a circular wavefront rather than a spherical shell). Hence we
THE PHYSICS TEACHER ◆ Vol. 56, December 2018
587
ered the paper clip toward a point (P) on the spandex, watching for the moment when it was first disturbed by contact with
a passing wave. Its height y was then noted and subtracted
from the equilibrium height y0 to give the wave amplitude,
h = y0 – y. To find y0, we measured the height of the paper clip
upon contact with the spandex when the drill was turned off,
averaging over a representative sample of nine crossbar orientation angles θ (Fig. 5, left) and repeating this procedure each
time the value of r or R was changed.
A second person held the drill steady horizontally by
means of a mark on the spandex at the center of the binary
Fig. 5. Experimental setup to measure spandex wave amplitude.
(point B) and vertically by means of a laser mounted on a
Left: a top view showing how the crossbar (B) is oriented in difstand and aimed at a mark (L) on the side of the drill (Fig. 5,
ferent directions with the drill turned off to obtain an equilibrium
right). Frequency was held constant by means of the strobe
reading at P. Right: a side view showing how a reading is obtained
light. We performed 10 runs for each data point (Fig. 6). Rewith the drill turned on. The drill is held steady with the help of
the laser (L) and the paper clip is lowered until disturbed by a
sults are plotted in Fig. 7, where error bars correspond to stawave at P.
tistical uncertainties,
and y0 and
y are standard deviations.
rather than 1/r. But how should h deFigures 7(a) and (b) suggest that the amplitude of two-diexpect that
pend on R or f ? We decided to attempt to answer this question
mensional spandex waves depends linearly, not quadratically,
experimentally.
on both R and f. Given our statistical uncertainties (2 to 3 mm,
compared to typical amplitudes of 5 to 15 mm), quadratic fits
Experiment
(red lines) are also marginally consistent with the data, but
After testing several different ways to measure the amplilinear ones (green lines) provide a better match.
tude of our spandex waves, we settled on the simple setup deThis is an interesting finding in light of Fig. 7(c), which
picted schematically in Fig. 5. The idea is similar to that found
confirms (albeit weakly) that amplitude falls off as 1/√r (green
in the garages of many car owners: a hanging ball is positioned
curve) rather than 1/r (red curve), as expected for two-dimento touch the windshield when the car reaches the desired spot.
sional waves. Our results here may have been more suscepIn our case we hung a light paper clip from a heavy weight (for
tible to systematic as well as statistical errors. For example, a
stability). The weight in turn was suspended from the ceiling
drawback of our measurement technique is that it registers
by a string, which passed over a pulley and back down to a
only the largest of any group of waves to pass through point P
spool. Near the spool, the string ran beside a vernier caliper
in the spandex. Thus the anomalously large value of h at
with a digital readout. A mark on the string allowed one perr = 625 mm was likely due to reflections from the boundary
at r = 760 mm. These sometimes interfered constructively
son to monitor the vertical displacement of the paper clip to a
with the outgoing waves (Fig. 4), and our detector would have
precision of tenths of a millimeter (Fig. 6). That person lowbeen “triggered” by these artificially amplified
cases, leading to an overestimate of the amplitude. We therefore excluded this data point
from our analysis (dashed lines). A similar
though smaller effect may have come into
play at the shortest distances. Small mass imbalances sometimes produced a slight wobble
in the drill, leading to a pattern of alternating
larger and smaller waves in the fabric. This
asymmetry was more pronounced at small
distances, where it may have “steepened” the
relationship between h and r. Air currents
near the crossbar could conceivably have had
a similar effect. More work could certainly be
done to identify and compensate for sources of
systematic as well as statistical uncertainty. But
given the caveats expressed above, our results
suggest that the amplitude of two-dimensional
Fig. 6. Measuring wave amplitude with the help of a laser guide (red circle) and spandex waves may be proportional to Rf /√r
paper clip detector (green rectangle). The inset shows a close-up view of the spool rather than R2 f 2/r (as for gravitational waves
and vernier caliper used to adjust and measure the height of the paper clip above in three-dimensional space).
the fabric.
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THE PHYSICS TEACHER ◆ Vol. 56, December 2018
Acknowledgments
We thank the referees for comments and suggestions that
considerably strengthened the paper. Thanks also go to the
Maryland Space Grant Consortium and the Fisher College of
Science and Mathematics at Towson University for summer
research support, and K. Takeno and T. K. Overduin for the
photographs in Figs. 1-4 and Fig. 6, respectively.
References
1.
(a)
(b)
(c)
Fig. 7. Experimental measurements (data points) of wave amplitude vs. orbital radius [Fig. 7(a), top], frequency [Fig. 7(b), middle],
and distance [Fig. 7(c), bottom], plotted together with best-fit
power-law functions (red and green curves).
In closing, we reemphasize that any demonstration of this
kind is classical, not relativistic, in nature. While entertaining
and even educational (within its limits), it can only hope at
best to capture some aspects of gravitational waves produced
during the inspiral phase, not the more spectacular merger
and ringdown phases. To understand those, students will need
to learn about general relativity in its full glory.
G. D. White and M. Walker, “The shape of ‘the Spandex’ and
orbits upon its surface,” Am. J. Phys. 70, 48 (Jan. 2002).
2. D. S. Lemons and T. C. Lipscombe, “Comment on ‘The shape
of “the Spandex” and orbits upon its surface,’” Am. J. Phys. 70,
1056 (Oct. 2002).
3. E. Baldy, “A new educational perspective for teaching gravity,”
Int. J. Sci. Edu. 29, 1767 (2007).
4. G. D. White, “On trajectories of rolling marbles in cones and
other funnels,” Am. J. Phys. 81, 890 (Dec. 2013).
5. C. A. Middleton and M. Langston, “Circular orbits on a warped
spandex fabric,” Am. J. Phys. 82, 287 (April 2014).
6. J. Ford, J. Stang, and C. Anderson, “Simulating gravity: Dark
matter and gravitational lensing in the classroom,” Phys. Teach.
53, 557 (Dec. 2015).
7. C. A. Middleton and D. Weller, “Elliptical-like orbits on a
warped spandex fabric: A theoretical/ experimental undergraduate research project,” Am. J. Phys. 84, 284 (April 2016).
8. T. Kaur et al., “Teaching Einsteinian physics at schools: Part
1, models and analogies for relativity,” Phys. Educ. 52, 065012
(2017).
9. R. H. Price, “Spatial curvature, spacetime curvature, and gravity,” Am. J. Phys. 84, 588 (Aug. 2016).
10. A. I. Janis, “On mass, spacetime curvature, and gravity,” Phys.
Teach. 56, 12 (Jan. 2018).
11. One way to try and correct this misinterpretation without abandoning demonstrations altogether is to appeal to the Newtonian
limit. The premise of the spandex-sheet demonstration is that
a marble follows a circular path around the central mass even
though no force acts “through the fabric.” But general relativity,
whatever else it may say, must reduce to Newton’s laws for weak
fields like those prevailing in the solar system. Newton’s first law
states that planets and marbles alike move on straight lines when
no force acts. What then causes the motion of a marble (or planet) to deviate so strongly from straightness, if no force is acting?
The answer is that its path is very close to straight in spacetime,
not in space. The Earth, for instance, travels along a helix whose
radius in space is only one astronomical unit, but each of whose
spiral turns stretch across a light-year in time (i.e., the duration
of one orbit, expressed in units of distance). The straightness
of this trajectory can be conveyed to non-physicists by asking
them to imagine a Slinky with 100 coils, stretched out until its
length is 3 million times greater than its width (i.e., the ratio of
100 light-years to two astronomical units). There is still some
curvature here, so the motion does violate Newton’s first law,
but only slightly. It is this slight curvature of spacetime, rather
than the gross spatial curvature suggested by the spandex, that
mimics the force of gravity in Einstein’s theory. Its smallness
reflects the weak gravitational field of the Sun.
THE PHYSICS TEACHER ◆ Vol. 56, December 2018
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12. G. W. Spetz, “Detection of gravity waves,” Phys. Teach. 22, 282
(May 1984).
13. L. J. Rubbo et al., “Gravitational waves: New observatories for
new astronomy,” Phys. Teach. 44, 420 (Oct. 2006).
14. B. Farr, G. Schelbert, and L. Trouille, “Gravitational wave science in the high school classroom,” Am. J. Phys. 80, 898 (Oct.
2012).
15. D. Lincoln and A. Stuver, “Ripples in reality,” Phys. Teach. 54,
398 (Oct. 2016) 398; erratum Phys. Teach. 55, 5 (Jan. 2017).
16. L. M. Burko, “Gravitational wave detection in the introductory
lab,” Phys. Teach. 55, 288 (May 2017).
17. H. Mathur, K. Brown, and A. Lowenstein, “An analysis of the
LIGO discovery based on introductory physics,” Am. J. Phys. 85,
676 (Sept. 2017).
18. Similar demonstrations can be seen online in several places,
including those by Steve Gould (https://www.youtube.com/
watch?v=dw7U3BYMs4U), LIGO-Caltech (https://www.youtube.com/watch?v=YfSyhcFu_MM), the Arvin Gottlieb Planetarium (https://www.youtube.com/watch?v=wnWmGr_523s),
and Benjamin Giblin and Ben Morton at the University of Edinburgh (https://www.youtube.com/watch?v=T6B1U-5oAp4).
19. Spandex fabric in the size used here can be ordered online, and
the other components for our demonstration found in any large
hardware store, for a total cost of less than $100.
20. We note that the sound speed cs found in this way is an average.
The actual speed of waves in the fabric may depend on distance
from the center. This could perhaps be determined using a
camera facing down onto the spandex from above, in combination with the tracking feature of PASCO Capstone. In general,
the propagation of two-dimensional elastic waves in a circular
membrane is mathematically challenging (more so than the
rectangular case, which is a simple extension of the linear string
model that most students are familiar with). Another, more indirect way to determine the mean sound speed might be to use
a PASCO force probe to measure the tension force per length
τ in the fabric (in N/m or kg/s2) and weigh a sample of known
area to obtain the surface mass density σ (in kg/m2). Then cs =
√( /σ).
21. LIGO Scientific and VIRGO Collaborations, “The basic physics
of the binary black hole merger GW150914,” Ann. Phys. (Berlin) 529, 1600209 (2017).
Towson University, Department of Physics, Astronomy and
Geosciences, Towson, MD 21252;
http://wp.towson.edu/joverdui/; joverduin@towson.edu
And the Survey Says ...
Susan C. White, Column Editor
American Institute of Physics
Statistical Research Center
College Park, MD 20740; swhite@aip.org
Who’s hiring physics bachelors?
A
Initial Employment* Sectors of New Physics Bachelors,
bout half of physics bachelor’s degree recipiClasses of 2015 & 2016 Combined
ents accept a job offer after graduation. About
half enroll in graduate study—about two-thirds
College &
of those in physics or astronomy. Very few job ads
University
indicate having a physics bachelor’s degree as a re9%
quirement for employment. So, one might wonder
High School
8%
where these degree recipients find jobs. We have
Other
a web resource that helps answer that question:
6%
Who’s Hiring Physics Bachelors? (www.aip.org/
statistics/whos-hiring-physics-bachelors).
Acve Military
6%
Private Sector
This resource provides a state-by-state listing of
66%
employers in each state who have recently hired
at least one physics bachelor’s degree recipient.
Civilian Gov't,
It also includes a listing of companies who have
Naonal Lab
recently hired three or more physics bachelor’s
5%
degree recipients.
A large proportion of physics bachelor’s degree
recipients enter the workforce, so it is important to
* 47% of physics bachelor’s degree recipients from these classes were employed in
the winter following the year in which they received their degree. 49% enrolled in
provide tools to help them.
graduate study – 29% in physics or astronomy and 20% in some other discipline.
Next month we will look at common titles for jobs
www.aip.org/statistics
physics bachelor’s degree recipients take upon
graduation. These job titles help degree recipients search job ads for jobs appropriate for their skill sets. In February,
we will explore the Careers Toolbox specifically designed to help recent physics bachelor’s degree recipients who
want to enter the workforce. Susan White works in the Statistical Research Center at the American Institute of Physics. She can be reached at swhite@aip.org.
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