Bending Spacetime
in the Basement: Is it Bogus?
Norman Scheinberg, Ph.D.
I. Introduction
An interesting and well written article
appeared in the 21 December 2001 Society for Amateur Scientist's
E-Bulletin under the title "Bending
Spacetime in the Basement," by John Walker. The article
came directly from his web
site. The quote below from the article's introduction
summarizes the article.
"This page presents a "basement
science" experiment which reveals the universality
of gravitation by demonstrating the gravitational attraction
between palpable objects on the human scale. The experiment
deliberately uses only the crudest and most commonplace
materials, permitting anybody who's so inclined to perform
it."
Basically, John Walker reproduces Cavendish's
torsion balance gravity experiment in his basement using
readily available materials and presents an amazing video
to show his results. Many other web sites about gravity
link to his web site or the SAS article. Read the article
and view the video, or go directly to his web site before
reading any more of this article. His web site also calculates
how long it would take a spaceship to travel to distant
locations using special relativity from the earth's and
the spaceship's reference frame. It plots orbits of objects
in extremely curved space as encountered near black holes
using general relativity, and shows how it differs from
Newton 's gravity. You will not be disappointed!
Now that you have read his article -
hold on. You haven't read it yet, have you? Go and do
it now, or you will not be permitted to read further!
After reading his article I decided to
duplicate his experiment and bend spacetime in MY
basement. After many tries I could not find
any gravity where I live. Could it be that there is no
gravity, and New Jersey just sucks? Upon hearing of my
results, my wife and kids, who were born in New Jersey,
insisted that there is nothing wrong with New Jersey.
Instead, they said, "Something must be wrong with John
Walker's experiment." Heeding their advice, I decided
to analyze his experiment, and the video for possible
sources of error.
My analysis of his video revealed many
problems. If you play the video one frame at a time and
read the time stamp in the video, the video shows that
it takes only three minutes for the pendulum to move from
one side to the other (to the first bounce). This is too
short a time for the pull of gravity to move the balls.
In case one might doubt the time stamp,
the time stamp shows that the frame rate settles down
to one frame every 30 seconds once the balls begin to
move. This is consistent with the frame rate stated in
the article as quoted below.
The following
time-lapse movies (about 30 seconds per frame)..."
Thus, the time stamp is correct.
Section II of this article follows below.
It is a lengthy discussion of why the videos that appear
in the SAS article do not show gravity. It includes numerous
calculations and computer simulations, so many, in fact,
that most readers will feel that it is overkill. But one
should not claim that someone else's work is flawed without
substantial evidence.
II. Calculations
and Simulations Based on the Video
To show that the video does not show
what it purports to show, we will calculate the time required
for the balls in the video to move under the pull of gravity.
A detailed calculation involves the solution of a nonlinear
differential equation, but if we assume that the restoring
force of the spring is zero, the damping is zero, and
gravity is constant, the solution can be found from two
simple equations using some algebra.
The equations are:
F/m = GM/r2 = a; from Newton 's law
of gravity (Eq. 1)
t = SQRT[2D/a]; from any elementary physics
book (Eq. 2)
where,
F is the force of gravity.
G = 6.7 x 10 -11; the
gravitational constant.
M is the mass of one
of the large fixed "weights" = 0.75 kg.
m is the mass of the
"weight" on one side of the torsion pendulum = 0.34 kg.
a is the acceleration
(due to gravity) of an object attracted to a mass M
at a distance r.
(Note: the acceleration of gravity is
independent of m.)
t is the time it takes
an object to move a distance D under
a constant acceleration a.
In order to find the time, t,
we must make some assumptions about the value of r
and D. The article states that
the center-to-center spacing of the masses (M and m),
when the pendulum is at equilibrium (midpoint), is 14
cm. We are also told that the supporting Styrofoam bar
is 5 cm wide and 30 cm long.
From this information, and the direct
overhead snapshot of the setup, we can estimate that the
closest the center-to-center spacing the masses (M and
m) can have is 6 cm. Let us assume that r = 6 cm, even
though the average distance between the masses is much
larger. Setting r = 6 in Equation1 will greatly overestimate
the acceleration of the pendulum and underestimate t.
The distance that the weights move to the midpoint point
is 8 cm (14cm - 6cm = 8cm). So we will assume that D =
8 cm, even though the pendulum moves about twice this
distance. This will further underestimate t. Calculating
this underestimated value for t yields:
a = (6.7x10 -11 )· (0.75)/(0.062) =
14x10 -9 m/sec2
t = SQRT[2(0.08)/14x10 -9 ] = 3400 sec
or 56 min
Thus the time that it would take the
pendulum to move from one side to the other is far longer
than 3 minutes!
I also programmed the nonlinear differential
equation into my circuit simulation program (ADS) and
got the result shown in Figs.1-4. In Fig. 1 the spring
constant was set to zero, and the damping was adjusted
to bounce the pendulum back half way. (The time to the
first bounce is a weak function of the damping.) The pendulum
starts at 0 cm (the mid-point) and hits the opposite side
at +8 cm and bounces. The first bounce occurs at over
120 minutes not 3 minutes.
Figure 1. Movement of the pendulum with the restoring
force of the wire set to zero and the pendulum starting
at the mid-point.
In the next simulation (Fig. 2) the pendulum
starts at -8 cm (the left side) as in the video. Now it
takes 270 minutes to reach the other side. The movement
of the pendulum shown in John Walker's the video is included
for comparison.
Figure 2. Movement of the pendulum with
the restoring force of the wire set to zero and the pendulum
starting at the left side as in the video. The simulation
and the video do not agree.
The simulation shown in Fig. 3 includes
the spring constant of the wire. Figure 3 shows that the
restoring force of the wire adds to the speed of the pendulum.
The restoring force is made as large as it can be and
still have the pendulum come to rest on the opposite side.
The restoring force reduced the time to the first bounce
from about 270 minutes (Fig. 2) to about 140 minutes (Fig.
3). This is still much longer than 3 minutes.
Figure 3: Movement of the pendulum with
the maximum restoring force that the wire can have and
still have the pendulum come to rest on the opposite side.
The simulations in Figs.1-3 used spring
constants that allowed the pendulum to move under the
pull of gravity to the extreme left and right sides as
shown in the videos. Unfortunately, the real spring constant
for the nylon fishing line described in the article would
generated such a strong restoring force that the pendulum
would move only a very small amount under the pull of
gravity, not the 8 cm shown in the video.
The restoring force of the wire, F
S, is given by Eq. 3. The spring constant of
the wire, k, can be calculated from the
mass of the weights attached to the pendulum, and the
natural period of oscillation of the pendulum (left to
right and back to the left). Thus:
F S = k·d = [4 p2 (2m)/T2 ]d (Eq. 3)
where d is the deflection
of the pendulum in meters from the midpoint, 2 m
is the total mass of the weights attached to
the pendulum, and T is the period of
the pendulum without damping and without the large fixed
weights near it.
I reproduced the pendulum in the video
and found the period to be about 1 hour. Thus, the restoring
force at the maximum deflection of the pendulum (8 cm)
is:
F S = [4 pi2 (0.68)/36002]0.08 = 161
x 10 -9 Newtons
The force of gravity is given by:
F G = 2GmM/r2 = 2GmM/(0.14-d)2 (Eq.
4)
where M is the mass
of one of the large fixed weights, r
is the center-to-center spacing of the weights, d
is the deflection of the pendulum from the midpoint,
and 14 cm is the center-to-center spacing of the weights
when the pendulum is at the midpoint. At maximum deflection,
r = (14 cm - 8 cm) = 6 cm and the force of gravity is:
F G = 2(6.7 x10 -11 )(0.34)(0.75)/0.06
2 = 18 x 10 -9 Newtons
Thus, gravity is about 10 times too weak
to maintain the pendulum deflected to the extreme ends
of its swing, contrary to what is shown in the videos.
By equating Equations 3 and 4 the theoretical
defection, d, can be calculated. The results of this calculation
show that the pendulum should be deflected only about
0.084 cm from the midpoint in this experiment. The video
shows 8 cm.
The simulation in Fig. 4, using the calculated
spring constant, shows that the pendulum comes to rest
very close to the midpoint as predicted above. The simulation
was started at the extreme left end of the swing to match
the initial conditions in the video even though this initial
condition could not occur by gravity alone.
Figure 4. The simulation of the pendulum
using the calculated spring constant of the wire shows
that it comes to rest near the midpoint.. The simulation
was started at the left end to reproduce the initial condition
shown in the video.
Now that we know that the video does
not show gravity, what does it show? All that I could
come up with is this.
There once was a gravity demonstration,
That clearly required another explanation.
The speed of the pendulum was far too quick,
Perhaps it was pushed by a gentle flick.
Were the weights moved by an electric force?
Could winds in the room create this bar's course?
In the end, I think, we will never know,
Just what it was that made this thing go.
