of 3
Correspondence:
Response of a gravimeter to an
instantaneous step in gravity
Thomas H. Heaton
1
Nature Communications
8
: 1348 doi:10.1038/s41467-017-01348-z (2017)
Montagner et al.
1
presented evidence that a signal observed
prior to P-waves on gravimeters operating in Japan during
the 2011 Tohoku-Oki earthquake was caused by gravitational
fi
eld changes propagating at light speed. In this comment,
I explore the expected response of gravimeters attached to
the Earth
s surface and subjected to an instantaneous acceleration
step,
gt
ðÞ¼
g
0
þ
Δ
gH t
ðÞ
, where
Ht
ðÞ
is a Heaviside step
function. I argue that at the time of the step, inertial accelerations
from the elastic response of the Earth exactly cancel the
gravitational step; in order to observe the gravity step using a
gravimeter, one must wait until the Earth deforms and begins
to re-equilibrate. This conclusion is an application of the
principle of equivalence
2
between gravitational and inertial
mass. In the case of the signal reported
1
for Tohoku-Oki, I
estimate that inclusion of the Earth
s elastic response should
signi
fi
cantly decrease the expected acceleration at the time of the
P-wave.
To determine a complete solution for the response of
a gravimeter attached to the Earth
s surface, I would need to
(1) calculate the spatial distribution of gravitational changes
caused by mass redistribution in an earthquake, (2) calculate
the motion at the gravimeter due to unbalanced elastic and
gravitational forces, and (3) sum the gravity change with
the inertial accelerations determined in step 2. Finding the
Earth
s elastic response is a complex problem that can be
approached by summing the responses of the Earth
s normal
modes. When the Earth response problem is decomposed into a
sum of modes, the time response of each mode simpli
fi
es into a
set of equivalent linear single-degree-of-freedom (sdof) oscillator
problems driven by a step function forcing function of the
appropriate amplitude. Alternatively, I could use a
fi
nite-element
code to calculate both the change in gravity and also the motion
of the Earth caused by those changes in gravitational body forces.
Performing those calculations would transform this brief com-
munication into a long treatise. However, I can provide insight
into this problem by introducing a whimsical gedanken
experiment.
Consider the response of a gravimeter attached to the
Earth
s surface, and that the gravitational constant is
suddenly transformed to zero, or
^
G
¼
1

Ht
ðÞ
½
G
. You
might guess that when this happened, we would all
instantly become weightless. However, what would actually
happen is that the Earth (previously compressed by gravity)
would instantaneously begin to expand with an outward
acceleration equal to gravity. That is, initially our weight
would be virtually unchanged until the Earth
s outward
acceleration decreased as it started to approach its
new uncompressed weightless state; only then would we truly
feel weightless.
This gedanken experiment illuminates the basic principle of
my comment about using gravimeters to detect changes in
gravity. Gravimeters are vertical accelerometers that measure the
sum of the vertical acceleration of the Earth
s surface plus the
vertical gravitational acceleration. When the Earth is in gravita-
tional/elastic equilibrium, then there is no inertial acceleration of
the Earth and then the gravimeter measures only the gravitational
acceleration.
A more quantitative description of the dynamics of an elastic
body that responds to gravitational changes comes from
considering the case of a linear sdof oscillator that is attached
to some stationary point; for a deformable body, it is the
center of mass. The solution of this sdof provides the time
history of motions of the Earth described by
0
S
0
(Earth
s
fundamental radial mode). It also provides a solution of the
gedanken problem (at least for the
fi
rst seconds when
the response is linear). Figure
1
shows parameters used for this
calculation. I assume that the downward gravitational force on
m
is
mg t
ðÞ¼
mg
0
þ
m
Δ
gH t
ðÞ
, and that prior to the gravity
change, the Earth is in static equilibrium, and that the Earth
s
elastic response is linear for small perturbations from
equilibrium.
The equation of the generalized motion of the Earth
s surface is
x
þ
2
β
_
x
þ
ω
2
0
x
¼
Δ
gH t
ðÞ
ð
1
Þ
where
ω
2
0
¼
k
=
m
, and
β
¼
b
=
2
m
.
ω
0
is the characteristic
DOI: 10.1038/s41467-017-01348-z
OPEN
1
Division of Geological and Planetary Sciences, Caltech, Pasadena, CA 91125, USA. Correspondence and requests for materials should be addressed to
T.H.H. (email:
heaton@caltech.edu
)
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1
frequency at which the Earth oscillates about its new
equilibrium (about 8.1 × 10
4
Hz), and 2
β
¼
ω
0
ffiffiffiffiffiffiffiffi
1
2
þ
Q
2
p

ω
0
Q
,
where
Q
0
S
0

5500 for
0
S
0
. The motions for small
times can be determined by recognizing that the third term in (1)
is negligible at short times (
x
is initially zero). That is, for short
times,
x

Δ
gH t
ðÞ
2
β
_
x
ð
2
Þ
Since the system is initially at rest (i.e., equilibrium),
_
xt
¼
0
ðÞ¼
0, and then
x

Δ
gH t
ðÞ!
0 when
t
!
0. The full
solution of (1) with at-rest initial conditions is
xt
ðÞ¼

Δ
gH t
ðÞ
ω
2
0
1

e

β
t
cos
ω
1
t
ðÞ
β
e

β
t
ω
1
sin
ω
1
t
ðÞ



Δ
gH t
ðÞ
ω
2
0
1

e

β
t
cos
ω
1
t
ðÞ
e

β
t
Q
sin
ω
1
t
ðÞ

ð
3
Þ
where
ω
1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω
2
0

β
2
p
. Taking time derivatives of (3),
_
xt
ðÞ¼

Δ
gH t
ðÞ
ω
0
e

β
t
sin
ω
1
t
ðÞ
ð
4
Þ
And
xt
ðÞ¼

Δ
gH t
ðÞ
ω
0
e

β
t

β
sin
ω
1
t
ðÞþ
ω
1
cos
ω
1
t
ðÞ
½


Δ
gH t
ðÞ
ω
0
e

β
t
ω
1
cos
ω
1
t
ðÞ
1
Q
sin
ω
1
t
ðÞ

ð
5
Þ
If the damping is small
ω
0

β
ðÞ
, then
ω
1

ω
0
and
xt
ðÞ

Δ
gH t
ðÞ
ω
2
0
1

cos
ω
0
t
ðÞ
½
ð
6
Þ
and
xt
ðÞ
Δ
gH t
ðÞ
cos
ω
0
t
ðÞð
7
Þ
The total vertical acceleration of the mass (inertial acceleration
plus gravity) is then
xt
ðÞ
Δ
gH t
ðÞ
Δ
gH t
ðÞ
1

cos
ω
0
t
ðÞ
½ð
8
Þ
Notice that as
t
!
0
þ
, the inertial response of the Earth,
xt
ðÞ
, exactly cancels the gravitational change
Δ
g
. This is
because inertial accelerations from unbalanced elastic forces
are indistinguishable from unbalanced gravitational accelerations.
Notice that this cancelling effect is lessened if the damping is
increased; there is a second term in (5) that increases as
1
Q
sin
ω
1
t
ðÞ
. This term is from viscous forces that are not part of
the static equilibrium problem, and this term grows linearly with
time. If the Earth was heavily damped, then the gravitational
forces would be easier to observe with an accelerometer.
However, since
Q
of the Earth is large (
>
500), this damping term
is very small.
Assuming that the elastic response of the Earth is approxi-
mately given by
0
S
0
, I used (6) to estimate that the acceleration
signal after 12 s would be about 2
́
10

3
Δ
g
. Even after 120 s, it
only grows to 0
:
19
Δ
g
.
Although this simple sdof gedanken problem is a
gross oversimpli
fi
cation of the Earth
s response, it does allow
us to guess that the acceleration response at a point will
include the inertial effects of changing the gravimeter
s initial
equilibrium position to a new equilibrium position appropriate
for the gravity perturbation. The gravimeter will then
oscillate about that new position until the motions damp
out (from inelastic and radiation damping). Nevertheless, the
initial acceleration of the gravimeter will be equal and
opposite to the gravity change. Considering the Tohoku
earthquake, there are only two plausible length/time scales to
determine this oscillation period; the
fi
rst is the Earth
s dimension
(as in
0
S
0
), and the second is the source
observer distance. For
the Tohoku earthquake, the travel time of elastic deformations
between the source and observer seems to be the natural scale. I
speculate that the effective period will be on the order of 500 s,
which is four times the S-wave travel time.
Q
S
for this defor-
mation may be about 500. That is, I hypothesize that the net
acceleration on a gravimeter is approximately
xt
ðÞ

Δ
gH t
ðÞ

Δ
gH t
ðÞ
1

cos
2
π
t
T
R


ð
9
Þ
Where
T
R

4
R
=
c
s
,
R
is the hypocentral distance and
c
S
is the
shear wave velocity. If the gravity signal is changing with time as
Δ
gt
ðÞ
, then the expected gravimeter signal is
xt
ðÞ
Δ
gt
ðÞ
Δ
_
gt
ðÞ
Ht
ðÞ
1

cos
π
c
S
t
2
R

hi
ð
10
Þ
Where

is the convolution operator. For short times after the
earthquake origin, a Taylor expansion of the cosine can be used to
approximate (10) as
xt
ðÞ
Δ
gt
ðÞ
Δ
_
gt
ðÞ
Ht
ðÞ
2
π
t
T
0

2
¼

2
π
T
2
R
ZZ
Δ
gt
ðÞ
dt
2
ð
11
Þ
Therefore, the expected gravimeter signal for the Tohoku
earthquake that is shown in Fig. 4a of Montagner et al.
1
, should
be modi
fi
ed as
gravimeter output

10

3
ZZ
Δ
gt
ðÞ
dt
2
ð
12
Þ
This means that the gravimeter response is very sluggish
with time and it is poorly suited to rapidly detect
earthquakes as is required for seismic early warning. Examination
of Fig. 4b from Montagner et al.
1
shows that their
predicted gravitational signal signi
fi
cantly overpredicts the
accelerations recorded by the gravimeter. Perhaps modi
fi
cation
of their predicted acceleration with the convolution of (11)
will produce better agreement with the observation.
Data sharing is not applicable to this article as no data sets were
generated or analyzed during the current study.
Received: 17 January 2017 Accepted: 12 September 2017
Centre of mass
F
k
= –
kx
x
(
t
)
F
d
= –
bx
·
Total acceleration =
x
(
t
) – [
g
0
+
Δ
gH
(
t
)]
Earth’s surface
··
F
g
=
–m
[
g
0
+
Δ
gH
(
t
)]
Fig. 1
Schematic showing the parameters used to describe the
acceleration of an elastically supported mass in the presence of a
downward gravitational acceleration with an instantaneous step of
Δ
g
.
x
is
upward motion with respect to elastic/gravitational equilibrium
CORRESPONDENCE
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2
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DOI: 10.1038/s41467-017-01348-z
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References
1. Montagner, J. P. et al. Prompt gravity signal induced by the 2011 Tohoku-Oki
earthquake.
Nat. Commun.
7
, 13349 (2016).
2. Roll, P., Krotkov, R. & Dicke, R. The equivalence of inertial and passive
gravitational mass.
Ann. Phys.
26
, 442
517 (1964).
Additional information
Competing interests:
The authors declare no competing
fi
nancial interests.
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© The Author(s) 2017
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3