NGR
05-002-256
Gravitational-Wave
Observations
as
a
Tool
for
Testing
Relativistic
Gravity
DOUGLAS
M.
EARDLEY,
DAVID
L.
LEE,t
and
ALAN
P.
LIGHTMAN
California
Institute
of
Technology,
Pasadena,
California
91109
ROBERT
V.
WAGONER
Center
for
Radiophysics
and
Space
Research,
Cornell
University,
Ithaca,
New
York
14850
CLIFFORD
M.
WILL
S
University
of
Chicago,
Chicago,
Illinois
60637
ABSTRACT
Gravitational-wave
observations
can
be
powerful
tools
in
the
testing
of
relativistic
theories
of
gravity.
Future
experiments
should
be
designed
to
search
for
six
different
types
of
polariza-
tion,
and
for
anomalies
in
the
propagation
speed
of
the
waves:
ICgrav.~
waves
-7
- c I >
10
c .
This
letter
grav.
waves
em
waves
10
-
cem
waves
outlines
the
nature
and
implications
of
such
measurements.
*Supported
in
part
by
the
National
Aeronautics
and
Space
Administration
[NGR
05-002-256]
and
the
National
Science
Foundation
[GP-36687X,
GP-28027]
at
Caltech;
by
the
National
Science
Foundation
[GP-26068]
at
Cornell;
and
by
the
National
Science
Foundation
[GP-34721X1
at
Chicago.
tImperial
Oil
Predoctoral
Fellow.,
SEnrico
Fermi
Fellow.
N73;2134
(NASA-C"--131
5
1 1
)
-GRAVITATIONAL-
AVE
OBSERVATIONS
AS
A
TOOL
FOR
TESTING
RELATIVISTIC
GRAVITY
(California
Inst.
of
Tech.)
9
p
HC
$3.00
CSCL
08N
G3/13
174
G3/13
17475
Several
viable
gravitation
theories
now
exist
that
differ
radically
when
describing
strong
gravitational
fields,
but
that
are
identical
to
each
other and
to
general
relativity
in
the
"post-Newtonian
limit."
During
the
next
twenty
years,
one
will
probably
not
be
able
to
distinguish
these
theories from
general
relativity
or
from
each
other
by
means
of
"solar-
system
experiments"
(gravitational
redshift,
perihelion
shift,
light
deflec-
tion,
time
delay,
gyroscope
precession,
lunar-laser
ranging,
gravimetry,
Earth
rotation,
...
).
However,
gravitational-wave
experiments
offer
hope:
These theories
differ
in
their
predictions
of
(i)
propagation
speed,
and
(ii)
polarization
properties
of
gravitational
waves.
Propagation
speed: Some
of
the
competing
theories
1
-
predict
the
same
propagation
speed
for
gravitational
waves
(Cg)
as
for
light
(Cem).
But
others
5
predict
a
difference
that,
in
weak
gravitational
fields,
is
typic-
ally
(cg-
Cem)/c
(1/c
2
)
X INewtonian potential|
10
-,
for
waves
travelling
in
our
region
of
the
Galaxy
or
in
the
field of
the
Virgo
cluster.
An
experimental
limit
of
S
10
-
8
would
disprove
most
such
theories
and
would
stringently
constrain
future
theory-building.
Perhaps
the
most
promising
way
to
obtain
such a
limit
is
by comparing
arrival
times
for
gravitational
waves
and
for
light that
come
from
the
onset
of
a
supernova,
or
from
some
other
discrete
event.
If
current
experimental
efforts
continue
unabated,
by
1980 one
may
detect
gravitational-wave
bursts
from
supernovae
in
the
Virgo cluster
(-
3
supernovae
per
year).
Then
a
limit
of
Ic
- c
/c
< 10
-
9
x
(time
lag
precision)/(l
week)
g em
will
be
possible.
Polarization:
All of
the
currently
viable
theories
fall
into
a
class
called "metric
theories
of
gravity.
"
Recently
we have
completed
an
1
analysis
of
the
polarization properties
of
the
most
general
weak,
plane,
null
gravitational wave
permitted
by
any
metric
theory.
(Details
will
be
published
elsewhere.
8
Our
considerations
also
apply
to
waves
which
are
approximately,
rather
than exactly,
null.)
We
find
that
the
most
general
wave
is
composed
of
six
modes
of
polarization
(general
relativity
has
only
two), as
follows.
Use
coordinates
txyz.
Let
the
wave
propagate
in
the
+z
direction.
The
wave
is
characterized
by
six
amplitudes
which
depend
only
on
"retarded
time"
u,
where
u
-
t-
z/c.
Our
analysis
describes
these
amplitudes
by
two
real
functions
Y
2
(u),
¢
2
2
(u)
and
the
real
and
imaginary
parts
of
two
com-
plex functions
y3
(u),
'
4
(u).
These
functions
are
related
to
those
components
9
of
the
Riemann
tensor
which
determine
the
action
of the
wave
on
a
detector
by
1
2 6
zozo
1
3 2
(
Rxozo
+
i
Ryozo)
4 = R
yoo
-
Rxoxo
+
2i
Rxoyo
22
=
(Roxo
+
Ryoyo)
Figure
1
shows
the
action
of
each
mode
on
a sphere
of
test
bodies.
Y4
and
022
are
purely
transverse,
T
2
is
purely
longitudinal,
and
Y3
is
mixed.
These
waves
can
be
classified
in
a
Lorentz-invariant
manner
according
to
the
vanishing
or
nonvanishing
of
certain
of the
amplitudes.
Imagine
many
observers
in
different
Lorentz
frames,
some
moving
with
respect
to
each
other,
but
all
measuring
the
same
4-momentum
of
the wave.
The
amplitudes
transform
between
observers
in
a
complicated
way
[cf. Eq.
(1)
below]
but
the
waves
fall
into these
invariant
classes:
2
Class
II6.
Y2
0.
All
observers
in
such
Lorentz
frames
measure
a
non-
zero
amplitude
in
the
2 mode,
and
agree
on the
value of
this
amplitude.
(But
they
will
generally
disagree
about
the
presence
or
absence
and ampli-
tude of
all
other
modes.)
Class
III
5
.
12
5
0
T
¶3.
All
observers
agree
on
the
absence
of
72
and
the
presence
of
Y3.
(But
they
generally
disagree
about
the
presence
or
absence
of
y4
and
022
)
Class
N
3
. 2 0
Si
04 - ° T
22
All
observers
agree
about
the
presence
or
absence
of
all
modes.
Class
N
2
.
`2
3 °
3;
\4
° =
22-
All
observers
agree.
Class 2
90
°
93;
4 = ° T
O22-
All observers
agree.
Class
II
6
is
the
most
general;
as
one
demands
that
successive
amplitudes
vanish
identically,
one
descends
to
less
and
less
general
classes.
The
class
of the
most
general
permitted
wave
in
some
currently
viable
metric
theories
is:
General
relativity,
N
2
;
Dicke-Brans-Jordan,
N3 ;
Will-Nordtvedt,
III$;
1J4
5
Ni's
new
theory,
II6;
and
Lightman-Lee,
II
6
.
All
these
but
Dicke-Brans-
Jordan
have
the
same
post-Newtonian
limit
as
general
relativity,
for
a
reasonable choice
of
cosmological
model.
We
see
that
measuring
the
polarization
of
gravitational
waves
provides
a
sharp
experimental
test
of
theories
of
gravity.
The
class
of
the
"correct"
theory
is
at
least
as
general
as
that of
any
observed
wave.
The
observation
of
a
wave
more
general
than
N2
would
contradict
general
relativity
but
would
2-5
10
be
consistent
with
other viable
theories.
Weber
has
initiated
such
experiments
by
searching
for
the
022
mode,
with
negative
results.
To
test
theories,
an
experimenter
musk
classify
the
waves
that
he
detects.
If
he
knows
the
direction
of
a
wave
a
priori
(e.g.,
from
a
partic-
ular
supernova),
he
can
directly
extract
the
amplitude
of
each
mode
from
his
3
data
and
determine
the
class. If
he
does
not
know
the
direction, he
cannot
extract
the
amplitudes
or
determine
the
direction without
applying
some
further
assumption
to
his
data
(e.g.,
that
the
wave
is
no
more
general
than
N3
and
is
therefore
purely
transverse).
But he
can
usually
say
something
definite
about
the
class
of
the wave:
(i)
If the
driving
forces
in
his
detector
are
not
in
any
one
plane,
the
wave
is
II6
or
III
5
.
(ii)
If
the
driving
forces
are
in
a
plane
and
are
"pure
monopole"
[as
in
Fig.
l(c)],
the
wave
is
not
N2 .
(iii)
If
the
driving
forces
are
in
a
plane
and
are
"pure
quadrupole"
[as
in
Fig.
l(a)],
the
wave
is
not
01.
(iv)
Otherwise
the
wave
is
either
II6,
III5, or
N3 .
We now
sketch
the
arguments
that lead
to
these
results
about
polar-
ization
of
gravitational
waves
in
metric
theories.
Consider
a
weak,
plane,
null
wave
described
by
a
linearized Riemann
tensor,
R
(u),
with
Vu.
Vu
=
0.
Work
in
an
approximately
constant
quasi-orthonormal
null
tetrad
1 1
(k,
2,
m,
m),
where
k =
vu.
The
Bianchi identities
imply
that
there are
six
functionally independent
real
components
of
the
Riemann
tensor;
take
them,
in the
notation
of
Newman
and Penrose,
to
be
j
2
',
T
3
12,,
of
Loren
tafa
4'
022'
as
above.
Consider
the
"little
group
"
of
Lorentz
transforma-
tions of the
tetrad
which
fix
k:
k'
=
k,
m'
=
ei'(m+a
k),
£'
=
£+am
+
am
+aYak,
where
a
is
complex
and
cp
is
a
real
phase.
The
action of
E(2)
on
the
Riemann
tensor
of
a
wave
is
4